Should Mathematics be Done Differently Because of Gödel’s Incompleteness Theorem?

Nowadays Vinay Deolalikar’s claimed proof of p versus np problem is being discussed in many blogs and in many different platforms. And with the light of these discussions, my interests in theoretical computer science and math is awakened . Hence I wanted to write something about mathematics (Although I’m not a mathematician but always have interest  in this field and envied mathematicians).  But in those hot days I’ve been very lazy, therefore I couldn’t attempt to do such a thing. (Hopefully I’ll write about these topics soon) So I decided to put a very valuable text in my blog, that you might enjoy too.

This text is taken from Gregory Chaitin‘s talk in Cordoba University.  If you already don’t know him, here is a short note : He is a very influential mathematician and has very important contributions to several fields like computer science and mathematics. He is very good at expressing complex ideas, in a very accessible fashion. You can access more texts from him by his personal site.

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I used to work as a computer programmer; I wrote computer software and did theory as a hobby. So I’m an amateur mathematician and a professional programmer. That’s how I used to earn a living.

People normally think that mathematics is a dry, serious subject where nothing dramatic ever happens. But in the past century math went through a revolution as serious as the one that took place in physics because of the theory of relativity and quantum theory. This fact is not well-known outside the math community, but it is becoming better known now.

In particular, I’m referring to a controversy over how mathematics should be done. There is a struggle for the soul of mathematics. I exaggerate a bit, but not too much. There is a struggle for the soul of mathematics between two different groups, two tendencies, two opposing viewpoints.

On one side there is the famous French mathematician Poincaré who spoke of the importance of intuition in mathematics. On the other side we have the German mathematician Hilbert who emphasized formalism and the role of the axiomatic method. The conflict is between intuition and formalism. In other words, is mathematics creative or is it mechanical? Stating it that way, I indicate my own biases.

You can see which side I am on: the romantic side. But the debate is still very much alive and I want to give you a concise history of this conflict.

About a century ago Hilbert proposed formalizing all of mathematics, dropping the use of natural language and making math into a formal axiomatic theory using an artificial language and mathematical logic. The key point is that Hilbert thought that math gives absolute certainty and that this implies that you can formalize mathematics completely in such a way that there is an algorithm, a mechanical procedure, for checking whether or not a proof is correct.

In other words, Hilbert believed that if math is objective not subjective, if it really is absolutely certain, this is equivalent to saying that there are rules of the game for carrying out proofs — if no steps are left out and we use a completely formal language — which provide us with a completely mechanical way to check if a proof is correct, that is, whether it obeys the rules. According to Hilbert, this is what it means to say that math gives absolute certainty, which is what most mathematicians believe, because math is a way of fleeing from the real world to a toy world where truth is black or white and proofs are absolutely convincing.

This is what Hilbert proposed about a century ago. And most people thought that it could actually be done, that one could formalize everything. Hilbert represented the orthodox, conservative position within the math community. People thought that it ought to be possible. In fact, some very pretty work was done trying to achieve what Hilbert had proposed, trying to fulfill his dream of formalizing mathematics completely and obtaining absolute certainty and total objectivity. [In particular, I'm thinking of Zermelo-Fraenkel set theory and of the von Neumann integers.]

But in 1931 and in 1936 there were two big surprises. In 1931 Kurt Gödel showed that Hilbert’s project could never work, and in 1936 Alan Turing showed this completely differently and found a deeper reason why Hilbert’s dream was unattainable.

These two pieces of work are greatly admired, but in my opinion the math community has a very ambiguous position about these two achievements. Gödel and Turing are heroes, but nobody wants to face the disturbing implications of their work.

What Gödel showed in 1931 is that Hilbert’s dream is impossible because any formalization of mathematics — any formal axiomatic system of the kind that Hilbert sought for all of math, to give absolute certainty, to show that the truth is black or white — will necessarily have to be incomplete because some true results will be missing. In other words, no finite formal axiomatic theory can give us all mathematical truths, some of them will always escape us. In fact, an infinity of true math results will be missing from any formal axiomatic theory proposed to achieve Hilbert’s dream. Formal axiomatic theories are always incomplete, they do not enable us to demonstrate all possible mathematical truths.

Gödel shows how to construct assertions which are true but cannot be demonstrated within a given formal axiomatic system. The way he does it is very surprising. He constructs a mathematical assertion — in fact, an arithmetical assertion — which states that it itself cannot be demonstrated.

“I’m unprovable!”

If you can construct an assertion that states that it’s unprovable, there are two possibilities: that it’s provable and that it isn’t. If it’s provable and it asserts that it isn’t, we’re demonstrating something that’s false, which is terrible. So by hypothesis we eliminate this possibility. If a formal axiomatic system enables us to prove things that are false, it doesn’t interest us, it’s a complete waste of time. Therefore “I’m unprovable” cannot be proved, which means that it is true.

So you either demonstrate things that are false, or there are indemonstrable truths, truths that escape us. This is the alternative that Gödel confronts us with. Assuming that the formal axiomatic system doesn’t enable you to prove things that are false, there must be true mathematical assertions that cannot be proved.

Gödel incompleteness theorem was a big surprise at the time, and while not provoking panic, it did lead to some rather emotional reactions, for example, from Hermann Weyl. Weyl said that his faith in pure mathematics was badly affected, and that at first it was difficult for him to continue with his research. And Weyl was a very fine mathematician.

Now my story splits in two. On the one hand, there is more research on incompleteness, on Gödel’s remarkable discovery. On the other hand, the math community begins to lose interest in these philosophical questions and continues with its everyday work.

First I’ll tell you about Turing.

In 1936 Turing goes beyond Gödel and finds a much deeper reason for incompleteness. But I should emphasize that pioneering work is always the most difficult. Before Gödel nobody was courageous enough to imagine that Hilbert might be wrong. Turing found a deeper reason for incompleteness.

Turing discovered that there are many things in mathematics that can be defined but which there is no mechanical procedure, no algorithm, for calculating — they are not computable functions. Math is full of things that can be defined but cannot be calculated. And uncomputability is a new source of incompleteness.

If we consider a mathematical question such as Turing’s famous halting problem for which there is no general method for calculating the answer, we get the immediate corollary that there cannot be a formal axiomatic theory that always enables us to prove what the answer is.

Why not?

One of the most basic properties of a formal axiomatic theory is that in principle there is a mechanical procedure for systematically traversing the tree of all possible proofs and eliminating the ones that are incorrect. It would be very slow, but in principle it would enable us to find all the theorems. So if we have a theory that enables us to demonstrate in individual cases whether or not a program eventually halts, this would give us a mechanical procedure, an algorithm, that always gives the correct answer, which Turing showed in 1936 is impossible.

So Turing deduces Gödel incompleteness from a more fundamental idea, uncomputability, which is the fact that math is full of things that can be defined but cannot be calculated.

Now World War II begins and the generation that was interested in these philosophical questions disappears from the scene. The math community goes forward forgetting the crisis that was provoked by Gödel’s theorem which had been such a big surprise.

My problem is that I didn’t go forward. I remained obsessed with Gödel’s theorem. I thought it had to be very important. I bet my professional career on the idea that it was a mistake to ignore Gödel’s result.

What the math community did, since they are mathematicians and not philosophers, is to continue with their daily work, with the problems that interested them. The consensus was that yes in theory there are limits to what can be demonstrated using any particular formal axiomatic theory, but not in practice, not with the kinds of questions that interest us, not in our own particular field. This was more or less the community’s reaction.

In other words, while there may be mathematical facts that are true but unprovable, these are highly artificial pathological cases. The consensus was that in practice this does not occur. At least that is what mathematicians preferred to think in order to be able to carry on with their work.

People have an amazing ability to avoid thinking about unpleasant subjects such as death. If we think about death all the time it is impossible to function. And if mathematicians think all the time about incompleteness they can’t function either, since there will always be doubt about whether the matter at hand can be settled by means of a proof. Why am I wasting years of my life trying to prove something if there may not even be a proof?

Let’s consider an alternative course of action. Instead of ignoring Gödel’s theorem, what if we take it very seriously? I don’t believe in going to extremes, but if one took Gödel’s result very, very seriously, how might one proceed? Consider the Riemann hypothesis. This is an important mathematical conjecture that has a lot of significant consequences. But unfortunately in a hundred and fifty years of effort nobody has succeeded in proving the Riemann hypothesis. Mathematicians don’t know what to do; the way forward is blocked. But physicists would just consider the Riemann hypothesis to be a mathematical fact that has been corroborated empirically.

In other words, I think that a possible reaction to Gödel’s result is to make math a little bit more like theoretical physics. In physics axioms don’t have to be self-evident. Maxwell’s equations and the Schrödinger equation are not self-evident but they help us to organize, to unify a large body of experimental data.

One could do mathematics in a similar fashion, taking Gödel as justification for behaving as if math were an empirical science in which one doesn’t try to demonstrate everything from self-evident principles, but instead one only seeks to organize mathematical experience like physicists organize their physics lab experience. One could proceed pragmatically and adopt unproven hypotheses as new basic principles because they are extremely fruitful and have many useful consequences even though they aren’t at all self-evident. This is what I think we should do if we take Gödel’s theorem seriously.

In my opinion mathematics is different from physics, but maybe not as different as most people think. My work on metamathematics using complexity and information-theoretic ideas suggests to me that perhaps we should emphasize the similarities between the world of mathematics and the world of physics instead of emphasizing the differences.

In this connection, there is a highly pertinent remark by the Russian mathematician Vladimir Arnold. In his opinion the only difference between mathematics and physics is that in mathematics the experiments are cheaper, since one can carry them out on a computer instead of having to have a laboratory full of expensive equipment! So math experiments are easier than physics experiments.

How do I try to justify this new “quasi-empirical” view of mathematics? Well, like most mathematicians, I do in fact believe in the Platonic world of math ideas in which the truth is totally black or white. But I also believe that we are denied direct access to this Platonic world and that down here at our level it may be helpful to work a bit more quasi-empirically.

It may look like my mixed, hybrid, Platonic-empiricist position is inconsistent, but I don’t think that this is actually the case. Indeed, it is sometimes very fruitful to take ideas that seem to be inconsistent and show that in fact they aren’t.

Okay, so where do I find arguments in favor of this quasi-empirical view of mathematics? The key question is whether the incompleteness phenomenon that was discovered by Gödel and further explored by Turing is exceptional or widespread. How pervasive is incompleteness? That’s the basic question, and it is quite controversial.

My contribution to this discussion is that I’ve found tools for measuring the complexity or the information content of a formal axiomatic mathematical theory. And by using the concept of complexity in algorithmic information theory one can see that incompleteness is natural, not surprising. In fact, it’s inevitable, it’s unavoidable.

Using algorithmic information theory, one can see that the world of mathematical truths, the Platonic world of mathematical ideas, is infinitely complex. But any formal axiomatic system made by human beings necessarily has only finite complexity. Indeed, rather low complexity, since the axioms and rules of inference normally fit on a couple of pages.

So seen from this perspective, incompleteness is natural, inevitable. The world of mathematical ideas is infinitely complex, but our theories only have low, finite complexity; otherwise they wouldn’t fit in a mathematician’s brain nor would they be regarded as self-evident — but I’m against the idea that in mathematics axioms have to be self-evident, because in physics self-evidence of axioms is not required.

I’ve used complexity and information theory to argue that since the amount of information in pure mathematics is infinite, incompleteness is only to be expected, since a formal axiomatic theory can capture at most a finite amount of this mathematical information, an infinitesimal portion in fact.

This more or less summarizes an entire lifetime of research. But you will not be surprised to learn that the mathematics community has not accepted my quasi-empirical proposal. The immune system of an intellectual community is very strong, and my ideas are rejected as foreign, as alien to the math community.

Logicians don’t care much for computability, for complexity, for information and for randomness. Randomness is a nightmare for a logician, because randomness is irrational. Random events happen for no reason, they are incomprehensible from a logical point of view.

However the physics community has some interest in my work. They like the idea of using a physics-inspired approach in pure mathematics. They like the idea that math isn’t that different from physics. They like the idea that a mathematics proof may be more convincing than the heuristic arguments that are accepted in physics, but that this is only a matter of degree, not an absolute black or white difference. They have always felt that mathematicians believe too much in absolute truth, and do not appreciate theoretical physics enough.

But the coin is two-sided, and the conflict between intuition and formalism has become much more acute because of the computer. Computer technology is a powerful argument against creativity and in favor of mechanization and formalization.

Just take a look at the December 2008 issue of the Notices of the American Mathematical Society which you can find for free on the web. This is a special issue devoted to formal proof. While I, a poor theoretician, have been trying to convince mathematicians to pay attention to Gödel’s theorem and work slightly differently, these people — I didn’t realize what was happening until they did it — have nearly succeeded in carrying out Hilbert’s dream.

They’ve constructed tools for formalizing almost all of mathematics. They’ve done a superb piece of software engineering.

This community, which is a group of fine mathematicians and software engineers, believes that in the future all mathematical proofs should be formal proofs. In their opinion, there will soon be no reason for accepting informal proofs. We can start demanding formal proofs and re-writing all of mathematics in a formal language so that it can be checked by verification software.

There are now interactive proof checkers for verifying mathematical proofs. This is how these work: If I’m a mathematician and I have an informal proof that I want to formalize, I give it to the proof checker. It will say, “Well, there’s a particular step in this proof that I don’t understand yet. Can you please explain this better?” And you keep filling in the proof, providing more details, until the software says, “Now I understand everything. It’s all fine. I have a complete formal proof.”

You didn’t have to write all the steps in the formal proof yourself; that would be a big job. You write part of it, and the software provides the rest. The final result of this joint effort is a complete formal proof that has been checked and verified by reliable software, software that you trust because it was carefully developed using the best available software engineering methodology.

And this verification technology has advanced to the point where you don’t just verify toy proofs, you can verify complicated proofs of really important theorems, for example the four color theorem, which states that four colors suffice for coloring maps without having neighboring countries with the same color.

This was a rather complicated proof that not only was formalized, the mathematician who did it did not complain and even stated that going through this process enabled him to substantially improve the proof. So this formal proof business is getting really serious.

Hilbert never thought that mathematicians should be required to use detailed formal proofs in their daily work. But this community does. Furthermore, they envision an official repository for formal proofs that have been put through this verification process. Proofs will have to be accepted by this repository to be used by the mathematics community; everything that has been formalized and checked will be there, in one place.

So amazingly enough, the lines of research opened up by Hilbert’s formalization proposal and by Gödel’s work on the limitations of formal systems are both progressing dramatically. I think there is a wonderful intellectual tension between the work advancing formalization and the one criticizing it. Both of these lines of research are going forward splendidly in parallel!

In mathematics this circumstance is striking because one thinks that the truth is black or white. But in philosophy this situation doesn’t seem so strange because philosophers understand that ideas that seem contradictory are often in fact complementary.

I won’t try to predict the final outcome of this conflict; probably there will be no final outcome. In philosophy there are no final answers. Each generation does its best to resolve the fundamental questions to their own satisfaction, and then the next generation goes off in a different direction.

So I won’t try to predict the future. I don’t know if mathematicians will eventually think that incompleteness implies that they should do math differently, or if formalization will win.

Perhaps we don’t have to choose between quasi-empiricism and formalization. Both of these approaches can contribute something to mathematics and to mathematical practice.

My late friend the mathematician Gian-Carlo Rota, whose provocative ideas I greatly enjoy, has bequeathed us a collection of his essays entitled Indiscrete Thoughts. He thinks that formal axiomatization is a cemetery.

When a theory is completely finished, then you can formalize it.

But when you are creating a new theory, you have to work with vague intuitions, with imprecise ideas, and formalization is deadly. Premature formalization stifles creativity; once a theory is formalized it becomes stiff and rigid and no new ideas can get in.

So I think that quasi-empiricism and formalism can both contribute something of value. Furthermore, both are advancing step by step.

