Diamond Theory, the author's 1976 monograph on pure mathematics, deals with how symmetries in small binary finite geometries can be visualized as symmetries in ordinary Euclidean space.
As a synopsis of an authoritative 1991 book notes, finite geometries in general "find wide application in coding theory, algebraic geometry, design theory, graph theory, and group theory." The synopsis does not mention physics. Occasionally, however, speculation by physicists has invoked the pure mathematics of finite geometry (i.e., Galois geometry).
This web page supplies a few links to such speculation.
The most direct connection to diamond theory is via the following figure:
This figure appears in
Discrete phase space based on finite fields,
Jan.-May 2004 (Citebase abstract), by Kathleen S. Gibbons, Matthew J. Hoffman, and William K. Wootters
and in
Picturing qubits in phase space
(pdf), Aug. 9, 2003, by William K. Wootters.
For more on the "striations" pictured above, see
Discrete Wigner functions and quantum computational speed-up
(pdf), May 13, 2004, by Ernesto F. Galvao (in Citebase).
These striations are one of the sets of five mutually orthogonal 4x4
arrays described in my note of December 1978 as equivalent to
"spreads" of mutually skew lines in the projective binary space
For details, see
Orthogonal Latin squares as skew lines
(html), 1978-2001, by Steven H. Cullinane.
(Burkard Polster has another way of visualizing such a spread. For differences between my approach and Polster's, see Geometry of the 4x4 square.)
For some other references to finite fields in physics, see
John Baez, sci.physics.research newsgroup letter of Sept. 2002, and
Some remarks on arithmetic physics (pdf), by V. S. Varadaran, ca. 1999.
Page created July 18, 2004; updated July 19, 2004.