Pentagonal icositetrahedron

Pentagonal icositetrahedron
Click ccw or cw for spinning versions.
Type	Catalan
Conway notation	gC
Coxeter diagram
Face polygon	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, ½BC3, [4,3]^+, 432
Dual polyhedron	snub cube
Properties	convex, face-transitive, chiral
Net

In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron^[1] is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.^[2][3]

It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

Geometry

Denote the tribonacci constant by t, approximately 1.8393. (See snub cube for a geometric explanation of the tribonacci constant.) Then the pentagonal faces have four angles of $\cos^{-1}\left(\frac{1-t}{2}\right)\approx$ 114.8° and one angle of $\cos^{-1}(2-t)\approx$ 80.75°. The pentagon has three short edges of unit length each, and two long edges of length $\frac{t+1}{2}\approx1.42$. The acute angle is between the two long edges.

If its dual snub cube has unit edge length, its surface area is $\scriptstyle{3}\sqrt{\tfrac{22(5t-1)}{4t-3}} \scriptstyle{\approx 19.29994}$ and its volume is $\sqrt{\tfrac{11(t-4)}{2(20t-37)}} \scriptstyle{\approx 7.4474}$.^[4]

Orthogonal projections

The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.

Orthogonal projections

Projective symmetry	[3]	[4]^+	[2]
Image
Dual image

Related polyhedra and tilings

Spherical pentagonal icositetrahedron

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.∞
Gryro 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.∞

The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n

Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	242	342	442	542	642	742	842	∞42
Snub figures
Config.	3.3.4.3.2	3.3.4.3.3	3.3.4.3.4	3.3.4.3.5	3.3.4.3.6	3.3.4.3.7	3.3.4.3.8	3.3.4.3.∞
Gyro figures
Config.	V3.3.4.3.2	V3.3.4.3.3	V3.3.4.3.4	V3.3.4.3.5	V3.3.4.3.6	V3.3.4.3.7	V3.3.4.3.8	V3.3.4.3.∞

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

Uniform octahedral polyhedra  Symmetry: [4,3], (*432) [4,3]^+ (432) [1^+,4,3] = [3,3] (*332) [3^+,4] (3*2) {4,3} t{4,3} r{4,3} r{3^1,1} t{3,4} t{3^1,1} {3,4} {3^1,1} rr{4,3} s2{3,4} tr{4,3} sr{4,3} h{4,3} {3,3} h2{4,3} t{3,3} s{3,4} s{3^1,1} = = = = or = or = Duals to uniform polyhedra V4^3 V3.8^2 V(3.4)^2 V4.6^2 V3^4 V3.4^3 V4.6.8 V3^4.4 V3^3 V3.6^2 V3^5

References

1. Conway, Symmetries of things, p.284
2. http://www.metafysica.nl/turing/promorph_crystals.html
3. http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm
4. Eric W. Weisstein, Pentagonal icositetrahedron (Catalan solid) at MathWorld

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208 (The thirteen semiregular convex polyhedra and their duals, Page 28, Pentagonal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)

External links

• Pentagonal Icositetrahedron – Interactive Polyhedron Model