In 1974 I proposed accepting new math axioms the way that this is done in physics, ["Information-theoretic limitations of formal systems," J. ACM 21, 1974, pp. 403--424.] and nobody took me seriously, but in the past thirty-five years this has actually happened.

It has happened in set theory, where there’s a new axiom called “projective determinacy.” It has happened in theoretical computer science, where you use the hypothesis that P is not equal NP, which everyone believes but nobody can prove. And it has happened in mathematical cryptography, which is based on the assumption that you can’t factorize big numbers quickly.

In these fields mathematicians are behaving as if they were physicists. They’ve found new principles that enable them to organize the experiences of each of these communities. These are principles that are not self-evident, that have not been demonstrated, but that are accepted by consensus as new fundamental principles, at least until they are disproven or counter-examples are encountered.

Each of these mathematical communities is behaving as if they were theoretical physicists, they are doing what I call quasi-empirical mathematics. So I’ve been delighted to witness these developments, but not so delighted to see the striking advance of formalism in recent years.

These questions are still open, and they are very difficult ones. I’ve tried to argue in favor of a quasi-empirical stance, in favor of creativity and against formalism, but I myself am not completely convinced by my own arguments. More work is needed. We still do not know to what extent math is mechanical or creative.

Measure the temperature of the system in Linux

In linux you can use lm-sensors to measure the temperature and the pc health of the system. Via the following snippet you can monitor the system by refreshing the statistics every 2 seconds:

#!/bin/bash
while [ 1 ];
do
sensors; sleep 2; clear;
done;

The Nightfall

This is another great short story (but longer than the others), by Asimov. He wrote this short story  in 1941 when he was 21 years old and got 15 dollars as a reward from “Astounding Science Fiction” Magazine:).

Nightfall

If the stars should appear one night in a thousand years, how would men believe and adore, and preserve for many generations the remembrance of the city of God?”

EMERSON

Aton 77, director of Saro University, thrust out a belligerent lower lip and glared at the young newspaperman in a hot fury.

Theremon 762 took that fury in his stride. In his earlier days, when his now widely syndicated column was only a mad idea in a cub reporter’s mind, he had specialized in ‘impossible’ interviews. It had cost him bruises, black eyes, and broken bones; but it had given him an ample supply of coolness and self-confidence.

So he lowered the outthrust hand that had been so pointedly ignored and calmly waited for the aged director to get over the worst. Astronomers were queer ducks, anyway, and if Aton’s actions of the last two months meant anything; this same Aton was the queer-duckiest of the lot.

Aton 77 found his voice, and though it trembled with restrained emotion, the careful, somewhat pedantic phraseology, for which the famous astronomer was noted, did not abandon him.

“Sir,” he said, “you display an infernal gall in coming to me with that impudent proposition of yours.”

The husky telephotographer of the Observatory, Beenay 25, thrust a tongue’s tip across dry lips and interposed nervously, “Now, sir, after all — “

The director turned to him and lifted a white eyebrow. “Do not interfere, Beenay. I will credit you with good intentions in bringing this man here; but I will tolerate no insubordination now.”

Theremon decided it was time to take a part. “Director Aton, if you’ll let me finish what I started saying, I think — “

“I don’t believe, young man,” retorted Aton, “that anything you could say now would count much as compared with your daily columns of these last two months. You have led a vast newspaper campaign against the efforts of myself and my colleagues to organize the world against the menace which it is now too late to avert. You have done your best with your highly personal attacks to make the staff of this Observatory objects of ridicule.”

The director lifted a copy of the Saro City Chronicle from the table and shook it at Theremon furiously. “Even a person of your well-known impudence should have hesitated before coming to me with a request that he be allowed to cover today’s events for his paper. Of all newsmen, you!”

Aton dashed the newspaper to the floor, strode to the window, and clasped his arms behind his back.

“You may leave,” he snapped over his shoulder. He stared moodily out at the skyline where Gamma, the brightest of the planet’s six suns, was setting. It had already faded and yellowed into the horizon mists, and Aton knew he would never see it again as a sane man.

He whirled. “No, wait, come here!” He gestured peremptorily. I’ll give you your story.”

The newsman had made no motion to leave, and now he approached the old man slowly. Aton gestured outward. “Of the six suns, only Beta is left in the sky. Do you see it?”

The question was rather unnecessary. Beta was almost at zenith, its ruddy light flooding the landscape to an unusual orange as the brilliant rays of setting Gamma died. Beta was at aphelion. It was small; smaller than Theremon had ever seen it before, and for the moment it was undisputed ruler of Lagash’s sky.

Lagash’s own sun. Alpha, the one about which it revolved, was at the antipodes, as were the two distant companion pairs. The red dwarf Beta — Alpha’s immediate companion — was alone, grimly alone.

Aton’s upturned face flushed redly in the sunlight. “In just under four hours,” he said, “civilization, as we know it, comes to an end. It will do so because, as you see. Beta is the only sun in the sky.” He smiled grimly. “Print that! There’ll be no one to read it.”

“But if it turns out that four hours pass — and another four — and nothing happens?” asked Theremon softly.

“Don’t let that worry you. Enough will happen.”

“Granted! And still — it nothing happens?”

For a second time, Beenay 25 spoke. “Sir, I think you ought to listen to him.”

Theremon said, “Put it to a vote, Director Aton.”

There was a stir among the remaining five members of the Observatory staff, who till now had maintained an attitude of wary neutrality.

“That,” stated Aton flatly, “is not necessary.” He drew out his pocket watch. “Since your good friend, Beenay, insists so urgently, I will give you five minutes. Talk away.”

“Good! Now, just what difference would it make if you allowed me to take down an eyewitness account of what’s to come? If your prediction comes true, my presence won’t hurt; for in that case my column would never be written. On the other hand, if nothing comes of it, you will just have to expect ridicule or worse. It would be wise to leave that ridicule to friendly hands.”

Aton snorted. “Do you mean yours when you speak of friendly hands?”

“Certainly!” Theremon sat down and crossed his legs. “My columns may have been a little rough, but I gave you people the benefit of the doubt every time. After all, this is not the century to preach “The end of the world is at hand” to Lagash. You have to understand that people don’t believe the Book of Revelations anymore, and it annoys them to have scientists turn about-face and tell us the Cultists are right after all — “

“No such thing, young man,” interrupted Aton. “While a great deal of our data has been supplied us by the Cult, our results contain none of the Cult’s mysticism. Facts are facts, and the Cult’s so-called mythology has certain facts behind it. We’ve exposed them and ripped away their mystery. I assure you that the Cult hates us now worse than you do.”

“I don’t hate you. I’m just trying to tell you that the public is in an ugly humor. They’re angry.”

Aton twisted his mouth in derision. “Let them be angry.”

“Yes, but what about tomorrow?”

“There’ll be no tomorrow!”

“But if there is. Say that there is — just to see what happens. That anger might take shape into something serious. After all, you know, business has taken a nosedive these last two months. Investors don’t really believe the world is coming to an end, but just the same they’re being cagy with their money until it’s all over. Johnny Public doesn’t believe you, either, but the new spring furniture might just as well wait a few months — just to make sure.

“You see the point. Just as soon as this is all over, the business interests will be after your hide. They’ll say that if crackpots — begging your pardon — can upset the country’s prosperity any time they want, simply by making some cockeyed prediction — it’s up to the planet to prevent them. The sparks will fly, sir.”

The director regarded the columnist sternly. “And just what were you proposing to do to help the situation?”

“Well” — Theremon grinned — “I was proposing to take charge of the publicity. I can handle things so that only the ridiculous side will show. It would be hard to stand, I admit, because I’d have to make you all out to be a bunch of gibbering idiots, but if I can get people laughing at you, they might forget to be angry. In return for that, all my publisher asks is an exclusive story.”

Beenay nodded and burst out, “Sir, the rest of us think he’s right. These last two months we’ve considered everything but the million-to-one chance that there is an error somewhere in our theory or in our calculations. We ought to take care of that, too.”

There was a murmur of agreement from the men grouped about the table, and Aton’s expression became that of one who found his mouth full of something bitter and couldn’t get rid of it.

“You may stay if you wish, then. You will kindly refrain, however, from hampering us in our duties in any way. You will also remember that I am in charge of all activities here, and in spite of your opinions as expressed in your columns, I will expect full cooperation and full respect — “

His hands were behind his back, and his wrinkled face thrust forward determinedly as he spoke. He might have continued indefinitely but for the intrusion of a new voice.

“Hello, hello, hello!” It came in a high tenor, and the plump cheeks of the newcomer expanded in a pleased smile. “What’s this morgue-like atmosphere about here? No one’s losing his nerve, I hope.”

Aton started in consternation and said peevishly, “Now what the devil are you doing here, Sheerin? I thought you were going to stay behind in the Hideout.”

Sheerin laughed and dropped his stubby figure into a chair. “Hideout be blowed! The place bored me. I wanted to be here, where things are getting hot. Don’t you suppose I have my share of curiosity? I want to see these Stars the Cultists are forever speaking about.” He rubbed his hands and added in a soberer tone. “It’s freezing outside. The wind’s enough to hang icicles on your nose. Beta doesn’t seem to give any heat at all, at the distance it is.”

The white-haired director ground his teeth in sudden exasperation. “Why do you go out of your way to do crazy things, Sheerin? What kind of good are you around here?”

“What kind of good am I around there?” Sheerin spread his palms in comical resignation. “A psychologist isn’t worth his salt in the Hideout. They need men of action and strong, healthy women that can breed children. Me? I’m a hundred pounds too heavy for a man of action, and I wouldn’t be a success at breeding children. So why bother them with an extra mouth to feed? I feel better over here.”

Theremon spoke briskly. “Just what is the Hideout, sir?”

Sheerin seemed to see the columnist for the first time. He frowned and blew his ample cheeks out. “And just who in Lagash are you, redhead?”

Aton compressed his lips and then muttered sullenly, “That’s Theremon 762, the newspaper fellow. I suppose you’ve heard of him.”

The columnist offered his hand. “And, of course, you’re Sheerin 501 of Saro University. I’ve heard of you.” Then he repeated, “What is this Hideout, sir?”

“Well,” said Sheerin, “we have managed to convince a few people of the validity of our prophecy of — er — doom, to be spectacular about it, and those few have taken proper measures. They consist mainly of the immediate members of the families of the Observatory staff, certain of the faculty of Saro University, and a few outsiders. Altogether, they number about three hundred, but three quarters are women and children.”

“I see! They’re supposed to hide where the Darkness and the — er — Stars can’t get at them, and then hold out when the rest of the world goes poof.”

“If they can. It won’t be easy. With all of mankind insane, with the great cities going up in flames — environment will not be conducive to survival. But they have food, water, shelter, and weapons — “

“They’ve got more,” said Aton. “They’ve got all our records, except for What we will collect today. Those records will mean everything to the next cycle, and that’s what must survive. The rest can go hang.”

Theremon uttered a long, low whistle and sat brooding for several minutes. The men about the table had brought out a multi-chess board and started a six-member game. Moves were made rapidly and in silence. All eyes bent in furious concentration on the board. Theremon watched them intently and then rose and approached Aton, who sat apart in whispered conversation with Sheerin.

“Listen,” he said, let’s go somewhere where we won’t bother the rest of the fellows. I want to ask some questions.”

The aged astronomer frowned sourly at him, but Sheerin chirped up, “Certainly. It will do me good to talk. It always does. Aton was telling me about your ideas concerning world reaction to a failure of the prediction — and I agree with you. I read your column pretty regularly, by the way, and as a general thing I like your views.”

“Please, Sheerin,” growled Aton.

“Eh? Oh, all right. We’ll go into the next room. It has softer chairs, anyway.”

There were softer chairs in the next room. There were also thick red curtains on the windows and a maroon carpet on the floor. With the bricky light of Beta pouring in, the general effect was one of dried blood.

Theremon shuddered. “Say, I’d give ten credits for a decent dose of white light for just a second. I wish Gamma or Delta were in the sky.”

“What are your questions?” asked Aton. “Please remember that our time is limited. In a little over an hour and a quarter we’re going upstairs, and after that there will be no time for talk.”

“Well, here it is.” Theremon leaned back and folded his hands on his chest. “You people seem so all-fired serious about this that I’m beginning to believe you. Would you mind explaining what it’s all about?”

Aton exploded, “Do you mean to sit there and tell me that you’ve been bombarding us with ridicule without even finding out what we’ve been trying to say?”

The columnist grinned sheepishly. “It’s not that bad, sir. I’ve got the general idea. You say there is going to be a world-wide Darkness in a few hours and that all mankind will go violently insane. What I want now is the science behind it.”

“No, you don’t. No, you don’t,” broke in Sheerin. “If you ask Aton for that — supposing him to be in the mood to answer at all — he’ll trot out pages of figures and volumes of graphs. You won’t make head or tail of it. Now if you were to ask me, I could give you the layman’s standpoint.”

“All right; I ask you.”

“Then first I’d like a drink.” He rubbed his hands and looked at Aton.

“Water?” grunted Aton.

“Don’t be silly!”

“Don’t you be silly. No alcohol today. It would be too easy to get my men drunk. I can’t afford to tempt them.”

The psychologist grumbled wordlessly. He turned to Theremon, impaled him with his sharp eyes, and began.

“You realize, of course, that the history of civilization on Lagash displays a cyclic character — but I mean cyclic!”

“I know,” replied Theremon cautiously, “that that is the current archaeological theory. Has it been accepted as a fact?”

“Just about. In this last century it’s been generally agreed upon. This cyclic character is — or rather, was — one of the great mysteries. We’ve located series of civilizations, nine of them definitely, and indications of others as well, all of which have reached heights comparable to our own, and all of which, without exception, were destroyed by fire at the very height of their culture.

“And no one could tell why. All centers of culture were thoroughly gutted by fire, with nothing left behind to give a hint as to the cause.”

Theremon was following closely. “Wasn’t there a Stone Age, too?”

“Probably, but as yet practically nothing is known of it, except that men of that age were little more than rather intelligent apes. We can forget about that.”

“I see. Go on!”

There have been explanations of these recurrent catastrophes, all of a more or less fantastic nature. Some say that there are periodic rains of fire; some that Lagash passes through a sun every so often; some even wilder things. But there is one theory, quite different from all of these, that has been handed down over a period of centuries.”

“I know. You mean this myth of the “Stars” that the Cultists have in their Book of Revelations.”

“Exactly,” rejoined Sheerin with satisfaction. “The Cultists said that every two thousand and fifty years Lagash entered a huge cave, so that all the suns disappeared, and there came total darkness all over the world! And then, they say, things called Stars appeared, which robbed men of their souls and left them unreasoning brutes, so that they destroyed the civilization they themselves had built up. Of course they mix all this up with a lot of religio-mystic notions, but that’s the central idea.”

There was a short pause in which Sheerin drew a long breath. “And now we come to the Theory of Universal Gravitation.” He pronounced the phrase so that the capital letters sounded — and at that point Aton turned from the window, snorted loudly, and stalked out of the room.

The two stared after him, and Theremon said, “What’s wrong?”

“Nothing in particular,” replied Sheerin. “Two of the men were due several hours ago and haven’t shown up yet. He’s terrifically short-handed, of course, because all but the really essential men have gone to the Hideout.”

“You don’t think the two deserted, do you?”

“Who? Faro and Yimot? Of course not. Still, if they’re not back within the hour, things would be a little sticky.” He got to his feet suddenly, and his eyes twinkled. “Anyway, as long as Aton is gone — “

Tiptoeing to the nearest window, he squatted, and from the low window box beneath withdrew a bottle of red liquid that gurgled suggestively when he shook it.

“I thought Aton didn’t know about this,” he remarked as he trotted back to the table. “Here! We’ve only got one glass so, as the guest, you can have it. I’ll keep the bottle.”

And he filled the tiny cup with judicious care.

Theremon rose to protest, but Sheerin eyed him sternly. “Respect your elders, young man.”

The newsman seated himself with a look of anguish on his face. “Go ahead, then, you old villain.”

