View source for Pentagonal icositetrahedron

{{Short description|Catalan polyhedron}}
{| class=wikitable align="right" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentagonal icositetrahedron
|-
|align=center colspan=2|[[Image:pentagonalicositetrahedronccw.jpg|120px|Pentagonal icositetrahedron, anticlockwise twist]][[Image:pentagonalicositetrahedroncw.jpg|120px|Pentagonal icositetrahedron]]<br>''(Click [[:image:pentagonalicositetrahedronccw.gif|''ccw'']] or [[:image:pentagonalicositetrahedroncw.gif|''cw'']] for rotating models.)''
|-
|bgcolor=#e7dcc3|Type||[[Catalan solid|Catalan]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||gC
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_fh|4|node_fh|3|node_fh}}
|-
|bgcolor=#e7dcc3|Face polygon||[[File:DU12 facets.png|60px]]<BR>[[pentagon|irregular pentagon]]
|-
|bgcolor=#e7dcc3|Faces||24
|-
|bgcolor=#e7dcc3|Edges||60
|-
|bgcolor=#e7dcc3|Vertices||38 = 6 + 8 + 24
|-
|bgcolor=#e7dcc3|[[Face configuration]]||V3.3.3.3.4
|-
|bgcolor=#e7dcc3|[[Dihedral angle]]||136° 18' 33'
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Octahedral symmetry|O]], {{sfrac|1|2}}BC<sub>3</sub>, [4,3]<sup>+</sup>, 432
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[snub cube]]
|-
|bgcolor=#e7dcc3|Properties||[[convex set|convex]], [[face-transitive]], [[chirality (mathematics)|chiral]]
|-
|align=center colspan=2|[[Image:Pentagonalicositetrahedron net.svg|240px|Pentagonal icositetrahedron]]<br>[[Net (polyhedron)|Net]]
|}[[File:TRIBONACCI.jpg|thumb|A geometric construction of the Tribonacci constant (AC), with compass and marked ruler, according to the method described by Xerardo Neira.]][[File:Pentagonal icositetrahedron.stl|thumb|3d model of a pentagonal icositetrahedron]]
In [[geometry]], a '''pentagonal icositetrahedron''' or '''pentagonal icosikaitetrahedron'''<ref>Conway, Symmetries of things, p.284</ref>  is a [[Catalan solid]] which is the [[Dual polyhedron|dual]] of the [[snub cube]]. In [[crystallography]] it is also called a '''gyroid'''.<ref>{{cite web |url=http://www.metafysica.nl/turing/promorph_crystals.html |title = Promorphology of Crystals I}}</ref><ref>{{cite web |url=http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm |url-status=dead |archive-url=https://web.archive.org/web/20030823194958/http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm |archive-date=2003-08-23 |title=Crystal Form, Zones, & Habit}}</ref>

It has two distinct forms, which are [[mirror image]]s (or "[[Chirality (mathematics)|enantiomorphs]]") of each other.

== Construction ==
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 coordinates ==

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 permutation]]s of {{math|(±1, ±(2''t''+1), ±''t''<sup>2</sup>)}} with an even number of minus signs
*the 12 [[odd permutation]]s of {{math|(±1, ±(2''t''+1), ±''t''<sup>2</sup>)}} with an odd number of minus signs
*the 6 points {{math|(±''t''<sup>3</sup>, 0, 0)}}, {{math|(0, ±''t''<sup>3</sup>, 0)}} and {{math|(0, 0, ±''t''<sup>3</sup>)}}
*the 8 points {{math|(±''t''<sup>2</sup>, ±''t''<sup>2</sup>, ±''t''<sup>2</sup>)}}

The [[Convex hull|convex hulls]] for these vertices<ref>{{cite journal
|title=Catalan Solids Derived From 3D-Root Systems and Quaternions
|first1=Mehmet
|last1=Koca
|first2=Nazife
|last2=Ozdes Koca
|first3=Ramazon
|last3=Koc
|year=2010
|journal=Journal of Mathematical Physics
|volume=51
|issue=4
|doi=10.1063/1.3356985 
|arxiv=0908.3272
}}</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:

:[[File:Pentagonal Icositetrahedron.svg|400px|Combining an octahedron and snub cube to form the Pentagonal Icositetrahedron ]]

== Geometry==
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>{{Mathworld2 |urlname=PentagonalIcositetrahedron |title=Pentagonal icositetrahedron |urlname2=CatalanSolid |title2=Catalan solid}}</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 projections==
The ''pentagonal icositetrahedron'' has three symmetry positions, two centered on vertices, and one on midedge.
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<BR>symmetry
|[3]
|[4]<sup>+</sup>
|[2]
|-
!Image
|[[Image:Dual snub cube_A2.png|120px]]
|[[Image:Dual snub cube_B2.png|120px]]
|[[Image:Dual snub cube_e1.png|120px]]
|-
!Dual<BR>image
|[[Image:Snub cube_A2.png|120px]]
|[[Image:Snub cube_B2.png|120px]]
|[[Image:Snub cube_e1.png|120px]]
|}
=== Variations===
[[Isohedral figure|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.
{| class=wikitable width=480
|- align=center valign=top
|[[File:Pentagonal_icositetrahedron_variation0.png|160px]]<BR>[[Snub cube]] with augmented pyramids and merged faces
|[[File:Pentagonal_icositetrahedron_variation.png|160px]]<BR>Pentagonal icositetrahedron
|[[File:Pentagonal_icositetrahedron_variation_net.png|160px]]<BR>Net
|}

==Related polyhedra and tilings==
[[File:Spherical pentagonal icositetrahedron.png|160px|thumb|Spherical pentagonal icositetrahedron]]
This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with [[face configuration]]s (V3.3.3.3.''n''). (The sequence progresses into tilings the hyperbolic plane to any ''n''.) These [[face-transitive]] figures have (n32) rotational [[Orbifold notation|symmetry]].
{{Snub table}}

The ''pentagonal icositetrahedron '' is second in a series of dual snub polyhedra and tilings with [[face configuration]] V3.3.4.3.''n''.
{{Snub4 table}}

The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
{{Octahedral truncations}}

==References==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{Citation |last=Wenninger |first=Magnus |authorlink=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |mr=730208 |year=1983 |doi=10.1017/CBO9780511569371}} (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]], {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)

==External links==
*[https://web.archive.org/web/20080714211254/http://polyhedra.org/poly/show/38/pentagonal_icositetrahedron Pentagonal Icositetrahedron] – Interactive Polyhedron Model

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]
[[Category:Chiral polyhedra]]