In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a snub is an operation applied to a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
. The term originates from
Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
's names of two
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s, for the
snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
() and
snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most ...
(). In general, snubs have
chiral symmetry
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, ...
with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an
expansion
Expansion may refer to:
Arts, entertainment and media
* ''L'Expansion'', a French monthly business magazine
* ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004
* ''Expansions'' (McCoy Tyner album), 1970
* ''Expansio ...
of a
regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
: 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
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, with a slightly different definition, for a wider set of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
s.
Conway snubs
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
explored generalized polyhedron operators, defining what is now called
Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
Conway and Hart extended the idea of using o ...
, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a ''semi-snub''.
In this notation,
snub
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub.
In Awards and Lists
For awards, the term "snub" ...
is defined by the dual and
gyro
Gyro may refer to:
Science and technology
* GYRO, a computer program for tokamak plasma simulation
* Gyro Motor Company, an American aircraft engine manufacturer
* ''Gyrodactylus salaris'', a parasite in salmon
* Gyroscope, an orientation-sta ...
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
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular face ...
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
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell.
There are 3 unique degrees of runcinations of the 24-cell including with permutations truncati ...
. It is instead actually an alternated
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24- ...
.
[Conway, 2008, p.401 Gosset's Semi-snub Polyoctahedron]
Coxeter's snubs, regular and quasiregular
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
's snub terminology is slightly different, meaning an
alternated truncation, deriving the
snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
as a ''snub
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
'', and the
snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most ...
as a ''snub
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
''. This definition is used in the naming of two
Johnson solids: the
snub disphenoid
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vert ...
and the
snub square antiprism, and of higher dimensional polytopes, such as the 4-dimensional
snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular face ...
, with extended Schläfli symbol s, and Coxeter diagram .
A
regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
(or tiling), with Schläfli symbol
, and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, has
truncation defined as
, and , and has snub defined as an
alternated truncation
, and . This alternated construction requires ''q'' to be even.
A
quasiregular polyhedron
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the se ...
, with Schläfli symbol
or ''r'', and Coxeter diagram or , has quasiregular
truncation defined as
or ''tr'', and or , and has quasiregular snub defined as an
alternated truncated rectification
or ''htr'' = ''sr'', and or .
For example, Kepler's
snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
is derived from the quasiregular
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, with a vertical
Schläfli symbol , and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol
, and Coxeter diagram . The snub cuboctahedron is the alternation of the ''truncated cuboctahedron'',
, and .
Regular polyhedra with even-order vertices can also be snubbed as alternated truncations, like the ''snub octahedron'', as
, , is the alternation of the
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
,
, and . The ''snub octahedron'' represents the
pseudoicosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, a regular
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
with
pyritohedral symmetry
image:tetrahedron.jpg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that c ...
.
The ''snub tetratetrahedron'', as
, and , is the alternation of the truncated tetrahedral symmetry form,
, and .
Coxeter's snub operation also allows n-
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
s to be defined as
or
, based on n-prisms
or
, while
is a regular n-
hosohedron
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune hav ...
, a degenerate polyhedron, but a valid tiling on the sphere with
digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
or
lune
Lune may refer to:
Rivers
*River Lune, in Lancashire and Cumbria, England
*River Lune, Durham, in County Durham, England
*Lune (Weser), a 43 km-long tributary of the Weser in Germany
* Lune River (Tasmania), in south-eastern Tasmania, Australia
P ...
-shaped faces.
The same process applies for snub tilings:
Examples
Nonuniform snub polyhedra
Nonuniform polyhedra with all even-valance vertices can be snubbed, including some infinite sets; for example:
Coxeter's uniform snub star-polyhedra
Snub star-polyhedra are constructed by their
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
(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 polychoron with
Schläfli symbol , and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, has a snub with
extended Schläfli symbol , and .
A rectified polychoron
= r, and has snub symbol
= sr, and .
Examples
There is only one uniform convex snub in 4-dimensions, the
snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular face ...
. The regular
24-cell has
Schläfli symbol,
, and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, and the snub 24-cell is represented by
,
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
. It also has an index 6 lower symmetry constructions as
or s and , and an index 3 subsymmetry as
or sr, and or .
The related
snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...
can be seen as a
or s, and , and lower symmetry
or sr and or , and lowest symmetry form as
or s and .
A Euclidean honeycomb is an
alternated hexagonal slab honeycomb, s, and or sr, and or sr, and .
:
Another Euclidean (scaliform) honeycomb is an
alternated square slab honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs are known:
* the familiar cubic honeycomb and 7 tr ...
, s, and or sr and :
:
The only uniform snub hyperbolic uniform honeycomb is the ''snub hexagonal tiling honeycomb'', as s and , which can also be constructed as an
alternated hexagonal tiling honeycomb, h, . It is also constructed as s and .
Another hyperbolic (scaliform) honeycomb is a
snub order-4 octahedral honeycomb, s, and .
See also
*
Snub polyhedron
In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub poly ...
References
*
*
Coxeter, H.S.M. ''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', (3rd edition, 1973), Dover edition, (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,
Googlebook
** (Paper 17)
Harold Scott MacDonald Coxeter, Coxeter, ''The Evolution of Coxeter–Dynkin diagrams'',
ieuw Archief voor Wiskunde 9 (1991) 233–248** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380–407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559–591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3–45*
Harold Scott MacDonald Coxeter, Coxeter, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (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
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008,
* {{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
Geometry
Snub tilings