Operation Snub

Two flat-nosed Archimedean bodies
Flat-nosed cube or plane-nosed cuboctahedron	Flat nosed dodecahedron or nasal icosododecahedron

Two chiral copies of a flat-nosed cube as an alternation of (red and green) vertices of a truncated cuboctahedron.

A flat-nosed cube can be constructed by transforming the rhombocuboctahedron by rotating 6 blue square faces until 12 white squares become pairs of equilateral triangles.

The snub operation or clipping vertices is an operation applied to polyhedra. The term appeared from the names given by Kepler to two Archimedean bodies - a flat - nosed cube (cubus simus) and a flat-nosed dodecahedron (dodecaedron simum) ^[1] . In the general case, flat-nosed forms have a chiral symmetry of two types, with clockwise and counterclockwise orientations. According to Kepler’s names, cutting off vertices can be considered as stretching a regular polyhedron, when the original faces move away from the center and rotate relative to the centers, polygons with centers at these vertices are added instead of the original vertices, and pairs of triangles fill the space between the original edges.

Coxeter generalized the terminology with a slightly different definition for a wider variety of .

Operation Conway's snub

John Conway investigated generalized operations on polyhedra, defining what is now called the Conway notation for polyhedra , which can be applied to polyhedra and mosaics. Conway called the Coxeter operation semi-snub (semi-snub) ^[2] .

In this notation, snub is defined as a composition of dual and gyro operators, $s=dg$ ${\ displaystyle s = dg}$ $s=dg$ , and this is equivalent to a sequence of , truncation, and ambo operators. Conway's notation avoids the alternation operation, since it only applies to polyhedra with faces that have an even number of sides.

Flat nosed regular figures
	Polyhedra			Euclidean Mosaics		Hyperbolic mosaics
Notation Conway	sT	sC = sO	sI = sD	sQ	sH = sΔ	sΔ ₇
Flat-nosed polyhedron	Tetrahedron	Cube or Octahedron	Icosahedron or Dodecahedron	Square mosaic	Hexagonal mosaic or Triangular mosaic	Heptagonal mosaic or
Flat-nosed polyhedron
Picture

In 4-dimensional spaces, Conway believes that a should be called a half-plane 24-cell , since it does not represent an alternate , as its counterpart in 3-dimensional space. Instead, it is an alternate, ^[3] .

Coxeter snub operations, right and quasi-correct

A flat-nosed cube derived from a cube or cuboctahedron
Source body	Fully truncated polyhedron r	Truncated polyhedron t	h
Cube	Cuboctahedron Fully truncated cube	Truncated cuboctahedron Beveled truncated cube	Plane nosed cuboctahedron Flat-nosed full-truncated cube
C	CO rC	tCO trC or trO	htCO = sCO htrC = srC
{4.3}	${\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ or r {4,3}	$t{\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ or tr {4,3}	$ht{\begin{Bmatrix}4\\3\end{Bmatrix}}=s{\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle ht {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}} = s {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ htr {4.3} = sr {4.3}
	or	or	or

The terminology of “Coxeter's snub” (cutting off the vertices) is somewhat different and means truncation , according to which the plane-nosed cube is obtained by the operation snub (cutting off the vertices) from the cuboctahedron , and the plane-nosed dodecahedron is obtained from the icosododecahedron . This definition is used in the names of Johnson’s two bodies - the flat-faced duclinoid and the flat- nosed square antiprism , as well as in the names of higher-dimensional polyhedra, such as the 4-dimensional , or s {3,4,3}.

The correct polyhedron (or mosaic) with the Shlefly symbol, ${\begin{Bmatrix}p,q\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} p, q \ end {Bmatrix}}}$ and Coxeter chart has a truncation defined as $t{\begin{Bmatrix}p,q\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} p, q \ end {Bmatrix}}}$ with chart , and a nasal shape defined as an truncation $ht{\begin{Bmatrix}p,q\end{Bmatrix}}=s{\begin{Bmatrix}p,q\end{Bmatrix}}$ ${\ displaystyle ht {\ begin {Bmatrix} p, q \ end {Bmatrix}} = s {\ begin {Bmatrix} p, q \ end {Bmatrix}}}$ with Coxeter chart . This construction requires q to be even.

