The icosahedron

Statistics

genus ^c	0, orientable
Schläfli formula ^c	{3,5}
V / F / E ^c	12 / 20 / 30
notes
vertex, face multiplicity ^c	1, 1
Petrie polygons holes 2nd-order Petrie polygons	6, each with 10 edges 12, each with 5 edges 10, each with 6 edges
antipodal sets	6 of ( 2v, 2h2; p1 ), 10 of ( 2f; p2 ), 15 of ( 2e )
rotational symmetry group	A5, with 60 elements
full symmetry group	A5×C2, with 120 elements
its presentation ^c	< r, s, t \| r², s², t², (rs)³, (st)⁵, (rt)² >
C&D number ^c	R0.3
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 2-hole derivative is S4:{5,5}.

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

Type II

Other Regular Maps

General Index