Disdyakis dodecahedron

Disdyakis dodecahedron
(rotating and 3D 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]) or d48, 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 resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid results in the Kleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.^[a] The net of the rhombic dodecahedral pyramid also shares the same topology.

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	Deltoidal icositetrahedron	Rhombic dodecahedron	Hexahedron	Octahedron

Spherical polyhedron

(see rotating model)	Orthographic projections from 2-, 3- and 4-fold axes

The edges of a spherical disdyakis dodecahedron belong to 9 great circles. Three of them form a spherical octahedron (gray in the images below). The remaining six form three square hosohedra (red, green and blue in the images below). They all correspond to mirror planes - the former in dihedral [2,2], and the latter in tetrahedral [3,3] symmetry. A spherical disdyakis dodecahedron can be thought of as the barycentric subdivision of the spherical cube or of the spherical octahedron.^[3]

Stereographic projections
	2-fold	3-fold	4-fold

Cartesian coordinates

Let $~a={\frac {1}{1+2{\sqrt {2}}}}~{\color {Gray}\approx 0.261},~~b={\frac {1}{2+3{\sqrt {2}}}}~{\color {Gray}\approx 0.160},~~c={\frac {1}{3+3{\sqrt {2}}}}~{\color {Gray}\approx 0.138}$ .
Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are:

● permutations of (±a, 0, 0) (vertices of an octahedron)
● permutations of (±b, ±b, 0) (vertices of a cuboctahedron)
● (±c, ±c, ±c) (vertices of a cube)

Convex hulls
Combining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron. The convex hulls for these vertices^[4] scaled by $1/a$ result in Cartesian coordinates of unit circumradius, which are visualized in the figure below:

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 $\arccos {\biggl (}{\frac {1}{6}}-{\frac {1}{12}}{\sqrt {2}}{\biggr )}~{\color {Gray}\approx 87.201^{\circ }}$ , $\arccos {\biggl (}{\frac {3}{4}}-{\frac {1}{8}}{\sqrt {2}}{\biggr )}~{\color {Gray}\approx 55.024^{\circ }}$ and $\arccos {\biggl (}{\frac {1}{12}}+{\frac {1}{2}}{\sqrt {2}}{\biggr )}~{\color {Gray}\approx 37.773^{\circ }}$ .

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 of triangular faces.^[5]

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

Uniform octahedral polyhedra v t e
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.∞

Notes

↑ Despite their resemblance, no subset of the disdyakis dodecahedron's vertices forms a rhombic dodecahedron (see #Cartesian coordinates), and therefore, the former is not the Kleetope of the latter. The "rhombic" bases of the pyramids of the disdyakis dodecahedron are in fact not even planar; for example, the vertices of one such rhombus are (a, 0, 0), (0, a, 0), (c, c, c), (c, c, -c) (again, see #Cartesian coordinates for the values of a and c), with diagonal midpoints (√2)×(a, a, 0) and (c, c, 0), which do not coincide.

References

↑ "Keyword: "forms" | ClipArt ETC".
↑ Conway, Symmetries of things, p.284
↑ Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR 2781856
↑ Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions". Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.
↑ 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-Strauss, ISBN 978-1-56881-220-5 (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

[3] Despite their resemblance, no subset of the disdyakis dodecahedron's vertices forms a rhombic dodecahedron (see #Cartesian coordinates), and therefore, the former is not the Kleetope of the latter. The "rhombic" bases of the pyramids of the disdyakis dodecahedron are in fact not even planar; for example, the vertices of one such rhombus are (a, 0, 0), (0, a, 0), (c, c, c), (c, c, -c) (again, see #Cartesian coordinates for the values of a and c), with diagonal midpoints (√2)×(a, a, 0) and (c, c, 0), which do not coincide.

[1] "Keyword: "forms" | ClipArt ETC".

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

[4] Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR 2781856

[5] Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions". Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.

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

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