# Snub (geometry)

 Snub cube or Snub cuboctahedron Snub dodecahedron or Snub icosidodecahedron
Two chiral copies of the snub cube, as alternated (red or green) vertices of the truncated cuboctahedron.
A snub cube can be constructed from a transformed rhombicuboctahedron by rotating the 6 blue square faces until the 12 white square become pairs of equilateral triangles.

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 (cubus simus) and snub dodecahedron (dodecaedron simum).[1] In general, snubs have chiral symmetry with two forms, with clockwise or counterclockwise orientations. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron, with the faces moved apart, and twists on their centers, adding new polygons centered on the original vertices, and 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.[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.

Snubbed regular figures
Form Polyhedra Euclidean Hyperbolic
Conway
notation
sT sC = sO sI = sD sQ sH = sΔ 7
Snubbed
polyhedra
Tetrahedron Cube or
octahedron
Icosahedron or
dodecahedron
Square tiling Hexagonal tiling or
Triangular tiling
Heptagonal tiling or
Order-7 triangular tiling
Image

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.[3]

## Coxeter's snubs, regular and quasiregular

Coxeter's snub terminology is slightly different, meaning an alternated truncation.

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

A quasiregular polyhedron $\begin{Bmatrix} p \\ q \end{Bmatrix}$ or r{p,q}, with Coxeter diagram or has a quasiregular truncation defined as $t\begin{Bmatrix} p \\ q \end{Bmatrix}$ or tr{p,q}, and Coxeter diagram or and quasiregular snub defined as an alternated truncated rectification $ht\begin{Bmatrix} p \\ q \end{Bmatrix} = s\begin{Bmatrix} p \\ q \end{Bmatrix}$ 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 $\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$, and Coxeter diagram , and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol $s\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$ and Coxeter diagram . The snub cuboctahedron is the alternation of the truncated cuboctahedron, $t\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$ and .

Snub cube, derived from cube or cuboctahedron
Seed Rectified
r
Truncated
t
Alternated
h

Cube
Cuboctahedron
Rectified cube
Truncated cuboctahedron
Cantitruncated cube
Snub cuboctahedron
Snub rectified cube
C CO
rC
tCO
trC
htCO = sCO
htrC = srC
{4,3} $\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$
r{4,3}
$t \begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$
tr{4,3}
$ht \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} = s \begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$
htr{4,3} = sr{4,3}

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

Seed Truncated
t
Alternated
h
Octahedron
O
Truncated octahedron
tO
Snub octahedron
htO or sO
{3,4} t{3,4} ht{3,4} = s{3,4}

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

 Image Coxeter Schläfli Conway diagrams symbols notation ... ... s{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} $s \begin{Bmatrix} 2 \\ 2 \end{Bmatrix}$ sr{2,3} $s \begin{Bmatrix} 2 \\ 3 \end{Bmatrix}$ sr{2,4} $s \begin{Bmatrix} 2 \\ 4 \end{Bmatrix}$ sr{2,5} $s \begin{Bmatrix} 2 \\ 5 \end{Bmatrix}$ sr{2,6} $s \begin{Bmatrix} 2 \\ 6 \end{Bmatrix}$ sr{2,7} $s \begin{Bmatrix} 2 \\ 7 \end{Bmatrix}$ sr{2,8}... $s \begin{Bmatrix} 2 \\ 8 \end{Bmatrix}$... sr{2,∞} $s \begin{Bmatrix} 2 \\ \infin \end{Bmatrix}$ A2 = T A3 = O A4 A5 A6 A7 A8... A∞

The same process applies for snub tilings:

Triangular tiling
Δ
Truncated triangular tiling
Snub triangular tiling
htΔ = sΔ
{3,6} t{3,6} ht{3,6} = s{3,6}

### Examples

Snubs based on {p,4}
Space Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
s{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}
Conway
notation
Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
sr{2,3} sr{3,3} sr{4,3} sr{5,3} sr{6,3} sr{7,3} sr{8,3} ...sr{∞,3}
$s\begin{Bmatrix} 2 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 3 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 4 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 5 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 6 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 7 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} 8 \\3 \end{Bmatrix}$ $s\begin{Bmatrix} \infin \\3 \end{Bmatrix}$
Conway
notation
A3 sT sC or sO sD or sI sΗ or sΔ
Quasiregular snubs based on r{p,4}
Space Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
sr{2,4} sr{3,4} sr{4,4} sr{5,4} sr{6,4} sr{7,4} sr{8,4} ...sr{∞,4}
$s\begin{Bmatrix} 2 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 3 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 4 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 5 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 6 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 7 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} 8 \\4 \end{Bmatrix}$ $s\begin{Bmatrix} \infin \\4 \end{Bmatrix}$
Conway
notation
A4 sC or sO sQ

### Nonuniform snubs polyhedra

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

 Image Schläfli symbols ... ss{2,4} ss{2,6} ss{2,8} ss{2,10}... ssr{2,2} $ss \begin{Bmatrix} 2 \\ 2 \end{Bmatrix}$ ssr{2,3} $ss \begin{Bmatrix} 2 \\ 3 \end{Bmatrix}$ ssr{2,4} $ss \begin{Bmatrix} 2 \\ 4 \end{Bmatrix}$ ssr{2,5}... $ss \begin{Bmatrix} 2 \\ 5 \end{Bmatrix}$

## 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.

 s{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)} s{3/2,5/3}

## Coxeter's higher-dimensional snubbed polytopes and honeycombs

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

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

### Examples

Orthogonal projection of snub 24-cell

There is only one uniform snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol, $\begin{Bmatrix} 3 , 4, 3 \end{Bmatrix}$, and Coxeter diagram , and the snub 24-cell is represented by $s\begin{Bmatrix} 3 , 4, 3 \end{Bmatrix}$, Coxeter diagram . It also has an index 6 lower symmetry constructions as $s\left\{\begin{array}{l}3\\3\\3\end{array}\right\}$ or s{31,1,1} and , and an index 3 subsymmetry as $s\begin{Bmatrix} 3 \\ 3 , 4 \end{Bmatrix}$ or sr{3,3,4}, and or .

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

Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}

## References

1. ^ Kepler, Harmonices Mundi, 1619
2. ^ Conway, (2008) p.287 Coxeter's semi-snub operation
3. ^ Conway, 2008, p.401 Gosset's Semi-snub Polyoctahedron
• Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences (The Royal Society) 246 (916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446.
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 154–156 8.6 Partial truncation, or alternation)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1], Googlebooks [2]
• (Paper 17) Coxeter, The Evolution of Coxeter–Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233–248]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• Richard Klitzing, Snubs, alternated facetings, and Stott–Coxeter–Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329–344, (2010) [3]