Rhombic dodecahedron

Rhombic dodecahedron
(Click here for rotating model)
Type	Catalan solid
Coxeter diagram
Conway notation	jC
Face type	V3.4.3.4 rhombus
Faces	12
Edges	24
Vertices	14
Vertices by type	8{3}+6{4}
Symmetry group	Oh, B3, [4,3], (*432)
Rotation group	O, [4,3]+, (432)
Dihedral angle	120°
Properties	convex, face-transitive isohedral, isotoxal, parallelohedron
Cuboctahedron (dual polyhedron)	Net

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

Properties

The rhombic dodecahedron is a

angles on each face measure arccos(1/3), or approximately 70.53°.

Being the dual of an

of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube.

The rhombic dodecahedron is one of the nine

.

The rhombic dodecahedron can be used to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane.

This

The graph of the rhombic dodecahedron is nonhamiltonian.

A rhombic dodecahedron can be

.

The collections of the

• 
  Rhombic dodecahedron dissected into 4 rhombohedra
• 
  Hexagon dissected into 3 rhombi
• 
  A garnet crystal
• 
  This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube.

Dimensions

Denoting by a the edge length of a rhombic dodecahedron,

• the radius of its inscribed sphere (tangent to each of the rhombic dodecahedron's faces) is

${\displaystyle r_{\mathrm {i} }={\frac {\sqrt {6}}{3}}~a\approx 0.816\,496\,5809~a\quad }$ (OEIS: A157697),

• the radius of its midsphere is

${\displaystyle r_{\mathrm {m} }={\frac {2{\sqrt {2}}}{3}}~a\approx 0.942\,809\,041\,58~a\quad }$ (OEIS: A179587),

• the radius of the sphere passing through the six order 4 vertices, but not through the eight order 3 vertices, is

${\displaystyle r_{\mathrm {o} }={\frac {2{\sqrt {3}}}{3}}~a\approx 1.154\,700\,538~a\quad }$ (OEIS: A020832),

• the radius of the sphere passing through the eight order 3 vertices is exactly equal to the length of the sides

${\displaystyle r_{\mathrm {t} }=a}$

Area and volume

The surface area A and the volume V of the rhombic dodecahedron with edge length a are:

${\displaystyle A=8{\sqrt {2}}~a^{2}\approx 11.313\,7085~a^{2}}$

${\displaystyle V={\frac {16{\sqrt {3}}}{9}}~a^{3}\approx 3.079\,201\,44~a^{3}}$

Orthogonal projections

The rhombic dodecahedron has four special

.

Orthogonal projections

Projective symmetry	[4]	[6]	[2]	[2]
Rhombic dodecahedron
Cuboctahedron (dual)

Cartesian coordinates

Pyritohedron variations between a cube and rhombic dodecahedron

Expansion of a rhombic dodecahedron

For edge length √3, the eight vertices where three faces meet at their obtuse angles have

:

(±1, ±1, ±1)

The coordinates of the six vertices where four faces meet at their acute angles are:

(±2, 0, 0), (0, ±2, 0) and (0, 0, ±2)

The rhombic dodecahedron can be seen as a degenerate limiting case of a

, with permutation of coordinates (±1, ±1, ±1) and (0, 1 + h, 1 − h²) with parameter h = 1.

These coordinates illustrate that a rhombic dodecahedron can be seen as a cube with a square pyramid attached to each face, and that the six square pyramids could fit together to a cube of the same size, i.e the rhombic dodecahedron has twice the volume of the inscribed cube with edges equal to the short diagonals of the rhombi.^[3]

Topologically equivalent forms

Parallelohedron

The rhombic dodecahedron is a

.

stellated rhombic dodecahedron .

The rhombic dodecahedron can be constructed with 4 sets of 6 parallel edges.

Dihedral rhombic dodecahedron

Other symmetry constructions of the rhombic dodecahedron are also space-filling, and as

For example, with 4 square faces, and 60-degree rhombic faces, and D_4h dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids augmented on the top and bottom.

Net

Coordinates (0, 0, ±2) (±1, ±1, 0) (±1, 0, ±1) (0, ±1, ±1)

Bilinski dodecahedron

Bilinski dodecahedron with edges and front faces colored by their symmetry positions.

Bilinski dodecahedron colored by parallel edges

In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio.^[5][6]

Faces

First form	Second form
√2:1	√5 + 1/2:1

Deltoidal dodecahedron

Another topologically equivalent variation, sometimes called a deltoidal dodecahedron,

(deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (a,b), where b and a depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution. As a approaches 1/2, b approaches infinity. It always holds that 1/a + 1/b = 2, with a, b > 1/2.

(±2, 0, 0), (0, ±2, 0), (0, 0, ±2)

(a, a, a), (−a, −a, a), (−a, a, −a), (a, −a, −a)

(−b, −b, −b), (−b, b, b), (b, −b, b), (b, b, −b)

(1,1)	(7/8,7/6)	(3/4,3/2)	(2/3,2)	(5/8,5/2)	(9/16,9/2)

Related polyhedra

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

When projected onto a sphere (see right), it can be seen that the edges make up the edges of two tetrahedra arranged in their dual positions (the stella octangula). This trend continues on with the deltoidal icositetrahedron and deltoidal hexecontahedron for the dual pairings of the other regular polyhedra (alongside the triangular bipyramid if improper tilings are to be considered), giving this shape the alternative systematic name of deltoidal dodecahedron.

