Finite Relativity

Today, Feb. 20, 2004, is the 18th birthday of my note "The Relativity Problem in Finite Geometry" shown below.

That note begins with a quotation from Weyl:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
-- Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

"... is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory."
-- Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note below.

Related material:

Invariants 

"What modern painters are trying to do,
if they only knew it, is paint invariants."

-- James J. Gibson in Leonardo
(Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978)

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn.  Taken as a set, these three line diagrams describe the structure of the bottom colored figure.  After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

For another sort of invariance of the colored figure, try applying a symmetry of the square to each of the set of four diagonally-divided squares from which the figure's entries are drawn, and observe the induced effect on the figure itself.

A more remarkable invariance -- that of symmetry itself -- is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2x2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry. The cause of this symmetry-invariance in the colored patterns is the symmetry-invariance of the line diagrams under a group of 322,560 binary affine transformations.

For more details on the above invariant structures, see Geometry of the 4x4 Square, the Diamond 16 Puzzle, and Diamond Theory.

Also perhaps relevant:

For another perspective, see Block Designs in Art and Mathematics.

For a large downloadable folder
containing this and many related web pages,
see Notes on Finite Geometry.

Page created Feb. 21, 2004