Snub (geometry)

The two snubbed Archimedean solids

Error creating thumbnail: Snub cube or Snub cuboctahedron

Error creating thumbnail: Snub dodecahedron or Snub icosidodecahedron

File:Snubcubes in grCO.svg

Two chiral copies of the snub cube, as alternated (red or green) vertices of the truncated cuboctahedron.

Error creating thumbnail:

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 (Template:Lang) and snub dodecahedron (Template:Lang).

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.

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
Names	Tetrahedron	Cube or octahedron	Icosahedron or dodecahedron	Square tiling	Hexagonal tiling or Triangular tiling	Heptagonal tiling or Order-7 triangular tiling
Images	File:Uniform polyhedron-33-t0.pngFile:Uniform polyhedron-33-t2.png	File:Uniform polyhedron-43-t0.svg	Error creating thumbnail: File:Uniform polyhedron-53-t2.svg	File:Uniform tiling 44-t0.svgFile:Uniform tiling 44-t2.svg	File:Uniform tiling 63-t0.svg	Error creating thumbnail:
Snubbed form Conway notation	sT	sC = sO	sI = sD	sQ	sH = sΔ	sΔ7
Image	File:Coxeter's snub octahedron from octahedron.gif				File:Uniform tiling 63-snub.svg	File:Snub triheptagonal tiling.svg

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.

Coxeter's snubs, regular and quasiregular

Snub cube, derived from cube or cuboctahedron

	Seed	Rectified r	Truncated t	Alternated h
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}	$\left\{\begin{array}{c}4\\ 3\end{array}\right\}$ or r{4,3}	$t\left\{\begin{array}{c}4\\ 3\end{array}\right\}$ or tr{4,3}	$ht\left\{\begin{array}{c}4\\ 3\end{array}\right\}=s\left\{\begin{array}{c}4\\ 3\end{array}\right\}$ htr{4,3} = sr{4,3}
Coxeter diagram	Template:CDD	Template:CDD or Template:CDD	Template:CDD or Template:CDD	Template:CDD or Template:CDD
Image	File:Uniform polyhedron-43-t0.svg	File:Uniform polyhedron-43-t1.svg	File:Uniform polyhedron-43-t012.png	Error creating thumbnail:

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 Template:CDD.

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

A quasiregular polyhedron, with Schläfli symbol $\left\{\begin{array}{c}p\\ q\end{array}\right\}$ or r{p,q}, and Coxeter diagram Template:CDD or Template:CDD, has quasiregular truncation defined as $t\left\{\begin{array}{c}p\\ q\end{array}\right\}$ or tr{p,q}, and Template:CDD or Template:CDD, and has quasiregular snub defined as an alternated truncated rectification $ht\left\{\begin{array}{c}p\\ q\end{array}\right\}=s\left\{\begin{array}{c}p\\ q\end{array}\right\}$ or htr{p,q} = sr{p,q}, and Template:CDD or Template:CDD.

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

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

The snub tetratetrahedron, as $s\left\{\begin{array}{c}3\\ 3\end{array}\right\}$, and Template:CDD, is the alternation of the truncated tetrahedral symmetry form, $t\left\{\begin{array}{c}3\\ 3\end{array}\right\}$, and Template:CDD.

	Seed	Truncated t	Alternated h
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	Template:CDD	Template:CDD	Template:CDD
Image	Error creating thumbnail:	File:Uniform polyhedron-43-t12.svg	Error creating thumbnail:

Coxeter's snub operation also allows n-antiprisms to be defined as $s\left\{\begin{array}{c}2\\ n\end{array}\right\}$ or $s\left\{\begin{array}{c}2,2n\end{array}\right\}$, based on n-prisms $t\left\{\begin{array}{c}2\\ n\end{array}\right\}$ or $t\left\{\begin{array}{c}2,2n\end{array}\right\}$, while $\left\{\begin{array}{c}2,n\end{array}\right\}$ 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	Error creating thumbnail:	File:Trigonal antiprism.png	File:Square antiprism.png	Error creating thumbnail:	File:Hexagonal antiprism.png	Error creating thumbnail:	Error creating thumbnail:	File:Infinite antiprism.svg
Coxeter diagrams	Template:CDD Template:CDD	Template:CDD Template:CDD	Template:CDD Template:CDD	Template:CDD Template:CDD	Template:CDD Template:CDD	Template:CDD Template:CDD	Template:CDD... Template:CDD...	Template:CDD Template:CDD
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} $s\left\{\begin{array}{c}2\\ 2\end{array}\right\}$	sr{2,3} $s\left\{\begin{array}{c}2\\ 3\end{array}\right\}$	sr{2,4} $s\left\{\begin{array}{c}2\\ 4\end{array}\right\}$	sr{2,5} $s\left\{\begin{array}{c}2\\ 5\end{array}\right\}$	sr{2,6} $s\left\{\begin{array}{c}2\\ 6\end{array}\right\}$	sr{2,7} $s\left\{\begin{array}{c}2\\ 7\end{array}\right\}$	sr{2,8}... $s\left\{\begin{array}{c}2\\ 8\end{array}\right\}$...	sr{2,∞} $s\left\{\begin{array}{c}2\\ \mathrm{\infty }\end{array}\right\}$
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
{3,6}	t{3,6}	ht{3,6} = s{3,6}
Template:CDD	Template:CDD	Template:CDD
	Error creating thumbnail:	File:Uniform tiling 63-h12.svg

