Small triambic icosahedron

Small triambic icosahedron
File:DU30 small triambic icosahedron.png
Type	Dual uniform polyhedron
Index	DU30, 2/59, W26
Elements (As a star polyhedron)	F = 20, E = 60 V = 32 (χ = −8)
Symmetry group	icosahedral (Ih)
Dual polyhedron	small ditrigonal icosidodecahedron
Stellation diagram Stellation core Convex hull File:Small triambic icosahedron stellation facets.svg File:Icosahedron.png Icosahedron File:Pentakis dodecahedron.png Pentakis dodecahedron

File:Small triambic icosahedron.stl

3D model of a small triambic icosahedron

In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric to each other. Branko Grünbaum has conjectured that it is the only Euclidean isohedron with convex faces of six or more sides, but the small hexagonal hexecontahedron is another example.

Geometry

The faces are equilateral hexagons, with alternating angles of $\arccos (-\frac{1}{4})\approx 104.47751218593^{\circ }$ and $\arccos (\frac{1}{4})+60^{\circ }\approx 135.52248781407^{\circ }$. The dihedral angle equals $\arccos (-\frac{1}{3})\approx 109.47122063449$.

Related shapes

The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are hidden by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with the first (B) stellation of the icosahedron. If instead, after removing the surrounded parts of each face, each resulting triple of coplanar triangles is considered to be three separate faces, then the result is one form of the triakis icosahedron, formed by adding a triangular pyramid to each face of an icosahedron.

The dual polyhedron of the small triambic icosahedron is the small ditrigonal icosidodecahedron. As this is a uniform polyhedron, the small triambic icosahedron is a uniform dual. Other uniform duals whose exterior surfaces are stellations of the icosahedron are the medial triambic icosahedron and the great triambic icosahedron.

Further reading

• (p. 46, Model W₂₆, triakis icosahedron)
• (pp. 42–46, dual to uniform polyhedron W₇₀)
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Template:Isbn, 3.6 6.2 Stellating the Platonic solids, pp.96-104

External links

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
File:Zeroth stellation of icosahedron.svg	File:First stellation of icosahedron.png	File:Ninth stellation of icosahedron.png		File:First compound stellation of icosahedron.png	File:Second compound stellation of icosahedron.png	File:Third compound stellation of icosahedron.png	File:Sixteenth stellation of icosahedron.png	File:Third stellation of icosahedron.svg	File:Seventeenth stellation of icosahedron.png
File:Stellation diagram of icosahedron.svg	File:Small triambic icosahedron stellation facets.svg	File:Great triambic icosahedron stellation facets.svg		File:Compound of five octahedra stellation facets.svg	File:Compound of five tetrahedra stellation facets.svg	File:Compound of ten tetrahedra stellation facets.svg	File:Great icosahedron stellation facets.svg	File:Excavated dodecahedron stellation facets.svg	File:Echidnahedron stellation facets.svg
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.