1.
Snub (geometry)
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In geometry, a snub is an operation applied to a polyhedron. The term originates from Keplers names of two Archimedean solids, for the cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms, with clockwise or counterclockwise orientations, the terminology was generalized by Coxeter, with a slightly different definition, for a wider set of uniform polytopes. John Conway explored generalized polyhedron operators, defining what is now called Conway polyhedron notation, Conway calls Coxeters operation a semi-snub. In this notation, snub is defined by the dual and gyro operators, as s = dg, conways notation itself avoids Coxeters alternation operation since it only applies for polyhedra with only even-sided faces. In 4-dimensions, Conway suggests the snub 24-cell should be called a semi-snub 24-cell because it doesnt represent an alternated omnitruncated 24-cell like his 3-dimensional polyhedron usage and it is instead actually an alternated truncated 24-cell. Coxeters snub terminology is different, meaning an alternated truncation, deriving the snub cube as a snub cuboctahedron. This definition is used in the naming two Johnson solids, snub disphenoid, and snub square antiprism, as well as higher dimensional polytopes such as the 4-dimensional snub 24-cell, or s. A regular polyhedron with Schläfli symbol, and Coxeter diagram, has defined as t, and and snub defined as an alternated truncation h t = s. This construction requires q to be even, the snub cuboctahedron is the alternation of the truncated cuboctahedron, t and. Regular polyhedra with even-order vertices to also be snubbed as alternated trunction, like a snub octahedron, s, represents the pseudoicosahedron, the snub octahedron is the alternation of the truncated octahedron, t and, or tetrahedral symmetry form, t and. In general, a regular polychora with Schläfli symbol, and Coxeter diagram, has a snub with extended Schläfli symbol s, a rectified polychora = r, and has snub symbol s = sr, and. There is only one uniform snub in 4-dimensions, the snub 24-cell, the regular 24-cell has Schläfli symbol, and Coxeter diagram, and the snub 24-cell is represented by s, Coxeter diagram. It also has an index 6 lower symmetry constructions as s or s and, and an index 3 subsymmetry as s or sr, and or. The related snub 24-cell honeycomb can be seen as a s or s, and, and lower symmetry s or sr and or, a Euclidean honeycomb is an alternated hexagonal slab honeycomb, s, and or sr, and or sr, and. It is also constructed as s and, another hyperbolic honeycomb is an snub order-4 octahedral honeycomb, s, and. Snub polyhedron Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, philosophical Transactions of the Royal Society of London. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Kaleidoscopes, Selected Writings of H. S. M, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C

2.
Snub polyhedron
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A snub polyhedron is a polyhedron obtained by alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some but not all authors include antiprisms as snub polyhedra, as obtained by this construction from a degenerate polyhedron with only two faces. Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two forms which are reflections of each other. Their symmetry groups are all point groups, for example, the snub cube, Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3. p.3. q.3. r. Retrosnub polyhedra still have this form of Wythoff symbol, but their vertex configurations are instead /2, the tetrahedron, octahedron, icosahedron, and great icosahedron appear commonly in non-prismatic uniform 4-polytopes, but not in their snub constructions. Every snub polyhedron however can appear in the polyhedral prism based on them, when the Schwarz triangle of the snub polyhedron is isosceles, the snub polyhedron is not chiral. This is the case for the antiprisms, the icosahedron, great icosahedron, small snub icosicosidodecahedron, in the pictures of the snub derivation where green is not present, the faces derived from alternation are coloured red and yellow, while the snub triangles are blue. Where green is present, the derived from alternation are red, yellow. Notes, The icosahedron, snub cube and snub dodecahedron are the three convex ones. They are obtained by snubification of the octahedron, truncated cuboctahedron. The only snub polyhedron with the chiral octahedral group of symmetries is the snub cube, only the icosahedron and the great icosahedron are also regular polyhedra. There is also the set of antiprisms. They are formed from prisms, which are truncated hosohedra, degenerate regular polyhedra and those up to hexagonal are listed below. In the pictures showing the snub derivation, the derived from alternation are coloured red. Two Johnson solids are snub polyhedra, the snub disphenoid and the square antiprism. 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 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