# The octahedron

### Statistics

 genus c 0, orientable Schläfli formula c {3,4} V / F / E c 6 / 8 / 12 notes vertex, face multiplicity c 1, 1 Petrie polygonsholes2nd-order Petrie polygons 4, each with 6 edges6, each with 4 edges6, each with 4 edges antipodal sets 3 of ( 2v, 2h2 ), 4 of ( 2f; p1 ), 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)3, (st)4, (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 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.

### Underlying Graph

Its skeleton is K2,2,2.

This is one of the five "Platonic solids".

 D6

 S4