Small triambic icosahedron

In geometry, the small triambic icosahedron is the dual to the uniform small ditrigonal icosidodecahedron. It is composed of 20 intersecting isogonal hexagon (triambus) faces. It has 60 edges and 32 vertices, and Euler characteristic of 8. Its external surface also represents the B stellation of the icosahedron.

Small triambic icosahedron
TypeDual uniform polyhedron
IndexDU30, 2/59, W26
Elements
(As a star polyhedron)
F = 20, E = 60
V = 32 (χ = 8)
Symmetry groupicosahedral (Ih)
Dual polyhedronsmall ditrigonal icosidodecahedron
Stellation diagramStellation coreConvex hull

Icosahedron

Pentakis dodecahedron

If the intersected hexagonal faces are divided and new edges created, this figure becomes the triakis icosahedron. The descriptive name triakis icosahedron represents a topological construction starting with an icosahedron and attaching tetrahedra to each face (not necessarily regular tetrahedra). With the proper height of each such tetrahedron above the triangular base, this figure becomes a Catalan solid by the same name and the dual of the truncated dodecahedron.

The nonconvex uniform polyhedra great stellated dodecahedron and great dodecahedron, as viewed as surface topologies, can also be constructed as icosahedron with pyramids, the first with much taller pyramids, and the second with inverted ones.

It is also a uniform dual, and is the dual of the small ditrigonal icosidodecahedron. Other uniform duals which are also stellations of the icosahedron are the medial triambic icosahedron and the great triambic icosahedron.

As a stellation

This figure is also the first stellation of the icosahedron, and given as Wenninger model index 26.

This stellation is a popular subject for construction in modular origami, often made of thirty Sonobe units.

References

  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. (p. 46, Model W26, triakis icosahedron)
  • Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8. (pp. 4246, dual to uniform polyhedron W70)
  • 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 6.2 Stellating the Platonic solids, pp.96-104
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
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.
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