# The cube

### Statistics

 genus c 0, orientable Schläfli formula c {4,3} V / F / E c 8 / 6 / 12 notes vertex, face multiplicity c 1, 1 Petrie polygons 4, each with 6 edges antipodal sets 4 of ( 2v; p1 ), 3 of ( 2f ), 6 of ( 2e ) rotational symmetry group S4, with 24 elements full symmetry group S4×C2, with 48 elements its presentation c < r, s, t | r2, s2, t2, (rs)4, (st)3, (rt)2 > C&D number c R0.2′ The statistics marked c are from the published work of Professor Marston Conder.

### Relations to other Regular Maps

Its dual is the octahedron.

Its Petrie dual is {6,3}(2,2).

It is a 2-fold cover of the hemicube.

It can be rectified to give the cuboctahedron.

It can be Eppstein tunnelled to give S4:{12,3}.

It can be obtained by truncating the 4-hosohedron.

It can be pyritified (type 4/3/5/3) to give the dodecahedron.
It is the result of pyritifying (type 2/3/4/3) the 3-hosohedron.

Its half shuriken is C4:{4,6}3.

It can be stellated (with path <1>) to give the tetrahedron . The multiplicity of the stellation is 2 and the total density is 2.

### Underlying Graph

Its skeleton is cubic graph.

This is one of the five "Platonic solids".

This regular map features in Jarke J. van Wijk's movie Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps, 0:10 seconds from the start. It is shown as a "wireframe diagram", on K1. The wireframe is arranged as the skeleton of the edgeless map.

### Cayley Graphs based in this Regular Map

#### Type I

 C4×C2
 D8
 D8
 C2×C2×C2

 S4

 A4×C2

 S4×C2