Snub (geometry)

+ The two snubbed Archimedean solidsSnub cube orSnub cuboctahedronSnub dodecahedron orSnub icosidodecahedron

In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube () and snub dodecahedron ().Kepler, Harmonices Mundi, 1619

In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

The terminology was generalized by Coxeter, with a slightly different definition, for a wider set of uniform polytopes.

Conway snubs John Conway explored generalized polyhedron operators, defining what is now called Conway polyhedron notation, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a semi-snub.Conway, (2008) p.287 Coxeter's semi-snub operation

In this notation, snub is defined by the dual and gyro operators, as s = dg, and it is equivalent to an alternation of a truncation of an ambo operator. Conway's notation itself avoids Coxeter's alternation (half) operation since it only applies for polyhedra with even-sided faces.

+ Snubbed regular figuresForms to snubPolyhedraEuclidean tilingsHyperbolic tilingsNamesTetrahedronCube oroctahedronIcosahedron ordodecahedronSquare tilingHexagonal tiling orTriangular tilingHeptagonal tiling orOrder-7 triangular tilingImagesSnubbed form ConwaynotationsTsC = sOsI = sDsQsH = sΔsΔ7Image

In 4-dimensions, Conway suggests the snub 24-cell should be called a semi-snub 24-cell because, unlike 3-dimensional snub polyhedra are alternated omnitruncated forms, it is not an alternated omnitruncated 24-cell. It is instead actually an alternated truncated 24-cell.Conway, 2008, p.401 Gosset's Semi-snub Polyoctahedron

Coxeter's snubs, regular and quasiregular + Snub cube, derived from cube or cuboctahedronSeedRectifiedrTruncatedtAlternatedhNameCubeCuboctahedronRectified cubeTruncated cuboctahedronCantitruncated cubeSnub cuboctahedronSnub rectified cubeConway notationCCOrCtCOtrC or trOhtCO = sCOhtrC = srCSchläfli symbol{4,3} or r{4,3} or tr{4,3}htr{4,3} = sr{4,3}Coxeter diagram or or or Image

Coxeter's snub terminology is slightly different, meaning an alternated truncation, deriving the snub cube as a snub cuboctahedron, and the snub dodecahedron as a snub icosidodecahedron. This definition is used in the naming of two Johnson solids: the snub disphenoid and the snub square antiprism, and of higher dimensional polytopes, such as the 4-dimensional snub 24-cell, with extended Schläfli symbol s{3,4,3}, and Coxeter diagram .

A regular polyhedron (or tiling), with Schläfli symbol , and Coxeter diagram , has truncation defined as , and , and has snub defined as an alternated truncation , and . This alternated construction requires q to be even.

A quasiregular polyhedron, with Schläfli symbol or r{p,q}, and Coxeter diagram or , has quasiregular truncation defined as or tr{p,q}, and or , and has quasiregular snub defined as an alternated truncated rectification or htr{p,q} = sr{p,q}, and or .

For example, Kepler's snub cube is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol , and Coxeter diagram , and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol , and Coxeter diagram . The snub cuboctahedron is the alternation of the truncated cuboctahedron, , and .

Regular polyhedra with even-order vertices can also be snubbed as alternated truncations, like the snub octahedron, as , , is the alternation of the truncated octahedron, , and . The snub octahedron represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry.

The snub tetratetrahedron, as , and , is the alternation of the truncated tetrahedral symmetry form, , and .

