Compound of ten tetrahedra

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
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
Compound of ten tetrahedra stellation facets.svg 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. Pentagrams can be seen on the compound where the pentagonal faces of the dodecahedron are positioned.

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
  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
  • Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 0676126. (1st Edn University of Toronto (1938))
  • H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104

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
Stellation diagram of icosahedron.svg Small triambic icosahedron stellation facets.svg Great triambic icosahedron stellation facets.svg Compound of five octahedra stellation facets.svg Compound of five tetrahedra stellation facets.svg Compound of ten tetrahedra stellation facets.svg Great icosahedron stellation facets.svg Excavated dodecahedron stellation facets.svg Echidnahedron stellation facets.svg
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.