|(quoted from The Time Axis)|
Chapter XIX, p. 65, column 2:
The whole is never larger than the sum of its parts, and the sum of the parts always equals the whole."
"Then you never heard of Banach and Tarski," I said.
"Once I was assigned to write a feature science story on their experiment. I did plenty of research,
because I had to find human interest in it somewhere and it was pure mathematics. The Banach-Tarski
paradox, it was called--a way of dividing a solid into pieces and reassembling them to form a solid of
"I should remember that," Belem said, "since I have all your memories. It was only theoretical, wasn't it?"
He searched my memory. I felt uncomfortable as though, under partial anaesthesia, I watched a
surgeon investigating my digestive tract.
"Theoretical, sure," I said. "But I did a repeat on the subject later. It took twenty-three years before
somebody figured out how to apply the trick to a physical solid. I forget the details."
Chapter XX, p. 66, column 1:
"California," I thought and something clicked and swung open and I saw a page open before me--a
page I had first read thousands of years ago--and the fine print swam into remembered visibility.
"'Professor Raphael M. Robinson of the University of California now shows that it is possible to divide a
solid sphere into a minimum of five pieces and reassemble them to form two spheres of the same size
as the original one. Two of the pieces are used to form one of the new spheres and three to form the
"'Some of the pieces must necessarily be of such complicated structure that it is impossible to assign
volume to them. Otherwise the sum of the volumes of the five pieces would have to be equal both to
the volume of the original sphere and to the sum of the volumes of the two new spheres,
which is twice as great.'"
Chapter XX, p. 66, column 2:
I was able to watch the first stages of Belem's experiments. He knocked down the problem of lenses
and lights upon which he'd spent so much time and began setting up theoretical paradoxes in three
dimensions, following the Banach-Tarski geometric plan. I watched him playing with ghostly spheres
and angles of light until my head began to ache from following the changing shapes.
What he was attempting was clearly impossible.
Chapter XX, p. 67, column 1:
He had a sphere about the size of a grapefruit, floating in mid-air above his table. He did things to it
with quick flashes of light that acted exactly like knives, in that it fell apart wherever the lights touched,
but I got the impression that those divisions were much less simple than knife-cuts would be. The light
shivered as it slashed and the cuts must have been very complex, dividing molecules with a selective
precision beyond my powers of comprehension.
The sphere floated apart. It changed shape under the lights. I am pretty sure it changed shape in four
dimensions, because after a while I literally could watch any more. The shape did agonizing things to
my eyes when I tried to focus on it.
When I heard a long sigh go up simultaneously from the watchers I risked a look again.
There were two spheres floating where one had floated before.
"Amoebas can do it," I said. "What's so wonderful about reproduction by fission?"