The cube


genus c0, orientable
Schläfli formula c{4,3}
V / F / E c 8 / 6 / 12
notesreplete singular is a polyhedral map permutes its vertices evenly
vertex, face multiplicity c1, 1
Petrie polygons
4, each with 6 edges
antipodal sets4 of ( 2v; p1 ), 3 of ( 2f ), 6 of ( 2e )
rotational symmetry groupS4, with 24 elements
full symmetry groupS4×C2, with 48 elements
its presentation c< r, s, t | r2, s2, t2, (rs)4, (st)3, (rt)2 >
C&D number cR0.2′
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.

List of regular maps in orientable genus 0.

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


Type II


Type IIa


Type III


Other Regular Maps

General Index

The images on this page are copyright © 2010 N. Wedd