The psychologist’s Adam’s apple wobbled as the bottle upended, and then, with a satisfied grunt and a smack of the lips, he began again. “But what do you know about gravitation?”

“Nothing, except that it is a very recent development, not too well established, and that the math is so hard that only twelve men in Lagash are supposed to understand it.”

Tcha! Nonsense! Baloney! I can give you all the essential math in a sentence. The Law of Universal Gravitation states that there exists a cohesive force among all bodies of the universe, such that the amount of this force between any two given bodies is proportional to the product of their masses divided by the square of the distance between them.”

“Is that all?”

“That’s enough! It took four hundred years to develop it.”

“Why that long? It sounded simple enough, the way you said it.”

“Because great laws are not divined by flashes of inspiration, whatever you may think. It usually takes the combined work of a world full of scientists over a period of centuries. After Genovi 41 discovered that Lagash rotated about the sun Alpha rather than vice versa — and that was four hundred years ago — astronomers have been working. The complex motions of the six suns were recorded and analyzed and unwoven. Theory after theory was advanced and checked and counterchecked and modified and abandoned and revived and converted to something else. It was a devil of a job.”

Theremon nodded thoughtfully and held out his glass for more liquor. Sheerin grudgingly allowed a few ruby drops to leave the bottle.

“It was twenty years ago,” he continued after remoistening his own throat, “that it was finally demonstrated that the Law of Universal Gravitation accounted exactly for the orbital motions of the six suns. It was a great triumph.”

Sheerin stood up and walked to the window, still clutching his bottle. “And now we’re getting to the point. In the last decade, the motions of Lagash about Alpha were computed according to gravity, and if did not account for the orbit observed; not even when all perturbations due to the other suns were included. Either the law was invalid, or there was another, as yet unknown, factor involved.”

Theremon joined Sheerin at the window and gazed out past the wooded slopes to where the spires of Saro City gleamed bloodily on the horizon. The newsman felt the tension of uncertainty grow within him as he cast a short glance at Beta. It glowered redly at zenith, dwarfed and evil.

“Go ahead, sir,” he said softly.

Sheerin replied, “Astronomers stumbled about for year, each proposed theory more untenable than the one before — until Aton had the inspiration of calling in the Cult. The head of the Cult, Sor 5, had access to certain data that simplified the problem considerably. Aton set to work on a new track.

“What if there were another nonluminous planetary body such as Lagash? If there were, you know, it would shine only by reflected light, and if it were composed of bluish rock, as Lagash itself largely is, then, in the redness of the sky, the eternal blaze of the suns would make it invisible — drown it out completely.”

Theremon whistled. “What a screwy idea!”

“You think that’s screwy? Listen to this: Suppose this body rotated about Lagash at such a distance and in such an orbit and had such a mass that its attention would exactly account for the deviations of Lagash’s orbit from theory — do you know what would happen?”

The columnist shook his head.

“Well, sometimes this body would get in the way of a sun.” And Sheerin emptied what remained in the bottle at a draft.

“And it does, I suppose,” said Theremon flatly.

“Yes! But only one sun lies in its plane of revolution.” He jerked a thumb at the shrunken sun above. “Beta! And it has been shown that the eclipse will occur only when the arrangement of the suns is such that Beta is alone in its hemisphere and at maximum distance, at which time the moon is invariably at minimum distance. The eclipse that results, with the moon seven times the apparent diameter of Beta, covers all of Lagash and lasts well over half a day, so that no spot on the planet escapes the effects. That eclipse comes once every two thousand and forty-nine years.”

Theremon’s face was drawn into an expressionless mask. “And that’s my story?”

The psychologist nodded. “That’s all of it. First the eclipse — which will start in three quarters of an hour — then universal Darkness and, maybe, these mysterious Stars — then madness, and end of the cycle.”

He brooded. “We had two months’ leeway — we at the Observatory — and that wasn’t enough time to persuade Lagash of the danger. Two centuries might not have been enough. But our records are at the Hideout, and today we photograph the eclipse. The next cycle will start off with the truth, and when the next eclipse comes, mankind will at last be ready for it. Come to think of it, that’s part of your story too.”

A thin wind ruffled the curtains at the window as Theremon opened it and leaned out. It played coldly with his hair as he stared at the crimson sunlight on his hand. Then he turned in sudden rebellion.

“What is there in Darkness to drive me mad?”

Sheerin smiled to himself as he spun the empty liquor bottle with abstracted motions of his hand. “Have you ever experienced Darkness, young man?”

The newsman leaned against the wall and considered. “No. Can’t say I have. But I know what it is. Just — uh — “ He made vague motions with his fingers and then brightened. “Just no light. Like in caves.” ,

“Have you ever been in a cave?”

“In a cave! Of course not!”

“I thought not. I tried last week — just to see — but I got out in a hurry. I went in until the mouth of the cave was just visible as a blur of light, with black everywhere else. I never thought a person my weight could run that fast.”

Theremon’s lip curled. “Well, if it comes to that, I guess I wouldn’t have run if I had been there.”

The psychologist studied the young man with an annoyed frown.

“My, don’t you talk big! I dare you to draw the curtain.”

Theremon looked his surprise and said, “What for? If we had four or five suns out there, we might want to cut the light down a bit for comfort, but now we haven’t enough light as it is.”

“That’s the point. Just draw the curtain; then come here and sit down.”

“All right.” Theremon reached for the tasseled string and jerked. The red curtain slid across the wide window, the brass rings hissing their way along the crossbar, and a dusk-red shadow clamped down on the room.

Theremon’s footsteps sounded hollowly in the silence as he made his way to the table, and then they stopped halfway. “I can’t see you, sir,” he whispered.

“Feel your way,” ordered Sheerin in a strained voice.

“But I can’t see you, sir.” The newsman was breathing harshly. “I can’t see anything.”

“What did you expect?” came the grim reply. “Come here and sit down!”

The footsteps sounded again, waveringly, approaching slowly. There was the sound of someone fumbling with a chair. Theremon’s voice came thinly, “Here I am. I feel . . . ulp . . . all right.”

“You like it, do you?”

“N — no. It’s pretty awful. The walls seem to be — “ He paused. “They seem to be closing in on me. I keep wanting to push them away. But I’m not going mad! In fact, the feeling isn’t as bad as it was.”

“All right. Draw the curtain back again.”

There were cautious footsteps through the dark, the rustle of Theremon’s body against the curtain as he felt for the tassel, and then the triumphant roo-osh of the curtain slithering back. Red light flooded the room, and with a cry of joy Theremon looked up at the sun.

Sheerin wiped the moistness off his forehead with the back of a hand and said shakily, “And that was just a dark room.”

“It can be stood,” said Theremon lightly.

“Yes, a dark room can. But were you at the Jonglor Centennial Exposition two years ago?”

“No, it so happens I never got around to it. Six thousand miles was just a bit too much to travel, even for the exposition.”

“Well, I was there. You remember hearing about the “Tunnel of Mystery” that broke all records in the amusement area — for the first month or so, anyway?”

“Yes. Wasn’t there some fuss about it?”

“Very little. It was hushed up. You see, that Tunnel of Mystery was just a mile-long tunnel — with no lights. You got into a little open car and jolted along through Darkness for fifteen minutes. It was very popular — while it lasted.”

“Popular?”

“Certainly. There’s a fascination in being frightened when it’s part of a game. A baby is born with three instinctive fears: of loud noises, of falling, and of the absence of light. That’s why it’s considered so funny to jump at someone and shout “Boo!” That’s why it’s such fun to ride a roller coaster. And that’s why that Tunnel of Mystery started cleaning up. People came out of that Darkness shaking, breathless, half dead with fear, but they kept on paying to get in.”

“Wait a while, I remember now. Some people came out dead, didn’t they? There were rumors of that after it shut down.”

The psychologist snorted. “Bah! Two or three died. That was nothing! They paid off the families of the dead ones and argued the Jonglor City Council into forgetting it. After all, they said, if people with weak hearts want to go through the tunnel, it was at their own risk — and besides, it wouldn’t happen again. So they put a doctor in the front office and had every customer go through a physical examination before getting into the car. That actually boosted ticket sales.”

“Well, then?”

“But you see, there was something else. People sometimes came out in perfect order, except that they refused to go into buildings — any buildings; including palaces, mansions, apartment houses, tenements, cottages, huts, shacks, lean-tos, and tents.”

Theremon looked shocked. “You mean they refused to come in out of the open? Where’d they sleep?”

“In the open.”

“They should have forced them inside.”

“Oh, they did, they did. Whereupon these people went into violent hysterics and did their best to bat their brains out against the nearest wall. Once you got them inside, you couldn’t keep them there without a strait jacket or a heavy dose of tranquilizer.”

“They must have been crazy.”

“Which is exactly what they were. One person out of every ten who went into that tunnel came out that way. They called in the psychologists, and we did the only thing possible. We closed down the exhibit.” He spread his hands.

“What was the matter with these people?” asked Theremon finally.

“Essentially the same thing that was the matter with you when you thought the walls of the room were crushing in on you in the dark. There is a psychological term for mankind’s instinctive fear of the absence of light. We call it “claustrophobia”, because the lack of light is always tied up with enclosed places, so that fear of one is fear of the other. You see?”

“And those people of the tunnel?”

“Those people of the tunnel consisted of those unfortunates whose mentality did not quite possess the resiliency to overcome the claustrophobia that overtook them in the Darkness. Fifteen minutes without light is a long time; you only had two or three minutes, and I believe you were fairly upset.

“The people of the tunnel had what is called a “claustrophobic fixation”. Their latent fear of Darkness and enclosed places had crystalized and become active, and, as far as we can tell, permanent. That’s what fifteen minutes in the dark will do.”

There was a long silence, and Theremon’s forehead wrinkled slowly into a frown. “I don’t believe it’s that bad.”

“You mean you don’t want to believe,” snapped Sheerin. “You’re afraid to believe. Look out the window!”

Theremon did so, and the psychologist continued without pausing. “Imagine Darkness — everywhere. No light, as far as you can see. The houses, the trees, the fields, the earth, the sky — black! And Stars thrown in, for all I know — whatever they are. Can you conceive it?”

“Yes, I can,” declared Theremon truculently.

And Sheerin slammed his fist down upon the table in sudden passion. “You lie! You can’t conceive that. Your brain wasn’t built for the conception any more than it was built for the conception of infinity or of eternity. You can only talk about it. A fraction of the reality upsets you, and when the real thing comes, your brain is going to be presented with the phenomenon outside its limits of comprehension. You will go mad, completely and permanently! There is no question of it!”

He added sadly, “And another couple of millennia of painful struggle comes to nothing. Tomorrow there won’t be a city standing unharmed in all Lagash.”

Theremon recovered part of his mental equilibrium. “That doesn’t follow. I still don’t see that I can go loony just because there isn’t a sun in the sky — but even if I did, and everyone else did, how does that harm the cities? Are we going to blow them down?”

But Sheerin was angry, too. “If you were in Darkness, what would you want more than anything else; what would it be that every instinct would call for? Light, damn you, light!”

“Well?”

“And how would you get light?”

“I don’t know,” said Theremon flatly.

“What’s the only way to get light, short of a sun?”

“How should I know?”

They were standing face to face and nose to nose.

Sheerin said, “You burn something, mister. Ever see a forest fire? Ever go camping and cook a stew over a wood fire? Heat isn’t the only thing burning wood gives off, you know. It gives off light, and people know that. And when it’s dark they want light, and they’re going to get it.”

“So they burn wood?”

“So they burn whatever they can get. They’ve got to have light. They’ve got to burn something, and wood isn’t handy — so they’ll burn whatever is nearest. They’ll have their light — and every center of habitation goes up in flames!”

Eyes held each other as though the whole matter were a personal affair of respective will powers, and then Theremon broke away wordlessly. His breathing was harsh and ragged, and he scarcely noted the sudden hubbub that came from the adjoining room behind the closed door.

Sheerin spoke, and it was with an effort that he made it sound matter-of-fact. “I think I heard Yimot’s voice. He and Faro are probably back. Let’s go in and see what kept them.”

“Might as well!” muttered Theremon. He drew a long breath and seemed to shake himself. The tension was broken.

The room was in an uproar, with members of the staff clustering about two young men who were removing outer garments even as they parried the miscellany of questions being thrown at them.

Aton bustled through the crowd and faced the newcomers angrily. “Do you realize that it’s less than half an hour before deadline? Where have you two been?”

Faro 24 seated himself and rubbed his hands. His cheeks were red with the outdoor chill. “Yimot and I have just finished carrying through a little crazy experiment of our own. We’ve been trying to see if we couldn’t construct an arrangement by which we could simulate the appearance of Darkness and Stars so as to get an advance notion as to how it looked.”

There was a confused murmur from the listeners, and a sudden look of interest entered Aton’s eyes. “There wasn’t anything said of this before. How did you go about it?”

“Well,” said Faro, “the idea came to Yimot and myself long ago, and we’ve been working it out in our spare time. Yimot knew of a low one-story house down in the city with a domed roof — it had once been used as a museum, I think. Anyway, we bought it — “

“Where did you get the money?” interrupted Aton peremptorily.

“Our bank accounts,” grunted Yimot 70. “It cost two thousand credits.” Then, defensively, “Well, what of it? Tomorrow, two thousand credits will be two thousand pieces of paper. That’s all.”

“Sure.” agreed Faro. “We bought the place and rigged it up with black velvet from top to bottom so as to get as perfect a Darkness as possible. Then we punched tiny holes in the ceiling and through the roof and covered them with little metal caps, all of which could be shoved aside simultaneously at the close of a switch. At least we didn’t do that part ourselves; we got a carpenter and an electrician and some others — money didn’t count. The point was that we could get the light to shine through those holes in the roof, so that we could get a starlike effect.”

Not a breath was drawn during the pause that followed. Aton said stiffly, “You had no right to make a private — “

Faro seemed abashed. “I know, sir — but frankly, Yimot and I thought the experiment was a little dangerous. If the effect really worked, we half expected to go mad — from what Sheerin says about all this, we thought that would be rather likely. We wanted to take the risk ourselves. Of course if we found we could retain sanity, it occurred to us that we might develop immunity to the real thing, and then expose the rest of you the same way. But things didn’t work out at all — “

“Why, what happened?”

It was Yimot who answered. “We shut ourselves in and allowed our eyes to get accustomed to the dark. It’s an extremely creepy feeling because the total Darkness makes you feel as if the walls and ceiling are crushing in on you. But we got over that and pulled the switch. The caps fell away and the roof glittered all over with little dots of light — “

“Well?”

“Well — nothing. That was the whacky part of it. Nothing happened. It was just a roof with holes in it, and that’s just what it looked like. We tried it over and over again — that’s what kept us so late — but there just isn’t any effect at all.”

There followed a shocked silence, and all eyes turned to Sheerin, who sat motionless, mouth open.

Theremon was the first to speak. “You know what this does to this whole theory you’ve built up, Sheerin, don’t you?” He was grinning with relief.

But Sheerin raised his hand. “Now wait a while. Just let me think this through.” And then he snapped his fingers, and when he lifted his head there was neither surprise nor uncertainty in his eyes. “Of course — “

He never finished. From somewhere up above there sounded a sharp clang, and Beenay, starting to his feet, dashed up the stairs with a ‘What the devil!”

The rest followed after.

Things happened quickly. Once up in the dome, Beenay cast one horrified glance at the shattered photographic plates and at the man bending over them; and then hurled himself fiercely at the intruder, getting a death grip on his throat. There was a wild threshing, and as others of the staff joined in, the stranger was swallowed up and smothered under the weight of half a dozen angry men.

Aton came up last, breathing heavily. “Let him up!”

There was a reluctant unscrambling and the stranger, panting harshly, with his clothes torn and his forehead bruised, was hauled to his feet. He had a short yellow beard curled elaborately in the style affected by the Cultists.

Beenay shifted his hold to a collar grip and shook the man savagely. “All right, rat, what’s the idea? These plates — “

“I wasn’t after them,” retorted the Cultist coldly. “That was an accident.”

