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 , 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.
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 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
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.
- Vertex radius:
- Edge radius:
- Face radius:
Where is the golden ratio.
- Dihedral angle:
The vertex coordinates of a dodecahedron of side 2 are:
- (±φ^2,±1,0) and all even permutations
where φ = (1+√5)/2.
|...||Trigonal hosohedron||Tetrahedron||Cube||Dodecahedron||Hexagonal tiling||Order-3 heptagonal tiling||Order-3 octagonal tiling||...||Order-3 apeirogonal tiling|
|...||Pentagonal dihedron||Dodecahedron||Order-4 pentagonal tiling||Order-5 pentagonal tiling||Order-6 pentagonal tiling||...||Infinite-order pentagonal tiling|
|Regular ||Rectified ||Birectified ||Truncated ||Bitruncated ||Cantellated ||Cantitruncated ||Dodecahedron||Icosidodecahedron||Icosahedron||Truncated dodecahedron||Truncated icosahedron||Rhombicosidodecahedron||Great rhombicosidodecahedron|