Gal-Wa was about to conclude his lecture for this week. He couldn't wait to see the reaction that his thesis would generate. If he was lucky, he might even earn himself a place in the Hall of Thinkers. His ideas would be permanently recorded so that future aspirants would marvel at his brilliance and perhaps build on his ideas.

"Suppose then that you continued to create polygons with ever more and more sides. No doubt you are all familiar with U-Clid's recent demonstration of a method of dividing evenly the sides of a quadragon to produce an octagon. Let us suppose that you could perform a similar operation on the resultant octagon, creating a new object with 16 sides. The sides of the new object would be very much shorter than those of the original quadragon. Now consider what would happen if you could repeat this process infinitely. Eventually you would create an object that had so many sides, each of such small dimensions, that they would all run together and appear to be one continuous side. Such a shape would have many unusual properties worth studying. Enough in fact, that I have decided to give this shape a name. I call it a Unigon."

As he completed this remarkable conjecture, the room grew silent. After a long moment, to Gal-Wa's horror, the audience erupted in peals of laughter.

"A Unigon!" cried out one thinker from the back row, "next thing you know he'll be telling us about numbers that don't exist!"

As the laughter increased at this remark, Gal-Wa could not bear it. Hanging his head in shame, he fled the lecture hall and retired to his private quarters. How could this be? Why wouldn't anyone believe him? He was so sure he was right about this. He'd been perfecting his thesis for months.

That night, the Moderator came to see him.

"Gal-Wa," he said, "How could you have been so foolish. You know very well that our sect is known for a profound reluctance to accept new ideas without absolute proof. Did you really think they would accept this notion of yours?"

"But I've checked and rechecked everything so thoroughly. I'm sure I'm right about this. I just don't know how to convince them."

"But I would have thought that would be obvious, even to a scatterbrained novice like yourself. You have to produce an example of the object. Even the most conservative of thinkers will not fail to believe evidence that can be seen and touched."

Gal-Wa was briefly miffed. The Moderator had accused him of being "scatterbrained."

But after a moment reason prevailed. At least the Moderator seemed willing to accept that perhaps there was some value in Gal-Wa's ideas. He just needed more proof. Besides, if Gal-Wa was so smart, why hadn't he thought of trying to make a Unigon already?

"Very well," Gal-Wa replied. "I will undertake the task of creating the object myself. Perhaps then the closed minds in the Hall of Thinkers will finally be opened to new ideas."

"That is all well and good, Gal-Wa, but I must insist that you do not devote so much time to this nonsense of yours that you neglect your other work. If I discover that you are neglecting your assigned studies, I will forbid this line of inquiry altogether."

"I understand, sir." Gal-Wa really had no choice but to accept the Moderator's conditions. He knew that without the support of the Moderator, he would never be allowed to address the Hall of Thinkers again.

The next day, Gal-Wa completed his study of indivisible numbers between 100 and 1000 in record time. It was actually quite an interesting field of study. There were four such numbers between one and 10, 10 such numbers between 10 and 100, and 25 such numbers between 100 and 1000. His tutor was trying to find out if there was some formula that could predict how many such numbers there were between 1000 and 10000. Or if a formula could be found for any similarly regular system of categorizing the indivisible numbers.

Still, Gal-Wa sensed that perhaps his Unigon might lead to even more fundamental discoveries concerning the nature of reality. So he plowed through the material and rushed back to his quarters to begin work on his Unigons.

He started with a quadragon of parchment and folded it in half. Next he folded it in half again so that he had a new quadragon. Then he unfolded the parchment, and cut along the visible creases so that he had four new quadragons of parchment. He set three of them off to one side of the bench, and applied U-Clid's method to fold the corners of the fourth parchment such that the resulting shape had eight equal length sides. When he repeated the process on this parchment, he was rewarded with a new object containing 16 equal length sides.

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By now the sides were starting to get very small. Gal-Wa was faced with a dilemma. He could see that his parchment was indeed beginning to look very much like the Unigon his theories predicted. But the parchment was getting to be almost impossible to fold. He decided that what he really needed to do was to start with a bigger parchment. Reluctantly, he set his experiment aside and went to sleep.

