The octahedron

Statistics

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
holes
4, each with 6 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.

Comments

This is one of the five "Platonic solids".

Cayley Graphs based in this Regular Map


Type I

D6

Type II

S4

Other Regular Maps

General Index

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