Compound of ten tetrahedra

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Compound of ten tetrahedra
Compound of ten tetrahedra.png
Type regular compound
Coxeter symbol 2{5,3}[10{3,3}]2{3,5}[1]
Index UC6, W25
Elements
(As a compound)
10 tetrahedra:
F = 40, E = 60, V = 20
Dual compound Self-dual
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent chiral tetrahedral (T)

The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.

It can be seen as a faceting of a regular dodecahedron.

As a compound[edit]

It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry (Ih). It is one of five regular compounds constructed from identical Platonic solids.

It shares the same vertex arrangement as a dodecahedron.

The compound of five tetrahedra represents two chiral halves of this compound.

It can be made from the compound of five cubes by replacing each cube with a stella octangula on the cube's vertices.

As a stellation[edit]

This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25.

Stellation diagram Stellation core Convex hull
Third compound stellation of icosahedron facets.png Icosahedron.png
Icosahedron
Dodecahedron.png
Dodecahedron

As a facetting[edit]

Ten tetrahedra in a dodecahedron.

It is also a facetting of the dodecahedron, as shown at left.


As a simple polyhedron[edit]

If it is treated as a simple non-convex polyhedron without self-intersecting surfaces, it has 180 faces (120 triangles and 60 concave quadrilaterals), 122 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, and 20 with degree 12), and 300 edges, giving an Euler characteristic of 122-300+180 = +2.

See also[edit]

References[edit]

  1. ^ Regular polytopes, p.98

External links[edit]

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.