SeedTruncatedtAlternatedhNameOctahedronTruncated octahedronSnub octahedronConway notationOtOhtO or sOSchläfli symbol{3,4}t{3,4}ht{3,4} = s{3,4}Coxeter diagramImage

Coxeter's snub operation also allows n-antiprisms to be defined as or , based on n-prisms or , while is a regular n-hosohedron, a degenerate polyhedron, but a valid tiling on the sphere with digon or lune-shaped faces. + Snub hosohedra, {2,2p}Image Coxeterdiagrams......Schläflisymbolss{2,4}s{2,6}s{2,8}s{2,10}s{2,12}s{2,14}s{2,16}...s{2,∞}sr{2,2}sr{2,3}sr{2,4}sr{2,5}sr{2,6}sr{2,7}sr{2,8}......sr{2,∞}ConwaynotationA2 = TA3 = OA4A5A6A7A8...A∞

The same process applies for snub tilings: Triangular tilingΔTruncated triangular tilingtΔSnub triangular tilinghtΔ = sΔ{3,6}t{3,6}ht{3,6} = s{3,6}

Examples

+ Snubs based on {p,4}SpaceSphericalEuclideanHyperbolicImageCoxeterdiagram...Schläflisymbols{2,4}s{3,4}s{4,4}s{5,4}s{6,4}s{7,4}s{8,4}...s{∞,4}

+ Quasiregular snubs based on r{p,3}ConwaynotationSphericalEuclideanHyperbolicImageCoxeterdiagram...Schläflisymbolsr{2,3}sr{3,3}sr{4,3}sr{5,3}sr{6,3}sr{7,3}sr{8,3}...sr{∞,3}ConwaynotationA3sTsC or sOsD or sIsΗ or sΔ

+ Quasiregular snubs based on r{p,4}SpaceSphericalEuclideanHyperbolicImageCoxeterdiagram...Schläflisymbolsr{2,4}sr{3,4}sr{4,4}sr{5,4}sr{6,4}sr{7,4}sr{8,4}...sr{∞,4}ConwaynotationA4sC or sOsQ

Nonuniform snub polyhedra Nonuniform polyhedra with all even-valance vertices can be snubbed, including some infinite sets; for example:

+ Snub bipyramids sdt{2,p}Snub square bipyramidSnub hexagonal bipyramid

+ Snub rectified bipyramids srdt{2,p}

+ Snub antiprisms s{2,2p}Image...Schläflisymbolsss{2,4}ss{2,6}ss{2,8}ss{2,10}...ssr{2,2}ssr{2,3}ssr{2,4}ssr{2,5}...

Coxeter's uniform snub star-polyhedra Snub star-polyhedra are constructed by their Schwarz triangle (p q r), with rational ordered mirror-angles, and all mirrors active and alternated.

+ Snubbed uniform star-polyhedrasr{3/2,3/2}s{(3,3,5/2)}sr{5,5/2}s{(3,5,5/3)}sr{5/2,3}sr{5/3,5}s{(5/2,5/3,3)}sr{5/3,3}s{(3/2,3/2,5/2)}sr{3/2,5/3}

The great dirhombicosidodecahedron is also a snub, but it relates to a spherical quadrilateral (3/2 5/3 3 5/2) instead.

Coxeter's higher-dimensional snubbed polytopes and honeycombs In general, a regular polychoron with Schläfli symbol , and Coxeter diagram , has a snub with extended Schläfli symbol , and .

A rectified polychoron = r{p,q,r}, and has snub symbol = sr{p,q,r}, and .

Examples

There is only one uniform convex 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 , Coxeter diagram . It also has an index 6 lower symmetry constructions as or s{31,1,1} and , and an index 3 subsymmetry as or sr{3,3,4}, and or .

The related snub 24-cell honeycomb can be seen as a or s{3,4,3,3}, and , and lower symmetry or sr{3,3,4,3} and or , and lowest symmetry form as or s{31,1,1,1} and .

A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and or sr{2,3,6}, and or sr{2,3}, and .

Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and or sr{2,41,1} and :

The only uniform snub hyperbolic uniform honeycomb is the snub hexagonal tiling honeycomb, as s{3,6,3} and , which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, . It is also constructed as s{3[3,3]} and .

Another hyperbolic (scaliform) honeycomb is a snub order-4 octahedral honeycomb, s{3,4,4}, and .

Snub (geometry)

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