## FANDOM

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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.

## 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.

## 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 = $3{ l }^{ 2 }\sqrt { 5(5+2\sqrt { 5 }) }$
• surcell volume = $\frac { 15+7\sqrt { 5 } }{ 4 } { l }^{ 3 }$

### Subfacets

$\{2,3\}$ $\{3,3\}$ $\{4,3\}$ $\{5,3\}$ $\{6,3\}$ $\{7,3\}$ $\{8,3\}$ ... $\{\aleph_0,3\}$
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling
$\{5,2\}$ $\{5,3\}$ $\{5,4\}$ $\{5,5\}$ $\{5,6\}$ ... $\{5,\aleph_0\}$
Pentagonal dihedron Dodecahedron Order-4 pentagonal tiling Order-5 pentagonal tiling Order-6 pentagonal tiling ... Infinite-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