Rhombicuboctahedron
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The first ever printed version of the rhombicuboctahedron, by Leonardo da Vinci as appeared in the Divina Proportione |
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.
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Rhombicuboctahedron_flat.png |
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.
Cartesian coordinates for a rhombicuboctahedron are all permutations of : (±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.
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 T
h symmetry, so they are invariant under the same rotations as the
tetrahedron but different reflections.
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.
The rhombicuboctahedron is space filling in combination with cubes and tetrahedra.
The polyhedron in the portrait of
Luca Pacioli is a glass rhombicuboctahedron half-filled with water.
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Spinning rhombicuboctahedron*
cube*
cuboctahedron*
octahedron*
rhombicosidodecahedron*
truncated cuboctahedron (great rhombicuboctahedron)
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elongated square gyrobicupola*
Rubik's Snake - puzzle that can form a Rhombicuboctahedron "ball"
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Archimedes and the Rhombicuboctahedron by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
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The Uniform Polyhedra*
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
