There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry (with fundamental domain Möbius triangle (5 3 2)). (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have reflective vertex symmetry.
| Vertex arrangement (Convex hull) |
Nonconvex forms | |||||||
|---|---|---|---|---|---|---|---|---|
Icosahedron |
{5,5/2} |
{5/2,5} |
{3,5/2} |
|||||
Nonuniform truncated icosahedron 2 5 |3 |
U37 2 5/2 | 5 |
U61 5/2 3 | 5/3 |
U67 5/3 3 | 2 |
U73 2 5/3 (3/2 5/4) | |
||||
Nonuniform truncated icosahedron 2 5 |3 |
U38 5/2 5 | 2 |
U44 5/3 5 | 3 |
U56 2 3 (5/4 5/2) | |
|||||
Nonuniform truncated icosahedron 2 5 |3 |
U32 | 5/2 3 3 |
|||||||
Icosidodecahedron 2 | 3 5 |
U49 3/2 3 | 5 |
U51 5/4 5 | 5 |
U54 2 | 3 5/2 |
U70 5/3 5/2 | 5/3 |
U71 3 3 | 5/3 |
U36 2 | 5 5/2 |
U62 5/3 5/2 | 3 |
U65 5/4 5 | 3 |
Truncated dodecahedron 2 3 | 5 |
U42 |
U48 |
U63 |
|||||
Nonuniform truncated dodecahedron |
U72 |
|||||||
Dodecahedron |
{5/2,3} |
U30 |
U41 |
U47 |
||||
Rhombicosidodecahedron |
U33 |
U39 |
U58 |
|||||
Beveled Dodecahedron |
U55 |
|||||||
Nonuniform rhombicosidodecahedron |
U31 |
U43 |
U50 |
U66 |
||||
Nonuniform rhombicosidodecahedron |
U75 |
U64 |
Skilling's figure (see below) |
|||||
Nonuniform truncated icosidodecahedron |
U45 |
|||||||
Nonuniform truncated icosidodecahedron |
U59 |
|||||||
Nonuniform truncated icosidodecahedron |
U68 |
|||||||
Nonuniform snub dodecahedron |
U40 |
U46 |
U57 |
U69 |
U60 |
U74 |
||
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)