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In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60
edges
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and 32 vertices, and Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of −8. It is an isohedron
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
, 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
In geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular fac ...
is another example.
Geometry
The faces are equilateral hexagons, with alternating angles of and . The dihedral angle equals .Related shapes
The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron. (1st Edn University of Toronto (1938)) 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 anicosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
.
The dual polyhedron of the small triambic icosahedron is the small ditrigonal icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfl ...
. 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
In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. The exterior surface also represents the De2f2 stellation of the icosahedron. These figure ...
and the great triambic icosahedron.
References
Further reading
* (p. 46, Model ''W''26, triakis icosahedron) * (pp. 42–46, dual to uniform polyhedron ''W''70) *H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, '' Regular Polytopes'', (3rd edition, 1973), Dover edition, , 3.6 6.2 ''Stellating the Platonic solids'', pp.96-104
External links
* {{mathworld , urlname = SmallTriambicIcosahedron, title = Small triambic icosahedron Polyhedral stellation Dual uniform polyhedra