Snub (geometry)

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.^[2]

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, 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.^[3]

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        .

Aregular polyhedron (or tiling), with Schläfli symbol ${\displaystyle {\begin{Bmatrix}p,q\end{Bmatrix}}}$ , and Coxeter diagram      , has truncation defined as ${\displaystyle t{\begin{Bmatrix}p,q\end{Bmatrix}}}$ , and      , and has snub defined as an alternated truncation ${\displaystyle ht{\begin{Bmatrix}p,q\end{Bmatrix}}=s{\begin{Bmatrix}p,q\end{Bmatrix}}}$ , and      . This alternated construction requires q to be even.

Aquasiregular polyhedron, with Schläfli symbol ${\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}$ orr{p,q}, and Coxeter diagram    or     , has quasiregular truncation defined as ${\displaystyle t{\begin{Bmatrix}p\\q\end{Bmatrix}}}$ ortr{p,q}, and    or     , and has quasiregular snub defined as an alternated truncated rectification ${\displaystyle ht{\begin{Bmatrix}p\\q\end{Bmatrix}}=s{\begin{Bmatrix}p\\q\end{Bmatrix}}}$ orhtr{p,q} = sr{p,q}, and    or     .

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

Regular polyhedra with even-order vertices can also be snubbed as alternated truncations, like the snub octahedron, as ${\displaystyle s{\begin{Bmatrix}3,4\end{Bmatrix}}}$ ,      , is the alternation of the truncated octahedron, ${\displaystyle t{\begin{Bmatrix}3,4\end{Bmatrix}}}$ , and      . The snub octahedron represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry.

The snub tetratetrahedron, as ${\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}$ , and    , is the alternation of the truncated tetrahedral symmetry form, ${\displaystyle t{\begin{Bmatrix}3\\3\end{Bmatrix}}}$ , and    .

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

The same process applies for snub tilings:

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

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

In general, a regular polychoron with Schläfli symbol ${\displaystyle {\begin{Bmatrix}p,q,r\end{Bmatrix}}}$ , and Coxeter diagram        , has a snub with extended Schläfli symbol ${\displaystyle s{\begin{Bmatrix}p,q,r\end{Bmatrix}}}$ , and        .

A rectified polychoron ${\displaystyle {\begin{Bmatrix}p\\q,r\end{Bmatrix}}}$  = r{p,q,r}, and         has snub symbol ${\displaystyle s{\begin{Bmatrix}p\\q,r\end{Bmatrix}}}$  = sr{p,q,r}, and        .

There is only one uniform convex snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol, ${\displaystyle {\begin{Bmatrix}3,4,3\end{Bmatrix}}}$ , and Coxeter diagram        , and the snub 24-cell is represented by ${\displaystyle s{\begin{Bmatrix}3,4,3\end{Bmatrix}}}$ , Coxeter diagram        . It also has an index 6 lower symmetry constructions as ${\displaystyle s\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}$  or s{3^1,1,1} and     , and an index 3 subsymmetry as ${\displaystyle s{\begin{Bmatrix}3\\3,4\end{Bmatrix}}}$  or sr{3,3,4}, and        or     .

The related snub 24-cell honeycomb can be seen as a ${\displaystyle s{\begin{Bmatrix}3,4,3,3\end{Bmatrix}}}$  or s{3,4,3,3}, and          , and lower symmetry ${\displaystyle s{\begin{Bmatrix}3\\3,4,3\end{Bmatrix}}}$  or sr{3,3,4,3} and          or       , and lowest symmetry form as ${\displaystyle s\left\{{\begin{array}{l}3\\3\\3\\3\end{array}}\right\}}$  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 a snub order-4 octahedral honeycomb, s{3,4,4}, and        .

• Snub polyhedron

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