Quasiregular Polyhedron ${\begin{Bmatrix}p\\q\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} p \\ q \ end {Bmatrix}}}$ or r { p , q }, with a Coxeter diagram or has a quasi-correct truncation defined as $t{\begin{Bmatrix}p\\q\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} p \\ q \ end {Bmatrix}}}$ or tr { p , q } (with Coxeter diagram or ) and a quasiregular plane-nosed shape defined as an truncation of full truncation $ht{\begin{Bmatrix}p\\q\end{Bmatrix}}=s{\begin{Bmatrix}p\\q\end{Bmatrix}}$ ${\ displaystyle ht {\ begin {Bmatrix} p \\ q \ end {Bmatrix}} = s {\ begin {Bmatrix} p \\ q \ end {Bmatrix}}}$ or htr { p , q } = sr { p , q } (with Coxeter diagram or )

For example, a Kepler's flat-nosed cube is obtained from a quasiregular cuboctahedron with a vertical Shlefli symbol ${\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ (and Coxeter chart ) and more precisely called the plane-nosed cuboctahedron , which is expressed by the Shlefli symbol $s{\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ (with Coxeter chart ) The plane-nosed cuboctahedron is an alternative to the truncated cuboctahedron $t{\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$ ( )

Regular polyhedra with an even order of vertices can also be reduced to a plane-nosed shape as an alternate truncation, similar to a plane-nosed octahedron $s{\begin{Bmatrix}3,4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3.4 \ end {Bmatrix}}}$ ( ) (and the flat-nosed tetrahedron $s{\begin{Bmatrix}3\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3 \\ 3 \ end {Bmatrix}}}$ , ) represents a pseudo- icosahedron , a regular icosahedron with pyritohedral symmetry . The flat-nosed octahedron is an alternate form of a truncated octahedron , $t{\begin{Bmatrix}3,4\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 3.4 \ end {Bmatrix}}}$ ( ), or in the form of tetrahedral symmetry: $t{\begin{Bmatrix}3\\3\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 3 \\ 3 \ end {Bmatrix}}}$ and .

	Truncated t	Alternate h
Octahedron O	Truncated octahedron tO	Ploskosa octahedron htO or sO
{3,4}	t {3,4}	ht {3,4} = s {3,4}

The operation of cutting off the vertices (noses) of Coxeter also makes it possible to determine the n- antiprism as $s{\begin{Bmatrix}2\\n\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ n \ end {Bmatrix}}}$ or $s{\begin{Bmatrix}2,2n\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2,2n \ end {Bmatrix}}}$ based on n-prisms $t{\begin{Bmatrix}2\\n\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 2 \\ n \ end {Bmatrix}}}$ or $t{\begin{Bmatrix}2,2n\end{Bmatrix}}$ ${\ displaystyle t {\ begin {Bmatrix} 2,2n \ end {Bmatrix}}}$ , but ${\begin{Bmatrix}2,n\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} 2, n \ end {Bmatrix}}}$ is a regular osohedron , a degenerate polyhedron, which is an admissible mosaic on a sphere with two-sided or moon-like faces.

Flat Nose Osohedra, {2,2p}
Picture
Charts Coxeter							... ...
Symbol Loafs	s {2,4}	s {2,6}	s {2,8}	s {2,10}	s {2,12}		...
Symbol Loafs	sr {2,2} $s{\begin{Bmatrix}2\\2\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 2 \ end {Bmatrix}}}$	sr {2,3} $s{\begin{Bmatrix}2\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 3 \ end {Bmatrix}}}$	sr {2,4} $s{\begin{Bmatrix}2\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 4 \ end {Bmatrix}}}$	sr {2,5} $s{\begin{Bmatrix}2\\5\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 5 \ end {Bmatrix}}}$	sr {2,6} $s{\begin{Bmatrix}2\\6\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 6 \ end {Bmatrix}}}$	sr {2.7} $s{\begin{Bmatrix}2\\7\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 7 \ end {Bmatrix}}}$	sr {2,8} ... $s{\begin{Bmatrix}2\\8\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 8 \ end {Bmatrix}}}$ ...	sr {2, ∞} $s{\begin{Bmatrix}2\\\infty \end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\\ infty \ end {Bmatrix}}}$
Notation Conway	A2 = T	A3 = O	A4	A5	A6	A7	A8 ...	A∞

The same process applies to flat-nosed mosaics:

Triangular mosaic Δ	Truncated Triangular Mosaic tΔ	Flat nosed triangular mosaic htΔ = sΔ
{3,6}	t {3,6}	ht {3,6} = s {3,6}

Examples

Flat-faced figures on {p, 4}
Space	Spherical		Euclidean
Picture
Diagram Coxeter				...
Symbol Loafs	s {2,4}	s {3,4}	s {4,4}	...

