The Problem of Galaxa: Infinite Area versus Finite Volume.

Galaxa was a nice Terran girl who at 2.587 years of age was looking forward to her first intelligence test. This would be her first chance to join the exalted ranks of the Perfs (those students who had obtained perfect scores on at least one examination). Most children take the test at age 2.589, but the Teachers had discovered that if a bet ter-than-average student was given the test a few hours early, unusual and even inge nious solutions frequently resulted. Natu rally, Galaxa had no idea what specific task she would be given, but she was reasonably sure she would need a good deal of mathe matics and all of her skill with Enlargers. The Enlarger was, of course, the most important and far-reaching invention of the twenty-second century (old style). Be fore Day 1 it was known that space was in finite but that matter was finite. What a dull universe it must have been! As any modern two-year-old knows, the Enlarger made infinite amounts of matter a reality. Of course, no Enlarger can take a finite chunk of matter and straightforwardly en large it to infinite size. What it can do is take any finite amount of matter, however small, and enlarge it to any finite size, how ever large. Further, while it can do nothing in no time, it can perform any simple task in as short a time as desired (much like modern Terran boys and girls). It required only a mildly clever computer to construct the universe's first infinitely long object?a wire?which was only one meter in length at the beginning of the experiment. The computer had the Enlarger double the first half-meter of wire in half a second, then quadruple the next quarter meter in a quarter of a second, octuple the next eighth meter in an eighth of a second, and so on. Of course, by the end of one second, an in finitely long piece of wire was present in the universe, and the Enlarger, which had necessarily been attached to the growing part of the wire, was absent. (Modern En largers, and Terrans, are equipped with safety devices that allow them to do their work without leaving the universe.) Terran poets are fond of saying that the wire reached the "end of the universe" and that the fabled Enlarger is "out of this uni verse," and perhaps they are right. The effects of the Enlarger and the ad vent of infinite matter were felt in all as pects of Terran life. Their present dating system, for example, is based on what ap peared to be the spontaneous creation of matter. Then there was the unbalanced in dividual who, believing himself God, at tempted to enlarge himself infinitely in all directions at once. In the resulting ex plosion, parts of him were exposed to his Enlargers longer than others. Travelers still return from that sector of space with tales of intelligent bacteria living on blood cells the size of planets, and vice versa. But we digress. Galaxa faced the Teachers with a mix ture of confidence, anticipation, and a dash of fear. She knew there was no real chance that she would fail the test; a century of ge netic engineering had seen to that. But she could embarrass herself and her Teacher with a dismally low score. Further, she wouldn't have another chance to become a