Dodecahedron

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Dodecahedron
Dodecahedron.png
Rank3
TypeRegular
SpaceSpherical
Bowers style acronymDoe
Coxeter diagramx5o3o (CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol{5,3}
Elements
Vertex figureTriangle, edge length (1+5)/2
Dodecahedron vertfig.png
Faces12 pentagons
Edges30
Vertices20
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density1
Euler characteristic2
Flag count120
Number of pieces12
Level of complexity1
Related polytopes
ArmyDoe
RegimentDoe
DualIcosahedron
Petrie dualPetrial dodecahedron
ConjugateGreat stellated dodecahedron
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

The dodecahedron, or doe, is one of the five Platonic solids. It has 12 pentagons as faces, joining 3 to a vertex.

It is the only Platonic solid that does not appear as the vertex figure in one of the convex regular polychora. It does, however, appear as the vertex figure of the nonconvex small stellated hecatonicosachoron and the hyperbolic icosahedral honeycomb. It also appears as a cell of the hecatonicosachoron.

Vertex coordinates[edit | edit source]

The vertices of a dodecahedron of edge length 1, centered at the origin, are given by:

along with all even permutations of:

The first set of vertices corresponds to a cube of edge length (1+5)/2 which can be inscribed into the dodecahedron's vertices.

Representations[edit | edit source]

A regular dodecahedron has the following Coxeter diagrams:

  • x5o3o (full symmetry)
  • x4oo5oo4x&#xt (H2 axial, face-first)
  • ofxfoo3oofxfo&#xt (A2 axial, vertex-first)
  • xfoFofx ofFxFxo&#xt (A1×A1 axial, edge-first)
  • oxfF xFfo Fofx&#zx (A1×A1×A1 symmetry)

In vertex figures[edit | edit source]

Dodecahedra in vertex figures
Name Picture Schläfli symbol Edge length
Small stellated hecatonicosachoron
Sishi.png
{5/2,5,3}
Icosahedral honeycomb
H3 353 CC center.png
{3,5,3}

Variations[edit | edit source]

The dodecahedron has a number of variations that retain its face-transitivity:

  • Pyritohedron - has 12 mirror-symmetric pentagonal faces
  • Tetartoid - has 12 generally irregular pentagonal faces, chiral tetrahedral symmetry

Related polyhedra[edit | edit source]

Several Johnson solids can be formed by augmenting the faces of the dodecahedron with pentagonal pyramids:

The dodecahedron has three regular stellations, namely the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron. It also has an uncounted number of stellations with pyritohedral or chiral-tetrahedral symmetry.

The dodecahedron can be constructed by augmenting a cube with 6 specifically-proportioned wedges, such that adjacent triangular and trapezoidal faces of the wedges combine into regular pentagons.

o5o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Dodecahedron doe {5,3} x5o3o
Uniform polyhedron-53-t0.png
Truncated dodecahedron tid t{5,3} x5x3o
Uniform polyhedron-53-t01.png
Icosidodecahedron id r{5,3} o5x3o
Uniform polyhedron-53-t1.png
Truncated icosahedron ti t{3,5} o5x3x
Uniform polyhedron-53-t12.png
Icosahedron ike {3,5} o5o3x
Uniform polyhedron-53-t2.png
Small rhombicosidodecahedron srid rr{5,3} x5o3x
Uniform polyhedron-53-t02.png
Great rhombicosidodecahedron grid tr{5,3} x5x3x
Uniform polyhedron-53-t012.png
Snub dodecahedron snid sr{5,3} s5s3s
Uniform polyhedron-53-s012.png

Stellations[edit | edit source]

A dodecahedron has three stellations:

External links[edit | edit source]

  • Klitzing, Richard. "Doe".