Great stellated dodecahedron

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Great stellated dodecahedron
Great stellated dodecahedron.png
Bowers style acronymGissid
Coxeter diagramx5/2o3o (CDel node 1.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol
Faces12 pentagrams
Vertex figureTriangle, edge length (5–1)/2
Great stellated dodecahedron vertfig.png
Measures (edge length 1)
Edge radius
Dihedral angle
Central density7
Number of pieces60
Level of complexity3
Related polytopes
DualGreat icosahedron
Petrie dualPetrial great stellated dodecahedron
Convex coreDodecahedron
Abstract properties
Flag count120
Euler characteristic2
Schläfli type{5,3}
Topological properties
SymmetryH3, order 120

The great stellated dodecahedron, or gissid, is one of the four Kepler–Poinsot solids. It has 12 pentagrams as faces, joining 3 to a vertex.

It is the last stellation of the dodecahedron, from which its name is derived. It is also the only Kepler-Poinsot solid to share its vertices with the dodecahedron as opposed to the icosahedron. It has the smallest circumradius of any uniform polyhedron.

Vertex coordinates[edit | edit source]

The vertices of a great stellated dodecahedron of edge length 1, centered at the origin, are all sign changes of

along with all even permutations and all sign changes of

The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

In vertex figures[edit | edit source]

The great stellated dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.

Name Picture Schläfli symbol Edge length
Great grand hecatonicosachoron

Related polyhedra[edit | edit source]

o3o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great icosahedron gike {3,5/2} x3o5/2o (CDel node 1.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node.png)
Great icosahedron.png
Truncated great icosahedron tiggy t{3,5/2} x3x5/2o (CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great truncated icosahedron.png
Great icosidodecahedron gid r{3,5/2} o3x5/2o (CDel node.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great icosidodecahedron.png
Truncated great stellated dodecahedron (degenerate, ike+2gad) t{5/2,3} o3x5/2x (CDel node.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Small complex icosidodecahedron.png
Great stellated dodecahedron gissid {5/2,3} o3o5/2x (CDel node.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Great stellated dodecahedron.png
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom) sicdatrid rr{3,5/2} x3o5/2x (CDel node 1.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Compound of small ditrigonal icosidodecahedron and the compound of five cubes.png
Truncated great icosidodecahedron (degenerate, ri+12(10/2)) tr{3,5/2} x3x5/2x (CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Great snub icosidodecahedron gosid sr{3,5/2} s3s5/2s (CDel node h.pngCDel 3.pngCDel node h.pngCDel 5-2.pngCDel node h.png)
Great snub icosidodecahedron.png

External links[edit | edit source]