Rhombicuboctahedron

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Rhombicuboctahedron
Rhombicuboctahedron.jpg
Type Archimdean
Uniform polyhedron
Faces 8 equilateral triangles
18 squares
Edges 48
Vertices 24
Vertex configuration
Schläfli symbol
Symmetry group Octahedral symmetry
Pyritohedral symmetry
Dihedral angle (degrees)square-to-square: 135°
square-to-triangle: 144.7°
Dual polyhedron Deltoidal icositetrahedron
Vertex figure
Polyhedron small rhombi 6-8 vertfig.svg
Net
Polyhedron small rhombi 6-8 net.svg

In geometry, rhombicuboctahedron is an Archimedean solid with 26 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]

Contents

The rhombicuboctahedron is an Archimedean solid, and it has Catalan solid as its dual, deltoidal icositetrahedron. The elongated square gyrobicupola is a 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. The rhombicuboctahedron is found in diverse cultures in architecture, toys, the arts, and elsewhere.

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]

Process of expanding the rhombicuboctahedron. P2-A5-P3.gif
Process of expanding the rhombicuboctahedron.

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] Relatedly, the rhombicuboctahedron may also be constructed by cutting all edges and vertices of either cube or a regular octahedron, a process known as rectification. [5]

Cartesian coordinates of a rhombicuboctahedron with an edge length 2 are the permutations of . [6]

Properties

Measurement and metric properties

The surface area of a rhombicuboctahedron can be determined by adding the area of all faces: 8 equilateral triangles and 18 squares. The volume of a rhombicuboctahedron can be determined by slicing it into two square cupolas and one octagonal prism. Given that the edge length , its surface area and volume is: [7]

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, van Roij & Dijkstra (2011). [8] 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: [9]

A rhombicuboctahedron has the Rupert property, meaning there is a polyhedron with the same or larger size that can pass through its hole. [10]

Symmetry and its classification family

3D model of a rhombicuboctahedron Rhombicuboctahedron.stl
3D model of a rhombicuboctahedron

The rhombicuboctahedron has the same symmetry as a cube and regular octahedron, the octahedral symmetry . [11] 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. [12] 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. [13]

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. [14] The polygonal faces that meet for every vertex are one equilateral triangle and three squares, and the vertex figure is denoted as . Its dual is deltoidal icositetrahedron, a Catalan solid, shares the same symmetry as the rhombicuboctahedron. [15]

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 not vertex-transitive, categorizing it as the Johnson solid instead. [16]

Graph

The graph of a rhombicuboctahedron Rhombicuboctahedral graph.png
The graph of a rhombicuboctahedron

The skeleton of a rhombicuboctahedron can be described as a graph. 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. [17]

Appearances

90-letie osnovaniia Natsional'noi biblioteki Belarusi.jpg
Diamond cube.jpg
Pacioli.jpg
De divina proportione - Vigintisex Basium Planum Vacuum.jpg
Many rhombicuboctahedral objects such as National Library in Minsk in the commemorative image (top left) and Rubik's cube variation (top right). As well as the rhombicuboctahedron may appear in art, as in Portrait of Luca Pacioli (bottom left) and Leonardo da Vinci's 1509 illustration in Divina proportione (bottom right).

The rhombicuboctahedron appears in the architecture, with an example of the building being the National Library located at Minsk. [18] The Wilson House is another example of the rhombicuboctahedron building, although its module was depicted as a truncated cube in which the edges are all cut off. It was built during the Second World War and Operation Breakthrough in the 1960s. [19]

The rhombicuboctahedron may also be found in toys. For example, 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) [20] [21] Another example may be found in dice from Corfe Castle, each of which square faces have marks of pairs of letters and pips. [22]

The rhombicuboctahedron may also appear in 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. [23] The first printed version of the rhombicuboctahedron was by Leonardo and appeared in Pacioli's Divina proportione (1509).

Related Research Articles

<span class="mw-page-title-main">Archimedean solid</span> Polyhedra in which all vertices are the same

In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids, and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as a rectangular cuboid with six square faces, or a parallelepiped with equal edges. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.

<span class="mw-page-title-main">Regular icosahedron</span> Polyhedron with 20 regular triangular faces

In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

<span class="mw-page-title-main">Snub cube</span> Archimedean solid with 38 faces

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol , and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol .

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron.

<span class="mw-page-title-main">Truncated octahedron</span> Archimedean solid

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

<span class="mw-page-title-main">Truncated cube</span> Archimedean solid with 14 regular faces

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

<span class="mw-page-title-main">Truncated cuboctahedron</span> Archimedean solid in geometry

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Truncated dodecahedron</span> Archimedean solid with 32 faces

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.

<span class="mw-page-title-main">Square cupola</span> Cupola with octagonal base

In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.

<span class="mw-page-title-main">Elongated square gyrobicupola</span> 37th Johnson solid

In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It was once mistakenly considered a rhombicuboctahedron by many mathematicians. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron. For this reason, it is also known as pseudo-rhombicuboctahedron, Miller solids, or Miller–Askinuze solid.

<span class="mw-page-title-main">Elongated triangular cupola</span> Polyhedron with triangular cupola and hexagonal prism

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

In geometry, the term semiregular polyhedron is used variously by different authors.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.

References

Notes

    • Hartshorne (2000), p.  463
    • 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.
  1. Viana et al. (2019), p. 1123, See Fig. 6.
  2. Linti (2013), p.  41.
  3. Shepherd (1954).
  4. Berman (1971), p. 336, See table IV, the Properties of regular-faced convex polyhedra, line 13..
  5. de Graaf, van Roij & Dijkstra (2011).
  6. Johnson (1966).
  7. Cromwell (1997), p.  386. See Table 10.21, Classes of vertex-transitive polyhedra..
  8. Diudea (2018), p.  39.
  9. Williams (1979), p.  80.
  10. Read & Wilson (1998), p. 269.
  11. Gabriel (1997), p.  105109.
  12. "Soviet Puzzle Ball". TwistyPuzzles.com. Retrieved 23 December 2015.
  13. "Diamond Style Puzzler". Jaap's Puzzle Page. Retrieved 31 May 2017.
  14. Cromwell (1997), p.  45.
  15. MacKinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette . 77 (479): 143. doi:10.2307/3619717. JSTOR   3619717. S2CID   195006163.

Works cited

See also

Further reading