Snub (geometry)

Short description: Geometric operation applied to a polyhedron

The two snubbed Archimedean solids
Snub cube or
Snub cuboctahedron  |
Snub dodecahedron or
Snub icosidodecahedron

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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 rhombicuboctahedron by rotating the 6 blue square faces until the 12 white square faces become pairs of equilateral triangle faces.

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 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.

## Contents

* 1 Conway snubs
* 2 Coxeter's snubs, regular and quasiregular
* 2.1 Examples
* 2.2 Nonuniform snub polyhedra
* 2.3 Coxeter's uniform snub star-polyhedra
* 2.4 Coxeter's higher-dimensional snubbed polytopes and honeycombs
* 2.5 Examples
* 3 See also
* 4 References

## 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  Forms to snub  | Polyhedra  | Euclidean tilings  | Hyperbolic tilings

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Names  | Tetrahedron | Cube or
octahedron | Icosahedron or
dodecahedron | Square tiling | Hexagonal tiling or
Triangular tiling | Heptagonal tiling or
Order-7 triangular tiling
Images  |  |  |  |  |  |
Snubbed form Conway
notation | sT  | sC = sO  | sI = sD  | sQ  | sH = sΔ | sΔ7
Image  |  |  |  |  |  | Error creating thumbnail: Unable to save thumbnail to destination

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 snubs, regular and quasiregular

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

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Name  | Cube  | Cuboctahedron
Rectified cube  | Truncated cuboctahedron
Cantitruncated cube  | Snub cuboctahedron
Snub rectified cube
Conway notation | C  | CO
rC  | tCO
trC or trO  | htCO = sCO
htrC = srC
Schläfli symbol  | {4,3}  | {43} or r{4,3}  | t{43} or tr{4,3}  | ht{43}=s{43}
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 {p,q}, and Coxeter diagram , has truncation defined as t{p,q}, and , and has snub defined as an alternated truncation ht{p,q}=s{p,q}, and . This alternated construction requires q to be even.

A quasiregular polyhedron, with Schläfli symbol {pq} or r{p,q}, and Coxeter diagram  or , has quasiregular truncation defined as t{pq} or tr{p,q}, and  or , and has quasiregular snub defined as an alternated truncated rectification ht{pq}=s{pq} 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 {43}, and Coxeter diagram , and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol s{43}, and Coxeter diagram . The snub cuboctahedron is the alternation of the truncated cuboctahedron, t{43}, and .

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

The snub tetratetrahedron, as s{33}, and , is the alternation of the truncated tetrahedral symmetry form, t{33}, and .

| Seed  | Truncated
t  | Alternated

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Name  | Octahedron  | Truncated octahedron  | Snub octahedron
Conway notation | O  | tO  | htO or sO
Schläfli symbol  | {3,4}  | t{3,4}  | ht{3,4} = s{3,4}
Coxeter diagram  |  |  |
Image  |  |  |

Coxeter's snub operation also allows n-antiprisms to be defined as s{2n} or s{2,2n}, based on n-prisms t{2n} or t{2,2n}, while {2,n} 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  |  |  |  |  |  |  |  |

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Coxeter
diagrams  |

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...  |

Schläfli
symbols | 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{22} | sr{2,3}
s{23} | sr{2,4}
s{24} | sr{2,5}
s{25} | sr{2,6}
s{26} | sr{2,7}
s{27} | sr{2,8}...
s{28}...  | sr{2,∞}
s{2∞}
Conway
notation  | A2 = T  | A3 = O  | A4  | A5  | A6  | A7  | A8...  | A∞

The same process applies for snub tilings:

Triangular tiling
Δ | Truncated triangular tiling
tΔ | Snub triangular tiling
htΔ = sΔ

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{3,6}  | t{3,6}  | ht{3,6} = s{3,6}
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### Examples

Snubs based on {p,4}  Space  | Spherical  | Euclidean  | Hyperbolic

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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

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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{23} | s{33} | s{43} | s{53} | s{63} | s{73} | s{83} | s{∞3}
Conway
notation  | A3  | sT  | sC or sO  | sD or sI  | sΗ or sΔ |  |  |
Quasiregular snubs based on r{p,4}  Space  | Spherical  | Euclidean  | Hyperbolic

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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{24} | s{34} | s{44} | s{54} | s{64} | s{74} | s{84} | s{∞4}
Conway
notation  | A4  | sC or sO  | sQ  |  |  |  |  |

### 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 bipyramid
Snub hexagonal bipyramid
Snub rectified bipyramids srdt{2,p}  Error creating thumbnail: Unable to save thumbnail to destination

Snub antiprisms s{2,2p}  Image  |  |  |  | ...

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Schläfli
symbols | ss{2,4} | ss{2,6} | ss{2,8} | ss{2,10}...
ssr{2,2}
ss{22} | ssr{2,3}
ss{23} | ssr{2,4}
ss{24} | ssr{2,5}...
ss{25}

### 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-polyhedra
s{3/2,3/2}

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

sr{5,5/2}

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

sr{5/2,3}

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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 polychoron with Schläfli symbol {p,q,r}, and Coxeter diagram , has a snub with extended Schläfli symbol s{p,q,r}, and .

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

### Examples

Orthogonal projection of snub 24-cell

There is only one uniform convex 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{333} or s{31,1,1} and , and an index 3 subsymmetry as s{33,4} 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{33,4,3} or sr{3,3,4,3} and  or , and lowest symmetry form as s{3333} 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 a snub order-4 octahedral honeycomb, s{3,4,4}, and .

## See also

* Snub polyhedron

## 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. Bibcode: 1954RSPTA.246..401C.
* 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
* Weisstein, Eric W.. "Snubification". http://mathworld.wolfram.com/Snubification.html.
* Richard Klitzing, Snubs, alternated facetings, and Stott–Coxeter–Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329–344, (2010) [3]

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