geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the rhombicuboctahedron, or small rhombicuboctahedron, is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

with eight

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, ...

, six

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, and twelve

rectangular In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...

faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square (equivalently, all the edges are the same length, ensuring the triangles are

equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

), it is an

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

. The

has

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhed ...

, like the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

and

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...

. Its dual is called the

deltoidal icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...

or trapezoidal icositetrahedron, although its faces are not really true

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eucli ...

Names

Johannes Kepler in

Harmonices Mundi ''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...

(1618) named this polyhedron a ''rhombicuboctahedron'', being short for ''truncated cuboctahedral rhombus'', with ''cuboctahedral rhombus'' being his name for a

rhombic dodecahedron 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 dodecahed ...

. There are different truncations of a rhombic dodecahedron into a

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

rhombicuboctahedron: Prominently its

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

(left), the one that creates the uniform solid (center), and the rectification of the dual

cuboctahedron 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 ...

(right), which is the core of the

dual compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common Centroid, centre. They are the three-dimensional analogs of star polygon#Regular compounds, polygonal compounds such as the hexagram. The outer ...

. It can also be called an '' expanded'' or '' cantellated''

, from truncation operations on either

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

. Since its inclusion in

Wings 3D Wings 3D is a free and open-source subdivision modeler inspired by Nendo and Mirai from Izware. Wings 3D is named after the winged-edge data structure it uses internally to store coordinate and adjacency data, and is commonly referred to by it ...

as an "octotoad" this unofficial moniker is spreading.

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 In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular ...

or the tetrahedral distortions of the

. 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 T_h symmetry, so they are invariant under the same rotations as the

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

but different reflections. The lines along which a

Rubik's Cube The Rubik's Cube is a 3-D combination puzzle originally invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by Pentangle Puzzles in t ...

can be turned are, projected onto a sphere, similar,

topologically In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

identical, to a rhombicuboctahedron's edges. In fact, variants using the Rubik's Cube mechanism have been produced which closely resemble the rhombicuboctahedron. The rhombicuboctahedron is used in three uniform space-filling tessellations: the

cantellated cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a reg ...

, the runcitruncated cubic honeycomb, and the

runcinated alternated cubic honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...

Dissection

The rhombicuboctahedron can be dissected into two

square cupola In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in t ...

e and a central

octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also ...

. A rotation of one cupola by 45 degrees creates the ''pseudorhombicuboctahedron''. Both of these polyhedra have the same vertex figure: 3.4.4.4. Comparison_of_rhombicuboctahedron_and_pseudorhombicuboctahedron

Comparison_of_rhombicuboctahedron_and_pseudorhombicuboctahedron

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 cupolae, which count among the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s; it is thus an ''elongated square ortho

bicupola In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together ...

''. These pieces can be reassembled to give a new solid called the

elongated square gyrobicupola In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pat ...

or ''pseudorhombicuboctahedron'', with the symmetry of a square antiprism. In this the vertices are all locally the same as those of a rhombicuboctahedron, with one triangle and three squares meeting at each one, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others.

Orthogonal projections

The ''rhombicuboctahedron'' has six special

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...

s, centered, on a vertex, on two types of edges, and three types of faces: triangles, and two squares. The last two correspond to the B₂ and A₂

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

Spherical tiling

The rhombicuboctahedron can also be represented as a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...

, and projected onto the plane via a

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter th ...

. This projection is

conformal Conformal may refer to: * Conformal (software), in ASIC Software * Conformal coating in electronics * Conformal cooling channel, in injection or blow moulding * Conformal field theory in physics, such as: ** Boundary conformal field theory ...

, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Pyritohedral symmetry

A half symmetry form of the rhombicuboctahedron, , exists with

pyritohedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

, ,3⁺ (3*2) as

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

s₂, and can be called a ''cantic snub octahedron''. This form can be visualized by alternatingly coloring the edges of the 6

s. These squares can then be distorted into rectangles, while the 8 triangles remain equilateral. The 12 diagonal square faces will become

isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defi ...

s. In the limit, the rectangles can be reduced to edges, and the trapezoids become triangles, and an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...

is formed, by a ''snub octahedron'' construction, , s. (The

compound of two icosahedra This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry ''Oh''. As a holosnub, it is represented by Schläfli symbol β and Coxeter diagram . The triangles in this compound decompose into two orbits unde ...

is constructed from both alternated positions.)

Algebraic properties

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

for the vertices of a rhombicuboctahedron centred at the origin, with edge length 2 units, are all the

even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...

s of :(±1, ±1, ±(1 + )). If the original rhombicuboctahedron has unit edge length, its dual

strombic icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosite ...

has edge lengths :

\frac\sqrt \quad \text \quad \sqrt.

Area and volume

The area ''A'' and the volume ''V'' of the rhombicuboctahedron of edge length ''a'' are: :

\begin
A &= \left(18+2\sqrt\right)a^2 &&\approx 21.464\,1016a^2 \\
V &= \frac a^3 &&\approx 8.714\,045\,21a^3.
\end

Close-packing density

The optimal packing fraction of rhombicuboctahedra is given by :

\eta = \tfrac \left( 4\sqrt - 5 \right)

. It was noticed that this optimal value is obtained in a

Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...

by . Since the rhombicuboctahedron is contained in a

whose

inscribed sphere In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and i ...

is identical to its own inscribed sphere, the value of the optimal packing fraction is a corollary of the

Kepler conjecture The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling ...

