what is this figure based on a cuboctahedral concept

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what is this figure based on a cuboctahedral concept

Postby Sphericality » Thu Nov 03, 2016 12:08 pm

It occurred to me that in the same way a cuboctahedron
can be conceptualised as a net of triangles expanding out from one triangle
to three which are joined to three others
and then contracting to meet at the eighth triangle -
it might be possible for 10 tetrahedrons to wrap around in 4D.

This image is a 3d print of the resulting concept.
Its a little hard to see the central tetrahedron
which is extended to form 4 more
which are connected to four larger ones
which connect to the large outer tetrahedron.

Can anyone clarify for me what 4D form this is a projection of
Image?
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Re: what is this figure based on a cuboctahedral concept

Postby wendy » Thu Nov 03, 2016 12:38 pm

Looks like xo3ox3oo&#xt.
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Re: what is this figure based on a cuboctahedral concept

Postby Klitzing » Thu Nov 03, 2016 10:35 pm

wendy wrote:Looks like xo3ox3oo&#xt.

No, Wendy. Rather it happens to be spid = x3o3o3x.

In fact, yours is rap = o3x3o3o = xo3ox3oo&#x, which uses octahedra besides of the tetrahedra. Whereas mine uses trigonal prisms as addendum instead, which can be spotted quite clearly in "sphericality"s picture...

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Re: what is this figure based on a cuboctahedral concept

Postby wendy » Fri Nov 04, 2016 1:15 am

You are right, it's a 4spc or x3o3o3x. I see it now.
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Re: what is this figure based on a cuboctahedral concept

Postby Sphericality » Sun Nov 06, 2016 6:19 am

Thanks Klitzing - with your help I have tracked down a very similar diagram on Wikipedia here
https://en.wikipedia.org/wiki/File:Schl ... 5-cell.png

I can also now see that a tetrahedron can be truncated to product a cuboctahedron.
This has helped me to find the equivalent 4D shape of the stella octangula (star tetrahedron).
Adding another point at the center of each of the 10 tetrahedra so that they each become projections of the 5cell seems to be the exact equivalent of drawing the stellaoctangula in 2D by extending each of the triangles of a projected cuboctahedron outwards.
Here is what I believe to be the exact equivalent 2D projection of the 3D stella octangula.
In both cases the centerpoint is apparently shared. (is there a technical term for this where projections share vertexes? I see it happen often).
Image
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Re: what is this figure based on a cuboctahedral concept

Postby Sphericality » Sun Nov 06, 2016 6:20 am

Can you help me with another question related to the simplex ... (Im not sure if I should create a new thread for this but it is a related question).

If two simplexes are interlocked so that each ones vertex penetrates one of the other ones tetrahedra -
and then the 10 vertexes are joined -
what is the resulting 4D form?
I believe this is the two dimensional projection of it
Image
It seems all 10 points must be equidistant from the 4D center.
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Re: what is this figure based on a cuboctahedral concept

Postby wendy » Sun Nov 06, 2016 6:53 am

The figure described would be the bi-apiculated pentachoron o3m3m3o. This is bounded by thirty disphenoid tetrahedra, three around the edges of each of the pentachoron's edges. The dual of this is o3x3x3o, bitruncated pentachoron, bounded by ten truncated tetrahedra.

The real magic is the similar one in five dimensions, which has 12 vertices, bounded by 20 bi-triangular tegums, its called the bisurtegmic hexapetan.
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