"hermans" <sas### [at] telenet_invalidbe> wrote in message
news:4241c745$1@news.povray.org...
> James Buddenhagen wrote:
> > "Le Forgeron" <jgr### [at] freefr> wrote in message
> > news:42411a22$1@news.povray.org...
> >
> >>-----BEGIN PGP SIGNED MESSAGE-----
> >>Hash: SHA1
> >>
> >>James Buddenhagen wrote:
> >>
> >>>I'm interested in artistic images with a math flavor.
> >>>Here is a simple one (at this point it has to be simple
> >>>or I can't create it!) which had its inspiration in
> >>>an arrangement Kottwitz found around 1995 for 20 points
> >>>on a sphere whose convex hull consists entirely of
> >>>triangles, and those triangles are as close to equilateral
> >>>as possible.
> >>
> >>20 points on a sphere... sound like a dodecahedron to me...
> >
> >
> > Except, the faces of the regular dodecahedron are not triangles...
> >
> > In the picture the 20 points are the centers of the red
> > circular caps and the yellow polyhedron (which has 20 faces)
> > is the dual of the convex hull.
> >
> > Jim
> >
> How the 20 points are arranged? The yellow polyhedron looks rather
> strange: hexagons and pentagons as faces and at a first sight rather
> arbitrary ones. Can you give a link to the paper of Kottwitz?
>
> http://cage.ugent.be/~hs

There was no paper.  Kottwitz and I looked at looked at the problem of
point arragments on spheres giving all triangular faces for the convex
hull, with the triangles "as close to equilateral as possible" for a
while and then abandoned it.  I don't even remember how we defined "as
close to equilateral as possible".  In any case, the 20 points of this
example are given approximately by:

p1:=[  0.4579,   0.0000,   0.8890 ];
p2:=[ -0.4579,   0.0000,   0.8890 ];
p3:=[  0.0000,  -0.6915,   0.7224 ];
p4:=[  0.0000,   0.6915,   0.7224 ];
p5:=[ -0.6684,  -0.6414,   0.3766 ];
p6:=[  0.6684,  -0.6414,   0.3766 ];
p7:=[  0.6684,   0.6414,   0.3766 ];
p8:=[ -0.6684,   0.6414,   0.3766 ];
p9:=[  0.9897,   0.0000,   0.1435 ];
p10:=[ -0.9897,   0.0000,   0.1435 ];
p11:=[  0.0000,  -0.9897,  -0.1435 ];
p12:=[  0.0000,   0.9897,  -0.1435 ];
p13:=[ -0.6414,  -0.6684,  -0.3766 ];
p14:=[  0.6414,  -0.6684,  -0.3766 ];
p15:=[  0.6414,   0.6684,  -0.3766 ];
p16:=[ -0.6414,   0.6684,  -0.3766 ];
p17:=[  0.6915,   0.0000,  -0.7224 ];
p18:=[ -0.6915,   0.0000,  -0.7224 ];
p19:=[  0.0000,  -0.4579,  -0.8890 ];
p20:=[  0.0000,   0.4579,  -0.8890 ];

I think it has S4 point symmetry, but better check.

Jim Buddenhagen ( jbuddenh at gmail dot com)

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