Finite
Geometry Notes |

by Steven H. Cullinane

September 8, 2009

Passages from a classic story:
... he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads.... Tesseract "Your mind has been conditioned to Euclid," Holloway said. "So this-- thing-- bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees." "Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child-- especially a baby-- might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas." "Hardening of the thought-arteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only--" "Well, look. Let's suppose there are two kinds of geometry-- we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid--" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis
of Euclid. Alphabet blocks. Math, geometry, algebra-- they come much
later. We're familiar with that development. On the other hand, start
the baby with the basic principles of our x logic--""Blocks? What
kind?"Holloway looked at the abacus. "It wouldn't make much sense
to us. But we've been conditioned to Euclid."-- "Mimsy Were the Borogoves," Lewis Padgett, 1943 |

Padgett (pseudonym of a husband-and-wife
writing team) says that alphabet blocks are the intuitive "basis of
Euclid." *Au contraire*; they are the basis of Gutenberg.

For the intuitive basis of one type of*non*-Euclidean*
geometry--
*finite*
geometry over the two-element Galois field-- see the work of...

For the intuitive basis of one type of

His "third gift" --

© 2005 The Institute for Figuring

Photo by Norman Brosterman

fom the*Inventing Kindergarten*

exhibit at The Institute for Figuring

**Go
figure.**

fom the

exhibit at The Institute for Figuring

* I.e., *other than* Euclidean.
The phrase "non-Euclidean" is usually applied to only *some*
of the geometries that are not Euclidean. The geometry illustrated by
the blocks in question is not Euclidean, but is also, in the jargon
used by most mathematicians, not "*non*-Euclidean."

Page written September 8, 2009.