Re: Transforming Solids...

Posted by Reilly Atkinson on November 02, 19100 at 11:38:03:

In Reply to: Transforming Solids... posted by Matthew Brown on October 31, 19100 at 05:37:27:

Mr. Brown -- An excellent question indeed,
and one to which there are many answers.
The simplest one, due to physicists, is called
"Inversion in a Sphere", and dates from the mid
nineteenth century, and is used, commonly, in the study
of Electricity and Magnetism to solve certain
complicated problems in determining electric potentials
and fields. The transformation converts a solid sphere
into all of three dimensional space with a hole where
the inverted sphere used to be. If the sphere has radius
R then the inversion formula is r * s = R squared,
where r is the radial distance of a point in the sphere, and
s is the radial distance of the inverted point.
These distances are measured from the center of the
sphere, and both points are on the same radial.

This may not be exactly what you want, but it does
indeed map the inside of a solid sphere to space
outside the sphere. Note also. that the points very
near to the center of the sphere map to points very
distant from the center.

To go beyond this requires the tools and concepts
of topology, some of which are very tricky. A key point
is that, belive it or not, the solid sphere, the spherical
shell, the 2-D plane, a 3-D space, an 11-D space all
have the same number of points--Cantor's uncountable
infinity. So the sphere can be mapped into any number
of things, squares, cubes, lines, 4D spheres,
you name it. The issue becomes, how "smooth" are the
transformations, how many "tears" do you need
(as in think of covering an orange with a napkin).

I'm quite sure that there are lot's of non-technical
accounts of all this, but offhand I don't know any.A
secondary school text dealing with E&M might discuss
the physicists approach to inversion.

Regards, hope this helps.
Reilly Atkinson

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