In geometry, rhombicuboctahedron is an
Archimedean solid with 24 faces, consisting of 8 equilateral triangles and 18 squares. It is named by
Johannes Kepler in his 1618
Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a
rhombic dodecahedron.[1]
The rhombicuboctahedron is an
Archimedean solid, and it has
Catalan solid as its dual,
deltoidal icositetrahedron. The
elongated square gyrobicupola is the polyhedron that is similar to a rhombicuboctahedron, but it is not an Archimedean solid because it is not
vertex-transitive. The skeleton of a rhombicuboctahedron can be represented as a graph known as rhombicuboctahedral graph. The rhombicuboctahedron may be found in many popular cultures as in architecture, toys, and arts.
Construction
The rhombicuboctahedron may be constructed from a
cube by drawing a smaller one in the middle of each face, parallel to the cube's edges. After removing the edges of a cube, the squares may be joined by adding more squares adjacent between them, and the corners may be filled by the
equilateral triangles. Another way to construct the rhombicuboctahedron is by attaching two regular
square cupolas into the bases of a regular
octagonal prism.[2]
A rhombicuboctahedron may also be known as an expanded octahedron or expanded cube. This is because the rhombicuboctahedron may also be constructed by separating and pushing away the faces of a cube or a
regular octahedron from their centroid (in blue or red, respectively, in the animation), and filling between them with the squares and equilateral triangles. This construction process is known as
expansion.[3] By using all of these methods above, the rhombicuboctahedron has 8 equilateral triangles and 16 squares as its faces.[4]
One way to construct the rhombicuboctahedron by
Cartesian coordinates with an edge length 2 is. . [5]
Properties
Measurement and metric properties
A rhombicuboctahedron with edge length has a surface area:[6]
by adding the area of 8 equilateral triangles and 10 squares. Its volume can be calculated by slicing it into two square cupolas and one octagonal prism:[6]
It was noticed that this optimal value is obtained in a
Bravais lattice by
de Graaf, van Roij & Dijkstra (2011).[7] Since the rhombicuboctahedron is contained in a
rhombic dodecahedron whose
inscribed sphere is identical to its 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.[citation needed]
The dihedral angle of a rhombicuboctahedron can be determined by adding the dihedral angle of a square cupola and an octagonal prism:[8]
the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom are that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a
regular octagon 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola square-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 45° + 90° = 135°. Therefore, the dihedral angle of a rhombicuboctahedron for every two adjacent squares is 135°.
the dihedral angle of a rhombicuboctahedron square-to-triangle is that of a square cupola between those 144.7°. The dihedral angle between square-to-triangle, on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola triangle-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 54.7° + 90° = 144.7°. Therefore, the dihedral angle of a rhombicuboctahedron for every square-to-triangle is 144.7°.
Symmetry and its classification family
The rhombicuboctahedron has the same symmetry as a cube and regular octahedron, the
octahedral symmetry.[9] 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
pyritohedral symmetry, so they are invariant under the same rotations as the tetrahedron but different reflections.[10] It is
centrosymmetric, meaning its symmetric is interchangeable by the appearance of
inversion center. It is also non-
chiral; that is, it is congruent to its own mirror image.[11]
The rhombicuboctahedron is an
Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different
regular polygonal faces meet in a vertex.[12] Its dual is
deltoidal icositetrahedron, a
Catalan solid, shares the same symmetry as the rhombicuboctahedron.[citation needed] A rhombicuboctahedron is one of the Archimedean solids with the
Rupert property, meaning there is a polyhedron with the same or larger size that can pass through its hole.[13]
The
elongated square gyrobicupola is the only polyhedron resembling the rhombicuboctahedron. The difference is that the elongated square gyrobicupola is constructed by twisting one of its cupolae. It was once considered as the 14th Archimedean solid, until it was discovered that it is
vertex-transitive, categorizing it as the
Johnson solid instead.[14]
Rhombicuboctahedral graph
The rhombicuboctahedral graph is a graph representing the
skeleton of a rhombicuboctahedron. It is
polyhedral graph, meaning that it is
planar and
3-vertex-connected. In other words, the edges of a graph are not crossed while being drawn, and removing any two of its vertices leaves a connected subgraph. It has 24
vertices and 48 edges. It is a
quartic, meaning each of its vertices is connected by four vertices. This graph is classified as
Archimedean graph, because it resembles the graph of Archimedean solid.[15]
The rhombicuboctahedron may also be found in toys. An example is the lines along which a
Rubik's Cube can be turned are, projected onto a sphere, similar,
topologically identical, to a rhombicuboctahedron's edges. Variants using the Rubik's Cube mechanism have been produced, which closely resemble the rhombicuboctahedron. During the Rubik's Cube craze of the 1980s, at least two twisty puzzles sold had the form of a rhombicuboctahedron (the mechanism was similar to that of a Rubik's Cube)[17][18] Another example may be found in dice from
Corfe Castle, each of which square faces have marks of pairs of letters and
pips.[19]
The rhombicuboctahedron may also appear in the art. An example is the 1495 Portrait of Luca Pacioli, traditionally attributed to
Jacopo de' Barbari, which includes a glass rhombicuboctahedron half-filled with water, which may have been painted by
Leonardo da Vinci.[21]
The first printed version of the rhombicuboctahedron was by Leonardo and appeared in
Pacioli's Divina proportione (1509).
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 printed separately and cut with scissors while leaving some flaps for assembly with glue.[22]
Berman (1971), p. 336, See table IV, the Properties of regular-faced convex polyhedra, line 13. Here, represents the octagonal prism and represents the square cupola.
Kepler, Johannes (1997),
Harmony of the World, American Philosophical Society. This is translated into English by Aiton E. J., Duncan E. M., Field J. V.
Viana, Vera; Xavier, João Pedro; Aires, Ana Paula; Campos, Helena (2019), "Interactive Expansion of Achiral Polyhedra", in Cocchiarella, Luigi (ed.), ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018, Springer,
doi:
10.1007/978-3-319-95588-9.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc.
ISBN0-486-23729-X. (Section 3–9)
Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids.
ISBN0-521-55432-2.