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| !bgcolor=#e7dcc3 colspan=2|Pentagonal icositetrahedron
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| |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 spinning versions.''
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| |bgcolor=#e7dcc3|Type||[[Catalan solid|Catalan]]
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| |bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_fh|4|node_fh|3|node_fh}}
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| |bgcolor=#e7dcc3|Face polygon||[[pentagon|irregular pentagon]]
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| |bgcolor=#e7dcc3|Faces||24
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| |bgcolor=#e7dcc3|Edges||60
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| |bgcolor=#e7dcc3|Vertices||38 = 6 + 8 + 24
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| |bgcolor=#e7dcc3|[[Face configuration]]||V3.3.3.3.4
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| |bgcolor=#e7dcc3|[[Dihedral angle]]||136° 18' 33'
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| |bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Octahedral symmetry|O]], ½BC<sub>3</sub>, [4,3]<sup>+</sup>, 432
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| |bgcolor=#e7dcc3|[[Dual polyhedron]]||[[snub cube]]
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| |bgcolor=#e7dcc3|Properties||[[convex set|convex]], [[face-transitive]], [[chirality (mathematics)|chiral]]
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| |align=center colspan=2|[[Image:Pentagonalicositetrahedron net.png|240px|Pentagonal icositetrahedron]]<br>[[Net (polyhedron)|Net]]
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| In [[geometry]], a '''pentagonal icositetrahedron''' is a [[Catalan solid]] which is the [[Dual polyhedron|dual]] of the [[snub cube]]. It has two distinct forms, which are [[mirror image]]s (or "[[Chirality (mathematics)|enantiomorphs]]") of each other.
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| If it has unit edge length, its surface area is <math>\scriptstyle{3}\sqrt{\tfrac{22(5t-1)}{4t-3}} \scriptstyle{\approx 19.29994}</math> and its volume is <math>\sqrt{\tfrac{11(t-4)}{2(20t-37)}} \scriptstyle{\approx 7.4474}</math>. Here ''t'' is the [[tribonacci constant]] (see [[snub cube]]).
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| ==Related polyhedra and tilings==
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| 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]].
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| {{Snub table}}
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| The ''pentagonal icositetrahedron '' is second in a series of dual snub polyhedra and tilings with [[face configuration]] V3.3.4.3.''n''.
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| {{Snub4 table}}
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| The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
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| {{Octahedral truncations}}
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| ==References==
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{Citation |last=Wenninger |first=Magnus |authorlink=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |id={{MathSciNet|id=730208}} |year=1983}} (The thirteen semiregular convex polyhedra and their duals, Page 28, Pentagonal icositetrahedron)
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| *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [http://www.akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)
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| ==External links==
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| *{{Mathworld2 |urlname=PentagonalIcositetrahedron |title=Pentagonal icositetrahedron |urlname2=CatalanSolid |title2=Catalan solid}}
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| *[http://polyhedra.org/poly/show/38/pentagonal_icositetrahedron Pentagonal Icositetrahedron] – Interactive Polyhedron Model
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| {{Polyhedron navigator}}
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| [[Category:Catalan solids]]
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| {{Polyhedron-stub}}
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