A full snub cubic antiprism or omnisnub cubic antiprism can be defined as an alternation of an truncated cuboctahedral prism, represented by ht0,1,2,3{4,3,2}, or 
, although it cannot be constructed as a uniform polychoron. It has 76 cells: 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedrons in the alternated gaps. There are 48 vertices, 192 edges, and 220 faces (12 squares, and 16+192 triangles). It has [4,3,2]+ symmetry, order 48.
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.
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.
In geometry, a tetrahedron, 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 ordinary convex polyhedra and the only one that has fewer than 5 faces.
A construction exists with two uniform snub cubes in snub positions with two edge lengths in a ratio of around 1 : 1.138.
Vertex figure for the omnisnub cubic antiprism
Also related is the bialternatosnub octahedral hosochoron, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has 40 cells: 2 rhombicuboctahedra (with Th symmetry), 6 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 8 octahedra (as triangular antiprisms), 24 triangular prisms (as Cs-symmetry wedges) filling the gaps, and 48 vertices. It has [4,(3,2)+] symmetry, order 48. Its vertex figure is a chiral hexahedron topologically identical to the tetragonal antiwedge.
In geometry, 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. 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.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.
Vertex figure for the bialternatosnub octahedral hosochoron
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.
In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are 2 (polygon) or higher.
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.
In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation 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 regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.
In geometry, a truncated 24-cell is a uniform 4-polytope formed as the truncation of the regular 24-cell.
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope, which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.
In four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation of the regular 24-cell.
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.
In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells It has 64 faces, and 96 edges and 48 vertices.
In geometry, a rhombicuboctahedral prism is a convex uniform polychoron.
In geometry, a snub cubic prism or snub cuboctahedral prism is a convex uniform polychoron.
In geometry, a truncated icosidodecahedral prism or great rhombicosidodecahedral prism is a convex uniform 4-polytope.
In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball, radius r1 and a line segment of length 2r2:
In the geometry of hyperbolic 3-space, the square tiling honeycomb, is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, has three square tilings, {4,4} around each edge, and 6 square tilings around each vertex in a cubic {4,3} vertex figure.
In geometry of 4 dimensions, a 4-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a square and an octagon.
In geometry of 4 dimensions, a 6-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a hexagon and an octagon.
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