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The Time Axis (1949)
Henry Kuttner
(click on names to see more mathematical fiction by the same author)

Contributed by Fred Galvin

This was published as an Ace paperback in 1965. I don't think I have a copy of the paperback in my collection, but I have the original magazine publication, in the January 1949 issue of Startling Stories. This is a story of time travel, as you might guess from the title; it also involves various science-fiction gadgets such as matter transmitters and "Mechandroids". The point of mathematical interest is the Banach-Tarski paradox, and the silly idea that it can be applied to material bodies. Here are some extracts. Page numbers refer to the magazine version.

(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 
different volume."

"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?"

Note that the more recent writings of Eliot Fintushel also feature the Banach-Tarski Paradox.

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Works Similar to The Time Axis
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. Milo and Sylvie by Eliot Fintushel
  2. The Fairy Chessmen by Henry Kuttner
  3. The Cube Root of Conquest by Rog Phillips
  4. The Planiverse: computer contact with a two-dimensional world by A.K. Dewdney
  5. Brave New World by Aldous Huxley
  6. A Wrinkle in Time by Madeleine L'Engle
  7. Diaspora by Greg Egan
  8. The Time Machine by Herbert George Wells
  9. Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott
  10. The Grass and Tree by Eliot Fintushel
Ratings for The Time Axis:
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Mathematical Content:
3/5 (1 votes)
Literary Quality:
3/5 (1 votes)

GenreScience Fiction,
MotifTime Travel,
MediumNovels, Available Free Online,

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