Examples

Snubs based on {p,4}

Space	Spherical		Euclidean	Hyperbolic
Image	Error creating thumbnail:		File:Uniform tiling 44-h01.svg		File:Uniform tiling 64-h02.png	Error creating thumbnail:	Error creating thumbnail:	File:Uniform tiling i42-h01.png
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	...Template:CDD
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	File:Trigonal antiprism.png	File:Uniform polyhedron-33-s012.svg		Error creating thumbnail:	File:Uniform tiling 63-snub.svg	File:Snub triheptagonal tiling.svg	Error creating thumbnail:	Error creating thumbnail:
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	...Template:CDD
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}
	$s\left\{\begin{array}{c}2\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}3\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}4\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}5\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}6\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}7\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}8\\ 3\end{array}\right\}$	$s\left\{\begin{array}{c}\mathrm{\infty }\\ 3\end{array}\right\}$
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	File:Square antiprism.png	Error creating thumbnail:	Error creating thumbnail:	File:H2-5-4-snub.svg	File:Uniform tiling 64-snub.png	Error creating thumbnail:	File:Uniform tiling 84-snub.png	Error creating thumbnail:
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	...Template:CDD
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}
	$s\left\{\begin{array}{c}2\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}3\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}4\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}5\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}6\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}7\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}8\\ 4\end{array}\right\}$	$s\left\{\begin{array}{c}\mathrm{\infty }\\ 4\end{array}\right\}$
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}

File:Snub square bipyramid sequence.svg
Snub square bipyramid
File:Snub hexagonal bipyramid sequence.png
Snub hexagonal bipyramid

Snub rectified bipyramids srdt{2,p}

Snub antiprisms s{2,2p}

Image	File:Snub digonal antiprism.png	File:Snub triangular antiprism.png	File:Snub square antiprism colored.png	File:Snub pentagonal antiprism.png...
Schläfli symbols	ss{2,4}	ss{2,6}	ss{2,8}	ss{2,10}...
	ssr{2,2} $ss\left\{\begin{array}{c}2\\ 2\end{array}\right\}$	ssr{2,3} $ss\left\{\begin{array}{c}2\\ 3\end{array}\right\}$	ssr{2,4} $ss\left\{\begin{array}{c}2\\ 4\end{array}\right\}$	ssr{2,5}... $ss\left\{\begin{array}{c}2\\ 5\end{array}\right\}$

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

File:Retrosnub tetrahedron.png s{3/2,3/2} Template:CDD	File:Small snub icosicosidodecahedron.png s{(3,3,5/2)} Template:CDD	File:Snub dodecadodecahedron.png sr{5,5/2} Template:CDD	File:Snub icosidodecadodecahedron.png s{(3,5,5/3)} Template:CDD	File:Great snub icosidodecahedron.png sr{5/2,3} Template:CDD
File:Inverted snub dodecadodecahedron.png sr{5/3,5} Template:CDD	File:Great snub dodecicosidodecahedron.png s{(5/2,5/3,3)} Template:CDD	sr{5/3,3} Template:CDD	File:Small retrosnub icosicosidodecahedron.png s{(3/2,3/2,5/2)}	File:Great retrosnub icosidodecahedron.png s{3/2,5/3} Template:CDD

Coxeter's higher-dimensional snubbed polytopes and honeycombs

In general, a regular polychoron with Schläfli symbol $\left\{\begin{array}{c}p,q,r\end{array}\right\}$, and Coxeter diagram Template:CDD, has a snub with extended Schläfli symbol $s\left\{\begin{array}{c}p,q,r\end{array}\right\}$, and Template:CDD.

A rectified polychoron $\left\{\begin{array}{c}p\\ q,r\end{array}\right\}$ = r{p,q,r}, and Template:CDD has snub symbol $s\left\{\begin{array}{c}p\\ q,r\end{array}\right\}$ = sr{p,q,r}, and Template:CDD.

Examples

File:Ortho solid 969-uniform polychoron 343-snub.png

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, $\left\{\begin{array}{c}3,4,3\end{array}\right\}$, and Coxeter diagram Template:CDD, and the snub 24-cell is represented by $s\left\{\begin{array}{c}3,4,3\end{array}\right\}$, Coxeter diagram Template:CDD. It also has an index 6 lower symmetry constructions as $s\left\{\begin{array}{c}3\\ 3\\ 3\end{array}\right\}$ or s{3^1,1,1} and Template:CDD, and an index 3 subsymmetry as $s\left\{\begin{array}{c}3\\ 3,4\end{array}\right\}$ or sr{3,3,4}, and Template:CDD or Template:CDD.

The related snub 24-cell honeycomb can be seen as a $s\left\{\begin{array}{c}3,4,3,3\end{array}\right\}$ or s{3,4,3,3}, and Template:CDD, and lower symmetry $s\left\{\begin{array}{c}3\\ 3,4,3\end{array}\right\}$ or sr{3,3,4,3} and Template:CDD or Template:CDD, and lowest symmetry form as $s\left\{\begin{array}{c}3\\ 3\\ 3\\ 3\end{array}\right\}$ or s{3^1,1,1,1} and Template:CDD.

A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and Template:CDD or sr{2,3,6}, and Template:CDD or sr{2,3^[3]}, and Template:CDD.

Error creating thumbnail:

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

File:Alternated cubic slab honeycomb.png

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

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

See also

• Snub polyhedron

References

• Template:Cite journal
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:Isbn (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, Template:Isbn [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, Template:Isbn (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, Template:Isbn
• Template:Mathworld
• Richard Klitzing, Snubs, alternated facetings, and Stott–Coxeter–Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329–344, (2010) [3]