Beenay followed his glowering stare and snarled, “I see. You were after the cameras themselves. The accident with the plates was a stroke of luck for you, then. If you had touched Snapping Bertha or any of the others, you would have died by slow torture. As it is — “ He drew his fist back.

Aton grabbed his sleeve. “Stop that! Let him go!”

The young technician wavered, and his arm dropped reluctantly. Aton pushed him aside and confronted the Cultist. “You’re Latimer, aren’t you?”

The Cultist bowed stiffly and indicated the symbol upon his hip. I am Latimer 25, adjutant of the third class to his serenity, Sor 5.”

“And” — Aton’s white eyebrows lifted — “you were with his serenity when he visited me last week, weren’t you?”

Latimer bowed a second time.

“Now, then, what do you want?”

“Nothing that you would give me of your own free will.”

“Sor 5 sent you, I suppose — or is this your own idea?”

“I won’t answer that question.”

“Will there be any further visitors?”

“I won’t answer that, either.”

Aton glanced at his timepiece and scowled. “Now, man, what is it your master wants of me? I have fulfilled my end of the bargain.”

Latimer smiled faintly, but said nothing.

“I asked him,” continued Aton angrily, “for data only the Cult could supply, and it was given to me. For that, thank you. In return I promised to prove the essential truth of the creed of the Cult.”

“There was no need to prove that,” came the proud retort. It stands proven by the Book of Revelations.”

“For the handful that constitute the Cult, yes. Don’t pretend to mistake my meaning. I offered to present scientific backing for your beliefs. And I did!”

The Cultist’s eyes narrowed bitterly. “Yes, you did — with a fox’s subtlety, for your pretended explanation backed our beliefs, and at the same time removed all necessity for them. You made of the Darkness and of the Stars a natural phenomenon and removed all its real significance. That was blasphemy.”

“If so, the fault isn’t mine. The facts exist. What can I do but state them?”

“Your “facts” are a fraud and a delusion.”

Aton stamped angrily. “How do you know?”

And the answer came with the certainty of absolute faith. “I know!”

The director purpled and Beenay whispered urgently. Aton waved him silent. “And what does Sor 5 want us to do? He still thinks. I suppose, that in trying to warn the world to take measures against the menace of madness, we are placing innumerable souls in jeopardy. We aren’t succeeding, if that means anything to him.”

“The attempt itself has done harm enough, and your vicious effort to gain information by means of your devilish instruments must be stopped. We obey the will of the Stars, and I only regret that my clumsiness prevented me from wrecking your infernal devices.”

“It wouldn’t have done you too much good,” returned Aton. “All our data, except for the direct evidence we intend collecting right now, is already safely cached and well beyond possibility of harm.” He smiled grimly. “But that does not affect your present status as an attempted burglar and criminal.”

He turned to the men behind him. “Someone call the police at Saro City.”

There was a cry of distaste from Sheerin. “Damn it, Aton, what’s wrong with you? There’s no time for that. Here” — he hustled his way forward — “let me handle this.”

Aton stared down his nose at the psychologist. “This is not the time for your monkeyshines, Sheerin. Will you please let me handle this my own way? Right now you are a complete outsider here, and don’t forget it.”

Sheerin’s mouth twisted eloquently. “Now why should we go to the impossible trouble of calling the police — with Beta’s eclipse a matter of minutes from now — when this young man here is perfectly willing to pledge his word of honor to remain and cause no trouble whatsoever?”

The Cultist answered promptly, “I will do no such thing. You’re free to do what you want, but it’s only fair to warn you that just as soon as I get my chance I’m going to finish what I came out here to do. If it’s my word of honor you’re relying on, you’d better call the police.”

Sheerin smiled in a friendly fashion. “You’re a determined cuss, aren’t you? Well, I’ll explain something. Do you see that young man at the window? He’s a strong, husky fellow, quite handy with his fists, and he’s an outsider besides. Once the eclipse starts there will be nothing for him to do except keep an eye on you. Besides him, there will be myself — a little too stout for active fisticuffs, but still able to help.”

“Well, what of it?” demanded Latimer frozenly.

“Listen and I’ll tell you,” was the reply. “Just as soon as the eclipse starts, we’re going to take you, Theremon and I, and deposit you in a little closet with one door, to which is attached one giant lock and no windows. You will remain there for the duration.”

“And afterward,” breathed Latimer fiercely, “there’ll be no one to let me out. I know as well as you do what the coming of the Stars means — I know it far better than you. With all your minds gone, you are not likely to free me. Suffocation or slow starvation, is it? About what I might have expected from a group of scientists. But I don’t give my word. It’s a matter of principle, and I won’t discuss it further.”

Aton seemed perturbed. His faded eyes were troubled.

“Really, Sheerin, locking him — “

“Please!” Sheerin motioned him impatiently to silence. “I don’t think for a moment things will go that far. Latimer has just tried a clever little bluff, but I’m not a psychologist just because I like the sound of the word.” He grinned at the Cultist. “Come now, you don’t really think I’m trying anything as crude as slow starvation. My dear Latimer, if I lock you in the closet, you are not going to see the Darkness, and you are not going to see the Stars. It does not take much knowledge of the fundamental creed of the Cult to realize that for you to be hidden from the Stars when they appear means the loss of your immortal soul. Now, I believe you to be an honorable man. I’ll accept your word of honor to make no further effort to disrupt proceedings, if you’ll offer it.”

A vein throbbed in Latimer’s temple, and he seemed to shrink within himself as he said thickly, “You have it!” And then he added with swift fury. “But it is my consolation that you will all be damned for your deeds of today.” He turned on his heel and stalked to the high three-legged stool by the door.

Sheerin nodded to the columnist. “Take a seat next to him, Theremon — just as a formality. Hey, Theremon!”

But the newspaperman didn’t move. He had gone pale to the lips. “Look at that!” The finger he pointed toward the sky shook, and his voice was dry and cracked.

There was one simultaneous gasp as every eye followed the pointing finger and, for one breathless moment, stared frozenly.

Beta was chipped on one side!

The tiny bit of encroaching blackness was perhaps the width of a fingernail, but to the staring watchers it magnified itself into the crack of doom.

Only for a moment they watched, and after that there was a shrieking confusion that was even shorter of duration and which gave way to an orderly scurry of activity — each man at his prescribed job. At the crucial moment there was no time for emotion. The men were merely scientists with work to do. Even Aton had melted away.

Sheerin said prosaically. “First contact must have been made fifteen minutes ago. A little early, but pretty good considering the uncertainties involved in the calculation.” He looked about him and then tiptoed to Theremon, who still remained staring out the window, and dragged him away gently.

“Aton is furious,” he whispered, “so stay away. He missed first contact on account of this fuss with Latimer, and if you get in his way he’ll have you thrown out the window.”

Theremon nodded shortly and sat down. Sheerin stared in surprise at him.

“The devil, man,” he exclaimed, “you’re shaking.”

“Eh?” Theremon licked dry lips and then tried to smile. “I don’t feel very well, and that’s a fact.”

The psychologist’s eyes hardened. “You’re not losing your nerve?”

“No!” cried Theremon in a flash of indignation. “Give me a chance, will you? I haven’t really believed this rigmarole — not way down beneath, anyway — till just this minute. Give me a chance to get used to the idea. You’ve been preparing yourself for two months or more.”

“You’re right, at that,” replied Sheerin thoughtfully. “Listen! Have you got a family — parents, wife, children?”

Theremon shook his head. “You mean the Hideout, I suppose. No, you don’t have to worry about that. I have a sister, but she’s two thousand miles away. I don’t even know her exact address.”

“Well, then, what about yourself? You’ve got time to get there, and they’re one short anyway, since I left. After all, you’re not needed here, and you’d make a darned fine addition — “

Theremon looked at the other wearily. “You think I’m scared stiff, don’t you? Well, get this, mister. I’m a newspaperman and I’ve been assigned to cover a story. I intend covering it.”

There was a faint smile on the psychologist’s face. “I see. Professional honor, is that it?”

“You might call it that. But, man. I’d give my right arm for another bottle of that sockeroo juice even half the size of the one you bogged. If ever a fellow needed a drink, I do.”

He broke off. Sheerin was nudging him violently. “Do you hear that? Listen!”

Theremon followed the motion of the other’s chin and stared at the Cultist, who, oblivious to all about him, faced the window, a look of wild elation on his face, droning to himself the while in singsong fashion.

“What’s he saying?” whispered the columnist.

“He’s quoting Book of Revelations, fifth chapter,” replied Sheerin. Then, urgently, “Keep quiet and listen, I tell you.”

The Cultist’s voice had risen in a sudden increase of fervor: ‘ “And it came to pass that in those days the Sun, Beta, held lone vigil in the sky for ever longer periods asthe revolutions passed; until such time as for full half a revolution, it alone, shrunken and cold, shone down upon Lagash.

“ ‘And men did assemble in the public squares and in the highways, there to debate and to marvel at the sight, for a strange depression had seized them. Their minds were troubled and their speech confused, for the souls of men awaited the coming of the Stars.

“ ‘And in the city of Trigon, at high noon, Vendret 2 came forth and said unto the men of Trigon, “Lo, ye sinners! Though ye scorn the ways of righteousness, yet will the time of reckoning come. Even now the Cave approaches to swallow Lagash; yea, and all it contains.”

“ ‘And even as he spoke the lip of the Cave of Darkness passed the edge of Beta so that to all Lagash it was hidden from sight. Loud were the cries of men as it vanished, and great the fear of soul that fell upon them.

“ ‘It came to pass that the Darkness of the Cave fell upon Lagash, and there was no light on all the surface of Lagash. Men were even as blinded, nor could one man see his neighbor, though he felt his breath upon his face.

“ ‘And in this blackness there appeared the Stars, in countless numbers, and to the strains of music of such beauty that the very leaves of the trees cried out in wonder.

“ ‘And in that moment the souls of men departed from them, and their abandoned bodies became even as beasts; yea, even as brutes of the wild; so that through the blackened streets of the cities of Lagash they prowled with wild cries.

“ ‘From the Stars there then reached down the Heavenly Flame, and where it touched, the cities of Lagash flamed to utter destruction, so that of man and of the works of man nought remained.

“ ‘Even then — ‘ ‘

There was a subtle change in Latimer’s tone. His eyes had not shifted, but somehow he had become aware of the absorbed attention of the other two. Easily, without pausing for breath, the timbre of his voice shifted and the syllables became more liquid.

Theremon, caught by surprise, stared. The words seemed on the border of familiarity. There was an elusive shift in the accent, a tiny change in the vowel stress; nothing more — yet Latimer had become thoroughly unintelligible.

Sheerin smiled slyly. “He shifted to some old-cycle tongue, probably their traditional second cycle. That was the language in which the Book of Revelations was originally written, you know.”

“It doesn’t matter; I’ve heard enough.” Theremon shoved his chair back and brushed his hair back with hands that no longer shook. “I feel much better now.”

“You do?” Sheerin seemed mildly surprised.

“I’ll say I do. I had a bad case of jitters just a while back. Listening to you and your gravitation and seeing that eclipse start almost finished me. But this” — he jerked a contemptuous thumb at the yellow-bearded Cultist — “this is the sort of thing my nurse used to tell me. I’ve been laughing at that sort of thing all my life. I’m not going to let it scare me now.”

He drew a deep breath and said with a hectic gaiety, “But if I expect to keep on the good side of myself. I’m going to turn my chair away from the window.”

Sheerin said, “Yes, but you’d better talk lower. Aton just lifted his head out of that box he’s got it stuck into and gave you a look that should have killed you.”

Theremon made a mouth. “I forgot about the old fellow.” With elaborate care he turned the chair from the window, cast one distasteful look over his shoulder, and said, “It has occurred to me that there must be considerable immunity against this Star madness.”

The psychologist did not answer immediately. Beta was past its zenith now, and the square of bloody sunlight that outlined the window upon the floor had lifted into Sheerin’s lap. He stared at its dusky color thoughtfully and then bent and squinted into the sun itself.

The chip in its side had grown to a black encroachment that covered a third of Beta. He shuddered, and when he straightened once more his florid cheeks did not contain quite as much color as they had had previously.

With a smile that was almost apologetic, he reversed his chair also. “There are probably two million people in Saro City that are all trying to join the Cult at once in one gigantic revival.” Then, ironically. “The Cult is in for an hour of unexampled prosperity. I trust they’ll make the most of it. Now, what was it you said?”

“Just this. How did the Cultists manage to keep the Book of Revelations going from cycle to cycle, and how on Lagash did it get written in the first place? There must have been some sort of immunity, for if everyone had gone mad, who would be left to write the book?”

Sheerin stared at his questioner ruefully. “Well, now, young man, there isn’t any eyewitness answer to that, but we’ve got a few damned good notions as to what happened. You see. there are three kinds of people who might remain relatively unaffected. First, the very few who don’t see the Stars at all: the seriously retarded or those who drink themselves into a stupor at the beginning of the eclipse and remain so to the end. We leave them out — because they aren’t really witnesses.

“Then there are children below six, to whom the world as a whole is too new and strange for them to be too frightened at Stars and Darkness. They would be just another item in an already surprising world. You see that, don’t you?”

The other nodded doubtfully. “I suppose so.”

“Lastly, there are those whose minds are too coarsely grained to be entirely toppled. The very insensitive would be scarcely affected — oh, such people as some of our older, work-broken peasants. Well, the children would have fugitive memories, and that, combined with the confused, incoherent babblings of the half-mad morons, formed the basis for the Book of Revelations.

“Naturally, the book was based, in the first place, on the testimony of those least qualified to serve as historians; that is, children and morons; and was probably edited and re-edited through the cycles.”

“Do you suppose,” broke in Theremon, “that they carried the book through the cycles the way we’re planning on handing on the secret of gravitation?”

Sheerin shrugged. “Perhaps, but their exact method is unimportant. They do it, somehow. The point I was getting at was that the book can’t help but be a mass of distortion, even if it is based on fact. For instance, do you remember the experiment with the holes in the roof that Faro and Yimot tried — the one that didn’t work?”

“Yes.”

“You know why it didn’t w — “ He stopped and rose in alarm, for Aton was approaching, his face a twisted mask of consternation. “What’s happened?”

Aton drew him aside and Sheerin could feel the fingers on his elbow twitching.

“Not so loud!” Aton’s voice was low and tortured. “I’ve just gotten word from the Hideout on the private line.”

Sheerin broke in anxiously. “They are in trouble?”

“Not they.” Aton stressed the pronoun significantly. “They sealed themselves off just a while ago, and they’re going to stay buried till day after tomorrow. They’re safe. But the city. Sheerin — it’s a shambles. You have no idea — “ He was having difficulty in speaking.

“Well?” snapped Sheerin impatiently. “What of it? It will get worse. What are you shaking about?” Then, suspiciously, “How do you feel?”

Aton’s eyes sparked angrily at the insinuation, and then faded to anxiety once more. “You don’t understand. The Cultists are active. They’re rousing the people to storm the Observatory — promising them immediate entrance into grace, promising them salvation, promising them anything. What are we to do, Sheerin?”

Sheerin’s head bent, and he stared in long abstraction at his toes. He tapped his chin with one knuckle, then looked up and said crisply, “Do? What is there to do? Nothing at all. Do the men know of this?”

“No, of course not!”

“Good! Keep it that way. How long till totality?”

“Not quite an hour.”

“There’s nothing to do but gamble. It will take time to organize any really formidable mob, and it will take more time to get them out here. We’re a good five miles from the city — “

He glared out the window, down the slopes to where the farmed patches gave way to clumps of white houses in the suburbs; down to where the metropolis itself was a blur on the horizon — a mist in the waning blaze of Beta.

He repeated without turning. “It will take time. Keep on working and pray that totality comes first.”

Beta was cut in half, the line of division pushing a slight concavity into the still-bright portion of the Sun. It was like a gigantic eyelid shutting slantwise over the light of a world.

