The tetrahedron


genus c0, orientable
Schläfli formula c{3,3}
V / F / E c 4 / 4 / 6
notesreplete singular is a polyhedral map permutes its vertices evenly
vertex, face multiplicity c1, 1
Petrie polygons
3, each with 4 edges
antipodal sets4 of ( v, f ), 3 of ( 2e, p1 )
rotational symmetry groupA4, with 12 elements
full symmetry groupS4, with 24 elements
its presentation c< r, s, t | r2, s2, t2, (rs)3, (st)3, (rt)2 >
C&D number cR0.1
The statistics marked c are from the published work of Professor Marston Conder.

Relations to other Regular Maps

It is self-dual.

Its Petrie dual is the hemicube.

It can be 2-split to give {6,3}(2,2).

It can be rectified to give the octahedron.

It is the diagonalisation of the di-square.

It can be pyritified (type 3/3/5/3) to give the dodecahedron.

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

List of regular maps in orientable genus 0.

Underlying Graph

Its skeleton is K4.


This is one of the five "Platonic solids".

Cayley Graphs based in this Regular Map

Type I


Type II


Type III


Other Regular Maps

General Index

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