# Disdyakis dodecahedron

Disdyakis dodecahedron

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Spinning version

Type Catalan
Coxeter diagram
Face polygon
scalene triangle
Faces 48
Edges 72
Vertices 26 = 6 + 8 + 12
Face configuration V4.6.8
Symmetry group Oh, BC3, [4,3], *432
Dihedral angle 155° 4' 56"
$\arccos(-\frac{71 + 12\sqrt{2}}{97})$
Dual polyhedron truncated cuboctahedron
Properties convex, face-transitive

Net

In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It looks a bit like an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron.

## Symmetry

It has Oh octahedral symmetry. Its collective edges represent the reflection planes of the symmetry.

## Dimensions

If its smallest edges have length 1, its surface area is $\tfrac{6}{7}\scriptstyle{\sqrt{783+436\sqrt{2}}}$ and its volume is $\tfrac{1}{7}\scriptstyle{\sqrt{3(2194+1513\sqrt{2})}}$.

## Related polyhedra and tilings

The disdyakis dodecahedron 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

It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any $n \ge 7.$

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli

tr{2,3}

tr{3,3}

tr{4,3}

tr{5,3}

tr{6,3}

tr{7,3}

tr{8,3}

tr{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞
Dimensional family of omnitruncated polyhedra and tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure

4.8.4

4.8.6

4.8.8

4.8.10

4.8.12

4.8.14

4.8.16

4.8.∞
Coxeter
Schläfli

tr{2,4}

tr{3,4}

tr{4,4}

tr{5,4}

tr{6,4}

tr{7,4}

tr{8,4}

tr{∞,4}
Omnitruncated
duals

V4.8.4

V4.8.6

V4.8.8

V4.8.10

V4.8.12

V4.8.14

V4.8.16

V4.8.∞
Coxeter