# The dodecahedron

### Statistics

 genus c 0, orientable Schläfli formula c {5,3} V / F / E c 20 / 12 / 30 notes vertex, face multiplicity c 1, 1 Petrie polygons 6, each with 10 edges antipodal sets 10 of ( 2v ), 6 of ( 2f; p1 ), 15 of ( 2e ) rotational symmetry group A5, with 60 elements full symmetry group A5×C2, with 120 elements its presentation c < r, s, t | r2, s2, t2, (rs)5, (st)3, (rt)2 > C&D number c R0.3′ The statistics marked c are from the published work of Professor Marston Conder.

### Relations to other Regular Maps

Its dual is the icosahedron.

Its Petrie dual is C6:{10,3}5.

It is a 2-fold cover of the hemidodecahedron.

It can be 2-split to give S5:{10,3}.

It can be rectified to give the icosidodecahedron.

It is the result of pyritifying (type 3/3/5/3) the tetrahedron.
It is the result of pyritifying (type 4/3/5/3) the cube.

It can be stellated (with path <1>) to give the dodecahedron . The density of the stellation is 7.
It can be stellated (with path <1,-1>) to give the tetrahedron . The multiplicity of the stellation is 5 and the total density is 5.
It can be derived by stellation (with path <1>) from the dodecahedron. The density of the stellation is 7.

### Underlying Graph

Its skeleton is dodecahedron.

This is one of the five "Platonic solids".

 A5

 A5×C2