Icosahedral pyramid

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Icosahedral pyramid
Icosahedral pyramid.png
Schlegel diagram
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ {3,5}
Cells211 {3,5} Icosahedron.png
20 ( ) ∨ {3} Tetrahedron.png
Faces5020+30 {3}
Edges12+30
Vertices13
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.

Contents

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.

Dodecahedral pyramid.png

Configuration

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

k-facesfkf0f1f2f3k-verfs
( )f01*120300200{3,5}
( )*12155551{5}∨( )
( )∨( )f11112*5050{5}
{ }02*301221{ }∨( )
{ }∨( )f2122130*20{ }
{3}0303*2011( )∨( )
{3}∨( )f313333120*( )
{3,5}012030020*1( )

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<span class="mw-page-title-main">Rectified 600-cell</span>

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<span class="mw-page-title-main">Great dodecahedron</span> Kepler-Poinsot polyhedron

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<span class="mw-page-title-main">Octahedral pyramid</span>

In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids can be made with regular faces by computing the appropriate height.

<span class="mw-page-title-main">Dodecahedral-icosahedral honeycomb</span>

In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure.

In geometry, a Blind polytope is a convex polytope composed of regular polytope facets. The category was named after the German couple Gerd and Roswitha Blind, who described them in a series of papers beginning in 1979. It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions.

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In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, {3,3} + { }. Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices. A tetrahedral bipyramid can be seen as two tetrahedral pyramids augmented together at their base.

<span class="mw-page-title-main">Icosahedral bipyramid</span> Four dimensional pyramid

In 4-dimensional geometry, the icosahedral bipyramid is the direct sum of a icosahedron and a segment, {3,5} + { }. Each face of a central icosahedron is attached with two tetrahedra, creating 40 tetrahedral cells, 80 triangular faces, 54 edges, and 14 vertices. An icosahedral bipyramid can be seen as two icosahedral pyramids augmented together at their bases.

References

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