Family of derived polyhedra
In geometry , a tetrahedrally diminished[ a] (also tetrahedrally stellated icosahedron or propello tetrahedron [ 1] self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals ).[ 2]
A canonical form exists with two edge lengths at 0.849 : 1.057, assuming that the radius of the midsphere is 1. The kites remain isosceles.
It has chiral tetrahedral symmetry , and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated , or from the pyritohedron , with 4 vertices diminished . Within its tetrahedral symmetry , it has geometric varied proportions. By Dorman Luke dual construction , a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.633.
Topologically, the triangles are always equilateral, while the quadrilaterals are irregular, although the two adjacent edges that meet at the vertices of a tetrahedron are equal.
As a self-dual hexadecahedron , it is one of 302404 forms, 1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry.[ 3]
As a diminished regular dodecahedron , with 4 vertices removed, the quadrilaterals faces are trapezoids .
As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry. It has kite faces.[ 4]
In Conway polyhedron notation , it can be represented as pT , applying George W. Hart 's propeller operator to a regular tetrahedron .[ 5]
This polyhedron represents the vertex figure of a hyperbolic uniform honeycomb , the partially diminished icosahedral honeycomb , pd{3,5,3}, with 12 pentagonal antiprisms and 4 dodecahedron cells meeting at every vertex.
Vertex figure projected as Schlegel diagram
Notes
^ It is also less accurately called a tetrahedrally truncated dodecahedron
References
External links
Listed by number of faces and type
1–10 faces 11–20 faces >20 faces elemental things convex polyhedron non-convex polyhedron prismatoid s
