Pentagonal hexecontahedron | |
---|---|
(Click here for rotating model) | |
Type | Catalan solid |
Coxeter diagram | |
Conway notation | gD |
Face type | V3.3.3.3.5 irregular pentagon |
Faces | 60 |
Edges | 150 |
Vertices | 92 |
Vertices by type | 12 {5} 20+60 {3} |
Symmetry group | I, 1/2H3, [5,3]+, (532) |
Rotation group | I, [5,3]+, (532) |
Dihedral angle | 153°10′43″ |
Properties | convex, face-transitive chiral |
Snub dodecahedron (dual polyhedron) | Net |
In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.
Using the Icosahedral symmetry of the Weyl orbits of order 60 [1] gives the following Cartesian coordinates with is the golden ratio:
and
A group of two sets of twelve have 0 or 2 minus signs (i.e. 1 or 3 plus signs):
and another group of three sets of 12 have 0 or 2 plus signs (i.e. 1 or 3 minus signs):
Negating all vertices in both groups gives the mirror of the chiral snub dodecahedron, yet results in the same pentagonal hexecontahedron convex hull.
The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual. Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron. [2]
An alternate construction method uses quaternions and the icosahedral symmetry of the Weyl group orbits of order 60. [3] This is shown in the figure on the right.
Specifically, with quaternions from the binary Icosahedral group , where is the conjugate of and and , then just as the Coxeter group is the symmetry group of the 600-cell and the 120-cell of order 14400, we have of order 120. is defined as the even permutations of such that gives the 60 twisted chiral snub dodecahedron coordinates, where is one permutation from the first set of 12 in those listed above. The exact coordinate for is obtained by taking the solution to , with , and applying it to the normalization of .
The faces are irregular pentagons with two long edges and three short edges. Let be the real zero of the polynomial . Then the ratio of the edge lengths is given by:
The faces have four equal obtuse angles and one acute angle (between the two long edges). The obtuse angles equal , and the acute one equals . The dihedral angle equals . Note that the face centers of the snub dodecahedron cannot serve directly as vertices of the pentagonal hexecontahedron: the four triangle centers lie in one plane but the pentagon center does not; it needs to be radially pushed out to make it coplanar with the triangle centers. Consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron.
To find the volume and surface area of a pentagonal hexecontahedron, denote the shorter side of one of the pentagonal faces as , and set a constant t [4] .
Then the surface area (A) is:
.
And the volume (V) is:
.
Using these, one can calculate the measure of sphericity for this shape:
Isohedral variations can be constructed with pentagonal faces with 3 edge lengths.
This variation shown can be constructed by adding pyramids to 12 pentagonal faces and 20 triangular faces of a snub dodecahedron such that the new triangular faces are coparallel to other triangles and can be merged into the pentagon faces.
Snub dodecahedron with augmented pyramids and merged faces | Example variation | Net |
The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.
Projective symmetry | [3] | [5]+ | [2] |
---|---|---|---|
Image | |||
Dual image |
Family of uniform icosahedral polyhedra | |||||||
---|---|---|---|---|---|---|---|
Symmetry: [5,3], (*532) | [5,3]+, (532) | ||||||
{5,3} | t{5,3} | r{5,3} | t{3,5} | {3,5} | rr{5,3} | tr{5,3} | sr{5,3} |
Duals to uniform polyhedra | |||||||
V5.5.5 | V3.10.10 | V3.5.3.5 | V5.6.6 | V3.3.3.3.3 | V3.4.5.4 | V4.6.10 | V3.3.3.3.5 |
This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.
n32 symmetry mutations of snub tilings: 3.3.3.3.n | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry n32 | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 | |
Snub figures | ||||||||
Config. | 3.3.3.3.2 | 3.3.3.3.3 | 3.3.3.3.4 | 3.3.3.3.5 | 3.3.3.3.6 | 3.3.3.3.7 | 3.3.3.3.8 | 3.3.3.3.∞ |
Gyro figures | ||||||||
Config. | V3.3.3.3.2 | V3.3.3.3.3 | V3.3.3.3.4 | V3.3.3.3.5 | V3.3.3.3.6 | V3.3.3.3.7 | V3.3.3.3.8 | V3.3.3.3.∞ |
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that it is a dual of an Archimedean solid, in this case, the truncated icosahedron.
In geometry, the deltoidal icositetrahedron is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.
In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It also has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{5⁄2,3}, and Coxeter-Dynkin diagram .
In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr{5/3,5}.
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.
In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices.
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
In geometry, the great pentagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams.
In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.
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 faces.
In geometry, the medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.
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