Dodecahedron

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Dodecahedron
Rank3
TypeRegular
Notation
Bowers style acronymDoe
Coxeter diagramx5o3o ()
Schläfli symbol{5,3}
Conway notationD
Stewart notationD5
Elements
Faces12 pentagons
Edges30
Vertices20
Vertex figureTriangle, edge length (1+5)/2
Petrie polygons6 skew decagons
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density1
Number of external pieces12
Level of complexity1
Related polytopes
ArmyDoe
RegimentDoe
DualIcosahedron
Petrie dualPetrial dodecahedron
κ ?Petrial great stellated dodecahedron
ConjugateGreat stellated dodecahedron
Abstract & topological properties
Flag count120
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonDodecahedral graph
Properties
SymmetryH3, order 120
Flag orbits1
ConvexYes
Net count43380[1]
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 as well as the nonconvex grand 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)
  • xfoo5oofx&#xt (H2 axial, face-first)
  • ofxfoo3oofxfo&#xt (A2 axial, vertex-first)
  • xfoFofx ofFxFxo&#xt (K2 axial, edge-first)
  • oxfF xFfo Fofx&#zx (K3 symmetry)

Related polytopes[edit | edit source]

Alternative realizations[edit | edit source]

PointDodecahedronGreat stellated dodecahedronSkew pure dodecahedronHemidodecahedron (4-dimensional)Hemidodecahedron (5-dimensional)Hemidodecahedron (9-dimensional)Dodecahedron (6-dimensional)Dodecahedron (cross-polytope realization)
Symmetric realizations of {5,3}. Click on a node to be taken to the page for that realization.

The dodecahedron and the great stellated dodecahedron are conjugates. Thus they are realizations of the same underlying abstract regular polytope {5,3}. These are the only faithful symmetric realizations of this polytope in 3-dimensional Euclidean space, however there are many more skew faithful symmetric realizations. In total there are 28 faithful symmetric realizations, of which 3 are pure.

Faithful symmetric realizations of {5,3}
Dimension Components Name
3 Dodecahedron Dodecahedron
3 Great stellated dodecahedron Great stellated dodecahedron
4 Skew pure dodecahedron Skew pure dodecahedron
6 Dodecahedron
Great stellated dodecahedron
Dodecahedron (6-dimensional)
7 Dodecahedron
Skew pure dodecahedron
7 Great stellated dodecahedron
Skew pure dodecahedron
7 Dodecahedron
Hemidodecahedron (4-dimensional)
7 Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
8 Dodecahedron
Hemidodecahedron (5-dimensional)
8 Great stellated dodecahedron
Hemidodecahedron (5-dimensional)
8 Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
10 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Dodecahedron (cross-polytope realization)
10 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
11 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
11 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
11 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (5-dimensional)
12 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
12 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
12 Dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
12 Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
13 Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
14 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
15 Dodecahedron
Great stellated dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
15 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (5-dimensional)
16 Dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
16 Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)
19 Dodecahedron
Great stellated dodecahedron
Skew pure dodecahedron
Hemidodecahedron (4-dimensional)
Hemidodecahedron (5-dimensional)

Johnson solids[edit | edit source]

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

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.

Stellations[edit | edit source]

A dodecahedron has three regular stellations:

It also has an uncounted number of stellations with pyritohedral or chiral-tetrahedral symmetry.

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

In vertex figures[edit | edit source]

Dodecahedra in vertex figures
Name Picture Schläfli symbol Edge length
Small stellated hecatonicosachoron
{5/2,5,3}
Icosahedral honeycomb
{3,5,3}

External links[edit | edit source]

References[edit | edit source]

  1. Edkins, Jo (2007). "Dodecahedron". Solid shapes and their nets. Archived from the original on 2019-12-26.