1.
Crossed square cupola
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In geometry, the crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, as in all cupolae, the base polygon has twice as many edges and vertices as the top, in this case the base polygon is an octagram. It may be seen as a cupola with a square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola. The crossed square cupola may be seen as a part of uniform polyhedra. Rotating one of the cupolae in this results in the pseudo-great rhombicuboctahedron. To this may be added the great rhombihexahedron, as the exclusive or of all three of these octagrammic prisms which may be used to construct the nonconvex great rhombicuboctahedron. The pictures below show the excavation of the prism with crossed square cupolae taking place one step at a time. The crossed square cupolae are always red, while the sides of the octagrammic prism are in the other colours. All images are oriented approximately the same way for clarity and this also occurs for the dual uniform polyhedra known as the great pentakis dodecahedron and medial inverted pentagonal hexecontahedron. Jim McNeill, Cupola OR Semicupola Jim McNeill, Relation of Cupolas to Uniform Polyhedra Paper model of this polyhedron by Robert Webb

2.
Great cubicuboctahedron
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In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It shares the vertex arrangement with the truncated cube and two other nonconvex uniform polyhedra. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron. List of uniform polyhedra Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Weisstein, Eric W

3.
Great rhombihexahedron
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In geometry, the great rhombihexahedron is a nonconvex uniform polyhedron, indexed as U21. Its dual is the great rhombihexacron and its vertex figure is a crossed quadrilateral. There is some controversy on how to colour the faces of this polyhedron, although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the neo filling is used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, and it shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron and it may be constructed as the exclusive or of three octagrammic prisms. The great rhombihexacron is a nonconvex isohedral polyhedron and it is the dual of the uniform great rhombihexahedron. It has 24 identical bow-tie-shaped faces,18 vertices, and 48 edges and it has 12 outer vertices which have the same vertex arrangement as the cuboctahedron, and 6 inner vertices with the vertex arrangement of an octahedron. As a surface geometry, it can be seen as similar to a Catalan solid. List of uniform polyhedra Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 uniform polyhedra and duals Weisstein, Eric W

4.
Nonconvex great rhombicuboctahedron
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In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It is represented by Schläfli symbol t0,2 and Coxeter-Dynkin diagram of and its vertex figure is a crossed quadrilateral. This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron, an alternate name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym, querco and it shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron, and it has the same vertex figure as the pseudo great rhombicuboctahedron, which is not a uniform polyhedron. The great deltoidal icositetrahedron is the dual of the nonconvex great rhombicuboctahedron, wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Weisstein, Eric W. Weisstein, Eric W. Uniform great rhombicuboctahedron

5.
Truncated cube
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In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces,36 edges, and 24 vertices, if the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + √2. The area A and the volume V of a cube of edge length a are. The truncated cube has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated cube can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ, where ξ = √2 −1. The parameter ξ can be varied between ±1, a value of 1 produces a cube,0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces. The truncated cube can be dissected into a cube, with six square cupola around each of the cubes faces. This dissection can also be seen within the cubic honeycomb, with cube, tetrahedron. This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupola and the central cube and this excavated cube has 16 triangles,12 squares, and 4 octagons. It shares the vertex arrangement with three nonconvex uniform polyhedra, The truncated cube is related to polyhedra and tlings in symmetry. The truncated cube is one of a family of uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, and Coxeter group symmetry, and a series of polyhedra and tilings n.8.8. A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces and it is one of a sequence of alternate truncations of polyhedra and tiling. It has 24 vertices and 36 edges, and is a cubic Archimedean graph, spinning truncated cube Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, cromwell, P. Polyhedra, CUP hbk, pbk. Ch.2 p. 79-86 Archimedean solids Eric W. Weisstein, Weisstein, Eric W. Truncated cubical graph. 3D convex uniform polyhedra o3x4x - tic

6.
Uniform star polyhedron
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In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting, each polyhedron can contain either star polygon faces, star polygon vertex figures or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra,5 quasiregular ones, there are also two infinite sets of uniform star prisms and uniform star antiprisms. The nonconvex forms are constructed from Schwarz triangles, all the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements. Regular polyhedra are labeled by their Schläfli symbol, other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U. Note, For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, for example an nonuniform cantellated form may have rectangles created in place of the edges rather than squares. There is one form, the tetrahemihexahedron which has tetrahedral symmetry. There are two Schwarz triangles that generate unique nonconvex uniform polyhedra, one triangle, and one general triangle. The general triangle generates the octahemioctahedron which is given further on with its octahedral symmetry. There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry, there are four Schwarz triangles that generate nonconvex forms, two right triangles, and, and two general triangles. There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry, some of the nonconvex snub forms have reflective vertex symmetry. Coxeter identified a number of star polyhedra by the Wythoff construction method. It is counted as a uniform polyhedron rather than a uniform polyhedron because of its double edges. Star polygon List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, a proof of the completeness on the list of elementary homogeneous polyhedra, Ukrainskiui Geometricheskiui Sbornik, 139–156, MR0326550 Skilling, J. The complete set of polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences,278, 111–135, doi,10. 1098/rsta.1975.0022, ISSN 0080-4614, JSTOR74475, MR0365333 HarEl, zvi Har’El, Kaleido software, Images, dual images Mäder, R. E. Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals