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The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles while the other twelve share an edge. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids. Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
In geometry, a Wythoff construction is a method for constructing a uniform polyhedron. ...
// List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ...
A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
Harold Scott MacDonald Donald Coxeter (February 9, 1907 - March 31, 2003) is regarded as one of the great geometers of the 20th century. ...
This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ...
The Bowers style acronym or pet names are a short notation for refering to uniform polyhedron devised by Jonathan Bowers. ...
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...
A deltoidal icositetrahedron (or trapezoidal icositetrahedron) is a catalan solid which looks a bit like an overinflated cube. ...
Image File history File links Small_rhombicuboctahedron_vertfig. ...
In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex as the sequence of faces around a vertex. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...
In plane geometry, a square is a polygon with four equal sides and equal angles. ...
This article is about the geometric shape. ...
The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...
A deltoidal icositetrahedron (or trapezoidal icositetrahedron) is a catalan solid which looks a bit like an overinflated cube. ...
In acrobatics, the trapeze is a certain acrobatic device that is shaped like a trapezoid. ...
 Image File history File links Rhombicuboctahedron_flat. ...
The name rhombicuboctahedron refers the fact that 12 of the square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ...
A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
Cartesian coordinates for a rhombicuboctahedron are all permutations of Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
- (±1, ±1, ±(1+√2))
There are three pairs of parallel planes that each intersect the rhombicuboctahedron through eight edges in the form of a regular octagon. The rhombicuboctahedron may divided along any of these two obtain an octagonal prism with regular faces and two additional polyhedra called square cupolae, which count among the Johnson solids. These can be reassembled to give a new solid called the pseudorhombicuboctahedron (or gyroelongated square bicupola) 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, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others. The square cupola with regular faces In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of triangles and rectangles. ...
The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ...
In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ...
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 rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ...
A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
The lines along which a Rubik's Cube can be turned are, projected onto a sphere, similar, topologically identical, to a rhombicuboctahedron's edges. In fact, variants using the Rubik's Cube mechanism have been produced which closely resemble the rhombicuboctahedron. Rubiks Cube in scrambled state. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
The rhombicuboctahedron is space filling in combination with cubes and tetrahedra.
Rhombicuboctahedra in the arts
 The first ever printed version of the rhombicuboctahedron, by Leonardo da Vinci as appeared in the Divina Proportione The polyhedron in the portrait of Luca Pacioli is a glass rhombicuboctahedron half-filled with water. Image File history File links Download high resolution version (1000x1063, 362 KB) Summary The first ever printed version NOT of the icosidodecahedron, by Leonardo da Vinci as apeared in the Divina Proportione by Luca Pacioli 1509, Venise Correction: This is a rhombicuboctahedron not an icosidodecahedron. ...
Image File history File links Download high resolution version (1000x1063, 362 KB) Summary The first ever printed version NOT of the icosidodecahedron, by Leonardo da Vinci as apeared in the Divina Proportione by Luca Pacioli 1509, Venise Correction: This is a rhombicuboctahedron not an icosidodecahedron. ...
Leonardo da Vinci ( Vinci, Italy, April 15, 1452 — May 2, 1519, Cloux, Amboise, France [1]) was an Italian Renaissance polymath: an architect, anatomist, sculptor, engineer, inventor, geometer, musician, and painter. ...
Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (controversial attribution: see the hereinafter external links). ...
See also Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ...
The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ...
In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ...
A Rubiks Snake (also Rubiks Twist) is a toy with twenty-four wedges shaped like triangular prisms. ...
External links - Archimedes and the Rhombicuboctahedron by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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