Proper great stellated dodecahedron
Proper great 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+3 
Central density  4 
Related polytopes  
Convex core  Icosahedron 
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 great 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 great stellated dodecahedron.
Its Collins tag is I53a_1.
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 pentagrams. They are actual {5/3}s, having density and winding number equal to 3. That notation, however, has historically been used for a simple {5/2} of opposite orientation.
Gallery[edit  edit source]
Vertex coordinates[edit  edit source]
The vertices of a proper great stellated dodecahedron of edge length 1, centered at the origin, are
 ,
along with all even permutations of
 ,
 .
The outer vertices of a proper great stellated dodecahedron, centered at the origin and enclosed by a unit sphere, are located at
 ,
along with all even permutations of
 .
The inner vertices are at radius and are located at 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 great stellated dodecahedron shares the same vertex arrangement. The overlapped great stellated dodecahedron and great stellated dodecahedron share the same edge line configuration.
Dual[edit  edit source]
Because the proper great stellated dodecahedron has colinear edges, its dual has coincident edges. Combining coincident edges to create a tetradic edge figure turns the dual into the great complex icosidodecahedron.