Re: Hypercube

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Jack U's correction (of John D.'s usage of the term '3-sphere'):

> In the math literature the dimension of the maximal sphere is one
> less than the dimension in which it is embedded. Thus
>                1-sphere, a circle, can lie in a plane
>                2-sphere is the ordinary sphere in Euclidean 3-space
>                ...etc.

This is a good point.  In geometry the concept of a sphere, hypersphere,
or D-sphere, S^D, etc. refers to the outer edge or "boundary" set being
the D-dimensional hyper*surface* manifold which, when embedded in a
(D+1)-dimensional Euclidean space, is the set of points whose distance
from a given central point in the (D+1)-dimensional space is a fixed
constant.  The the interior points of the (D+1)-dimensional space that
happen to be inside of the D-dimensional (hyper)spherical boundary
surface are *not* considered to be part of the (hyper)sphere.

Confusion from this distinction can sometimes arise in a physics teaching
context when the cosmology of finite Friedmann-Robertson-Walker (FRW)
universes is being discussed.  In finite/bounded FRW universes the
topology (and coarse-scale geometry) of each spatial 3-section of
spacetime at each instant of cosmic time is that of a (homogeneous and
isotropic) 3-sphere S^3.  If we let R be the cosmic scale factor, (AKA
the radius of curvature of the universe) then the 3-dimensional spatial
volume of this universe is 2*([pi]^2)*R^3. This 3-D volume region is best
visualized as really the 3-D hyper*area* of a hyperspherical hypersurface
embedded in a Euclidean version of R^4.

Sometimes neophyte students learning about this kind of cosmology
mistakenly think of the S^3 sphere as being the 3-dimensional region
interior to a big 2-sphere S^2 that happens to be embedded in R^3.  When
this happens the student can't understand how the universe can be
spherical and finite--and yet homogeneous without an outer boundary
surface.  Such a student also thinks that the volume of of such a
universe ought to be (4*[pi]/3)*R^3 rather than the previous formula.

Often, when counteracting such a misconception a textbook or instructor
will use the inflating balloon analogy to explain how this kind of
universe can be homogeneous, have a finite extent--yet without a boundary
of its own, and can expand--yet without a center of expansion.  But
sometimes this cure for a misconception creates a collateral confusion.
The student then might not catch the crucial point that balloon model is
a *lower dimensional* 2-D *analog* the actual 3-D spatial sphere S^3.  In
this case the student would have a problem trying to mentally fit the
three spatial dimensions that (s)he knows must exist into a model that
only has room for 2 dimensions.  When we use such an analogy in class we
must emphasize the lower-dimensional nature of that analogy and emphasize
to the class that sthe spatial region of such a finite FRW universe is to
be imageined as restricted to a 3-D hypersurface being the boundary of a
region embedded in *4* Euclidean dimensions whose center is the center
of the sphere--yet is not apart of it.

But maybe this whole problem is becoming moot as evidence continues to
accumulate that our actual universe does not seem to be described by a
finite model anyway.

David Bowman
David_Bowman@georgetowncollege.edu