Re: [PSUBS-MAILIST] spherical axxproximation

[Lew Clayman <lew_clayman@yahoo.com>]

--- "Sean T. Stevenson" <ststev@uniserve.com> wrote:
> Here's a question to stew over:
> 
> For a given diameter of sphere, which inscribed uniform (vertices congruent, equal edge length)
> or regular (vertices, faces congruent, equal edge length) polyhedron has the shortest edge
> length?
> 
> -Sean

Not sure without going a little deeper into it, but oddly enough there are only five such figures.
 The "Platonic solids" which I knew I'd babbling about sooner or later are defined as those
regular polyhedra formed of congruent regular polygons.  Oddly enough, there are only five of
these, and some roll badly or effectively not at all.  They are:

tetrahedron, 4 equilateral triangles arranged to form a 3-sided pyramid... 3 faces at each vertex

cube, 6 squares arranged to form a box - three faces at each vertex

octahedron, 8 triangles arranged such that four faces meet at each vertex; arranged like two
"Egyptian" pyramids joined at the base

dodecahedron, 12 pentagons joined into a, well, a 12-sided spheroid, three faces meeting at each
vertex

icosohedron, 20 triangles in a ball, five meeting at each vertex (the most common "geodesic"
structure, btw).

The last two roll ok.  The middle two probably don't in this application; the 4-sided guy is
incredibly stable, all weight centered low on a large base.  The Mars lander used a variation on
this shape specifically because it *doesn't* roll easily.

Another consideration here, the seams and vertices will take a beating as they "roll" on the
bottom, momentarily taking the whole weight of the sphere directly on the weld.  Not to mention
literally being sanded away by sandy bottoms (or rocky, etc).

My suspicion is that fewer sides, out of these five solids, tracks to shortest total seam length. 
However we don't care about the first two or three shapes for this application.  Rolling a cube
from the inside, no thanks.  So if anyone wants to solve the thing, you could limit the question
to 12'ers and 20'ers.  IMHO, natch.

Ya see Sean, it's an easier problem than you probably thought, I'm guessing.  Only two shapes to
figger with, nothing infinite or continuous or hyperwhateverized.  Almost disappointing, really.

Now, if you mix more than one shape......

-Lew

PS I am very embarassed to mention this, but if anyone out there knows what D&D dice look like,
then you know these five shapes.  Ooh, very mystical, whooo-eeeee-oooo.



=====
"Yo no soy marinero / Soy capitan"
          - Traditional Mexican Lyric (La Bamba)
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