The hemi-icosahedron

Statistics

genus ^c	1, non-orientable
Schläfli formula ^c	{3,5}
V / F / E ^c	6 / 10 / 15
notes
vertex, face multiplicity ^c	1, 1
Petrie polygons holes 2nd-order Petrie polygons	6, each with 5 edges 6, each with 5 edges 10, each with 3 edges
antipodal sets	6 of ( v, p1, h2 ), 10 of ( f, p2 ), 5 of ( 3e )
rotational symmetry group	A5, with 60 elements
full symmetry group	A5, with 60 elements
its presentation ^c	< r, s, t \| r², s², t², (rs)³, (st)⁵, (rt)², (rsrsrt)² >
C&D number ^c	N1.2
The statistics marked ^c are from the published work of Professor Marston Conder.

Relations to other Regular Maps

Its dual is the hemidodecahedron.

Its Petrie dual is C5:{5,5}.

It can be 2-fold covered to give the icosahedron.

It can be 2-split to give N10.4′.

It can be rectified to give the hemi-icosidodecahedron.

Its 2-hole derivative is C5:{5,5}.

Its full shuriken is C7:{6,4}.

List of regular maps in non-orientable genus 1.

Underlying Graph

Its skeleton is K₆.

Comments

If you take a hemi-icosahedron and glue another one to each face, and bend them round so that three meet at each edge, you will find that the 11 of them form a regular polytope, the 11-cell, Schläfli symbol {3,5,3}. Its rotational symmetry group is PSL(2,11). Do not try this at home – it is not possible while you are embedded in 3-space.

Other Regular Maps

General Index