The next day, Gal-Wa again finished his assigned studies in record time. Then he rushed back to his quarters to continue working on his pet project. This time, he started by connecting four sheets of parchment together to make one big quadragon of parchment. Then he started applying U-Clid's technique again to create a new sheet with eight equal length sides. This time when he repeated U-Clid's technique to create a shape with 16 sides, the parchment was still big enough to work with. He continued to apply U-Clid's technique 12 more times, so that he now had a very small parchment with 65,536 equal length sides. Gal-Wa was very pleased with the result. He set his new creation on a small table at one end of the room and stepped across to the other side of the room to look at it from a distance. It did indeed appear to him that the new shape had only one side. He could almost imagine that if this process could be continued infinitely, he would indeed have a "Unigon." By now it was getting late, and since his tutors were likely to take a dim view of him napping during his studies, he went to sleep.

The next morning he checked his "Unigon" to make sure it hadn't been just a dream, then went happily off to his first lesson. When he came back to continue his project at the end of this day, he was dismayed to find he'd apparently reached some sort of fundamental limit with his Unigon. No matter how hard he tried, he could not make one with double 65,536 sides. Just as he was finishing the final fold in U-Clid's process, the shape would revert to its original 65,536 sided form.

Over the next several weeks, Gal-Wa spent many hours trying to find a way around this limit. But no matter how hard he tried he could never find a way to make a shape with more than 65,536 sides. He began to concoct experiments to test this limit. First he tried just cutting pieces of parchment in half, then cutting each of those in half, and so on. Each time he got to the 16th repetition, he found that it was impossible to perform the 17th such repetition. It was as if quantities greater than 65,536 could not be permitted to exist. He wanted to discuss this new discovery with his instructors, but he knew what their reaction would be. They already thought he was wasting his time with silly notions anyway. This would only provide them with more reason to ridicule his work.

Finally, he decided to take the advice of the Moderator. What was it he had said about the thinkers?

"Even the most conservative of thinkers could not fail to believe evidence he could see and feel."

Rather than continue to make an object with more and more sides, Gal-Wa decided to produce one 65,536-sided object for each thinker. He went to the Moderator and asked for permission to address them once again. At first the Moderator was reluctant to allow him another chance to make a fool of himself after his recent disastrous speech, but when Gal-Wa showed him his almost Unigon, he relented. Gal-Wa spent the next several weeks creating as many Unigons as he could.

Finally, the scheduled day arrived. The thinkers were assembled in the Hall, and Gal-Wa stepped up to the podium to begin his address. This time he felt certain that things would turn out differently. This time he had proof of the validity of his ideas.

"Erudite Thinkers...," he began.

"Always begin with flattery," the Moderator had told him. Very well, that was done.

"I am here today to report the progress I've made in reaching my goal of creating the Unigon, a shape whose existence I hypothesized recently."

"Not the crazy kid with the Unigon again. I can't believe we are wasting our time with this," interrupted one thinker.

Gal-Wa decided he'd better play his trump card.

"Wait. I have proof!" he shouted. He picked up the bag of Unigons he'd brought with him and stepped in front of the podium so the entire audience would have a clear view. As he untied the bag, he stepped off the platform. But he missed the first step and tripped. The Unigons in the bag came flying out and started to fall slowly to the ground.

Gal-Wa was shocked to see that they seemed to fall far too slowly. In fact, it suddenly seemed that everything in the room was moving more slowly than it should. He noticed the Moderator taking a step to come to his aid, but his foot seemed almost frozen above the first step. Then he looked back at the Unigons again. He was amazed to see that he could see each individual Unigon twisting slowly as it fell. First one, then the next would twist slightly and move just a little closer to the ground. It was as if time itself was being distorted. It was the most frightening thing he'd ever experienced. Then suddenly, his terror intensified. It was as if he was deprived of all his senses in one single instant. He was plunged into darkness. That wasn't the worst of it though. The worst thing was the sudden loss of any connection to himself. It was as if he'd been removed from his body. He could feel no sensation from his toes, his legs, his hands, his arms. Nothing. He endured a few seconds of this torture, and when he began to think he must be losing his mind, his senses returned to normal. But he was somewhere else.

Meanwhile, in the Hall of Thinkers, the Unigons finally started to hit the floor. As they did, the effects that Gal-Wa had observed began to lessen, until finally, when the last Unigon fell to the floor, time seemed to resume its normal state. There was an immediate uproar in the Hall of Thinkers.

"What magic is this?" cried one.

"Why weren't we told that Gal-Wa was dabbling in the manipulation of time as well as space?" cried another.

One of the thinkers noticed Gal-Wa's disappearance. "Look," he cried, "Gal-Wa is gone. Where did he go?".

The only response to his question was utter silence. Some thinkers were contemplating the significance of Gal-Wa's sudden disappearance. Others were wondering if there might not be something to this "Unigon" after all.