# Pentagonal icositetrahedron

Pentagonal icositetrahedron

Click ccw or cw for spinning versions.
Type Catalan
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 is a Catalan solid which is the dual of the snub cube. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

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).

## Related polyhedra and tilings

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{4,3}
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

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