clifford_a-_pickover_surfing_through_hyperspacebookfi-org

116 surfing through hyperspace
Figure 4.25 Circle packing, (left) In two dimensions, no more than four circles can
be placed so that each circle touches all the others, with every pair touching at a different
point. What happens in higher dimensions? (right) An attractive computergraphic
study of circle packing.
m V n. This means that if I were to hand you a three-foot-long thigh
bone and ask you to stuff it into a 9-D hypercube with edges one foot in
length, the bone would just fit because V 9 = 3. A dinosaur bone ten
feet long could fit diagonally in a 100-D cube with edges only foot in
length. A mile-long toothpick could fit inside an w-cube with edges the
same length as those of an ordinary sugar cube, if n is large! On the other
hand, a hypersphere behaves somewhat differently. An w-sphere can
never contain a toothpick longer than twice its radius, no matter how
large n becomes. As we've discussed, other odd things happen to spheres
as the dimension increases.
The number of edges of a cube of dimension nis n ~X 2"~ } . For example,
the number of corners of a 7-D cube is 2 7 = 128, and the number of
edges is 7 X 2 6 = 7 X 64 = 448. Another factoid: two perpendicular
planes in four-space can meet at a point.
A 4-D analog of a pyramid has a hypervolume one-fourth the volume of
its 3-D base multiplied by its height in the fourth direction. An n-
dimensional analog of a pyramid has a hypervolume IIn times the volume
of its (n — l)-dimensional base multiplied by its height in the nth
direction.