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Icosahedral pyramid

## Summary

Icosahedral pyramid

Schlegel diagram
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ {3,5}
Cells 21 1 {3,5}
20 ( ) ∨ {3}
Faces 50 20+30 {3}
Edges 12+30
Vertices 13
Dual Dodecahedral pyramid
Symmetry group H3, [5,3,1], order 120
Properties convex, regular-cells, Blind polytope

The icosahedral pyramid is a four-dimensional convex polytope, bounded by one icosahedron as its base and by 20 triangular pyramid cells which meet at its apex. Since an icosahedron's circumradius is less than its edge length,[1] the tetrahedral pyramids can be made with regular faces.

Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an icosahedral bipyramid which is also a Blind Polytope.

The regular 600-cell has icosahedral pyramids around every vertex.

The dual to the icosahedral pyramid is the dodecahedral pyramid, seen as a dodecahedral base, and 12 regular pentagonal pyramids meeting at an apex.

## Configuration

Seen in a configuration matrix, all incidence counts between elements are shown.[2]

k-faces fk f0 f1 f2 f3 k-verfs
( ) f0 1 * 12 0 30 0 20 0 {3,5}
( ) * 12 1 5 5 5 5 1 {5}∨( )
( )∨( ) f1 1 1 12 * 5 0 5 0 {5}
{ } 0 2 * 30 1 2 2 1 { }∨( )
{ }∨( ) f2 1 2 2 1 30 * 2 0 { }
{3} 0 3 0 3 * 20 1 1 ( )∨( )
{3}∨( ) f3 1 3 3 3 3 1 20 * ( )
{3,5} 0 12 0 30 0 20 * 1 ( )

## References

1. ^ Klitzing, Richard. "3D convex uniform polyhedra x3o5o - ike"., circumradius sqrt[(5+sqrt(5))/8 = 0.951057
2. ^ Klitzing, Richard. "ikepy".