1.
Cubitruncated cuboctahedron
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In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. Its convex hull is a truncated cuboctahedron. Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of The tetradyakis hexahedron is a nonconvex isohedral polyhedron and it has 48 intersecting scalene triangle faces,72 edges, and 20 vertices. It is the dual of the uniform cubitruncated cuboctahedron, list of uniform polyhedra Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 p.92 Weisstein, Eric W. Cubitruncated cuboctahedron

2.
Omnitruncated polyhedron
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In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra and they have Wythoff symbol p q r | and vertex figures as 2p. 2q. 2r. More generally an omnitruncated polyhedron is an operator in Conway polyhedron notation. They can be seen as red faces of one regular polyhedron, yellow or green faces of the dual polyhedron, there are 5 nonconvex uniform omnitruncated polyhedra. There are 7 nonconvex forms with mixed Wythoff symbols p q | and they are not true omnitruncated polyhedra, the true omnitruncates have coinciding 2r-gonal faces that must be removed to form a proper polyhedron. All these polyhedra are one-sided, i. e. non-orientable, the p q r | degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols. Omnitruncations are also called cantitruncations or truncated rectifications, and Conways bevel operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra, Uniform polyhedron Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, Uniform polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446 Wenninger, the complete set of uniform 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

3.
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