Disdyakis dodecahedron

Disdyakis dodecahedron
(Click here for rotating model)
Type	Catalan solid
Conway notation	mC
Coxeter diagram
Face polygon	scalene triangle
Faces	48
Edges	72
Vertices	26 = 6 + 8 + 12
Face configuration	V4.6.8
Symmetry group	O_h, B₃, [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, (also hexoctahedron^[1], hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron^[2]), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles 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. It is the net of a rhombic dodecahedral pyramid.

Symmetry

It has O_h octahedral symmetry. Its collective edges represent the reflection planes of the symmetry. It can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron.

Disdyakis dodecahedron	Spherical	Cubic	Octahedral	Rhombic dodecahedral

Seen in stereographic projection the edges of the disdyakis dodecahedron form 9 circles (or centrally radial lines) in the plane. The 9 circles can be divided into two groups of 3 and 6 (drawn in purple and red), representing in two orthogonal subgroups: [2,2], and [3,3]:

Orthogonal	Stereographic

	[4]	[3]	[2]

Dimensions

If its smallest edges have length a, its surface area and volume are

{\begin{aligned}A&={\tfrac {6}{7}}{\sqrt {783+436{\sqrt {2}}}}\,a^{2}\\V&={\tfrac {1}{7}}{\sqrt {3\left(2194+1513{\sqrt {2}}\right)}}a^{3}\end{aligned}}

The faces are scalene triangles. Their angles are $\arccos({\frac {1}{6}}-{\frac {1}{12}}{\sqrt {2}})\approx 87.201\,963\,767\,96^{\circ }$ , $\arccos({\frac {3}{4}}-{\frac {1}{8}}{\sqrt {2}})\approx 55.024\,696\,148\,90^{\circ }$ and $\arccos({\frac {1}{12}}+{\frac {1}{2}}{\sqrt {2}})\approx 37.773\,340\,083\,13^{\circ }$ .

Orthogonal projections

The truncated cuboctahedron and its dual, the disdyakis dodecahedron can be drawn in a number of symmetric orthogonal projective orientations. Between a polyhedron and its dual, vertices and faces are swapped in positions, and edges are perpendicular.

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

Related polyhedra and tilings


Polyhedra similar to the disdyakis dodecahedron are duals to the Bowtie octahedron and cube, containing extra pairs triangular faces .^[3]

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{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
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 ≥ 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.

n32 symmetry mutation of omnitruncated tilings: 4.6.2n* v t e
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Sym. *n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]	[3i,3]
Figures
Config.	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞	4.6.24i	4.6.18i	4.6.12i	4.6.6i
Duals
Config.	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞	V4.6.24i	V4.6.18i	V4.6.12i	V4.6.6i

n42 symmetry mutation of omnitruncated tilings: 4.8.2n* v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry *n42 [n,4]	*242 [2,4]	*342 [3,4]	*442 [4,4]	*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.∞
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.∞

References

^ https://etc.usf.edu/clipart/keyword/forms
^ Conway, Symmetries of things, p.284
^ Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan

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)
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

External links

Weisstein, Eric W., "Disdyakis dodecahedron" ("Catalan solid") at MathWorld.
Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model

[1] ttps://etc.usf.edu/clipart/keyword/forms

[2] Conway, Symmetries of things, p.284

[3] Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan

[1]

[2]

[3]