Rhombicuboctahedron
Rhombicuboctahedron | |
---|---|
Type | Archimdean Uniform polyhedron |
Faces | 26 |
Edges | 48 |
Vertices | 24 |
Vertex configuration | |
Symmetry group | Octahedral symmetry |
Dual polyhedron | Deltoidal icositetrahedron |
Vertex figure | |
Net | |
In
Names
It can also be called an expanded or cantellated cube or octahedron, from truncation operations on either uniform polyhedron.
Geometric relations
There are distortions of the rhombicuboctahedron that, while some of the faces are not regular polygons, are still vertex-uniform. Some of these can be made by taking a cube or octahedron and cutting off the edges, then trimming the corners, so the resulting polyhedron has six square and twelve rectangular faces. These have octahedral symmetry and form a continuous series between the cube and the octahedron, analogous to the distortions of the rhombicosidodecahedron or the tetrahedral distortions of the cuboctahedron. However, the rhombicuboctahedron also has a second set of distortions with six rectangular and sixteen trapezoidal faces, which do not have octahedral symmetry but rather Th symmetry, so they are invariant under the same rotations as the tetrahedron but different reflections.
The lines along which a
The rhombicuboctahedron is used in three
Dissection
The rhombicuboctahedron can be dissected into two square cupolae and a central octagonal prism. A rotation of one cupola by 45 degrees creates the pseudorhombicuboctahedron. Both of these polyhedra have the same vertex figure: 3.4.4.4.
There are three pairs of parallel planes that each intersect the rhombicuboctahedron in a regular octagon. The rhombicuboctahedron may be divided along any of these to obtain an octagonal prism with regular faces and two additional polyhedra called square
Rhombicuboctahedron | |
Pseudorhombicuboctahedron |
Orthogonal projections
The rhombicuboctahedron has six special
Centered by | Vertex | Edge 3-4 |
Edge 4-4 |
Face Square-1 |
Face Square-2 |
Face Triangle |
---|---|---|---|---|---|---|
Solid | ||||||
Wireframe | ||||||
Projective symmetry |
[2] | [2] | [2] | [2] | [4] | [6] |
Dual |
Spherical tiling
The rhombicuboctahedron can also be represented as a
(6) square-centered |
(6) square-centered |
(8) triangle-centered | |
Orthogonal projection
|
Stereographic projections |
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Pyritohedral symmetry
A half symmetry form of the rhombicuboctahedron, , exists with
Pyritohedral symmetry variations | |||||||||
---|---|---|---|---|---|---|---|---|---|
Uniform geometry |
Nonuniform geometry |
Nonuniform geometry |
In the limit, an icosahedron snub octahedron, , from one of two positions. |
Compound of two icosahedra from both alternated positions. |
Algebraic properties
Cartesian coordinates
- (±1, ±1, ±(1 + √2)).
If the original rhombicuboctahedron has unit edge length, its dual
Area and volume
The area A and the volume V of the rhombicuboctahedron of edge length a are:
Close-packing density
The optimal packing fraction of rhombicuboctahedra is given by
- .
It was noticed that this optimal value is obtained in a Bravais lattice by de Graaf (2011). Since the rhombicuboctahedron is contained in a rhombic dodecahedron whose inscribed sphere is identical to its own inscribed sphere, the value of the optimal packing fraction is a corollary of the Kepler conjecture: it can be achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it.
In the arts
The 1495 Portrait of Luca Pacioli, traditionally attributed to Jacopo de' Barbari, includes a glass rhombicuboctahedron half-filled with water, which may have been painted by Leonardo da Vinci.[5] The first printed version of the rhombicuboctahedron was by Leonardo and appeared in
A spherical 180° × 360° panorama can be projected onto any polyhedron; but the rhombicuboctahedron provides a good enough approximation of a sphere while being easy to build. This type of projection, called Philosphere, is possible from some panorama assembly software. It consists of two images that are printed separately and cut with scissors while leaving some flaps for assembly with glue.[6]
Objects
During the
-
Sundial (1596)
-
Sundial
-
Street lamp in Mainz
-
Die with 18 labelled faces
-
Cabela's shooting target
-
Rubik's Cube variation
-
Pyrite crystal
Related polyhedra
The rhombicuboctahedron is one of a family of 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
|
Symmetry mutations
This polyhedron is topologically related as a part of sequence of
*n32 symmetry mutation of expanded tilings: 3.4.n.4 | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paracomp. | ||||
*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. | 3.4.2.4 | 3.4.3.4 | 3.4.4.4 | 3.4.5.4 | 3.4.6.4 | 3.4.7.4 | 3.4.8.4 | 3.4.∞.4 |
*n42 symmetry mutation of expanded tilings: n.4.4.4 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry [n,4], (*n42) |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | |||||||
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4] |
*∞42 [∞,4] | |||||
Expanded figures |
|||||||||||
Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4
| ||||
Rhombic figures config.
|
V3.4.4.4 |
V4.4.4.4 |
V5.4.4.4 |
V6.4.4.4 |
V7.4.4.4 |
V8.4.4.4 |
V∞.4.4.4 |
Vertex arrangement
It shares its vertex arrangement with three
Rhombicuboctahedron |
Small cubicuboctahedron |
Small rhombihexahedron |
Stellated truncated hexahedron |
Rhombicuboctahedral graph | |
---|---|
Hamiltonian, regular | |
Table of graphs and parameters |
Rhombicuboctahedral graph
The rhombicuboctahedral graph is the
See also
- Compound of five rhombicuboctahedra
- Cube
- Cuboctahedron
- Nonconvex great rhombicuboctahedron
- Truncated rhombicuboctahedron
- Elongated square gyrobicupola
- Moravian star
- Octahedron
- Rhombicosidodecahedron
- Rubik's Snake – puzzle that can form a Rhombicuboctahedron "ball"
- National Library of Belarus – its architectural main component has the shape of a rhombicuboctahedron.
- Truncated cuboctahedron (great rhombicuboctahedron)
References
- ISBN 0-87169-209-0(page 119)
- ^ a b "Soviet Puzzle Ball". TwistyPuzzles.com. Retrieved 23 December 2015.
- ^ a b "Diamond Style Puzzler". Jaap's Puzzle Page. Retrieved 31 May 2017.
- ^ "RitrattoPacioli.it".
- S2CID 195006163.
- ^ "Philosphere".
- ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
Further reading
- ISBN 0-486-23729-X. (Section 3–9)
- Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
- S2CID 202575183.
- de Graaf, J.; van Roij, R.; Dijkstra, M. (2011), "Dense Regular Packings of Irregular Nonconvex Particles", S2CID 14041658
- Betke, U.; Henk, M. (2000), "Densest Lattice Packings of 3-Polytopes",
- Torquato, S.; Jiao, Y. (2009), "Dense packings of the Platonic and Archimedean solids", Nature, 460 (7257): 876–879, S2CID 52819935
External links
- Weisstein, Eric W., "Rhombicuboctahedron" ("Archimedean solid") at MathWorld.
- Klitzing, Richard. "3D convex uniform polyhedra x3o4x - sirco".
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Editable printable net of a rhombicuboctahedron with interactive 3D view
- Rhombicuboctahedron Star by Sándor Kabai, Wolfram Demonstrations Project.
- Rhombicuboctahedron: paper strips for plaiting