• Source: 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
⁡
(
−


1
4


)
≈
104.477

512

185


93

∘




{\displaystyle \arccos(-{\frac {1}{4}})\approx 104.477\,512\,185\,93^{\circ }}

and



arccos
⁡
(


1
4


)
+

60

∘


≈
135.522

487

814


07

∘




{\displaystyle \arccos({\frac {1}{4}})+60^{\circ }\approx 135.522\,487\,814\,07^{\circ }}

. The dihedral angle equals



arccos
⁡
(
−


1
3


)
≈
109.471

220

634

49


{\displaystyle \arccos(-{\frac {1}{3}})\approx 109.471\,220\,634\,49}

.




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. 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






Further reading


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. 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


Weisstein, Eric W. "Small triambic icosahedron". MathWorld.

Kata Kunci Pencarian:

• Daftar bentuk matematika
• Small triambic icosahedron
• Regular icosahedron
• Great triambic icosahedron
• Icosahedron
• Triakis icosahedron
• Great icosahedron
• Final stellation of the icosahedron
• List of Tron characters
• List of mathematical shapes
• List of polygons, polyhedra and polytopes