Snub (geometry)

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Short description: Geometric operation applied to a polyhedron
The two snubbed Archimedean solids
Uniform polyhedron-43-s012.png
Snub cube or
Snub cuboctahedron 		Uniform polyhedron-53-s012.png
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 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.

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
Names Tetrahedron Cube or
octahedron 		Icosahedron or
dodecahedron 		Square tiling Hexagonal tiling or
Triangular tiling 		Heptagonal tiling or
Order-7 triangular tiling
Images Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-43-t0.svgUniform polyhedron-43-t2.svg Uniform polyhedron-53-t0.svgUniform polyhedron-53-t2.svg Uniform tiling 44-t0.svgUniform tiling 44-t2.svg Uniform tiling 63-t0.svgUniform tiling 63-t2.svg Heptagonal tiling.svgOrder-7 triangular tiling.svg
Snubbed form Conway
notation 		sT sC = sO sI = sD sQ sH = sΔ sΔ7
Image Uniform polyhedron-33-s012.svg Uniform polyhedron-43-s012.png Uniform polyhedron-53-s012.png Uniform tiling 44-snub.svg Uniform tiling 63-snub.svg 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.[3]

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} [math]\displaystyle{ \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math] or r{4,3} [math]\displaystyle{ t \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math] or tr{4,3} [math]\displaystyle{ ht \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} = s \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math]
htr{4,3} = sr{4,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel split1-43.pngCDel nodes.png or CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel split1-43.pngCDel nodes 11.png or CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel split1-43.pngCDel nodes hh.png or CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Image Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png

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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.

A regular polyhedron (or tiling), with Schläfli symbol [math]\displaystyle{ \begin{Bmatrix} p , q \end{Bmatrix} }[/math], and Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png, has truncation defined as [math]\displaystyle{ t \begin{Bmatrix} p , q \end{Bmatrix} }[/math], and CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png, and has snub defined as an alternated truncation [math]\displaystyle{ ht \begin{Bmatrix} p , q \end{Bmatrix} = s \begin{Bmatrix} p , q \end{Bmatrix} }[/math], and CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.png. This alternated construction requires q to be even.

A quasiregular polyhedron, with Schläfli symbol [math]\displaystyle{ \begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] or r{p,q}, and Coxeter diagram CDel node 1.pngCDel split1-pq.pngCDel nodes.png or CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png, has quasiregular truncation defined as [math]\displaystyle{ t\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] or tr{p,q}, and CDel node 1.pngCDel split1-pq.pngCDel nodes 11.png or CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png, and has quasiregular snub defined as an alternated truncated rectification [math]\displaystyle{ ht\begin{Bmatrix} p \\ q \end{Bmatrix} = s\begin{Bmatrix} p \\ q \end{Bmatrix} }[/math] or htr{p,q} = sr{p,q}, and CDel node h.pngCDel split1-pq.pngCDel nodes hh.png or CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png.

For example, Kepler's snub cube is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol [math]\displaystyle{ \begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math], and Coxeter diagram CDel node 1.pngCDel split1-43.pngCDel nodes.png, and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol [math]\displaystyle{ s\begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math], and Coxeter diagram CDel node h.pngCDel split1-43.pngCDel nodes hh.png. The snub cuboctahedron is the alternation of the truncated cuboctahedron, [math]\displaystyle{ t\begin{Bmatrix} 4 \\ 3 \end{Bmatrix} }[/math], and CDel node 1.pngCDel split1-43.pngCDel nodes 11.png.

Regular polyhedra with even-order vertices can also be snubbed as alternated truncations, like the snub octahedron, as [math]\displaystyle{ s\begin{Bmatrix} 3 , 4 \end{Bmatrix} }[/math], CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png, is the alternation of the truncated octahedron, [math]\displaystyle{ t\begin{Bmatrix} 3 , 4 \end{Bmatrix} }[/math], and CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png. The snub octahedron represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry.

The snub tetratetrahedron, as [math]\displaystyle{ s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix} }[/math], and CDel node h.pngCDel split1.pngCDel nodes hh.png, is the alternation of the truncated tetrahedral symmetry form, [math]\displaystyle{ t\begin{Bmatrix} 3 \\ 3 \end{Bmatrix} }[/math], and CDel node 1.pngCDel split1.pngCDel nodes 11.png.

