# Proper small stellated dodecahedron

Proper small stellated dodecahedron | |
---|---|

Rank | 3 |

Type | Isotoxal |

Elements | |

Faces | 12 star pentambi |

Edges | 60 |

Vertices | 12+20 |

Vertex figure | |

Measures (edge length 1) | |

Dihedral angle | |

Vertex density | 1+2 |

Central density | 2 |

Related polytopes | |

Convex core | Dodecahedron |

Abstract & topological properties | |

Flag count | 240 |

Euler characteristic | –16 |

Orientable | Yes |

Genus | 9 |

Properties | |

Symmetry | I_{h},H_{3},(*532), order 120 |

Flag orbits | 2 |

Convex | No |

Nature | Wild |

The **proper small stellated dodecahedron** is an isotoxal polyhedron that is isohedral but not isogonal. It consists of 12 star pentambi in the same orientation as the 12 pentagrams found in the small stellated dodecahedron.

Its Collins tag is I53a_4.

The term "proper" indicates that this polyhedron can be produced by extending the faces of a regular Dodecahedron in their planes. Such an operation is the basis of stellation but cannot increase the density of any face. The usual interpretation of the corresponding Kepler's star, with pentagrams as face boundaries, cannot be produced by this operation.

The vertex figures at the inner vertices cannot be shown as planar polygons and are not ordinary triangles. They are actual {3/2}s, having density and winding number equal to 2. That notation, however, has historically been used for a simple {3} of opposite orientation.

## Vertex coordinates[edit | edit source]

The vertices of a proper small stellated dodecahedron of edge length 1, centered at the origin, are located at

- ,

along with all even permutations of

- ,
- .

The outer vertices of a proper small stellated dodecahedron, centered at the origin and enclosed by a unit sphere, are located at all even permutations of

- .

The inner vertices are at radius and are located at

- ,

along with all even permutations of

- .

In this case the edge length is .

## Face Angles[edit | edit source]

The decagons have alternating interior angles of and .

## Related polytopes[edit | edit source]

The overlapped small stellated dodecahedron shares the same vertex arrangement. The overlapped small stellated dodecahedron and small stellated dodecahedron share the same edge line configuration.

### Dual[edit | edit source]

Because the proper small stellated dodecahedron has colinear edges, its dual has coincident edges. Combining the coincident edges to produce a tetradic edge figure produces the small complex icosidodecahedron.