Dodecahedron

Dodecahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Notation
Bowers style acronym	Doe
Coxeter diagram	x5o3o ()
Schläfli symbol	{5,3}
Conway notation	D
Stewart notation	D5
Elements
Faces	12 pentagons
Edges	30
Vertices	20
Vertex figure	Triangle, edge length (1+√5)/2
Petrie polygons	6 skew decagons
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {3}}+{\sqrt {15}}}{4}}\approx 1.40126$
Edge radius	${\frac {3+{\sqrt {5}}}{4}}\approx 1.30902$
Inradius	${\sqrt {\frac {25+11{\sqrt {5}}}{40}}}\approx 1.11352$
Volume	${\frac {15+7{\sqrt {5}}}{4}}\approx 7.66312$
Dihedral angle	$\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	12
Level of complexity	1
Related polytopes
Army	Doe
Regiment	Doe
Dual	Icosahedron
Petrie dual	Petrial dodecahedron
Conjugate	Great stellated dodecahedron
Abstract & topological properties
Flag count	120
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Net count	43380
Nature	Tame

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

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

• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$,

along with all even permutations of:

• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right)$.

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

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 (K₂ axial, edge-first)
• oxfF xFfo Fofx&#zx (K₃ symmetry)

In vertex figures

Dodecahedra in vertex figures

Name	Picture	Schläfli symbol	Edge length
Small stellated hecatonicosachoron		{5/2,5,3}	${\frac {{\sqrt {5}}-1}{2}}$
Icosahedral honeycomb		{3,5,3}

Variations

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

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

• Augmented dodecahedron - One face is augmented
• Parabiaugmented dodecahedron - Two opposite faces are augmented
• Metabiaugmented dodecahedron - Two non-adjacent, non-opposite faces are augmented
• Triaugmented dodecahedron - Three mutually non-adjacent faces are augmented

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

A dodecahedron has three regular stellations:

• The small stellated dodecahedron
• The great dodecahedron
• The great stellated dodecahedron

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

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#4).

• Klitzing, Richard. "Doe".

• Quickfur. "The Dodecahedron".

• Nan Ma. "Dodecahedron {5, 3}".

• McCooey, David. "Dodecahedron"

• Hi.gher.Space Wiki Contributors. "Dodecahedron".

• Wikipedia contributors. "Dodecahedron".
• Hartley, Michael. "{5,3}*120".
• Wedd, N. The dodecahedron