Equatorial octahedral cells in 3d-proj run24-cell planar?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Equatorial octahedral cells in 3d-proj run24-cell planar?

Postby Flx » Mon Aug 26, 2019 11:17 am

In the octahedron centered projection of the runcinated 24-cell, are the 12 equatorial octahedra projected onto rhombi? In this case I guess the projected 3d-envelope shoud correspond to the polyhedron where these planar octahedra become flattened rhombi with diagonals 1 and Sqrt(2) as shown in the attached image. Is this correct? I do not see the planes clearly in the cell decomposition http://eusebeia.dyndns.org/4d/runci24cell, they could be two triangles slightly folded instead of a rhombus?
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Possible Projection Runci24-cell.JPG
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Re: Equatorial octahedral cells in 3d-proj run24-cell plan

Postby wendy » Tue Aug 27, 2019 8:00 am

The link actually shows a stereo projection of the polytope.

The decomposition of the middle section into 12 \(1:\sqrt{2}\) rhombs is correct. This is the process of rectifying. The 12 rhombuses are the faces of a rhombic dodecahedron, and the triangles and squares make a cuboctahedron. All of these are sections of octahedra.
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Re: Equatorial octahedral cells in 3d-proj run24-cell plan

Postby Flx » Wed Aug 28, 2019 10:45 am

Thank you, Wendy.
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