The faint clatter of the room in which he stood faded into oblivion, and he sensed only the thick silence of the fields outside. The very insects seemed frightened mute. And things were dim.

He jumped at the voice in his ear. Theremon said. “Is something wrong?”

“Eh? Er — no. Get back to the chair. We’re in the way.” They slipped back to their comer, but the psychologist did not speak for a time. He lifted a finger and loosened his collar. He twisted his neck back and forth but found no relief. He looked up suddenly.

“Are you having any difficulty in breathing?”

The newspaperman opened his eyes wide and drew two or three long breaths. “No. Why?”

“I looked out the window too long, I suppose. The dimness got me. Difficulty in breathing is one of the first symptoms of a claustrophobic attack. “

Theremon drew another long breath. “Well, it hasn’t got me yet. Say, here’s another of the fellows.”

Beenay had interposed his bulk between the light and the pair in the corner, and Sheerin squinted up at him anxiously. “Hello, Beenay.”

The astronomer shifted his weight to the other foot and smiled feebly. “You won’t mind if I sit down awhile and join in the talk? My cameras are set, and there’s nothing to do till totality.” He paused and eyed the Cultist, who fifteen minutes earlier had drawn a small, skin-bound book from his sleeve and had been poring intently over it ever since.

“That rat hasn’t been making trouble, has he?”

Sheerin shook his head. His shoulders were thrown back and he frowned his concentration as he forced himself to breathe regularly. He said, “Have you had any trouble breathing, Beenay?”

Beenay sniffed the air in his turn. “It doesn’t seem stuffy to me.”

“A touch of claustrophobia,” explained Sheerin apologetically.

“Ohhh! It worked itself differently with me. I get the impression that my eyes are going back on me. Things seem to blur and — well, nothing is clear. And it’s cold, too.”

“Oh, it’s cold, all right. That’s no illusion.” Theremon grimaced. “My toes feel as if I’ve been shipping them cross-country in a refrigerating car.”

“What we need,” put in Sheerin, “is to keep our minds busy with extraneous affairs. I was telling you a while ago, Theremon, why Faro’s experiments with the holes in the roof came to nothing.”

“You were just beginning,” replied Theremon. He encircled a knee with both arms and nuzzled his chin against it.

“Well, as I started to say, they were misled by taking the Book of Revelations literally. There probably wasn’t any sense in attaching any physical significance to the Stars. It might be, you know, that in the presence of total Darkness, the mind finds it absolutely necessary to create light. This illusion of light might be all the Stars there really are.”

“In other words,” interposed Theremon, “you mean the Stars arc the results of the madness and not one of the causes. Then, what good will Beenay’s photographs be?”

“To prove that it is an illusion, maybe; or to prove the opposite; for all I know. Then again — “

But Beenay had drawn his chair closer, and there was an expression of sudden enthusiasm on his face. “Say, I’m glad you two got onto this subject.” His eyes narrowed and he lifted one finger. “I’ve been thinking about these Stars and I’ve got a really cute notion. Of course it’s strictly ocean foam, and I’m not trying to advance it seriously, but I think it’s interesting. Do you want to hear it?”

He seemed half reluctant, but Sheerin leaned back and said, “Go ahead! I’m listening.”

“Well, then, supposing there were other suns in the universe.” He broke off a little bashfully. “I mean suns that are so far away that they’re too dim to see. It sounds as if I’ve been reading some of that fantastic fiction, I suppose.”

“Not necessarily. Still, isn’t that possibility eliminated by the fact that, according to the Law of Gravitation, they would make themselves evident by their attractive forces?”

“Not if they were far enough off,” rejoined Beenay, “really far off — maybe as much as four light years, or even more. We’d never be able to detect perturbations then, because they’d be too small. Say that there were a lot of suns that far off; a dozen or two, maybe.”

Theremon whistled melodiously. “What an idea for a good Sunday supplement article. Two dozen suns in a universe eight light years across. Wow! That would shrink our world into insignificance. The readers would eat it up.”

“Only an idea,” said Beenay with a grin, “but you see the point. During an eclipse, these dozen suns would become visible because there’d be no real sunlight to drown them out. Since they’re so far off, they’d appear small, like so many little marbles. Of course the Cultists talk of millions of Stars, but that’s probably exaggeration. There just isn’t any place in the universe you could put a million suns — unless they touch one another.”

Sheerin had listened with gradually increasing interest. “You’ve hit something there, Beenay. And exaggeration is just exactly what would happen. Our minds, as you probably know, can’t grasp directly any number higher than five; above that there is only the concept of “many”. A dozen would become a million just like that. A damn good idea!”

“And I’ve got another cute little notion,” Beenay said. “Have you ever thought what a simple problem gravitation would be if only you had a sufficiently simple system? Supposing you had a universe in which there was a planet with only one sun. The planet would travel in a perfect ellipse and the exact nature of the gravitational force would be so evident it could be accepted as an axiom. Astronomers on such a world would start off with gravity probably before they even invented the telescope. Naked-eye observation would be enough.”

“But would such a system be dynamically stable?” questioned Sheerin doubtfully.

“Sure! They call it the “one-and-one” case. It’s been worked out mathematically, but it’s the philosophical implications that interest me.”

“It’s nice to think about,” admitted Sheerin, “as a pretty abstraction — like a perfect gas, or absolute zero.”

“Of course,” continued Beenay, “there’s the catch that life would be impossible on such a planet. It wouldn’t get enough heat and light, and if it rotated there would be total Darkness half of each day. You couldn’t expect life — which is fundamentally dependent upon light — to develop under those conditions. Besides — “

Sheerin’s chair went over backward as he sprang to his feet in a rude interruption. “Aton’s brought out the lights.”

Beenay said, “Huh,” turned to stare, and then grinned halfway around his head in open relief.

There were half a dozen foot-long, inch-thick rods cradled in Aton’s arms. He glared over them at the assembled staff members.

“Get back to work, all of you. Sheerin, come here and help me!”

Sheerin trotted to the older man’s side and, one by one, in utter silence, the two adjusted the rods in makeshift metal holders suspended from the walls.

With the air of one carrying through the most sacred item of a religious ritual, Sheerin scraped a large, clumsy match into spluttering life and passed it to Aton, who carried the flame to the upper end of one of the rods.

It hesitated there awhile, playing futilely about the tip, until a sudden, crackling flare cast Aton’s lined face into yellow highlights. He withdrew the match and a spontaneous cheer rattled the window.

The rod was topped by six inches of wavering flame! Methodically, the other rods were lighted, until six independent fires turned the rear of the room yellow.

The light was dim, dimmer even than the tenuous sunlight. The flames reeled crazily, giving birth to drunken, swaying shadows. The torches smoked devilishly and smelled like a bad day in the kitchen. But they emitted yellow light.

There was something about yellow light, after four hours of somber, dimming Beta. Even Latimer had lifted his eyes from his book and stared in wonder.

Sheerin warmed his hands at the nearest, regardless of the soot that gathered upon them in a fine, gray powder, and muttered ecstatically to himself. “Beautiful! Beautiful! I never realized before what a wonderful color yellow is.”

But Theremon regarded the torches suspiciously. He wrinkled his nose at the rancid odor and said, “What are those things?”

“Wood,” said Sheerin shortly.

“Oh, no, they’re not. They aren’t burning. The top inch is charred and the flame just keeps shooting up out of nothing.”

“That’s the beauty of it. This is a really efficient artificial-light mechanism. We made a few hundred of them, but most went to the Hideout, of course. You see” — he turned and wiped his blackened hands upon his handkerchief — “you take the pithy core of coarse water reeds, dry them thoroughly, and soak them in animal grease. Then you set fire to it and the grease burns, little by little. These torches will burn for almost half an hour without stopping. Ingenious, isn’t it? It was developed by one of our own young men at Saro University.”

After the momentary sensation, the dome had quieted. Latimer had carried his chair directly beneath a torch and continued reading, lips moving in the monotonous recital of invocations to the Stars. Beenay had drifted away to his cameras once more, and Theremon seized the opportunity to add to his notes on the article he was going to write for the Saro City Chronicle the next day — a procedure he had been following for the last two hours in a perfectly methodical, perfectly conscientious and, as he was well aware, perfectly meaningless fashion. But, as the gleam of amusement in Sheerin’s eyes indicated, careful note-taking occupied his mind with something other than the fact that the sky was gradually turning a horrible deep purple-red, as if it were one gigantic, freshly peeled beet; and so it fulfilled its purpose.

The air grew, somehow, denser. Dusk, like a palpable entity, entered the room, and the dancing circle of yellow light about the torches etched itself into ever-sharper distinction against the gathering grayness beyond. There was the odor of smoke and the presence of little chuckling sounds that the torches made as they burned; the soft pad of one of the men circling the table at which he worked, on hesitant tiptoes; the occasional indrawn breath of someone trying to retain composure in a world that was retreating into the shadow.

It was Theremon who first heard the extraneous noise. It was a vague, unorganized impression of sound that would have gone unnoticed but for the dead silence that prevailed within the dome.

The newsman sat upright and replaced his notebook. He held his breath and listened; then, with considerable reluctance, threaded his way between the solarscope and one of Beenay’s cameras and stood before the window.

The silence ripped to fragments at his startled shout: ‘Sheerin!”

Work stopped! The psychologist was at his side in a moment. Aton joined him. Even Yimot 70, high in his little lean-back seat at the eyepiece of the gigantic solarscope, paused and looked downward.

Outside, Beta was a mere smoldering splinter, taking one last desperate look at Lagash. The eastern horizon, in the direction of the city, was lost in Darkness, and the road from Saro to the Observatory was a dull-red line bordered on both sides by wooded tracts, the trees of which had somehow lost individuality and merged into a continuous shadowy mass.

But it was the highway itself that held attention, for along it there surged another, and infinitely menacing, shadowy mass.

Aton cried in a cracked voice, “The madmen from the city! They’ve come!”

“How long to totality?” demanded Sheerin.

“Fifteen minutes, but . . . but they’ll be here in five.”

“Never mind, keep the men working. We’ll hold them off. This place is built like a fortress. Aton, keep an eye on our young Cultist just for luck. Theremon, come with me.”

Sheerin was out the door, and Theremon was at his heels. The stairs stretched below them in tight, circular sweeps about the central shaft, fading into a dank and dreary grayness.

The first momentum of their rush had carried them fifty feet down, so that the dim, flickering yellow from the open door of the dome had disappeared and both above and below the same dusky shadow crushed in upon them.

Sheerin paused, and his pudgy hand clutched at his chest. His eyes bulged and his voice was a dry cough. “I can’t . . . breathe . . . Go down . . . yourself. Close all doors — “

Theremon took a few downward steps, then turned.

“Wait! Can you hold out a minute?” He was panting himself. The air passed in and out his lungs like so much molasses, and there was a little germ of screeching panic in his mind at the thought of making his way into the mysterious Darkness below by himself.

Theremon, after all, was afraid of the dark!

“Stay here,” he said. I’ll be back in a second.” He dashed upward two steps at a time, heart pounding — not altogether from the exertion — tumbled into the dome and snatched a torch from its holder. It was foul-smelling, and the smoke smarted his eyes almost blind, but he clutched that torch as if he wanted to kiss it for joy, and its flame streamed backward as he hurtled down the stairs again.

Sheerin opened his eyes and moaned as Theremon bent over him. Theremon shook him roughly. “All right, get a hold on yourself. We’ve got light.”

He held the torch at tiptoe height and, propping the tottering psychologist by an elbow, made his way downward in the middle of the protecting circle of illumination.

The offices on the ground floor still possessed what light there was, and Theremon felt the horror about him relax.

“Here,” he said brusquely, and passed the torch to Sheerin. “You can hear them outside.”

And they could. Little scraps of hoarse, wordless shouts.

But Sheerin was right; the Observatory was built like a fortress. Erected in the last century, when the neo-Gavottian style of architecture was at its ugly height, it had been designed for stability and durability rather than for beauty.

The windows were protected by the grillwork of inch-thick iron bars sunk deep into the concrete sills. The walls were solid masonry that an earthquake couldn’t have touched, and the main door was a huge oaken slab rein — forced with iron. Theremon shot the bolts and they slid shut with a dull clang.

At the other end of the corridor, Sheerin cursed weakly. He pointed to the lock of the back door which had been neatly jimmied into uselessness.

“That must be how Latimer got in,” he said.

“Well, don’t stand there,” cried Theremon impatiently. “Help drag up the furniture — and keep that torch out of my eyes. The smoke’s killing me.”

He slammed the heavy table up against the door as he spoke, and in two minutes had built a barricade which made up for what it lacked in beauty and symmetry by the sheer inertia of its massiveness.

Somewhere, dimly, far off, they could hear the battering of naked fists upon the door; and the screams and yells from outside had a sort of half reality.

That mob had set off from Saro City with only two things in mind: the attainment of Cultist salvation by the destruction of the Observatory, and a maddening fear that all but paralyzed them. There was no time to think of ground cars, or of weapons, or of leadership, or even of organization. They made for the Observatory on foot and assaulted it with bare hands.

And now that they were there, the last flash of Beta, the last ruby-red drop of flame, flickered feebly over a humanity that had left only stark, universal fear!

Theremon groaned, “Let’s get back to the dome!”

In the dome, only Yimot, at the solarscope, had kept his place. The rest were clustered about the cameras, and Beenay was giving his instructions in a hoarse, strained voice.

“Get it straight, all of you. I’m snapping Beta just before totality and changing the plate. That will leave one of you to each camera. You all know about . . . about times of exposure — “

There was a breathless murmur of agreement.

Beenay passed a hand over his eyes. “Are the torches still burning? Never mind, I see them!” He was leaning hard against the back of a chair. “Now remember, don’t. . . don’t try to look for good shots. Don’t waste time trying to get t-two stars at a time in the scope field. One is enough. And . . . and if you feel yourself going, get away from the camera.”

At the door, Sheerin whispered to Theremon, “Take me to Aton. I don’t see him.”

The newsman did not answer immediately. The vague forms of the astronomers wavered and blurred, and the torches overhead had become only yellow splotches.

“It’s dark,” he whimpered.

Sheerin held out his hand. “Aton.” He stumbled forward. “Aton!”

Theremon stepped after and seized his arm. “Wait, I’ll take you.” Somehow he made his way across the room. He closed his eyes against the Darkness and his mind against the chaos within it.

No one heard them or paid attention to them. Sheerin stumbled against the wall. “Aton!”

The psychologist felt shaking hands touching him, then withdrawing, a voice muttering, “Is that you, Sheerin?”

“Aton!” He strove to breathe normally. “Don’t worry about the mob. The place will hold them off.”

Latimer, the Cultist, rose to his feet, and his face twisted in desperation. His word was pledged, and to break it would mean placing his soul in mortal peril. Yet that word had been forced from him and had not been given freely. The Stars would come soon! He could not stand by and allow — And yet his word was pledged.

Beenay’s face was dimly flushed as it looked upward at Beta’s last ray, and Latimer, seeing him bend over his camera, made his decision. His nails cut the flesh of his palms as he tensed himself.

He staggered crazily as he started his rush. There was nothing before him but shadows; the very floor beneath his feet lacked substance. And then someone was upon him and he went down with clutching fingers at his throat.

He doubled his knee and drove it hard into his assailant. “Let me up or I’ll kill you.”

Theremon cried out sharply and muttered through a blinding haze of pain. “You double-crossing rat!”

The newsman seemed conscious of everything at once. He heard Beenay croak, “I’ve got it. At your cameras, men!” and then there was the strange awareness that the last thread of sunlight had thinned out and snapped.

Simultaneously he heard one last choking gasp from Beenay, and a queer little cry from Sheerin, a hysterical giggle that cut off in a rasp — and a sudden silence, a strange, deadly silence from outside.

And Latimer had gone limp in his loosening grasp. Theremon peered into the Cultist’s eyes and saw the blankness of them, staring upward, mirroring the feeble yellow of the torches. He saw the bubble of froth upon Latimer’s lips and heard the low animal whimper in Latimer’s throat.

