There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry (with fundamental domain Möbius triangle (4 3 2)).
There are four Schwarz_triangles that generate nonconvex forms, two right triangles (3/2 4 2), and (4/3 3 2), and two general triangles: (4/3 4 3), (3/2 4 4).
| Vertex arrangement (Convex hull) |
Nonconvex forms | ||
|---|---|---|---|
Cube |
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Octahedron |
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Cuboctahedron |
(6.4/3.6.4) 4/3 4 | 3 |
||
Truncated cube |
(4.8/3.4/3.8/5) 2 4/3 (3/2 4/2) | |
(8/3.3.8/3.4) 3 4 | 4/3 |
(4.3/2.4.4) 3/2 4 | 2 |
Truncated octahedron |
|||
Rhombicuboctahedron |
(4.8.4/3.8) 2 4 (3/2 4/2) | |
(8.3/2.8.4) 3/2 4 | 4 |
(8/3.8/3.3) 2 3 | 4/3 |
Nonuniform truncated cuboctahedron |
(4.6.8/3) 2 3 4/3 | |
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Nonuniform truncated cuboctahedron |
(8/3.6.8) 3 4 4/3 | |
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Snub cube |
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Read more about this topic: Nonconvex Polyhedra
Famous quotes containing the word symmetry:
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)