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