small triambic icosahedron
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!bgcolor=#e7dcc3 colspan=2|Small triambic icosahedron | |
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| bgcolor=#e7dcc3|Type | Dual uniform polyhedron |
| bgcolor=#e7dcc3|Index | DU30, 2/59, W26 |
| bgcolor=#e7dcc3|Elements (As a star polyhedron) | F = 20, E = 60 V = 32 (χ = −8) |
| bgcolor=#e7dcc3|Symmetry group | icosahedral (Ih) |
| bgcolor=#e7dcc3|Dual polyhedron | small ditrigonal icosidodecahedron |
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{| !Stellation diagram!!Stellation core | Convex hull |
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File:Small triambic icosahedron.stl
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,{{cite book
| last = Grünbaum | first = Branko | authorlink = Branko Grünbaum
| contribution = Can every face of a polyhedron have many sides?
| hdl = 1773/4593
| mr = 2512345
| pages = 9–26
| publisher = Comap, Inc. | location = Bedford, Massachusetts
| title = Geometry, games, graphs and education: the Joe Malkevitch Festschrift
| year = 2008}} but the small hexagonal hexecontahedron 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.{{Cite book | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Du Val | first2=P. | last3=Flather | first3=H. T. | last4=Petrie | first4=J. F. | title=The fifty-nine icosahedra | publisher=Tarquin | edition=3rd | isbn=978-1-899618-32-3 |mr=676126 | year=1999 | postscript=}} (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 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.
References
{{reflist}}
Further reading
- {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }} (p. 46, Model W26, triakis icosahedron)
- {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | year=1983 | isbn=0-521-54325-8 }} (pp. 42–46, dual to uniform polyhedron W70)
- 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
External links
- {{mathworld | urlname = SmallTriambicIcosahedron| title = Small triambic icosahedron}}{{Icosahedron stellations}}