Pentagonal icositetrahedron
| Pentagonal icositetrahedron | |
|---|---|
| Pentagonal icositetrahedron, anticlockwise twistPentagonal icositetrahedron (Click ccw or cw for rotating models.) | |
| Type | Catalan |
| Conway notation | gC |
| Coxeter diagram | Template:CDD |
| Face polygon | File:DU12 facets.png irregular pentagon |
| Faces | 24 |
| Edges | 60 |
| Vertices | 38 = 6 + 8 + 24 |
| Face configuration | V3.3.3.3.4 |
| Dihedral angle | 136° 18' 33' |
| Symmetry group | O, Template:SfracBC3, [4,3]+, 432 |
| Dual polyhedron | snub cube |
| Properties | convex, face-transitive, chiral |
| Pentagonal icositetrahedron Net | |
In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron<ref>Conway, Symmetries of things, p.284</ref> is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.
ConstructionEdit
The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual. Square pyramids are added to the six square faces of the snub cube, and triangular pyramids are added to the eight triangular faces that do not share an edge with a square. The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube. The result is the pentagonal icositetrahedron.
Cartesian coordinatesEdit
Denote the tribonacci constant by <math>t\approx 1.839\,286\,755\,21</math>. (See snub cube for a geometric explanation of the tribonacci constant.) Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the origin, are as follows:
- the 12 even permutations of Template:Math with an even number of minus signs
- the 12 odd permutations of Template:Math with an odd number of minus signs
- the 6 points Template:Math, Template:Math and Template:Math
- the 8 points Template:Math
The convex hulls for these vertices<ref>Template:Cite journal</ref> scaled by <math>t^{-3}</math> result in a unit circumradius octahedron centered at the origin, a unit cube centered at the origin scaled to <math>R\approx0.9416969935</math>, and an irregular chiral snub cube scaled to <math>R</math>, as visualized in the figure below:
GeometryEdit
The pentagonal faces have four angles of <math>\arccos((1-t)/2)\approx 114.812\,074\,477\,90^{\circ}</math> and one angle of <math>\arccos(2-t)\approx 80.751\,702\,088\,39^{\circ}</math>. The pentagon has three short edges of unit length each, and two long edges of length <math>(t+1)/2\approx 1.419\,643\,377\,607\,08</math>. The acute angle is between the two long edges. The dihedral angle equals <math>\arccos(-1/(t^2-2))\approx 136.309\,232\,892\,32^{\circ}</math>.
If its dual snub cube has unit edge length, its surface area and volume are:<ref>Template:Mathworld2</ref>
- <math>\begin{align} A &= 3\sqrt{\frac{22(5t-1)}{4t-3}} &&\approx 19.299\,94 \\ V &= \sqrt{\frac{11(t-4)}{2(20t-37)}} &&\approx 7.4474 \end{align}</math>
Orthogonal projectionsEdit
The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.
| Projective symmetry |
[3] | [4]+ | [2] |
|---|---|---|---|
| Image | File:Dual snub cube A2.png | File:Dual snub cube B2.png | File:Dual snub cube e1.png |
| Dual image |
File:Snub cube A2.png | File:Snub cube B2.png | File:Snub cube e1.png |
VariationsEdit
Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths.
This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces.
| File:Pentagonal icositetrahedron variation0.png Snub cube with augmented pyramids and merged faces |
File:Pentagonal icositetrahedron variation.png Pentagonal icositetrahedron |
File:Pentagonal icositetrahedron variation net.png Net |
Related polyhedra and tilingsEdit
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. Template:Snub table
The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n. Template:Snub4 table
The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. Template:Octahedral truncations
ReferencesEdit
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
- Template:Citation (The thirteen semiregular convex polyhedra and their duals, Page 28, Pentagonal icositetrahedron)
- The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:Isbn [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)
External linksEdit
- Pentagonal Icositetrahedron – Interactive Polyhedron Model