clifford_a-_pickover_surfing_through_hyperspacebookfi-org

HYPERSPHERES AND TESSERACTS 109
4-D line or 4-D distance. Edward Kasner and James Neuman remarked in
1940 (the same year that Heinlein published his science-fiction tale about
the 4-D house):
Distance in four dimensions means nothing to the layman. Even fourdimensional
space is wholly beyond ordinary imagination. But the
mathematician is not called upon to struggle with the bounds of imagination,
but only with the limitations of his logical faculties.
Hyperspheres
We sail within a vast sphere, ever drifting in uncertainty, driven from
end to end.
—Blaise Pascal, Pensees
I would like to delve further into the fourth dimension by discussing hyperspheres
in greater detail. Let's start by considering some exciting experiments you
can conduct using a pencil and paper or calculator. My favorite 4-D object is not
the hypercube but rather its close cousin, the hypersphere. Just as a circle of radius
r can be define by the equation x 2 + y 2 = r 2 , and a sphere can be defined by x 2
+ y 2 + z 2 = r 2 , a hypersphere in four dimensions can be defined simply by
adding a fourth term: x 2 + y 2 + z 2 + w 2 = r 2 , where w is the fourth dimension.
I want to make it easy for you to experiment with the exotic properties of hyperspheres
by giving you the equation for their volume. (Derivations for the following
formulas are in the Apostol reference in Further Readings.) The formulas
permit you to compute the volume of a sphere of any dimension, and you'll find
that it's relatively easy to implement these formulas using a computer or hand
calculator. The volume of a ^-dimensional sphere is
for even dimensions k.
The exclamation point is the mathematical symbol for factorial. (Factorial is
the product of all the positive integers from one to a given number. For example,
5/=1X2X3X4X5 = 120.) The volume of a 6-D sphere of radius 1 is