The icosahedron

Statistics

genus c0, orientable
Schläfli formula c{3,5}
V / F / E c 12 / 20 / 30
notesreplete singular is a polyhedral map permutes its vertices evenly
vertex, face multiplicity c1, 1
Petrie polygons
holes
2nd-order Petrie polygons
6, each with 10 edges
12, each with 5 edges
10, each with 6 edges
antipodal sets6 of ( 2v, 2h2; p1 ), 10 of ( 2f; p2 ), 15 of ( 2e )
rotational symmetry groupA5, with 60 elements
full symmetry groupA5×C2, with 120 elements
its presentation c< r, s, t | r2, s2, t2, (rs)3, (st)5, (rt)2 >
C&D number cR0.3
The statistics marked c are from the published work of Professor Marston Conder.

Relations to other Regular Maps

Its dual is the dodecahedron.

Its Petrie dual is N14.3′.

It is a 2-fold cover of the hemi-icosahedron.

It can be 2-split to give R9.15′.

It can be rectified to give the icosidodecahedron.

Its full shuriken is C12{6,4}5.

It can be stellated (with path <>/2) to give S4:{5,5} . The density of the stellation is 3.
It can be stellated (with path <1/-1>) to give S4:{5,5} . The density of the stellation is 3.
It can be stellated (with path <1,-1>/2) to give the icosahedron . The density of the stellation is 7.
It can be derived by stellation (with path <1,-1>/2) from the icosahedron. The density of the stellation is 7.

List of regular maps in orientable genus 0.

Underlying Graph

Its skeleton is icosahedron.

Comments

This is one of the five "Platonic solids".

Cayley Graphs based in this Regular Map


Type I

A4

Type II

A5

Other Regular Maps

General Index

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