HOW MUCH DOES IT HOLD?
(C)1999 Alan M. Schwartz
(Uncle Al's itinerate meanders four ways perpendicular to reality
are generally restricted to 750 words. Sometimes something so
astounding, so revolting, so beyond the pale of human endeavor
surfaces that "excess is not enough!" rules.)
Space comes in three flavors, one flat and plain for common men
and two extra crunchy for mathematicians, navigators, surveyors,
and graduate students hot to fail a required course. Volumes and
surface areas are easy to calculate in flat space. Uncle Al
wondered what the surface to volume ratio of a sphere and a cube
would be in the other two cases. Bad Uncle Al, bad, bad!
Forget the intricacies of a cube. What is a sphere? In flat
space a sphere is as simple as you can get - the locus of all
points equidistant from a given point (the center). Here is the
input I got for what I thought was a simple question. But first,
let's get our feet on the ground (uneven as it may be).
Euclidean, plane, or zero-curvature space is the old high school
standby with simple formulas and easily visualized shapes. One
line is parallel to a given line and a triangle's internal angles
sum to exactly 180 degrees. A circle's circumference divided by
its diameter is pi. A sphere's surface (4(pi)r^2) to volume
((4(pi)r^3)/3) ratio is 3/r. The radius can be a fundamental
metric, the radian, and useful in other spaces.
Riemann, hyperbolic, or negative-curvature space is shaped like a
saddle surface. An infinite number of lines are parallel to a
given line and triangles' internal angles sum to less than 180
degrees. A circle's circumference divided by its diameter is
more than pi.
Bolyai-Lobechevsky, elliptic, or positive curvature space has the
properties of a sphere's surface. No lines are parallel to a
given line and triangles' internal angles sum to more than 180
degrees - up to 540 degrees. A circle's circumference divided by
its diameter is less than pi. Fasten your safety belts.
We begin by wondering what a sphere really is, then things go
positively (and negatively) ape.
"A zero curvature Riemann space is locally isometric to E^3, but
might not be homeomorphic to it; e.g. consider S^1 x E^2. One
might say a sphere's surface is 4(pi) squareradians and the
volume (4/3)(pi) cubicradians (restricted to S^3)."
Who could argue with that? Lest gentle reader die of boredom,
Uncle Al will violate 5000 years of academic discipline and cut
to the finish line (courtesy of psmith9626@aol.com),
k=+1 case: Consider a small ball of radius R,
Area(R)/Vol(R) = [sin(R)]^2 / [(R/2)-(1/4)sin(2R)]
k=-1 case, same story:
Area(R)/Vol(R) = [sin(R)]^2) / [(-R/4)+(1/2)sinh(2R)].
"Notice how in both cases the ratio depends on the size of the
ball. This is a fact in all non-Euclidean metric geometries."
This is the nice answer, a thing one might expect to be billeted
within a bubble gum wrapper (though hyperbolic sines probably
taste different from regular ones). Here is a rigorous answer:
Consider spheres in 3-D hyperbolic, Riemann, or negative
curvature space.
Usually denoted H^3, with the line elements (among others)
dx^2 + dy^2 + dz^2
ds^2 = ------------------ (upper half space coords)
z^2
ds^2 = dw^2 + sinh^2 w (du^2 + sin^2 u dv^2) (polar coords)
Consider spheres in 3-D elliptic, Bolyai-Lobachevsky, or positive
curvature space.
Usually denoted RP^3, the quotient of S^3 obtained by identifying
opposite points. Geometrically, it is locally isometric to S^3,
which has the line elements (among others)
ds^2 = dz^2 + cos^2 z du^2 + sin^2 z dv^2
ds^2 = dw^2 + sin^2 w (du^2 + sin^2 u dv^2)
(Note analogies with cylindrical and polar spherical coordinates
in E^3, respectively; try small z and w if these aren't obvious.)
Since RP^3 is locally isometric, with proper restrictions on the
coordinates these give local coordinate patches for RP^3 as well
as for S^3.)
Are there symbolic formulas for expressing the areas and volumes
of spheres in these curved spaces?
There are closed form formulas which are easily derived. Using
the line element
ds^2 = dw^2 + sin^2 w (du^2 + sin^2 u dv^2)
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