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.
--
Jim Buddenhagen

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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...
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"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

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James Buddenhagen wrote:
> "Le Forgeron" <jgr### [at] freefr> wrote in message
>>
>>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
> 

Are you sure it has 20 faces ? From here, I can see only 12 ...
Regis.

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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

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"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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"James Buddenhagen" <jbu### [at] REMOVEtexasnet> wrote in message
news:42403f71@news.povray.org...
> 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.
> --
> Jim Buddenhagen

In response to several people I made two images of the 20 vertex
polyhedron (whose dual was the yellow part of the original picture).
I think the rather large amount of perspective made it hard to
see there were 20 faces.  Anyway, here are the two images.
The one with orthogonal perspective shows the symmetry better.
If I recall correctly there are 4 different edge lengths.
--
Jim Buddenhagen

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Attachments:
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James Buddenhagen wrote:
> "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)
> 
Why don't you take the 20 points on the sphere as the vertices of a 
regular icosahedron? In that case all the triangles aren't "as close to 
equilateral as possible" but equilateral. The dual polyhedron is a 
regular dodecahedron (as can be seen on my site and on many other 
sites). For similar arrangments of more points in many cases one can use 
the theory of geodesic spheres (domes). Also in povray a lot of examples 
exist.
Herman Serras
http://cage.ugent.be/~hs

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"hermans" <sas### [at] telenet_invalidbe> wrote in message
news:4242fec1$1@news.povray.org...

[...snip...]

> Why don't you take the 20 points on the sphere as the vertices of a
> regular icosahedron?

Because the regular icosahedron has only 12 vertices.  It has 20 faces,
so one might try the centers of those faces for the 20 points, but that
fails also because the convex hull is then the reg dodecahedron which
does not have triangular faces.

> In that case all the triangles aren't "as close to
> equilateral as possible" but equilateral. The dual polyhedron is a
> regular dodecahedron (as can be seen on my site and on many other
> sites). For similar arrangments of more points in many cases one can use
> the theory of geodesic spheres (domes). Also in povray a lot of examples
> exist.
> Herman Serras
> http://cage.ugent.be/~hs

Jim Buddenhagen

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James Buddenhagen wrote:
> "hermans" <sas### [at] telenet_invalidbe> wrote in message
> news:4242fec1$1@news.povray.org...
> 
> [...snip...]
> 
> 
>>Why don't you take the 20 points on the sphere as the vertices of a
>>regular icosahedron?
> 
> 
> Because the regular icosahedron has only 12 vertices.  It has 20 faces,
> so one might try the centers of those faces for the 20 points, but that
> fails also because the convex hull is then the reg dodecahedron which
> does not have triangular faces.
> 
> 
>>In that case all the triangles aren't "as close to
>>equilateral as possible" but equilateral. The dual polyhedron is a
>>regular dodecahedron (as can be seen on my site and on many other
>>sites). For similar arrangments of more points in many cases one can use
>>the theory of geodesic spheres (domes). Also in povray a lot of examples
>>exist.
>>Herman Serras
>>http://cage.ugent.be/~hs
> 
> 
> Jim Buddenhagen
> 
Shame on me! I must have been very confused.
Herman

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