The hemi-icosahedron


genus c1, non-orientable
Schläfli formula c{3,5}
V / F / E c 6 / 10 / 15
notesreplete singular is a polyhedral map permutes its vertices evenly
vertex, face multiplicity c1, 1
Petrie polygons
2nd-order Petrie polygons
6, each with 5 edges
6, each with 5 edges
10, each with 3 edges
antipodal sets6 of ( v, p1, h2 ), 10 of ( f, p2 ), 5 of ( 3e )
rotational symmetry groupA5, with 60 elements
full symmetry groupA5, with 60 elements
its presentation c< r, s, t | r2, s2, t2, (rs)3, (st)5, (rt)2, (rsrsrt)2 >
C&D number cN1.2
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.


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

The image on this page is copyright © 2010 N. Wedd