Snub (geometry)

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.

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 only even-sided faces.

In 4-dimensions, Conway suggests the snub 24-cell should be called a semi-snub 24-cell because it doesn't represent an alternated omnitruncated 24-cell like his 3-dimensional polyhedron usage. It is instead actually an alternated truncated 24-cell.

Coxeter's snubs, regular and quasiregular

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 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{3,4,3}.

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

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

For example, Kepler's snub cube is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol { 4 3 } , and Coxeter diagram , and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol s { 4 3 } and Coxeter diagram . The snub cuboctahedron is the alternation of the truncated cuboctahedron, t { 4 3 } and .

Regular polyhedra with even-order vertices to also be snubbed as alternated trunction, like a snub octahedron, s { 3 , 4 } , (and snub tetratetrahedron, as s { 3 3 } , ) represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry. The snub octahedron is the alternation of the truncated octahedron, t { 3 , 4 } and , or tetrahedral symmetry form: t { 3 3 } and .

Coxeter's snub operation also allows n-antiprisms to be defined as s { 2 n } or s { 2 , 2 n } , based on n-prisms t { 2 n } or t { 2 , 2 n } , while { 2 , n } is a regular n-hosohedron, a degenerate polyhedron, but a valid tiling on the sphere with digon or lune-shaped faces.

The same process applies for snub tilings:

Nonuniform snub polyhedra

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

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.

Coxeter's higher-dimensional snubbed polytopes and honeycombs

In general, a regular polychora with Schläfli symbol, { p , q , r } , and Coxeter diagram , has a snub with extended Schläfli symbol s { p , q , r } , and .

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

Examples

There is only one uniform snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol, { 3 , 4 , 3 } , and Coxeter diagram , and the snub 24-cell is represented by s { 3 , 4 , 3 } , Coxeter diagram . It also has an index 6 lower symmetry constructions as s { 3 3 3 } or s{3^1,1,1} and , and an index 3 subsymmetry as s { 3 3 , 4 } or sr{3,3,4}, and or .

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

Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and or sr{2,4^1,1} 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 an snub order-4 octahedral honeycomb, s{3,4,4}, and .