Dodecahedron

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
}

{\displaystyle \{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

{\displaystyle 20}
* edge length =

30
l

{\displaystyle 30l}
* surface area =

3

5
(
5
2

5

)

l
2

{\displaystyle 3\sqrt { 5(5+2\sqrt { 5 }) }l^2}
* surcell volume =

15
7

5

4

l
3

{\displaystyle \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:

φ

3

2

l

{\displaystyle \frac{\varphi\sqrt{3}}{2}l}
* Edge radius:

φ
2

2

l

{\displaystyle \frac{\varphi^2}{2}l}
* Face radius:

25
11

5

10

2

l

{\displaystyle \frac{\sqrt{\frac{25+11\sqrt{5}}{10}}{2}}l}

Where

φ

{\displaystyle \varphi}

 is the golden ratio.

### Angles

* Dihedral angle:

arccos
⁡
(

5

5

)

{\displaystyle \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".

**Self-intersecting regular polyhedra:** gad · sissid · gike · gissid

{
2
,
3
}

{\displaystyle \{2,3\}}

{
3
,
3
}

{\displaystyle \{3, 3\}}

{
4
,
3
}

{\displaystyle \{4,3\}}

{
5
,
3
}

{\displaystyle \{5, 3\}}

{
6
,
3
}

{\displaystyle \{6,3\}}

{
7
,
3
}

{\displaystyle \{7,3\}}

{
8
,
3
}

{\displaystyle \{8,3\}}

{
∞
,
3
}

{\displaystyle \{\infty,3\}}

{

π
i

λ

,
3

{\displaystyle \{\frac{\pi i}{\lambda},3}

{
5
,
2
}

{\displaystyle \{5,2\}}

{
5
,
3
}

{\displaystyle \{5, 3\}}

{
5
,
4
}

{\displaystyle \{5,4\}}

{
5
,
5
}

{\displaystyle \{5,5\}}

{
5
,
6
}

{\displaystyle \{5,6\}}

{
5
,
7
}

{\displaystyle \{5,7\}}

{
5
,
8
}

{\displaystyle \{5,8\}}

{
5
,
∞
}

{\displaystyle \{5,\infty\}}

{
5
,

π
i

λ

}

{\displaystyle \{5,\frac{\pi i}{\lambda}\}}

t
0

{
5
,
3
}

{\displaystyle t\_0 \{5,3\}}

t
1

{
5
,
3
}

{\displaystyle t\_1 \{5,3\}}

t
2

{
5
,
3
}

{\displaystyle t\_2 \{5,3\}}

t

0
,
1

{
5
,
3
}

{\displaystyle t\_{0,1} \{5,3\}}

t

1
,
2

{
5
,
3
}

{\displaystyle t\_{1,2} \{5,3\}}

t

0
,
2

{
5
,
3
}

{\displaystyle t\_{0,2} \{5,3\}}

t

0
,
1
,
2

{
5
,
3
}

{\displaystyle t\_{0,1,2} \{5,3\}}