# Proper small stellated dodecahedron

Proper small stellated dodecahedron
Rank3
TypeIsotoxal
Elements
Faces12 star pentambi
Edges60
Vertices12+20
Vertex figure
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Vertex density1+2
Central density2
Related polytopes
Convex coreDodecahedron
Abstract & topological properties
Flag count240
Euler characteristic–16
OrientableYes
Genus9
Properties
SymmetryIh,H3,(*532), order 120
Flag orbits2
ConvexNo
NatureWild

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

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$,

along with all even permutations of

• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1-{\sqrt {5}}}{4}}\right)}$.

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

• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,0\right)}$.

The inner vertices are at radius ${\displaystyle {\sqrt {\frac {15-6{\sqrt {5}}}{5}}}\approx 0.56278}$ and are located at

• ${\displaystyle \left(\pm {\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right)}$,

along with all even permutations of

• ${\displaystyle \left(\pm {\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,0,\,\pm {\sqrt {\frac {25-11{\sqrt {5}}}{10}}}\right)}$.

In this case the edge length is ${\displaystyle {\sqrt {\frac {20-8{\sqrt {5}}}{5}}}\approx 0.64984}$.

## Face Angles

The decagons have alternating interior angles of ${\displaystyle 36^{\circ }}$ and ${\displaystyle 252^{\circ }}$.

## Related polytopes

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

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.