216 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures
(one can be moved onto any other by appropriately rotating the cube, and
this rotation maps all the other planes onto planes). It is also clear that
the planes formed by two opposite edges are alike, and the all-black plane
is like the all-light plane (reflecting the cube in its center will interchange
black and light vertices).
This was the easy part. Now we do a trickier transformation: We in-
terchangeEwithF andGwithH(Figure 14.7). What happens to the
planes?
A B
D C
E F
H G
A B
D C
E
H
F
G
FIGURE 14.7. Why different-looking planes are alike.Some of them are not changed (even though their points will change
places): The top, bottom, front, and back faces and the planesABGHand
CDEFare mapped onto themselves. The left faceADEHis mapped onto
the planeADF G, and vice versa. Similarly, the right faceBCFGis mapped
onto the planeBCEH, and vice versa. The all-black plane is mapped onto
the planeACEG, and vice versa. The all-light plane is mapped onto the
planeBDFH, and vice versa.
Thus all planes are accounted for. We make two observations:
—Every plane is mapped onto a plane, and so if we relabel the cube as
above, the Cubics won’t notice the difference!— There is a face-plane mapped onto an opposite-edge-plane, and there
is an opposite-edge-plane mapped onto the all-black plane. This im-
plies that the Cubics cannot see any difference between these three
types of planes.14.1.1Fanoan philosophers have long been troubled by the difference between
points and lines. There are many similarities (for example, there are 7 of each);
why are they different? Represent each line by a new point; for each old point,
take the 3 lines through it, and connect the 3 new points representing them by a
(new) line. What structure do you get?
14.1.2The Fanoans call a set of 3 points acircleif they are not on a line. For
example, the 3 vertices in figure 14.1 form a circle. They call a line atangentto