Great dodecahedron

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Great dodecahedron
Great dodecahedron.png
Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymGad
Coxeter diagramo5/2o5x (CDel node.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node 1.png)
Schläfli symbol
Elements
Faces12 pentagons
Edges30
Vertices12
Vertex figurePentagram, edge length (1+5)/2
Great dodecahedron vertfig.png
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density3
Number of pieces60
Level of complexity3
Related polytopes
ArmyIke
RegimentIke
DualSmall stellated dodecahedron
Petrie dualPetrial great dodecahedron
ConjugateSmall stellated dodecahedron
Convex coreDodecahedron
Abstract properties
Flag count120
Euler characteristic–6
Schläfli type{5,5}
Topological properties
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great dodecahedron, or gad, is one of the four Kepler–Poinsot solids. It has 12 pentagons as faces, joining 5 to a vertex in a pentagrammic fashion.

It is in the same regiment as the icosahedron, and comes from using the icosahedron's vertex figure pentagons as the faces.

It is the second stellation of the dodecahedron.

Vertex coordinates[edit | edit source]

Its vertices are the same as those of its regiment colonel, the icosahedron.

In vertex figures[edit | edit source]

The great dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name Picture Schläfli symbol Edge length
Grand stellated hecatonicosachoron
Gishi.png
{5/2,5,5/2}
Faceted hexacosichoron
Schlegel wireframe 600-cell vertex-centered.png
{3,5,5/2}

Related polyhedra[edit | edit source]

A fundamental domain of the great dodecahedron in {5,5}.

Abstractly the great dodecahedron is a quotient of the order-5 pentagonal tiling. Specifically it is , a tessellation of Bring's surface. It is also abstractly equivalent to its conjugate, the small stellated dodecahedron.

Two uniform polyhedron compounds are composed of great dodecahedra:

o5o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great dodecahedron gad {5,5/2} x5o5/2o (CDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node.png)
Great dodecahedron.png
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Great truncated dodecahedron.png
Dodecadodecahedron did r{5,5/2} o5x5/2o (CDel node.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node.png)
Dodecadodecahedron.png
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (CDel node.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Dodecahedron.png
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (CDel node.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Small stellated dodecahedron.png
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (CDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node 1.png)
Rhombidodecadodecahedron.png
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5-2.pngCDel node 1.png)
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (CDel node h.pngCDel 5.pngCDel node h.pngCDel 5-2.pngCDel node h.png)
Snub dodecadodecahedron.png

External links[edit | edit source]

  • Klitzing, Richard. "Gad".