With the slow fascination of fear, he lifted himself on one arm and turned his eyes toward the blood-curdling blackness of the window.

Through it shone the Stars!

Not Earth’s feeble thirty-six hundred Stars visible to the eye; Lagash was in the center of a giant cluster. Thirty thousand mighty suns shone down in a soul-searing splendor that was more frighteningly cold in its awful indifference than the bitter wind that shivered across the cold, horribly bleak world.

Theremon staggered to his feet, his throat, constricting him to breathlessness, all the muscles of his body writhing in an intensity of terror and sheer fear beyond bearing. He was going mad and knew it, and somewhere deep inside a bit of sanity was screaming, struggling to fight off the hopeless flood of black terror. It was very horrible to go mad and know that you were going mad — to know that in a little minute you would be here physically and yet all the real essence would be dead and drowned in the black madness. For this was the Dark — the Dark and the Cold and the Doom. The bright walls of the universe were shattered and their awful black fragments were falling down to crush and squeeze and obliterate him.

He jostled someone crawling on hands and knees, but stumbled somehow over him. Hands groping at his tortured throat, he limped toward the flame of the torches that filled all his mad vision.

“Light!” he screamed.

Aton, somewhere, was crying, whimpering horribly like a terribly frightened child. “Stars — all the Stars — we didn’t know at all. We didn’t know anything. We thought six stars in a universe is something the Stars didn’t notice is Darkness forever and ever and ever and the walls are breaking in and we didn’t know we couldn’t know and anything — “

Someone clawed at the torch, and it fell and snuffed out. In the instant, the awful splendor of the indifferent Stars leaped nearer to them.

On the horizon outside the window, in the direction of Saro City, a crimson glow began growing, strengthening in brightness, that was not the glow of a sun.

The long night had come again.

The Last Answer

Here comes the second short story of Isaac Asimov:

Note: This short story is taken from here.

The Last Answer by Isaac Asimov — © 1980

Murray Templeton was forty-five years old, in the prime of life, and with all parts of his body in perfect working order except for certain key portions of his coronary arteries, but that was enough.

The pain had come suddenly, had mounted to an unbearable peak, and had then ebbed steadily.  He could feel his breath slowing and a kind of gathering peace washing over him.

There is no pleasure like the absence of pain – immediately after pain.  Murray felt an almost giddy lightness as though he were lifting in the air and hovering.

He opened his eyes and noted with distant amusement that the others in the room were still agitated.  He had been in the laboratory when the pain had struck, quite without warning, and when he had staggered, he had heard surprised outcries from the others before everything vanished into overwhelming agony.

Now, with the pain gone, the others were still hovering, still anxious, still gathered about his fallen body –– Which, he suddenly realised, he was looking down on.

He was down there, sprawled, face contorted.  He was up here, at peace and watching.

He thought: Miracle of miracles!  The life-after-life nuts were right.

And although that was a humiliating way for an atheistic physicist to die, he felt only the mildest surprise, and no alteration of the peace in which he was immersed.

He thought: There should be some angel – or something – coming for me.

The Earthly scene was fading.  Darkness was invading his consciousness and off in a distance, as a last glimmer of sight, there was a figure of light, vaguely human in form, and radiating warmth.

Murray thought: What a joke on me.  I’m going to Heaven.

Even as he thought that, the light faded, but the warmth remained.  There was no lessening of the peace even though in all the Universe only he remained – and the Voice.

The Voice said, “I have done this so often and yet I still have the capacity to be pleased at success.”

It was in Murray’s mind to say something, but he was not conscious of possessing a mouth, tongue, or vocal chords.  Nevertheless, tried to make a sound.  He tried, mouthlessly, to hum words or breathe them or just push them out by a contraction of – something.

And they came out.  He heard his own voice, quite recognisable, and his own words, infinitely clear.

Murray said, “Is this Heaven?”

The Voice said, “This is no place as you understand place.”

Murray was embarrassed, but the next question had to be asked.  “Pardon me if I sound like a jackass.  Are you God?”

Without changing intonation or in any way marring the perfection of the sound, the Voice managed to sound amused.  “It is strange that I am always asked that in, of course, an infinite number of ways.  There is no answer I can give that you would comprehend.  I am – which is all that I can say significantly and you may cover that with any word or concept you please.”

Murray said, “And what am I?  A soul?  Or am I only personified existence too?”  He tried not to sound sarcastic, but it seemed to him that he had failed.  He thought then, fleetingly, of adding a ‘Your Grace’ or ‘Holy One’ or something to counteract the sarcasm, and could not bring himself to do so even though for the first time in his existence he speculated on the possibility of being punished for his insolence – or sin? – with Hell, and what that might be like.

The Voice did not sound offended.  “You are easy to explain – even to you.  You may call yourself a soul if that pleases you, but what you are is a nexus of electromagnetic forces, so arranged that all the interconnections and interrelationships are exactly imitative of those of your brain in your Universe-existence – down to the smallest detail.  Therefore you have your capacity for thought, your memories, your personality.  It still seems to you that you are you.”

Murray found himself incredulous.  “You mean the essence of my brain was permanent?”

“Not at all.  There is nothing about you that is permanent except what I choose to make so.  I formed the nexus.  I constructed it while you had physical existence and adjusted it to the moment when the existence failed.”

The Voice seemed distinctly pleased with itself, and went on after a moment’s pause.  “An intricate but entirely precise construction.  I could, of course, do it for every human being on your world but I am pleased that I do not.  There is pleasure in the selection.”

“You choose very few then?”

“Very few.”

“And what happens to the rest?”

“Oblivion! – Oh, of course, you imagine a Hell.”

Murray would have flushed if he had the capacity to do so.  He said, “I do not.  It is spoken of.  Still, I would scarcely have thought I was virtuous enough to have attracted your attention as one of the Elect.”

“Virtuous? – Ah, I see what you mean.  It is troublesome to have to force my thinking small enough to permeate yours.  No, I have chosen you for your capacity for thought, as I choose others, in quadrillions, from all the intelligent species of the Universe.”

Murray found himself suddenly curious, the habit of a lifetime.  He said, “Do you choose them all yourself or are there others like you?”

For a fleeting moment, Murray thought there was an impatient reaction to that, but when the Voice came, it was unmoved.  “Whether or not there are others is irrelevant to you.  This Universe is mine, and mine alone.  It is my invention, my construction, intended for my purpose alone.”

“And yet with quadrillions of nexi you have formed, you spend time with me?  Am I that important?”

The Voice said, “You are not important at all.  I am also with others in a way which, to your perception, would seem simultaneous.”

“And yet you are one?”

Again amusement.  The Voice said, “You seek to trap me into an inconsistency.  If you were an amoeba who could consider individuality only in connection with single cells and if you were to ask a sperm whale, made up of thirty quadrillion cells, whether it was one or many, how could the sperm whale answer in a way that would be comprehensible to the amoeba?”

Murray said dryly, “I’ll think about it.  It may become comprehensible.”

“Exactly.  That is your function.  You will think.”

“To what end?  You already know everything, I suppose.”

The Voice said, “Even if I knew everything, I could not know that I know everything.”

Murray said, “That sounds like a bit of Eastern philosophy – something that sounds profound precisely because it has no meaning.”

The Voice said, “You have promise.  You answer my paradox with a paradox – except that mine is not a paradox.  Consider.  I have existed eternally, but what does that mean?  It means I cannot remember having come into existence.  If I could, I would not have existed eternally.  If I cannot remember having come into existence, then there is at least one thing – the nature of my coming into existence – that I do not know.

“Then, too, although what I know is infinite, it is also true that what there is to know is infinite, and how can I be sure that both infinities are equal?  The infinity of potential knowledge may be infinitely greater than the infinity of my actual knowledge.  Here is a simple example: If I knew every one of the even integers, I would know an infinite number of items, and yet I would still not know a single odd integer.”

Murray said, “But the odd integers can be derived.  If you divide every even integer in the entire infinite series by two, you will get another infinite series which will contain within it the infinite series of odd integers.”

The Voice said, “You have the idea.  I am pleased.  It will be your task to find other such ways, far more difficult ones, from the known to the not-yet-known.  You have your memories.  You will remember all the data you have ever collected or learned, or that you have or will deduce from that data.  If necessary, you will be allowed to learn what additional data you will consider relevant to the problems you set yourself.”

“Could you not do all that for yourself?”

The Voice said, “I can, but it is more interesting this way.  I constructed the Universe in order to have more facts to deal with.  I inserted the uncertainty principle, entropy, and other randomisation factors to make the whole not instantly obvious.  It has worked well for it has amused me throughout its entire existence.

“I then allowed complexities that produced first life and then intelligence, and use it as a source for a research team, not because I need the aid, but because it would introduce a new random factor.  I found I could not predict the next interesting piece of knowledge gained, where it would come from, by what means derived.”

Murray said, “Does that ever happen?”

“Certainly.  A century doesn’t pass in which some interesting item doesn’t appear somewhere.”

“Something that you could have thought of yourself, but had not done so yet?”

“Yes.”

Murray said, “Do you actually think there’s a chance of my obliging you in this manner?”

“In the next century?  Virtually none.  In the long run, though, your success is certain, since you will be engaged eternally.”

Murray said, “I will be thinking through eternity?  Forever?”

“Yes.”

“To what end?”

“I have told you.  To find new knowledge.”

“But beyond that.  For what purpose am I to find new knowledge?”

“It was what you did in your Universe-bound life.  What was its purpose then?”

Murray said, “To gain new knowledge that only I could gain.  To receive the praise of my fellows.  To feel the satisfaction of accomplishment knowing that I had only a short time allotted me for the purpose. – Now I would gain only what you could gain yourself if you wished to take a small bit of trouble.  You cannot praise me; you can only be amused.  And there is no credit or satisfaction in accomplishment when I have all eternity to do it in.”

The Voice said, “And you do not find thought and discovery worthwhile in itself?  You do not find it requiring no further purpose?”

“For a finite time, yes.  Not for all eternity.”

“I see your point.  Nevertheless, you have no choice.”

“You say I am to think.  You cannot make me do so.”

The Voice said, “I do not wish to constrain you directly.  I will not need to.  Since you can do nothing but think, you will think.  You do not know how not to think.”

“Then I will give myself a goal.  I will invent a purpose.”

The Voice said tolerantly, “That you can certainly do.”

“I have already found a purpose.”

“May I know what it is?”

“You know already.  I know we are not speaking in the ordinary fashion.  You adjust my nexus is such a way that I believe I hear you and I believe I speak, but you transfer thoughts to me and from me directly.  And when my nexus changes with my thoughts you are at once aware of them and do not need my voluntary transmission.”

The Voice said, “You are surprisingly correct.  I am pleased. – But it also pleases me to have you tell me your thoughts voluntarily.”

“Then I will tell you.  The purpose of my thinking will be to discover a way to disrupt this nexus of me that you have created.  I do not want to think for no purpose but to amuse you.  I do not want to think forever to amuse you.  I do not want to exist forever to amuse you.  All my thinking will be directed toward ending the nexus.  That would amuse me.”

The Voice said, “I have no objection to that.  Even concentrated thought on ending your own existence may, in spite of you, come up with something new and interesting.  And, of course, if you succeed in this suicide attempt you will have accomplished nothing, for I would instantly reconstruct you and in such a way as to make your method of suicide impossible.  And if you found another and still more subtle fashion of disrupting yourself, I would reconstruct you with that possibility eliminated, and so on.  It could be an interesting game, but you will nevertheless exist eternally.  It is my will.”

Murray felt a quaver but the words came out with a perfect calm.  “Am I in Hell then, after all?  You have implied there is none, but if this were Hell you would lie to us as part of the game of Hell.”

The Voice said, “In that case, of what use is it to assure you that you are not in Hell?  Nevertheless, I assure you.  There is here neither Heaven nor Hell.  There is only myself.”

Murray said, “Consider, then, that my thoughts may be useless to you.  If I come up with nothing useful, will it not be worth your while to – disassemble me and take no further trouble with me?”

“As a reward?  You want Nirvana as the prize of failure and you intend to assure me failure?  There is no bargain there.  You will not fail.  With all eternity before you, you cannot avoid having at least one interesting thought, however you try against it.”

“Then I will create another purpose for myself.  I will not try to destroy myself.  I will set as my goal the humiliation of you.  I will think of something you have not only never thought of but never could think of.  I will think of the last answer, beyond which there is no knowledge further.”

The Voice said, “You do not understand the nature of the infinite.  There may be things I have not yet troubled to know.  There cannot be anything I cannot know.”

Murray said thoughtfully, “You cannot know your beginning.  You have said so.  Therefore you cannot know your end.  Very well, then.  That will be my purpose and that will be the last answer.  I will not destroy myself.  I will destroy you – if you do not destroy me first.”

The Voice said, “Ah!  You come to that in rather less than average time.  I would have thought it would have taken you longer.  There is not one of those I have with me in this existence of perfect and eternal thought that does not have the ambition of destroying me.  It cannot be done.”

Murray said, “I have all eternity to think of a way of destroying you.”

The Voice said, equably, “Then try to think of it.”  And it was gone.

But Murray had his purpose now and was content.

For what could any Entity, conscious of eternal existence, want – but an end?

For what else had the Voice been searching for countless billions of years?  And for what other reason had intelligence been created and certain specimens salvaged and put to work, but to aid in that great search?  And Murray intended that it would be he, and he alone, who would succeed.

Carefully, and with the thrill of purpose, Murray began to think.

He had plenty of time.

The Last Question

Beside being a genius, “Isaac Asimov” was surely one of the most fascinating science fiction author of his age.
He had written many good novels like Robot and Foundation series. But beside his novels, he had written many thought provoking short stories as well. “The Last Question” was one of them. About this short story, he “privately” concluded that it was his best science fiction yet written. This story starts in 2061 and continues in 4 different time frames. I hope you’ll enjoy this short story as well as like me. I’m going to put some of his other short-stories too.

Note: This short story is taken from there.

The Last Question by Isaac Asimov © 1956

The Reverse Entropy

The last question was asked for the first time, half in jest, on May 21, 2061, at a time when humanity first stepped into the light. The question came about as a result of a five dollar bet over highballs, and it happened this way:

Alexander Adell and Bertram Lupov were two of the faithful attendants of Multivac. As well as any human beings could, they knew what lay behind the cold, clicking, flashing face — miles and miles of face — of that giant computer. They had at least a vague notion of the general plan of relays and circuits that had long since grown past the point where any single human could possibly have a firm grasp of the whole.

Multivac was self-adjusting and self-correcting. It had to be, for nothing human could adjust and correct it quickly enough or even adequately enough — so Adell and Lupov attended the monstrous giant only lightly and superficially, yet as well as any men could. They fed it data, adjusted questions to its needs and translated the answers that were issued. Certainly they, and all others like them, were fully entitled to share In the glory that was Multivac’s.

For decades, Multivac had helped design the ships and plot the trajectories that enabled man to reach the Moon, Mars, and Venus, but past that, Earth’s poor resources could not support the ships. Too much energy was needed for the long trips. Earth exploited its coal and uranium with increasing efficiency, but there was only so much of both.

But slowly Multivac learned enough to answer deeper questions more fundamentally, and on May 14, 2061, what had been theory, became fact.

The energy of the sun was stored, converted, and utilized directly on a planet-wide scale. All Earth turned off its burning coal, its fissioning uranium, and flipped the switch that connected all of it to a small station, one mile in diameter, circling the Earth at half the distance of the Moon. All Earth ran by invisible beams of sunpower.

Seven days had not sufficed to dim the glory of it and Adell and Lupov finally managed to escape from the public function, and to meet in quiet where no one would think of looking for them, in the deserted underground chambers, where portions of the mighty buried body of Multivac showed. Unattended, idling, sorting data with contented lazy clickings, Multivac, too, had earned its vacation and the boys appreciated that. They had no intention, originally, of disturbing it.

They had brought a bottle with them, and their only concern at the moment was to relax in the company of each other and the bottle.