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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Image Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-h01.svg

Coxeter's snub operation also allows n-antiprisms to be defined as [math]\displaystyle{ s\begin{Bmatrix} 2 \\ n \end{Bmatrix} }[/math] or [math]\displaystyle{ s\begin{Bmatrix} 2 , 2n \end{Bmatrix} }[/math], based on n-prisms [math]\displaystyle{ t\begin{Bmatrix} 2 \\ n \end{Bmatrix} }[/math] or [math]\displaystyle{ t\begin{Bmatrix} 2 , 2n \end{Bmatrix} }[/math], while [math]\displaystyle{ \begin{Bmatrix} 2 , n \end{Bmatrix} }[/math] 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 Digonal antiprism.png Trigonal antiprism.png Square antiprism.png Pentagonal antiprism.png Hexagonal antiprism.png Antiprism 7.png Octagonal antiprism.png Infinite antiprism.svg
Coxeter
diagrams 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 10.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 12.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 14.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 7.pngCDel node h.png 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 16.pngCDel node.png...
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node h.png... 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node h.png
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}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 2 \end{Bmatrix} }[/math] 		sr{2,3}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 3 \end{Bmatrix} }[/math] 		sr{2,4}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 4 \end{Bmatrix} }[/math] 		sr{2,5}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 5 \end{Bmatrix} }[/math] 		sr{2,6}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 6 \end{Bmatrix} }[/math] 		sr{2,7}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 7 \end{Bmatrix} }[/math] 		sr{2,8}...
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ 8 \end{Bmatrix} }[/math]... 		sr{2,∞}
[math]\displaystyle{ s \begin{Bmatrix} 2 \\ \infin \end{Bmatrix} }[/math]
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}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
Uniform tiling 63-t2.svg Uniform tiling 63-t12.svg Uniform tiling 63-h12.png

Examples

Snubs based on {p,4}
Space Spherical Euclidean Hyperbolic
Image Digonal antiprism.png Uniform polyhedron-43-h01.svg Uniform tiling 44-h01.png Uniform tiling 542-h01.png Uniform tiling 64-h02.png Uniform tiling 74-h01.png Uniform tiling 84-h01.png Uniform tiling i42-h01.png
Coxeter
diagram 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node.png ...CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png
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 Trigonal antiprism.png Uniform polyhedron-33-s012.svg Uniform polyhedron-43-s012.png Uniform polyhedron-53-s012.png Uniform tiling 63-snub.svg Snub triheptagonal tiling.svg H2-8-3-snub.svg Uniform tiling i32-snub.png
Coxeter
diagram 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png ...CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
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}
[math]\displaystyle{ s\begin{Bmatrix} 2 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 3 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 4 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 5 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 6 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 7 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 8 \\3 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} \infin \\3 \end{Bmatrix} }[/math]
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 Square antiprism.png Uniform polyhedron-43-s012.png Uniform tiling 44-snub.svg H2-5-4-snub.svg Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
Coxeter
diagram 		CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png ...CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
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}
[math]\displaystyle{ s\begin{Bmatrix} 2 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 3 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 4 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 5 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 6 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 7 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} 8 \\4 \end{Bmatrix} }[/math] [math]\displaystyle{ s\begin{Bmatrix} \infin \\4 \end{Bmatrix} }[/math]
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 sequence.png
Snub square bipyramid
Snub hexagonal bipyramid sequence.png
Snub hexagonal bipyramid
Snub rectified bipyramids srdt{2,p}
Snub rectified hexagonal bipyramid sequence.png
Snub antiprisms s{2,2p}
Image Snub digonal antiprism.png Snub triangular antiprism.png Snub square antiprism colored.png Snub pentagonal antiprism.png...
Schläfli
symbols 		ss{2,4} ss{2,6} ss{2,8} ss{2,10}...
ssr{2,2}
[math]\displaystyle{ ss \begin{Bmatrix} 2 \\ 2 \end{Bmatrix} }[/math] 		ssr{2,3}
[math]\displaystyle{ ss \begin{Bmatrix} 2 \\ 3 \end{Bmatrix} }[/math] 		ssr{2,4}
[math]\displaystyle{ ss \begin{Bmatrix} 2 \\ 4 \end{Bmatrix} }[/math] 		ssr{2,5}...
[math]\displaystyle{ ss \begin{Bmatrix} 2 \\ 5 \end{Bmatrix} }[/math]

