The octahedron


genus c0, orientable
Schläfli formula c{3,4}
V / F / E c 6 / 8 / 12
notesreplete singular is a polyhedral map permutes its vertices oddly
vertex, face multiplicity c1, 1
Petrie polygons
2nd-order Petrie polygons
4, each with 6 edges
6, each with 4 edges
6, each with 4 edges
antipodal sets3 of ( 2v, 2h2 ), 4 of ( 2f; p1 ), 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)3, (st)4, (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 cube.

Its Petrie dual is C4:{6,4}3.

It is a 2-fold cover of the hemioctahedron.

It can be 2-split to give S3:{6,4}.
It can be 4-split to give R9.11′.
It can be 5-split to give R12.1′.
It can be 7-split to give R18.1′.
It can be 10-split to give R27.1′.
It can be 11-split to give R30.1′.
It can be 8-split to give R21.7′.

It can be rectified to give the cuboctahedron.
It is the result of rectifying the tetrahedron.

It can be obtained by triambulating the di-hexagon.

It is the result of pyritifying (type 2/4/3/4) the 4-hosohedron.

List of regular maps in orientable genus 0.

Underlying Graph

Its skeleton is K2,2,2.


This is one of the five "Platonic solids".

Cayley Graphs based in this Regular Map

Type I


Type II


Other Regular Maps

General Index

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