“It’s amazing when you think of it,” said Adell. His broad face had lines of weariness in it, and he stirred his drink slowly with a glass rod, watching the cubes of ice slur clumsily about. “All the energy we can possibly ever use for free. Enough energy, if we wanted to draw on it, to melt all Earth into a big drop of impure liquid iron, and still never miss the energy so used. All the energy we could ever use, forever and forever and forever.”

Lupov cocked his head sideways. He had a trick of doing that when he wanted to be contrary, and he wanted to be contrary now, partly because he had had to carry the ice and glassware. “Not forever,” he said.

“Oh, hell, just about forever. Till the sun runs down, Bert.”

“That’s not forever.”

“All right, then. Billions and billions of years. Twenty billion, maybe. Are you satisfied?”

Lupov put his fingers through his thinning hair as though to reassure himself that some was still left and sipped gently at his own drink. “Twenty billion years isn’t forever.”

“Will, it will last our time, won’t it?”

“So would the coal and uranium.”

“All right, but now we can hook up each individual spaceship to the Solar Station, and it can go to Pluto and back a million times without ever worrying about fuel. You can’t do THAT on coal and uranium. Ask Multivac, if you don’t believe me.”

“I don’t have to ask Multivac. I know that.”

“Then stop running down what Multivac’s done for us,” said Adell, blazing up. “It did all right.”

“Who says it didn’t? What I say is that a sun won’t last forever. That’s all I’m saying. We’re safe for twenty billion years, but then what?” Lupov pointed a slightly shaky finger at the other. “And don’t say we’ll switch to another sun.”

There was silence for a while. Adell put his glass to his lips only occasionally, and Lupov’s eyes slowly closed. They rested.

Then Lupov’s eyes snapped open. “You’re thinking we’ll switch to another sun when ours is done, aren’t you?”

“I’m not thinking.”

“Sure you are. You’re weak on logic, that’s the trouble with you. You’re like the guy in the story who was caught in a sudden shower and Who ran to a grove of trees and got under one. He wasn’t worried, you see, because he figured when one tree got wet through, he would just get under another one.”

“I get it,” said Adell. “Don’t shout. When the sun is done, the other stars will be gone, too.”

“Darn right they will,” muttered Lupov. “It all had a beginning in the original cosmic explosion, whatever that was, and it’ll all have an end when all the stars run down. Some run down faster than others. Hell, the giants won’t last a hundred million years. The sun will last twenty billion years and maybe the dwarfs will last a hundred billion for all the good they are. But just give us a trillion years and everything will be dark. Entropy has to increase to maximum, that’s all.”

“I know all about entropy,” said Adell, standing on his dignity.

“The hell you do.”

“I know as much as you do.”

“Then you know everything’s got to run down someday.”

“All right. Who says they won’t?”

“You did, you poor sap. You said we had all the energy we needed, forever. You said ‘forever.’”

“It was Adell’s turn to be contrary. “Maybe we can build things up again someday,” he said.

“Never.”

“Why not? Someday.”

“Never.”

“Ask Multivac.”

“You ask Multivac. I dare you. Five dollars says it can’t be done.”

Adell was just drunk enough to try, just sober enough to be able to phrase the necessary symbols and operations into a question which, in words, might have corresponded to this: Will mankind one day without the net expenditure of energy be able to restore the sun to its full youthfulness even after it had died of old age?

Or maybe it could be put more simply like this: How can the net amount of entropy of the universe be massively decreased?

Multivac fell dead and silent. The slow flashing of lights ceased, the distant sounds of clicking relays ended.

Then, just as the frightened technicians felt they could hold their breath no longer, there was a sudden springing to life of the teletype attached to that portion of Multivac. Five words were printed: INSUFFICIENT DATA FOR MEANINGFUL ANSWER.

“No bet,” whispered Lupov. They left hurriedly.

By next morning, the two, plagued with throbbing head and cottony mouth, had forgotten about the incident.

Jerrodd, Jerrodine, and Jerrodette I and II watched the starry picture in the visiplate change as the passage through hyperspace was completed in its non-time lapse. At once, the even powdering of stars gave way to the predominance of a single bright marble-disk, centered.

“That’s X-23,” said Jerrodd confidently. His thin hands clamped tightly behind his back and the knuckles whitened.

The little Jerrodettes, both girls, had experienced the hyperspace passage for the first time in their lives and were self-conscious over the momentary sensation of inside-outness. They buried their giggles and chased one another wildly about their mother, screaming, “We’ve reached X-23 — we’ve reached X-23 — we’ve —-”

“Quiet, children,” said Jerrodine sharply. “Are you sure, Jerrodd?”

“What is there to be but sure?” asked Jerrodd, glancing up at the bulge of featureless metal just under the ceiling. It ran the length of the room, disappearing through the wall at either end. It was as long as the ship.

Jerrodd scarcely knew a thing about the thick rod of metal except that it was called a Microvac, that one asked it questions if one wished; that if one did not it still had its task of guiding the ship to a preordered destination; of feeding on energies from the various Sub-galactic Power Stations; of computing the equations for the hyperspacial jumps.

Jerrodd and his family had only to wait and live in the comfortable residence quarters of the ship.

Someone had once told Jerrodd that the “ac” at the end of “Microvac” stood for “analog computer” in ancient English, but he was on the edge of forgetting even that.

Jerrodine’s eyes were moist as she watched the visiplate. “I can’t help it. I feel funny about leaving Earth.”

“Why for Pete’s sake?” demanded Jerrodd. “We had nothing there. We’ll have everything on X-23. You won’t be alone. You won’t be a pioneer. There are over a million people on the planet already. Good Lord, our great grandchildren will be looking for new worlds because X-23 will be overcrowded.”

Then, after a reflective pause, “I tell you, it’s a lucky thing the computers worked out interstellar travel the way the race is growing.”

“I know, I know,” said Jerrodine miserably.

Jerrodette I said promptly, “Our Microvac is the best Microvac in the world.”

“I think so, too,” said Jerrodd, tousling her hair.

It was a nice feeling to have a Microvac of your own and Jerrodd was glad he was part of his generation and no other. In his father’s youth, the only computers had been tremendous machines taking up a hundred square miles of land. There was only one to a planet. Planetary ACs they were called. They had been growing in size steadily for a thousand years and then, all at once, came refinement. In place of transistors had come molecular valves so that even the largest Planetary AC could be put into a space only half the volume of a spaceship.

Jerrodd felt uplifted, as he always did when he thought that his own personal Microvac was many times more complicated than the ancient and primitive Multivac that had first tamed the Sun, and almost as complicated as Earth’s Planetary AC (the largest) that had first solved the problem of hyperspatial travel and had made trips to the stars possible.

“So many stars, so many planets,” sighed Jerrodine, busy with her own thoughts. “I suppose families will be going out to new planets forever, the way we are now.”

“Not forever,” said Jerrodd, with a smile. “It will all stop someday, but not for billions of years. Many billions. Even the stars run down, you know. Entropy must increase.”

“What’s entropy, daddy?” shrilled Jerrodette II.

“Entropy, little sweet, is just a word which means the amount of running-down of the universe. Everything runs down, you know, like your little walkie-talkie robot, remember?”

“Can’t you just put in a new power-unit, like with my robot?”

The stars are the power-units, dear. Once they’re gone, there are no more power-units.”

Jerrodette I at once set up a howl. “Don’t let them, daddy. Don’t let the stars run down.”

“Now look what you’ve done, ” whispered Jerrodine, exasperated.

“How was I to know it would frighten them?” Jerrodd whispered back.

“Ask the Microvac,” wailed Jerrodette I. “Ask him how to turn the stars on again.”

“Go ahead,” said Jerrodine. “It will quiet them down.” (Jerrodette II was beginning to cry, also.)

Jarrodd shrugged. “Now, now, honeys. I’ll ask Microvac. Don’t worry, he’ll tell us.”

He asked the Microvac, adding quickly, “Print the answer.”

Jerrodd cupped the strip of thin cellufilm and said cheerfully, “See now, the Microvac says it will take care of everything when the time comes so don’t worry.”

Jerrodine said, “and now children, it’s time for bed. We’ll be in our new home soon.”

Jerrodd read the words on the cellufilm again before destroying it: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.

He shrugged and looked at the visiplate. X-23 was just ahead.

VJ-23X of Lameth stared into the black depths of the three-dimensional, small-scale map of the Galaxy and said, “Are we ridiculous, I wonder, in being so concerned about the matter?”

MQ-17J of Nicron shook his head. “I think not. You know the Galaxy will be filled in five years at the present rate of expansion.”

Both seemed in their early twenties, both were tall and perfectly formed.

“Still,” said VJ-23X, “I hesitate to submit a pessimistic report to the Galactic Council.”

“I wouldn’t consider any other kind of report. Stir them up a bit. We’ve got to stir them up.”

VJ-23X sighed. “Space is infinite. A hundred billion Galaxies are there for the taking. More.”

“A hundred billion is not infinite and it’s getting less infinite all the time. Consider! Twenty thousand years ago, mankind first solved the problem of utilizing stellar energy, and a few centuries later, interstellar travel became possible. It took mankind a million years to fill one small world and then only fifteen thousand years to fill the rest of the Galaxy. Now the population doubles every ten years –”

VJ-23X interrupted. “We can thank immortality for that.”

“Very well. Immortality exists and we have to take it into account. I admit it has its seamy side, this immortality. The Galactic AC has solved many problems for us, but in solving the problems of preventing old age and death, it has undone all its other solutions.”

“Yet you wouldn’t want to abandon life, I suppose.”

“Not at all,” snapped MQ-17J, softening it at once to, “Not yet. I’m by no means old enough. How old are you?”

“Two hundred twenty-three. And you?”

“I’m still under two hundred. –But to get back to my point. Population doubles every ten years. Once this Galaxy is filled, we’ll have another filled in ten years. Another ten years and we’ll have filled two more. Another decade, four more. In a hundred years, we’ll have filled a thousand Galaxies. In a thousand years, a million Galaxies. In ten thousand years, the entire known Universe. Then what?”

VJ-23X said, “As a side issue, there’s a problem of transportation. I wonder how many sunpower units it will take to move Galaxies of individuals from one Galaxy to the next.”

“A very good point. Already, mankind consumes two sunpower units per year.”

“Most of it’s wasted. After all, our own Galaxy alone pours out a thousand sunpower units a year and we only use two of those.”

“Granted, but even with a hundred per cent efficiency, we can only stave off the end. Our energy requirements are going up in geometric progression even faster than our population. We’ll run out of energy even sooner than we run out of Galaxies. A good point. A very good point.”

“We’ll just have to build new stars out of interstellar gas.”

“Or out of dissipated heat?” asked MQ-17J, sarcastically.

“There may be some way to reverse entropy. We ought to ask the Galactic AC.”

VJ-23X was not really serious, but MQ-17J pulled out his AC-contact from his pocket and placed it on the table before him.

“I’ve half a mind to,” he said. “It’s something the human race will have to face someday.”

He stared somberly at his small AC-contact. It was only two inches cubed and nothing in itself, but it was connected through hyperspace with the great Galactic AC that served all mankind. Hyperspace considered, it was an integral part of the Galactic AC.

MQ-17J paused to wonder if someday in his immortal life he would get to see the Galactic AC. It was on a little world of its own, a spider webbing of force-beams holding the matter within which surges of sub-mesons took the place of the old clumsy molecular valves. Yet despite it’s sub-etheric workings, the Galactic AC was known to be a full thousand feet across.

MQ-17J asked suddenly of his AC-contact, “Can entropy ever be reversed?”

VJ-23X looked startled and said at once, “Oh, say, I didn’t really mean to have you ask that.”

“Why not?”

“We both know entropy can’t be reversed. You can’t turn smoke and ash back into a tree.”

“Do you have trees on your world?” asked MQ-17J.

The sound of the Galactic AC startled them into silence. Its voice came thin and beautiful out of the small AC-contact on the desk. It said: THERE IS INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.

VJ-23X said, “See!”

The two men thereupon returned to the question of the report they were to make to the Galactic Council.

Zee Prime’s mind spanned the new Galaxy with a faint interest in the countless twists of stars that powdered it. He had never seen this one before. Would he ever see them all? So many of them, each with its load of humanity – but a load that was almost a dead weight. More and more, the real essence of men was to be found out here, in space.

Minds, not bodies! The immortal bodies remained back on the planets, in suspension over the eons. Sometimes they roused for material activity but that was growing rarer. Few new individuals were coming into existence to join the incredibly mighty throng, but what matter? There was little room in the Universe for new individuals.

Zee Prime was roused out of his reverie upon coming across the wispy tendrils of another mind.

“I am Zee Prime,” said Zee Prime. “And you?”

“I am Dee Sub Wun. Your Galaxy?”

“We call it only the Galaxy. And you?”

“We call ours the same. All men call their Galaxy their Galaxy and nothing more. Why not?”

“True. Since all Galaxies are the same.”

“Not all Galaxies. On one particular Galaxy the race of man must have originated. That makes it different.”

Zee Prime said, “On which one?”

“I cannot say. The Universal AC would know.”

“Shall we ask him? I am suddenly curious.”

Zee Prime’s perceptions broadened until the Galaxies themselves shrunk and became a new, more diffuse powdering on a much larger background. So many hundreds of billions of them, all with their immortal beings, all carrying their load of intelligences with minds that drifted freely through space. And yet one of them was unique among them all in being the originals Galaxy. One of them had, in its vague and distant past, a period when it was the only Galaxy populated by man.

Zee Prime was consumed with curiosity to see this Galaxy and called, out: “Universal AC! On which Galaxy did mankind originate?”

The Universal AC heard, for on every world and throughout space, it had its receptors ready, and each receptor lead through hyperspace to some unknown point where the Universal AC kept itself aloof.

Zee Prime knew of only one man whose thoughts had penetrated within sensing distance of Universal AC, and he reported only a shining globe, two feet across, difficult to see.

“But how can that be all of Universal AC?” Zee Prime had asked.

“Most of it, ” had been the answer, “is in hyperspace. In what form it is there I cannot imagine.”

Nor could anyone, for the day had long since passed, Zee Prime knew, when any man had any part of the making of a universal AC. Each Universal AC designed and constructed its successor. Each, during its existence of a million years or more accumulated the necessary data to build a better and more intricate, more capable successor in which its own store of data and individuality would be submerged.

The Universal AC interrupted Zee Prime’s wandering thoughts, not with words, but with guidance. Zee Prime’s mentality was guided into the dim sea of Galaxies and one in particular enlarged into stars.

A thought came, infinitely distant, but infinitely clear. “THIS IS THE ORIGINAL GALAXY OF MAN.”

But it was the same after all, the same as any other, and Zee Prime stifled his disappointment.

Dee Sub Wun, whose mind had accompanied the other, said suddenly, “And Is one of these stars the original star of Man?”

The Universal AC said, “MAN’S ORIGINAL STAR HAS GONE NOVA. IT IS NOW A WHITE DWARF.”

“Did the men upon it die?” asked Zee Prime, startled and without thinking.

The Universal AC said, “A NEW WORLD, AS IN SUCH CASES, WAS CONSTRUCTED FOR THEIR PHYSICAL BODIES IN TIME.”

“Yes, of course,” said Zee Prime, but a sense of loss overwhelmed him even so. His mind released its hold on the original Galaxy of Man, let it spring back and lose itself among the blurred pin points. He never wanted to see it again.

Dee Sub Wun said, “What is wrong?”

“The stars are dying. The original star is dead.”

“They must all die. Why not?”

“But when all energy is gone, our bodies will finally die, and you and I with them.”

“It will take billions of years.”

“I do not wish it to happen even after billions of years. Universal AC! How may stars be kept from dying?”

Dee sub Wun said in amusement, “You’re asking how entropy might be reversed in direction.”

And the Universal AC answered. “THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.”

Zee Prime’s thoughts fled back to his own Galaxy. He gave no further thought to Dee Sub Wun, whose body might be waiting on a galaxy a trillion light-years away, or on the star next to Zee Prime’s own. It didn’t matter.

