1. Conway snubs

2. Coxeter's snubs, regular and quasiregular
     Examples   Nonuniform snub polyhedra    Coxeter's uniform snub star-polyhedra    Coxeter's higher-dimensional snubbed polytopes and honeycombs    Examples 

3. See also

4. References

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

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.^[3]

Coxeter's snubs, regular and quasiregular

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 or trO	htCO = sCO htrC = srC
{4,3}	or r{4,3}	or tr{4,3}	htr{4,3} = sr{4,3}
node_1|4|node|3|node}}	node_1|split1-43|nodes}} or {{CDD|node|4|node_1|3|node}}	node_1|split1-43|nodes_11}} or {{CDD|node_1|4|node_1|3|node_1}}	node_h|split1-43|nodes_hh}} or {{CDD|node_h|4|node_h|3|node_h}}

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, {{CDD|node_h|3|node_h|4|node|3|node}} or s{3,4,3}.

A regular polyhedron (or tiling) with Schläfli symbol, , and Coxeter diagram {{CDD|node_1|p|node|q|node}}, has truncation defined as , and {{CDD|node_1|p|node_1|q|node}} and snub defined as an alternated truncation , and Coxeter diagram {{CDD|node_h|p|node_h|q|node}}. This construction requires q to be even.

A quasiregular polyhedron or r{p,q}, with Coxeter diagram {{CDD|node_1|split1-pq|nodes}} or {{CDD|node|p|node_1|q|node}} has a quasiregular truncation defined as or tr{p,q}, and Coxeter diagram {{CDD|node_1|split1-pq|nodes_11}} or {{CDD|node_1|p|node_1|q|node_1}} and quasiregular snub defined as an alternated truncated rectification or htr{p,q} = sr{p,q}, and Coxeter diagram {{CDD|node_h|split1-pq|nodes_hh}} or {{CDD|node_h|p|node_h|q|node_h}}.

For example, Kepler's snub cube is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol , and Coxeter diagram {{CDD|node_1|split1-43|nodes}}, and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol and Coxeter diagram {{CDD|node_h|split1-43|nodes_hh}}. The snub cuboctahedron is the alternation of the truncated cuboctahedron, and {{CDD|node_1|split1-43|nodes_11}}.

Regular polyhedra with even-order vertices to also be snubbed as alternated trunction, like a snub octahedron, , {{CDD|node_h|3|node_h|4|node}} (and snub tetratetrahedron, as , {{CDD|node_h|split1|nodes_hh}}) represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry. The snub octahedron is the alternation of the truncated octahedron, and {{CDD|node_1|3|node_1|4|node}}, or tetrahedral symmetry form: and {{CDD|node_1|split1|nodes_11}}.

Coxeter's snub operation also allows n-antiprisms to be defined as or , based on n-prisms or , while 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
Coxeter diagrams	node_h|2x|node_h|4|node}} {{CDD|node_h|2x|node_h|2x|node_h}}	node_h|2x|node_h|6|node}} {{CDD|node_h|2x|node_h|3|node_h}}	node_h|2x|node_h|8|node}} {{CDD|node_h|2x|node_h|4|node_h}}	node_h|2x|node_h|10|node}} {{CDD|node_h|2x|node_h|5|node_h}}	node_h|2x|node_h|12|node}} {{CDD|node_h|2x|node_h|6|node_h}}	node_h|2x|node_h|14|node}} {{CDD|node_h|2x|node_h|7|node_h}}	node_h|2x|node_h|16|node}}... {{CDD|node_h|2x|node_h|8|node_h}}...	node_h|2x|node_h|infin|node}} {{CDD|node_h|2x|node_h|infin|node_h}}
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}	sr{2,3}	sr{2,4}	sr{2,5}	sr{2,6}	sr{2,7}	sr{2,8}... ...	sr{2,∞}
Conway notation	A2 = T	A3 = O	A4	A5	A6	A7	A8...	A∞

The same process applies for snub tilings:

Examples

Quasiregular snubs based on r{p,3}

Conway notation	Spherical				Euclidean	Hyperbolic
Image
Coxeter diagram	node_h|2x|node_h|3|node_h}}	node_h|3|node_h|3|node_h}}	node_h|4|node_h|3|node_h}}	node_h|5|node_h|3|node_h}}	node_h|6|node_h|3|node_h}}	node_h|7|node_h|3|node_h}}	node_h|8|node_h|3|node_h}}	node_h|infin|node_h|3|node_h}}
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}
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	node_h|2x|node_h|4|node_h}}	node_h|3|node_h|4|node_h}}	node_h|4|node_h|4|node_h}}	node_h|5|node_h|4|node_h}}	node_h|6|node_h|4|node_h}}	node_h|7|node_h|4|node_h}}	node_h|8|node_h|4|node_h}}	node_h|infin|node_h|4|node_h}}
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}
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 antiprisms s{2,2p}

Image	...
Schläfli symbols	ss{2,4}	ss{2,6}	ss{2,8}	ss{2,10}...
	ssr{2,2}	ssr{2,3}	ssr{2,4}	ssr{2,5}...

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, , and Coxeter diagram {{CDD|node_1|p|node|q|node|r|node}}, has a snub with extended Schläfli symbol , and {{CDD|node_h|p|node_h|q|node|r|node}} .

A rectified polychora = r{p,q,r}, and {{CDD|node|p|node_1|q|node|r|node}} has snub symbol = sr{p,q,r}, and {{CDD|node_h|p|node_h|q|node_h|r|node}}.

Examples

There is only one uniform snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol, , and Coxeter diagram {{CDD|node_1|3|node|4|node|3|node}}, and the snub 24-cell is represented by , Coxeter diagram {{CDD|node_h|3|node_h|4|node|3|node}}. It also has an index 6 lower symmetry constructions as or s{3^1,1,1} and {{CDD|node_h|splitsplit1|branch3_hh|node_h}}, and an index 3 subsymmetry as or sr{3,3,4}, and {{CDD|node_h|3|node_h|3|node_h|4|node}} or {{CDD|node_h|split1|nodes_hh|4a|nodea}}.

The related snub 24-cell honeycomb can be seen as a or s{3,4,3,3}, and {{CDD|node_h|3|node_h|4|node|3|node|3|node}}, and lower symmetry or sr{3,3,4,3} and {{CDD|node_h|3|node_h|3|node_h|4|node|3|node}} or {{CDD|node_h|split1|nodes_hh|3a|nodea|4a|nodea}}, and lowest symmetry form as or s{3^1,1,1,1} and {{CDD|nodes_hh|split2|node_h|split1|nodes_hh}}.

A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and {{CDD|node_h|2x|node_h|6|node|3|node}} or sr{2,3,6}, and {{CDD|node_h|2x|node_h|3|node_h|6|node}} or sr{2,3^[3]}, and {{CDD|node_h|2x|node_h|split1|branch_hh}}.

Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and {{CDD|node_h|2x|node_h|4|node|4|node}} or sr{2,4^1,1} and {{CDD|node_h|2x|node_h|split1-44|nodes_hh}}:

The only uniform snub hyperbolic uniform honeycomb is the snub hexagonal tiling honeycomb, as s{3,6,3} and {{CDD|node_h|3|node_h|6|node|3|node}}, which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, {{CDD|node_h1|6|node|3|node|3|node}}. It is also constructed as s{3^[3,3]} and {{CDD|branch_hh|splitcross|branch_hh}}.

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

See also

• Snub polyhedron

References

• {{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 |issue=916 | pages=401–450 | publisher=The Royal Society | doi=10.1098/rsta.1954.0003 | ref=harv| 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}}  , Googlebooks [https://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PP1&dq=Kaleidoscopes%20Coxeter&pg=PP1#v=onepage&q=&f=false]
  • (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}}
• {{mathworld | urlname = Snubification | title = Snubification}}
• Richard Klitzing, Snubs, alternated facetings, and Stott–Coxeter–Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329–344, (2010)  

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