clifford_a-_pickover_surfing_through_hyperspacebookfi-org

198 appendix g
V(r) = S(D) X r°/D
where S is the so-called "solid angle" in dimension D, and r the radius. (Generally
speaking, the solid angle is the set of rays emanating from a point and passing
through a particular continuous surface. S(D) is the largest possible measure of a
solid angle in D dimensions. The measure of a solid angle is the surface that it
intersects of the unit sphere whose center is its vertex.) If we compare two spheres
of radius 1 and 1 — a, where a is very small, the difference is
(V(l)-V(\-a))IV(\)
= \-(\-a) D
This is essentially the fraction of the volume within 1 — a of the surface. For
example, if you have a 10-D sphere, then about 40 percent of the volume is within
0.05 X r of the surface {a = 0.05). In a 4-D sphere, about 34 percent of the volume
is within Q.I X r of the surface. In a 3-D sphere, this is 27 percent. For the D = 10
sphere, the fraction within 0.1 X r is 65 percent. (Note that these numbers are
much higher for nonspherical shapes.)
Also note that for a given volume V, the surface-area-to-volume ratio
increases when going from 3-D to 4-D creatures. This, in turn, implies a favorable
oxygen-exchange ratio for respiration and nutrient exchange; it also implies
that large animals may be stronger in the fourth dimension, partly because of
increased muscular attachment sites. This also means that higher-dimensional
beings might be bigger than their 3-D counterparts. In addition, warm-blooded
4-D creatures may need to have efficient means of temperature regulation if the
ambient environment has a greater effect on their bodies. If 4-D creatures had
different sizes and metabolic rates than us—with accompanying different lifespans
and sleep durations—this could make it difficult to communicate with
them. (These difficulties might be overcome with "asynchronous" communication
such as e-mail.)
Notice that the surface-area-to-volume ratio for a given spatial extent increases
as one goes from the third dimension to the fourth. We see this easily in the second
and third dimensions by using familiar formulas for area and volume for circles
and spheres:
Area Volume Area/Volume
2 X IT X r TT X r 2 21 r (circle)
4 X TT X r 2 (4/3) X TT X r" 3/r (sphere)