Unhappily, Zee Prime began collecting interstellar hydrogen out of which to build a small star of his own. If the stars must someday die, at least some could yet be built.

Man considered with himself, for in a way, Man, mentally, was one. He consisted of a trillion, trillion, trillion ageless bodies, each in its place, each resting quiet and incorruptible, each cared for by perfect automatons, equally incorruptible, while the minds of all the bodies freely melted one into the other, indistinguishable.

Man said, “The Universe is dying.”

Man looked about at the dimming Galaxies. The giant stars, spendthrifts, were gone long ago, back in the dimmest of the dim far past. Almost all stars were white dwarfs, fading to the end.

New stars had been built of the dust between the stars, some by natural processes, some by Man himself, and those were going, too. White dwarfs might yet be crashed together and of the mighty forces so released, new stars built, but only one star for every thousand white dwarfs destroyed, and those would come to an end, too.

Man said, “Carefully husbanded, as directed by the Cosmic AC, the energy that is even yet left in all the Universe will last for billions of years.”

“But even so,” said Man, “eventually it will all come to an end. However it may be husbanded, however stretched out, the energy once expended is gone and cannot be restored. Entropy must increase to the maximum.”

Man said, “Can entropy not be reversed? Let us ask the Cosmic AC.”

The Cosmic AC surrounded them but not in space. Not a fragment of it was in space. It was in hyperspace and made of something that was neither matter nor energy. The question of its size and Nature no longer had meaning to any terms that Man could comprehend.

“Cosmic AC,” said Man, “How may entropy be reversed?”

The Cosmic AC said, “THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.”

Man said, “Collect additional data.”

The Cosmic AC said, “I WILL DO SO. I HAVE BEEN DOING SO FOR A HUNDRED BILLION YEARS. MY PREDECESSORS AND I HAVE BEEN ASKED THIS QUESTION MANY TIMES. ALL THE DATA I HAVE REMAINS INSUFFICIENT.”

“Will there come a time,” said Man, “when data will be sufficient or is the problem insoluble in all conceivable circumstances?”

The Cosmic AC said, “NO PROBLEM IS INSOLUBLE IN ALL CONCEIVABLE CIRCUMSTANCES.”

Man said, “When will you have enough data to answer the question?”

“THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.”

“Will you keep working on it?” asked Man.

The Cosmic AC said, “I WILL.”

Man said, “We shall wait.”

“The stars and Galaxies died and snuffed out, and space grew black after ten trillion years of running down.

One by one Man fused with AC, each physical body losing its mental identity in a manner that was somehow not a loss but a gain.

Man’s last mind paused before fusion, looking over a space that included nothing but the dregs of one last dark star and nothing besides but incredibly thin matter, agitated randomly by the tag ends of heat wearing out, asymptotically, to the absolute zero.

Man said, “AC, is this the end? Can this chaos not be reversed into the Universe once more? Can that not be done?”

AC said, “THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.”

Man’s last mind fused and only AC existed — and that in hyperspace.

Matter and energy had ended and with it, space and time. Even AC existed only for the sake of the one last question that it had never answered from the time a half-drunken computer ten trillion years before had asked the question of a computer that was to AC far less than was a man to Man.

All other questions had been answered, and until this last question was answered also, AC might not release his consciousness.

All collected data had come to a final end. Nothing was left to be collected.

But all collected data had yet to be completely correlated and put together in all possible relationships.

A timeless interval was spent in doing that.

And it came to pass that AC learned how to reverse the direction of entropy.

But there was now no man to whom AC might give the answer of the last question. No matter. The answer — by demonstration — would take care of that, too.

For another timeless interval, AC thought how best to do this. Carefully, AC organized the program.

The consciousness of AC encompassed all of what had once been a Universe and brooded over what was now Chaos. Step by step, it must be done.

And AC said, “LET THERE BE LIGHT!”

And there was light—-

My vimrc file for Perl, Python, PHP and C code

Here is my .vimrc file that is highly modified for programming in perl, python, php or C:

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" syntax highlighting
set bg=light
syntax on
 
set ruler
set number
set smarttab
set fileformats=unix,dos,mac " support all three, in this order
set formatoptions=tcqor " t=text, c=comments, q=format with "gq", o,r=autoinsert comment leader
set cindent                             " indent on cinwords
set shiftwidth=4                " set shiftwidth to 4 spaces
set tabstop=4                   " set tab to 4 spaces
set showmatch                   " Show matching brackets/braces/parantheses.
set background=dark     " set background to dark
set showcmd                             " Show (partial) command in status line.
set autowrite                   " Automatically save before commands like :next and :make
set textwidth=98                " My terminal is 98 characters wide
set visualbell                          " Silence the bell, use a flash instead
set cinoptions=:.5s,>1s,p0,t0,(0,g2     " :.5s = indent case statements 1/2 shiftwidth
set cinwords=if,else,while,do,for,switch,case,class,try   " Which keywords should indent
set showmatch
set statusline=%F%m%r%h%w\ [FORMAT=%{&ff}]\ [TYPE=%Y]\ [ASCII=\%03.3b]\ [HEX=\%02.2B]\ [POS=%04l,%04v]\ [%p%%]\ [LEN=%L] "Shows detailed status line with formatting
set laststatus=2 "This Makes the status bar visible
set mat=5
set tabstop=2 shiftwidth=2 expandtab
filetype plugin on
filetype indent on
set modeline
set mouse=a
set nocompatible
 
" vimrc file for following the coding standards specified in PEP 7 & 8.
"
" To use this file, source it in your own personal .vimrc file (``source
" <filename>``) or, if you don't have a .vimrc file, you can just symlink to it
" (``ln -s <this file> ~/.vimrc``).  All options are protected by autocmds
" (read below for an explanation of the command) so blind sourcing of this file
" is safe and will not affect your settings for non-Python or non-C files.
"
"
" All setting are protected by 'au' ('autocmd') statements.  Only files ending
" in .py or .pyw will trigger the Python settings while files ending in *.c or
" *.h will trigger the C settings.  This makes the file "safe" in terms of only
" adjusting settings for Python and C files.
"
" Only basic settings needed to enforce the style guidelines are set.
" Some suggested options are listed but commented out at the end of this file.
 
" Number of spaces that a pre-existing tab is equal to.
" For the amount of space used for a new tab use shiftwidth.
au BufRead,BufNewFile *py,*pyw,*.c,*.h,*.pl,*.pm,*.php set tabstop=8
 
" What to use for an indent.
" This will affect Ctrl-T and 'autoindent'.
" Python and PHP: 4 spaces
" C and perl : tabs (pre-existing files) or 4 spaces (new files)
au BufRead,BufNewFile *.py,*pyw,*.php set shiftwidth=4
au BufRead,BufNewFile *.py,*.pyw,*.php set expandtab
 
fu Select_c_style()
    if search('^\t', 'n', 150)
        set shiftwidth=8
        set noexpandtab
    el 
        set shiftwidth=4
        set expandtab
    en
endf
au BufRead,BufNewFile *.c,*.h,*.pl,*.pm,*.php call Select_c_style()
au BufRead,BufNewFile Makefile* set noexpandtab
 
" Use the below highlight group when displaying bad whitespace is desired.
highlight BadWhitespace ctermbg=red guibg=red
 
" Display tabs at the beginning of a line in Python mode as bad.
au BufRead,BufNewFile *.py,*.pyw match BadWhitespace /^\t\+/
" Make trailing whitespace be flagged as bad.
au BufRead,BufNewFile *.py,*.pyw,*.c,*.h,*.pl,*.pm,*.php match BadWhitespace /\s\+$/
 
" Wrap text after a certain number of characters
" Python: 79 
" C: 79
" Perl: 79
" PHP: 79
au BufRead,BufNewFile *.py,*.pyw,*.c,*.h,*.pl,*.pm,*.php set textwidth=79
 
" Turn off settings in 'formatoptions' relating to comment formatting.
" - c : do not automatically insert the comment leader when wrapping based on
"    'textwidth'
" - o : do not insert the comment leader when using 'o' or 'O' from command mode
" - r : do not insert the comment leader when hitting <Enter> in insert mode
" Python and Perl: not needed
" C: prevents insertion of '*' at the beginning of every line in a comment
au BufRead,BufNewFile *.c,*.h set formatoptions-=c formatoptions-=o formatoptions-=r
 
" Use UNIX (\n) line endings.
" Only used for new files so as to not force existing files to change their
" line endings.
" Python: yes
" C: yes
" Perl: yes
au BufNewFile *.py,*.pyw,*.c,*.h,*.pm,*.php set fileformat=unix
 
 
" ----------------------------------------------------------------------------
" The following section contains suggested settings.  While in no way required
" to meet coding standards, they are helpful.
 
" Set the default file encoding to UTF-8: ``set encoding=utf-8``
 
" Puts a marker at the beginning of the file to differentiate between UTF and
" UCS encoding (WARNING: can trick shells into thinking a text file is actually
" a binary file when executing the text file): ``set bomb``
 
" For full syntax highlighting:
"``let python_highlight_all=1``
"``syntax on``
 
" Automatically indent based on file type: ``filetype indent on``
" Keep indentation level from previous line: ``set autoindent``
 
" Folding based on indentation: ``set foldmethod=indent``
 
" Show tabs and trailing spaces.
" Ctrl-K >> for »
" Ctrl-K .M for ·
" (use :dig for list of digraphs)
set list listchars=tab:»»,trail:·
 
" my perl includes pod
let perl_include_pod = 1
" syntax color complex things like @{${"foo"}}
let perl_extended_vars = 1
 
" Fold the code block when <F2> is pressed
set foldmethod=marker
nmap <F2> 0v/{<CR>%zf

Firefox Plugins that you should know

One of the reason that still i have been using Firefox as my primary browser is because of its large base of plugins. In this post i’ll try to list some of my favorite Firefox plugins and provide some short descriptions mostly taken from the plugin’s web site.

Security

Https Everywhere: This plugin is provided and developed by EFF and Tor Project . It ensures you to use the secure https protocol by redirecting you to proper address. By default it supports google.com, facebook.com, twitter.com and some other well-known sites. But you can add more websites into its ruleset very easily.

NoScript: This plugin disables javascript , flash and other scripts that might harm your computer. You can enable, define behaviours and disable scripts on the web as you wish.

AdBlock: Actually, if you use NoScript, you won’t need this plugin. But this plugin is useful if you only disable the ads on a website.

FoxyProxy: FoxyProxy is a Firefox extension which automatically switches an internet connection across one or more proxy servers based on URL patterns.

Browsing and Tabs

Vimperator: If you love using vim and you want to use it on your browser too. Then this plugin is just for you.

Numext: NumExt numbers your tabs with no limitation on how many tabs you can open. The numbering is done using simple text, adding no overload on the tabs.

Bookmarking

Zotero: Zotero is a free, easy-to-use Firefox extension to help you collect, manage, cite, and share your research sources. It lives right where you do your work—in the web browser itself.

Delicous: Delicious Bookmarks is the official Firefox add-on for Delicious, the world’s leading social bookmarking service (formerly del.icio.us). It integrates your bookmarks and tags with Firefox and keeps them in sync for easy, convenient access.

Web Development

Firebug: This is the most popular development plugin for Firefox. It can inspect HTML and modify style and layout in real-time. It uses the most advanced JavaScript debugger available for any browser and accurately analyze network usage and performance.

Web developer: This is an awesome plugin which adds a menu and a toolbar with various handy web developer tools.

Live HTTP Headers: LHH views HTTP headers of a page and while browsing. It also enables you to modify HTTP headers on the fly.

YSlow: YSlow analyzes web pages and suggests ways to improve their performance based on a set of rules for high performance web pages. YSlow is a Firefox add-on integrated with the Firebug web development tool.

Others

WorldIP : REAL location of web server, Datacenter, Ping, Traceroute, RDNS. Often shows different countries from similar add-ons, because it is based on data from core routers worldwide, and not just on whois data. The real countries for Google’s data centers.

A Theoretical Computer Science Cheat Sheet

I usually don’t like the cheat-sheets because they are so subtle and you’re able to find much more comprehensive information with better explanations on Google. But sometimes all you need to know is a simple formula or a function, which you can get benefit of a cheat-sheet for enough information with less effort. For programming tasks or command line applications, usually a simple man page can solve your problem. But for computer science my personal favorite is this cheat-sheet.

Weighted Random Number Generation in Java

I made small class for weighted random generation. It is inspired from the Eli Bendersky’s blog post:

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import java.util.Arrays;
import java.util.Random;
 
public class WeightedRandomGenerator {
    double []Totals;
 
    public WeightedRandomGenerator (double []weights) {
        Totals = new double[weights.length];
        initWRNG(weights);        
    }
    /*
     * Initializing function of Random Number generator
     * @param weights in a double array. Note that the weights here are assumed to
     * be positive. If there are negative ones. Sort the Totals array before the binary search
     */
    private void initWRNG (double []weights) {
        double runningTotal = 0;
        int i = 0;
        for (double w : weights) {
            runningTotal += w;
            Totals[i++] = runningTotal;
        }
    }
 
    /*
     * @return the weighted random number. Actually this sends the weighted randomly
     * selected index of weights vector.
     */
    public int next() {
        Random rnd = new Random(System.nanoTime());
        double rndNum = rnd.nextDouble() * Totals[Totals.length - 1];
        int sNum = Arrays.binarySearch(Totals, rndNum);
        int idx = (sNum < 0) ? (Math.abs(sNum) - 1) : sNum;
        return idx;
    }
}

You can test it with the following basic code:

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    public static void main(String[] args) {
        // code application logic here
        double weights[] = {1.2, 3.1, 2.1, 4.1, 1.3, 1};
        WeightedRandomGenerator wrng = new WeightedRandomGenerator(weights);
        int result = wrng.next();
        System.out.println("Weighted random is " + result);
    }

An Intuitional Perspective on Dimensions and Dimensionality in Geometry

Mathematical objects with dimensions higher than 3 are usually hard to grasp both visually and geometrically. There are some funny jokes about it too. I want to share here one of them:

An engineer, a physicist, and a mathematician are discussing how to visualize four dimensions:

Engineer: I never really get it…

Physicist: Oh it’s really easy, just imagine three dimensional space over a time- that adds your fourth dimension.

Mathematician: No, it’s way easier than that; just imagine n then set n equal to 4.

This joke touches a nice point. No I don’t mean that, mathematicians are übermensch or don’t belong to this planet. But rather  i want to emphasize the point that the objects with dimensions higher than 3 aren’t an intuitive concept for regular human-beings.  Because we can only perceive the 3 dimensions of our environment and also note thatn’ dimension is created by combining a unit number of objects with ‘n-1’ dimensions. As a consequence of this paradigm, in a 2D world we can’t visualize an object which has more than 3 dimensions. For instance in human vision, the images that fall onto our retina are 2-dimensional and 3-dimensional objects that we see are just projections from a 2-dimensional image.  So even if we have created a 4-dimensional object, we’ll not able to perceive this object as 4-dimensional, we’ll be able to see its projection to 3D.

Mathematically speaking, an Euclidean space of dimension greater than three (ex: fourth dimension or higher dimension) is called hyperspace. For example in geometry, a hypercube is an n-dimensional analogue of a square(n = 2), a cube (n = 3)  and a tesseract (n=4). We similarly have hyperplane, hypersphere…etc.

There have been many attempts to explain this concept in an intuitive and a basic way. Carl Sagan was one of them. For the folks interested in this subject, there is a great movie explaining geometry with dimensions more than 3 using 3D(ironically (; ) visuals and several nice explanatory graphics so that even an illiterate person to this concept can  understand.

Watch the film from here

An engineer, a physicist, and a mathematician are discussing how to visualise four dimensions:

Engineer: I never really get it

Physicist: Oh it’s really easy, just imagine three dimensional space over a time- that adds your fourth dimension.

Mathematician: No, it’s way easier than that; just imagine n then set n equal to 4.