"… the best way to understand a group is to
see it as the group of symmetries of something."
— John Baez, book review, p. 239,
Bulletin of the American Mathematical Society, April 2005
Update of Oct. 22, 2008:
Concepts of Space:
Euclid vs. Galois
From the Rankin Lectures
by John Baez at
the University of Glasgow,
September 15-19, 2008:
Baez's statement that "lines in the Fano plane correspond to planes through the origin [the vertex labeled '1'] in this cube" might be taken literally by some viewers of the Baez slides, with the planes regarded as cutting a cube in Euclidean 3-space. So interpreted, the statement would be false. But this is not what he meant.
Baez took the pictures in Fig. 7 from his article "The Octonions" in the AMS Bulletin of April 2002. As that article shows, Baez's "planes through the origin" are not to be regarded as planes in Euclidean 3-space, but rather as planes in the linear 3-space over the two-element Galois field GF(2). For another picture of this 3-space, with coordinates from GF(2), see Fig. 5 above. (See also the cubes in Diamonds and Whirls, 1984.)
That projective-plane lines correspond to planes through the origin in linear 3-space-- a standard definition of lines in projective geometry-- is trivially true both of the Baez cube and of the eightfold cube (Fig. 5 above).
Lest viewers of the Baez lectures confuse his remarks on the Fano plane and the cube model with my own remarks on the same subject, it should be emphasized that the point of the eightfold-cube model is-- unlike the Baez cube-- to exhibit the seven projective lines not as planes through the origin, but rather as sets of partitions of the eight subcubes (Fig. 3 above). This allows group actions on the space to be visualized as generated by simple permutations of 1x1x2 cube sections (Fig. 4 above and Diamonds and Whirls).
Although it is not a figure from Euclidean geometry, the eightfold cube (a Galois geometry) may be modeled by a concrete, palpable (Fig. 6 above) physical structure whose natural transformations are, unlike those of Euclidean geometry, non-continuous. As such, it may eventually, as Euclidean geometry has in the past, throw some light on the structure of the space we live in.