: it can be achieved by putting a rhombicuboctahedron in each cell of the

rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal ...

, 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 The ''Portrait of Luca Pacioli'' is a painting attributed to the Italian Renaissance artist Jacopo de' Barbari, dating to around 1500 and housed in the Capodimonte Museum, Naples, southern Italy. The painting portrays the Renaissance mathematicia ...

'', traditionally attributed to

Jacopo de' Barbari Jacopo de' Barbari, sometimes known or referred to as de'Barbari, de Barberi, de Barbari, Barbaro, Barberino, Barbarigo or Barberigo (c. 1460/70 – before 1516), was an Italian painter, printmaker and miniaturist with a highly individual style. H ...

, includes a glass rhombicuboctahedron half-filled with water, which may have been painted by

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially re ...

. The first printed version of the rhombicuboctahedron was by Leonardo and appeared in

Pacioli Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accountin ...

's ''

Divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da ...

'' (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 that are printed separately and cut with scissors while leaving some flaps for assembly with glue.

Objects

The

Freescape Freescape is a video game engine, an early 3D game engine used in video games such as 1987's '' Driller''. Graphics were composed mostly of solid geometry rendered without shading. History Developed in-house by Incentive Software, Freescape is c ...

games '' Driller'' and '' Dark Side'' both had a game map in the form of a rhombicuboctahedron. The "Hurry-Scurry Galaxy" and "Sea Slide Galaxy" in the videogame ''

Super Mario Galaxy is a 2007 platform game developed and published by Nintendo for the Wii. It is the third 3D game in the ''Super Mario'' series. As Mario, the player embarks on a quest to rescue Princess Peach, save the universe from Bowser, and collect ...

'' have planets in the similar shape of a rhombicuboctahedron. ''

Sonic the Hedgehog 3 is a 1994 platform game developed and published by Sega for the Sega Genesis, Genesis. Like previous ''Sonic the Hedgehog, Sonic'' games, players traverse side-scrolling levels while collecting Rings (Sonic the Hedgehog), rings and defeating en ...

s Icecap Zone features pillars topped with rhombicuboctahedra. During the

craze of the 1980s, at least two twisty puzzles sold had the form of a rhombicuboctahedron (the mechanism was similar to that of a

). File:Polyhedral sundial by Ludwig von Hohenfeld, with 17 different sudials for the region between Tubingen and Stuttgart, 1596, wood, paper, iron, brass - Landesmuseum Württemberg - Stuttgart, Germany - DSC03151.jpg, Sundial (1596) File:SchlossWesterholt14.jpg, Sundial File:Street lamp mainz, crop.jpg, Street lamp in

Mainz Mainz () is the capital and largest city of Rhineland-Palatinate, Germany. Mainz is on the left bank of the Rhine, opposite to the place that the Main joins the Rhine. Downstream of the confluence, the Rhine flows to the north-west, with Ma ...

File:18-sided dice from tomb of Dou Wan.jpg, Die with 18 labelled faces File:CabelaRhombicuboctahedronTarget.jpg,

Cabela's Cabela's Inc. is an American retailer that specializes in hunting, fishing, boating, camping, shooting and other outdoor recreation merchandise. The chain is based in Sidney, Nebraska. Cabela's was founded by Richard N. Cabela in 1961. Cabela' ...

shooting target File:Rubik's Snake 9.jpg, Rubik's Snake File:4x4 Dodecahedron solved cubemeister com.jpg, Rubik's Cube variation Pyrite-160710.jpg,

Pyrite The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Fe S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral. Pyrite's metallic luster and pale brass-yellow hue giv ...

crystal

Related polyhedra

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

Symmetry mutations

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.''n''.4), and continues as tilings of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. These

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...

figures have (*''n''32) reflectional symmetry.

Vertex arrangement

It shares its vertex arrangement with three nonconvex uniform polyhedra: the

stellated truncated hexahedron In geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices. It is represented by ...

, the

small rhombihexahedron In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram. Related polyhedra ...

(having the triangular faces and six square faces in common), and the

small cubicuboctahedron In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral. The small cubicuboctahedr ...

(having twelve square faces in common).

Rhombicuboctahedral graph

The rhombicuboctahedral graph is the graph of vertices and edges of the rhombicuboctahedron. It has 24 vertices and 48 edges, and is a quartic

Archimedean graph In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-verte ...

References

External links

* ** *
The Uniform Polyhedra
The Encyclopedia of Polyhedra

*'' ttp://demonstrations.wolfram.com/RhombicuboctahedronStar/ Rhombicuboctahedron Star' by Sándor Kabai,

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

. * ttp://www.hbmeyer.de/flechten/rhku/indexeng.htm Rhombicuboctahedron: paper strips for plaiting {{Polyhedron navigator Uniform polyhedra Archimedean solids