*n32 symmetry mutation of dual expanded tilings: V3.4.n.4

Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure Config.	V3.4.2.4	V3.4.3.4	V3.4.4.4	V3.4.5.4	V3.4.6.4	V3.4.7.4	V3.4.8.4	V3.4.∞.4

This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32	Spherical			Euclidean	Hyperbolic
	*332	*432	*532	*632	*732	*832...	*∞32
Tiling
Conf.	V(3.3)2	V(3.4)2	V(3.5)2	V(3.6)2	V(3.7)2	V(3.8)2	V(3.∞)2

*n42 symmetry mutations of quasiregular dual tilings: V(4.n)2
Symmetry *4n2 [n,4]	Spherical	Euclidean	Compact hyperbolic				Paracompact	Noncompact
	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]	[iπ/λ,4]
Tiling   Conf.	V4.3.4.3	V4.4.4.4	V4.5.4.5	V4.6.4.6	V4.7.4.7	V4.8.4.8	V4.∞.4.∞	V4.∞.4.∞

Similarly it relates to the infinite series of tilings with the

V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.

V3.4.3.4 (Drawn as a net)

V3.6.3.6 Euclidean plane tiling Rhombille tiling

V3.8.3.8 Hyperbolic plane tiling (Drawn in a Poincaré disk model)

Stellations

Like many convex polyhedra, the rhombic dodecahedron can be stellated by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke.^[8]

stellated rhombic dodecahedron

by inverting the center-face pyramids of a rhombic dodecahedron.

The first stellation, often simply called the

, is well known. It can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces:

• 
  The first stellation of the rhombic dodecahedron
• 
  3D model of decomposition into 12 pyramids and 4 half-cubes

Luke describes four more stellations: the second and third stellations (expanding outwards), one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.

Second	Third
Great stellated rhombic dodecahedron	Stellated rhombic dodecahedron

Related polytopes

The rhombic dodecahedron forms the hull of the vertex-first projection of a

, giving eight possible rhombohedra as projections of the tesseracts 8 cubic cells. One set of projective vectors are: u = (1,1,−1,−1), v = (−1,1,−1,1), w = (1,−1,−1,1).

The rhombic dodecahedron forms the maximal cross-section of a

meeting at a single vertex in the center; these form the images of six pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.

This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube into six congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using two tesseracts.^[9]

Architectural meaning and cultural freight

The architectural expert James D. Wenn has identified that philosophical and meanings coded into buildings connected with meanings associate with the rhombic dodecahedron by thinkers such as Plato.^[10]

Buildings identified as engaging with this kind of code include:^[11]

• Soulton Hall,^[12]
• Gonville and Caius College, Cambridge
• Verulam House, the now lost house of Francis Bacon
• Wollaton Hall

Practical usage

In spacecraft reaction wheel layout, a tetrahedral configuration of four wheels is commonly used. For wheels that perform equally (from a peak torque and max angular momentum standpoint) in both spin directions and across all four wheels, the maximum torque and maximum momentum envelopes for the 3-axis attitude control system (considering idealized actuators) are given by projecting the tesseract representing the limits of each wheel's torque or momentum into 3D space via the 3 × 4 matrix of wheel axes; the resulting 3D polyhedron is a rhombic dodecahedron.^[13] Such an arrangement of reaction wheels is not the only possible configuration (a simpler arrangement consists of three wheels mounted to spin about orthogonal axes), but it is advantageous in providing redundancy to mitigate the failure of one of the four wheels (with degraded overall performance available from the remaining three active wheels) and in providing a more convex envelope than a cube, which leads to less agility dependence on axis direction (from an actuator/plant standpoint). Spacecraft mass properties influence overall system momentum and agility, so decreased variance in envelope boundary does not necessarily lead to increased uniformity in preferred axis biases (that is, even with a perfectly distributed performance limit within the actuator subsystem, preferred rotation axes are not necessarily arbitrary at the system level).

The polyhedron is also the basis for the

.

See also

• Dodecahedron
• Rhombic triacontahedron
• Trapezo-rhombic dodecahedron
• Truncated rhombic dodecahedron
• 24-cell – 4D analog of rhombic dodecahedron
• Archimede construction systems
• Fully truncated rhombic dodecahedron

References

Further reading

• ISBN 0-486-23729-X
  . (Section 3-9)
• MR 0730208
  . (The thirteen semiregular convex polyhedra and their duals, Page 19, Rhombic dodecahedron)
• 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, p. 285, Rhombic dodecahedron )

External links

• Weisstein, Eric W. "Rhombic dodecahedron". MathWorld.
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra

Computer models

• Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron, Rhombic Dodecahedron 5-Compound and Rhombic Dodecahedron 5-Compound by Sándor Kabai,
  The Wolfram Demonstrations Project
  .

Paper projects

• Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue
• Another Rhombic Dodecahedron Calendar – made by plaiting paper strips

Practical applications

• Archimede Institute Examples of actual housing construction projects using this geometry