Pentagonal hexecontahedron

Last updated
Pentagonal hexacontahedron
Pentagonalhexecontahedron.jpg
Faces 60
Edges 150
Vertices 92
Symmetry group icosahedral symmetry
Dihedral angle (degrees)153.2°
Dual polyhedron snub dodecahedron
Net
Pentagonalhexecontahedron net.png

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.

Contents

Properties

3D model of a pentagonal hexecontahedron Pentagonal hexecontahedron.stl
3D model of a pentagonal hexecontahedron

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 [1]

Then the surface area () is: .

And the volume () is: .

Using these, one can calculate the measure of sphericity for this shape:

Construction

Combining a unit circumradius icosahedron (12) centered at the origin with a chiral snub dodecahedron (60) combined with a dodecahedron of the same non-unity circumradius (20) to construct the pentagonal hexecontahedron Pentagonal Hexacontahedron Convex Hulls.svg
Combining a unit circumradius icosahedron (12) centered at the origin with a chiral snub dodecahedron (60) combined with a dodecahedron of the same non-unity circumradius (20) to construct the pentagonal hexecontahedron

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 .

Cartesian coordinates

Using the Icosahedral symmetry in the orbits of the Weyl group of order 60 [4] 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.

Variations

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.

Pentagonal hexecontahedron variation0.png
Snub dodecahedron with augmented pyramids and merged faces
Pentagonal hexecontahedron variation.png
Example variation
Pentagonal hexecontahedron variation net.png
Net

Orthogonal projections

The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.

Orthogonal projections
Projective
symmetry
[3][5]+[2]
Image Dual snub dodecahedron A2.png Dual snub dodecahedron H2.png Dual snub dodecahedron e1.png
Dual
image
Snub dodecahedron A2.png Snub dodecahedron H2.png Snub dodecahedron e1.png
Spherical pentagonal hexecontahedron Spherical pentagonal hexecontahedron.png
Spherical pentagonal hexecontahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532)[5,3]+, (532)
Uniform polyhedron-53-t0.svg Uniform polyhedron-53-t01.svg Uniform polyhedron-53-t1.svg Uniform polyhedron-53-t12.svg Uniform polyhedron-53-t2.svg Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{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
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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 hyperbolicParacomp.
23233243253263273283232
Snub
figures
Spherical trigonal antiprism.svg Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.svg Snub triheptagonal tiling.svg H2-8-3-snub.svg Uniform tiling i32-snub.png
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
Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.svg Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg 7-3 floret pentagonal tiling.svg H2-8-3-floret.svg Order-3-infinite floret pentagonal tiling.png
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 V3.3.3.3.7V3.3.3.3.8V3.3.3.3.

See also

Related Research Articles

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<span class="mw-page-title-main">Great pentagrammic hexecontahedron</span> Polyhedron with 60 faces

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<span class="mw-page-title-main">Small hexagonal hexecontahedron</span> Polyhedron with 60 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.

<span class="mw-page-title-main">Medial hexagonal hexecontahedron</span> Polyhedron with 60 faces

In geometry, the medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

References

  1. "Pentagonal Hexecontahedron - Geometry Calculator". rechneronline.de. Retrieved 2020-05-26.
  2. Reference
  3. Koca, Mehmet; Ozdes Koca, Nazife; Al-Shu’eilic, Muna (2011). "Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions". arXiv: 1006.3149 [math-ph].
  4. Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions". Journal of Mathematical Physics. 51 (4). arXiv: 0908.3272 . doi:10.1063/1.3356985. S2CID   115157829.