Möbius Deltahedra
There are five acoptic deltahedra for which every edge line is on a symmetry plane. See the Möbius Deltahedra paper by Peter Messer.
Möbius Triangles are those that occur on the surface of a sphere has been divided its symmetry planes. See George Hart's Symmetry Planes Page.
For Tetrahedral symmetry there are three different cases depending on how many planes are present. There can be 0, 3 or 6 planes of symmetry. If 3 tetrahedral planes of symmetry alone would be considered, then the octahedron would be the one and only Möbius Deltahedron for that case.
For the purposes of this study only the case of 6 symmetry planes is considered. A sphere divided by the 6 symmetry planes results in 24 spherical triangles of 54-54-90 degrees. Note that the angles of Spherical Triangles can add up to more the 180 degrees.
While the following models are not spherical, they help depict the planes of symmetry. The Tetrahedral Symmetry Planes model shows the 6 planes. The tetrahedron, which has been divided by those planes has on its surface, has triangles which are 30-60-90 degrees. Also shown is the same tetrahedron but the vertices have been projected onto a sphere, and the triangles are still flat.
Interestingly, whenever we move the central vertices off the plane of the tetrahedron by the same vector, then octahedral symmetry results. This can be seen in the symmetry plane and sphere projection models.
Each figure in the following tables lists the symmetry (S) Dn - Dihedral, T - Tetrahedral, O - Octahedral, I - Icosahedral. The total Face, Edge and Vertex counts are given.
| Tetrahedral Symmetry and Mobius Triangles |
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| Mobius Triangle Tetrahedron SP | | S=O | F=24 E=36 V=14 |  | | off wrl switch |
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For Octahedral symmetry, a sphere divided by the 9 symmetry planes results in 48 spherical triangles of 45-60-90 degrees. The Octahedral Symmetry Planes model shows the 9 planes. The cube, which has been divided by those planes has on its surface, has triangles which are 45-45-90 degrees. Also shown is the same cube but the vertices have been projected onto a sphere.
| Octahedral Symmetry and Mobius Triangles |
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For Icosahedral symmetry, a sphere divided by the 15 symmetry planes results in 120 spherical triangles of 36-60-90 degrees. The Octahedral Symmetry Planes model shows the 15 planes. The dodecahedron which has been divided by those planes has on its surface, triangles which are 36-54-90 degrees. Also shown is the same dodecahedron but the vertices have been projected onto a sphere.
| Icosahedral Symmetry and Mobius Triangles |
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| Mobius Triangle Dodecahedron | | S=I | F=120 E=180 V=62 |  | | off wrl switch |
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| Mobius Triangle Dodecahedron SP | | S=I | F=120 E=180 V=62 |  | | off wrl switch |
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It turns out that Möbius Deltahedra are simply isomers of the Möbius Triangle versions of the tetrahedron, cube and dodecahedron above. Each one has two isomers, denoted by A and B. This is in keeping with Messer's notation. Notice there is no 24-Deltahedron B displayed, because, as noted by Messer, this isomer has faces which would be split by a symmetry plane. Also known as the Hexaugmented Cube, it is a biform deltahedra and can seen on the The Cundy Deltahedra page. Also, as noted above, once the vertices are raised off the tetrahedron's planes, the symmetry of the 24-Deltahedron isomers becomes octahedral.
