Dodecahedron

Dodecahedron
Dimensionality	3
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Full icosahedral symmetry (Ih)
Coxeter–Dynkin diagram
Bowers acronym	doe
Subfacets
Vertices	20 points
Edges	30 line segments
Faces	12 pentagons
Cells	1 dodecahedron
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000005-QINU`"'

A dodecahedron is a regular polyhedron, one of the five three-dimensional platonic solids, and has twelve congruent pentagonal faces. It has a schläfli symbol of $\{5, 3\}$, meaning that 3 pentagons join at each vertex. It is the dual of the icosahedron.

Under the elemental naming scheme, it is called a cosmogon.

Properties

It is possible to inscribe 5 cubes into a dodecahedron such that all of the edges of the cubes are diagonals of the dodecahedron's faces.

It is also the only platonic solid to have pentagonal faces.

Symbols

Symbols for the dodecahedron include:

• x5o3o (regular)
• xfoo5oofx&Ext (pentagon symmetry)
• ofxfoo3oofxfo&#xt (triangular symmetry)
• xfoFofx ofFxFxo&#xt (digonal symmetry)
• oxfF xFfo Fofx&#zx (block symmetry)

Subfacets and Structure

Structure

The dodecahedron has 12 pentagonal faces, joining three to a vertex. When seen face first, the layers start with one face, then the five surrounding it, then five more, and finally the opposite face.

Hypervolumes

• vertex count = $20$
• edge length = $30l$
• surface area = $3\sqrt { 5(5+2\sqrt { 5 }) }l^2$
• surcell volume = $\frac { 15+7\sqrt { 5 } }{ 4 }l^3$

Subfacets

• 20 points (0D)
• 30 line segments (1D)
• 12 pentagons (2D)
• 1 dodecahedron (3D)

Radii

• Vertex radius: $\frac{\varphi\sqrt{3}}{2}l$
• Edge radius: $\frac{\varphi^2}{2}l$
• Face radius: $\frac{\sqrt{\frac{25+11\sqrt{5}}{10}}{2}}l$

Where $\varphi$ is the golden ratio.

Angles

• Dihedral angle: $\arccos(\frac{\sqrt{5}}{5})$

vertex coordinates

The vertex coordinates of a dodecahedron of side 2 are:

• (±φ,±φ,±φ)
• (±φ^2,±1,0) and all even permutations

where φ = (1+√5)/2.

Related shapes

• dual: Icosahedron
• Vertex figure: Triangle, side length φ
• Conjugate: Great stellated dodecahedron

See Also

• Polytope Wiki. "Dodecahedron".
• Bowers, Jonathan. "Polyhedron Catgeory 1: Regulars".

$\{2,3\}$	$\{3, 3\}$	$\{4,3\}$	$\{5, 3\}$	$\{6,3\}$	$\{7,3\}$	$\{8,3\}$	...	$\{\infty,3\}$	$\{\frac{\pi i}{\lambda},3$
Trigonal hosohedron	Tetrahedron	Cube	Dodecahedron	Hexagonal tiling	Order-3 heptagonal tiling	Order-3 octagonal tiling	...	Order-3 apeirogonal tiling	Order-3 pseudogonal tiling

$\{5,2\}$	$\{5, 3\}$	$\{5,4\}$	$\{5,5\}$	$\{5,6\}$	$\{5,7\}$	$\{5,8\}$	...	$\{5,\infty\}$	$\{5,\frac{\pi i}{\lambda}\}$
Pentagonal dihedron	Dodecahedron	Order-4 pentagonal tiling	Order-5 pentagonal tiling	Order-6 pentagonal tiling	Order-7 pentagonal tiling	Order-8 pentagonal tiling	...	Infinite-order pentagonal tiling	Imaginary-order pentagonal tiling

Regular $t_0 \{5,3\}$	Rectified $t_1 \{5,3\}$	Birectified $t_2 \{5,3\}$	Truncated $t_{0,1} \{5,3\}$	Bitruncated $t_{1,2} \{5,3\}$	Cantellated $t_{0,2} \{5,3\}$	Cantitruncated $t_{0,1,2} \{5,3\}$
Dodecahedron	Icosidodecahedron	Icosahedron	Truncated dodecahedron	Truncated icosahedron	Rhombicosidodecahedron	Great rhombicosidodecahedron

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	...	Aleph null
Simplex $\{3^{n-1}\}$	Null polytope $)($ $\emptyset$	Point $()$ $\mathbb{B}^0$	Line segment $\{\}$ $\mathbb{B}^1$	Triangle $\{3\}$	Tetrahedron $\{3^2\}$	Pentachoron $\{3^3\}$	Hexateron $\{3^4\}$	Heptapeton $\{3^5\}$	Octaexon $\{3^6\}$	Enneazetton $\{3^7\}$	Decayotton $\{3^8\}$	Hendecaxennon $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	...	Omegasimplex
Cross $\{3^{n-2},4\}$				Square $\{4\}$	Octahedron $\{3, 4\}$	Hexadecachoron $\{3^2, 4\}$	Pentacross $\{3^3, 4\}$	Hexacross $\{3^4, 4\}$	Heptacross $\{3^5, 4\}$	Octacross $\{3^6, 4\}$	Enneacross $\{3^7, 4\}$	Dekacross $\{3^8, 4\}$	Hendekacross $\{3^9, 4\}$	Dodekacross $\{3^{10}, 4\}$	Tridekacross $\{3^{11}, 4\}$	Tetradekacross $\{3^{12}, 4\}$	Pentadekacross $\{3^{13}, 4\}$	Hexadekacross $\{3^{14}, 4\}$	...	Omegacross
Hydrotopes $\{3^{n-2}, 5\}$				Pentagon $\{5\}$	Icosahedron $\{3, 5\}$	Hexacosichoron $\{3^2, 5\}$	Order-5 pentachoric tetracomb $\{3^3, 5\}$	Order-5 hexateric pentacomb $\{3^4, 5\}$	...
Hypercube $\{4, 3^{n-2}\}$				Square $\{4\}$	Cube $\{4,3\}$	Tesseract $\{4, 3^2\}$	Penteract $\{4, 3^3\}$	Hexeract $\{4, 3^4\}$	Hepteract $\{4, 3^5\}$	Octeract $\{4, 3^6\}$	Enneract $\{4, 3^7\}$	Dekeract $\{4, 3^8\}$	Hendekeract $\{4, 3^9\}$	Dodekeract $\{4, 3^{10}\}$	Tridekeract $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	...	Omegeract
Cosmotopes $\{5, 3^{n-2}\}$				Pentagon $\{5\}$	Dodecahedron $\{5, 3\}$	Hecatonicosachoron $\{5, 3^2\}$	Order-3 hecatonicosachoric tetracomb $\{5, 3^3\}$	Order-3-3 hecatonicosachoric pentacomb $\{5, 3^4\}$	...
Hyperball $\mathbb B^n$				Disk $\mathbb B^2$	Ball $\mathbb B^3$	Gongol $\mathbb B^4$	Pentorb $\mathbb B^5$	Hexorb $\mathbb B^6$	Heptorb $\mathbb B^7$	Octorb $\mathbb B^8$	Enneorb $\mathbb B^9$	Dekorb $\mathbb B^{10}$	Hendekorb $\mathbb B^{11}$	Dodekorb $\mathbb B^{12}$	Tridekorb $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	...	Omegaball $\mathbb B^{\aleph_0}$