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