# 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 crystalography it is also called a gyroid.[2][3]

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

## Geometry

If it 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 $\sqrt{\tfrac{11(t-4)}{2(20t-37)}} \scriptstyle{\approx 7.4474}$. Here t is the tribonacci constant (see snub cube).

## Orthogonal projections

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

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

## 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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure

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.∞
Coxeter
Schläfli

sr{2,3}

sr{3,3}

sr{4,3}

sr{5,3}

sr{6,3}

sr{7,3}

sr{8,3}

sr{∞,3}
Snub
dual
figure

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.∞
Coxeter

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

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
242
[2,4]+
342
[3,4]+
442
[4,4]+
542
[5,4]+
642
[6,4]+
742
[7,4]+
842
[8,4]+...
∞42
[∞,4]+
Snub
figure

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.∞
Coxeter
Schläfli

sr{2,4}

sr{3,4}

sr{4,4}

sr{5,4}

sr{6,4}

sr{7,4}

sr{8,4}

sr{∞,4}
Snub
dual
figure

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.∞
Coxeter

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{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
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{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35