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
Retrosnub tetrahedron.png
s{3/2,3/2}
CDel node h.pngCDel 3x.pngCDel rat.pngCDel 2x.pngCDel node h.pngCDel 3x.pngCDel rat.pngCDel 2x.pngCDel node h.png 		Small snub icosicosidodecahedron.png
s{(3,3,5/2)}
CDel node h.pngCDel split1.pngCDel branch hh.pngCDel label5-2.png 		Snub dodecadodecahedron.png
sr{5,5/2}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 5-2.pngCDel node h.png 		Snub icosidodecadodecahedron.png
s{(3,5,5/3)}
CDel node h.pngCDel split1-53.pngCDel branch hh.pngCDel label5-3.png 		Great snub icosidodecahedron.png
sr{5/2,3}
CDel node h.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 3.pngCDel node h.png
Inverted snub dodecadodecahedron.png
sr{5/3,5}
CDel node h.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node h.pngCDel 5.pngCDel node h.png 		Great snub dodecicosidodecahedron.png
s{(5/2,5/3,3)}
CDel label5-3.pngCDel branch hh.pngCDel split2-p3.pngCDel node h.png 		Great inverted snub icosidodecahedron.png
sr{5/3,3}
CDel node h.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node h.pngCDel 3.pngCDel node h.png 		Small retrosnub icosicosidodecahedron.png
s{(3/2,3/2,5/2)} 		Great retrosnub icosidodecahedron.png
s{3/2,5/3}
CDel node h.pngCDel 3x.pngCDel rat.pngCDel 2x.pngCDel node h.pngCDel 5-3.pngCDel node h.png

Coxeter's higher-dimensional snubbed polytopes and honeycombs

In general, a regular polychoron with Schläfli symbol [math]\displaystyle{ \begin{Bmatrix} p , q, r \end{Bmatrix} }[/math], and Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png, has a snub with extended Schläfli symbol [math]\displaystyle{ s \begin{Bmatrix} p , q, r \end{Bmatrix} }[/math], and CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png.

A rectified polychoron [math]\displaystyle{ \begin{Bmatrix} p \\ q, r \end{Bmatrix} }[/math] = r{p,q,r}, and CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png has snub symbol [math]\displaystyle{ s\begin{Bmatrix} p \\ q , r \end{Bmatrix} }[/math] = sr{p,q,r}, and CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node.png.

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, [math]\displaystyle{ \begin{Bmatrix} 3 , 4, 3 \end{Bmatrix} }[/math], and Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, and the snub 24-cell is represented by [math]\displaystyle{ s\begin{Bmatrix} 3 , 4, 3 \end{Bmatrix} }[/math], Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png. It also has an index 6 lower symmetry constructions as [math]\displaystyle{ s\left\{\begin{array}{l}3\\3\\3\end{array}\right\} }[/math] or s{31,1,1} and CDel node h.pngCDel splitsplit1.pngCDel branch3 hh.pngCDel node h.png, and an index 3 subsymmetry as [math]\displaystyle{ s\begin{Bmatrix} 3 \\ 3 , 4 \end{Bmatrix} }[/math] or sr{3,3,4}, and CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel 4a.pngCDel nodea.png.

The related snub 24-cell honeycomb can be seen as a [math]\displaystyle{ s\begin{Bmatrix} 3 , 4, 3, 3 \end{Bmatrix} }[/math] or s{3,4,3,3}, and CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, and lower symmetry [math]\displaystyle{ s\begin{Bmatrix} 3 \\ 3 , 4, 3 \end{Bmatrix} }[/math] or sr{3,3,4,3} and CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png, and lowest symmetry form as [math]\displaystyle{ s\left\{\begin{array}{l}3\\3\\3\\3\end{array}\right\} }[/math] or s{31,1,1,1} and CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel split1.pngCDel nodes hh.png.

A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png or sr{2,3,6}, and CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png or sr{2,3[3]}, and CDel node h.pngCDel 2x.pngCDel node h.pngCDel split1.pngCDel branch hh.png.

Tetroctahedric semicheck.png

Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png or sr{2,41,1} and CDel node h.pngCDel 2x.pngCDel node h.pngCDel split1-44.pngCDel nodes hh.png:

Alternated cubic slab honeycomb.png

The only uniform snub hyperbolic uniform honeycomb is the snub hexagonal tiling honeycomb, as s{3,6,3} and CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png. It is also constructed as s{3[3,3]} and CDel branch hh.pngCDel splitcross.pngCDel branch hh.png.

Another hyperbolic (scaliform) honeycomb is a snub order-4 octahedral honeycomb, s{3,4,4}, and CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png.

See also


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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