clifford_a-_pickover_surfing_through_hyperspacebookfi-org

HYPERSPHERES AND TESSERACTS 115
answers question 3. Similarly, for question 4, you could fit an infinite number
of 3-D marbles into the 16-D sphere mentioned. Finally, just as a circular plate
in two dimensions has zero thickness—and hence no volume—the whale, the
earth, the Latting Observatory, and Einstein's brain have no "hypervolume" in
higher dimensions. (Please forgive me for giving so many similar examples. I
could have made my point by using two or three questions rather than six, but
I hope the repetition reinforced the concept.) For some interesting student
exercises, see note 4.
Hypersphere Packing
Now that we've discussed hyperspheres in depth, let's consider how hyperspheres
might pack together—like pool balls in a rack or oranges in a box.
On a plane, no more than four circles can be placed so that each circle
touches all others, with every pair touching at a different point. Figure 4,25
shows two examples of four intersecting circles. In general, for w-space, the
maximum number of mutually touching spheres is n + 2.
What is the largest number of spheres that can touch a single sphere
(assume that each sphere has the same radius)? For circles, we know the answer
is six (Fig. 4.26). For spheres, the largest number is twelve, but this fact was not
proved until 1874. In other words, the largest number of unit spheres that can
touch another unit sphere is twelve. For hyperspheres, it is not yet known if the
number is twenty-four, twenty-five, or twenty-six, nor is a solution known for
higher dimensions, as far as I know. Mathematicians do know that it is possible
for at least 306 equal spheres to touch another equal sphere in nine dimensions,
and 500 can touch another in ten dimensions. But mathematicians are
not sure if more can be packed!
Fact File
For those of you with a fondness for numbers, I close this section with a potpourri
of fascinating facts.
• A cube has diagonals of two different lengths: the shorter one lying on
the square faces and the longer one passing through the center of the
cube. The length of the longest diagonal of an «-cube of side length m is