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.
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[]
Subfacets[]
- 20 points (0D)
- 30 line segments (1D)
- 12 pentagons (2D)
- 1 dodecahedron (3D)
Radii[]
- Vertex radius:
- Edge radius:
- Face radius:
Where is the golden ratio.
Angles[]
- Dihedral angle:
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".
Regular polyhedra |
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Convex regular polyhedra: tet · cube · oct · doe · ike
Self-intersecting regular polyhedra: gad · sissid · gike · gissid |
... | |||||||||
---|---|---|---|---|---|---|---|---|---|
Trigonal hosohedron | Tetrahedron | Cube | Dodecahedron | Hexagonal tiling | Order-3 heptagonal tiling | Order-3 octagonal tiling | ... | Order-3 apeirogonal tiling | Order-3 pseudogonal tiling |
Regular |
Rectified |
Birectified |
Truncated |
Bitruncated |
Cantellated |
Cantitruncated |
---|---|---|---|---|---|---|
Dodecahedron | Icosidodecahedron | Icosahedron | Truncated dodecahedron | Truncated icosahedron | Rhombicosidodecahedron | Great rhombicosidodecahedron |