clifford_a-_pickover_surfing_through_hyperspacebookfi-org

82 surfing through hyperspace
Figure 4.1 A cross section of a hypersphere centered at C. The radius of the hypersphere
is seventeen feet.
space. But suppose we could move the string upsilon into the fourth
dimension. I could move the earring eight feet away from your finger in
our space, then turn at right angles, and then move fifteen feet upsilon
into hyperspace. Your ring would still be on the hypersphere."
"How did you determine that eight and fifteen were the correct numbers?"
"Do you remember the Pythagorean theorem, or distance formula? If
you move only in two directions, the distance d would be measured by x 2
+ f = d 2 . So for our example, 8 2 + 15 2 = 17 2 " (Fig. 4.1).
Sally nods. "Using your formula, this means that no matter in what
direction you move eight feet away from my finger, the additional fifteenfoot
move upsilon gives a point exactly seventeen feet away from my fingertip."
"Yes. That also means if we take all the points on an eight foot sphere
around your fingertip, and them move upsilon fifteen feet, we will get a
displaced sphere of points all belonging to the seventeen-foot hypersphere
around your fingertip."
Sally thinks for a few seconds. "Now I can understand why a hypersphere
consists of a series of spheres—spheres that grow smaller as one
moves upsilon or delta away from my fingertip at the sphere's center.
Also, the less I move away from the hypersphere's center in our space, the
more I can move upsilon or delta to be on the hypersphere's surface."
"That's right. All the spheres make up a 3-D hypersurface that's analogous
to a 2-D surface of a sphere. The hypersurface of a hypersphere is