Quasi-regular plane-nosed figures based on r {p, 3}
Space	Spherical				Euclidean	Hyperbolic
Picture
Diagram Coxetere								...
Symbol Loafs	sr {2,3}	sr {3,3}	sr {4,3}	sr {5,3}	sr {6,3}			...
Symbol Loafs	$s{\begin{Bmatrix}2\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}3\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}4\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 4 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}5\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 5 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}6\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 6 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}7\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 7 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}8\\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 8 \\ 3 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} \ infty \\ 3 \ end {Bmatrix}}}$
Notation Conway	A3	sT	sC or sO	sD or sI	sΗ or sΔ

Quasiregular planos based on r {p, 4}
Space	Spherical		Euclidean	Hyperbolic
Picture
Diagram Coxeter								...
Symbol Loafs	sr {2,4}	sr {3,4}	sr {4,4}					...
Symbol Loafs	$s{\begin{Bmatrix}2\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 2 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}3\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}4\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 4 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}5\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 5 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}6\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 6 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}7\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 7 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}8\\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 8 \\ 4 \ end {Bmatrix}}}$	$s{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} \ infty \\ 4 \ end {Bmatrix}}}$
Notation Conway	A4	sC or sO	sQ

Inhomogeneous flat-bottomed polyhedrons

In heterogeneous polyhedra, for which an even number of edges converge to the vertices, vertices can be cut off, including some infinite collections, for example:

Flat-nosed bipyramids sdt {2, p}
Flat Nose Square Bipyramid


Flat Hexagonal Bipyramid

Plane full truncated bipyramids srdt {2, p}

Flat Nose Antiprisms {2.2p}
Picture				...
Symbol Loafs	ss {2,4}	ss {2,6}	ss {2,8}	ss {2,10} ...
Symbol Loafs	ssr {2,2} $ss{\begin{Bmatrix}2\\2\end{Bmatrix}}$ ${\ displaystyle ss {\ begin {Bmatrix} 2 \\ 2 \ end {Bmatrix}}}$	ssr {2,3} $ss{\begin{Bmatrix}2\\3\end{Bmatrix}}$ ${\ displaystyle ss {\ begin {Bmatrix} 2 \\ 3 \ end {Bmatrix}}}$	ssr {2,4} $ss{\begin{Bmatrix}2\\4\end{Bmatrix}}$ ${\ displaystyle ss {\ begin {Bmatrix} 2 \\ 4 \ end {Bmatrix}}}$	ssr {2,5} ... $ss{\begin{Bmatrix}2\\5\end{Bmatrix}}$ ${\ displaystyle ss {\ begin {Bmatrix} 2 \\ 5 \ end {Bmatrix}}}$

Homogeneous flat-stellated stellate polyhedrons of Coxeter

Plane-star stellated polyhedra are constructed according to the Schwartz triangle (pqr) with rational mirrors, in which all the mirrors are active and alternate.

Flat nosed homogeneous stellate polyhedrons
s {3 / 2,3 / 2}

Flat-faced polyhedra and Coxeter honeycombs in high-dimensional spaces

In the general case, regular 4-dimensional polyhedra with the Shlefli symbol , ${\begin{Bmatrix}p,q,r\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} p, q, r \ end {Bmatrix}}}$ and Coxeter chart has a flat-nosed shape with an extended Shlefly symbol $s{\begin{Bmatrix}p,q,r\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} p, q, r \ end {Bmatrix}}}$ and chart .

Fully truncated polyhedron ${\begin{Bmatrix}p\\q,r\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} p \\ q, r \ end {Bmatrix}}}$ = r {p, q, r} , and has snub symbol $s{\begin{Bmatrix}p\\q,r\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} p \\ q, r \ end {Bmatrix}}}$ = sr {p, q, r} , and .

Examples

Orthogonal projection of the

There is only one homogeneous plane-nosed polyhedron in 4-dimensional space, the plane- . The correct twenty-four-cell has the Schleaf symbol , ${\begin{Bmatrix}3,4,3\end{Bmatrix}}$ ${\ displaystyle {\ begin {Bmatrix} 3,4,3 \ end {Bmatrix}}}$ and Coxeter chart , and the flat-nosed 24-cell is represented by $s{\begin{Bmatrix}3,4,3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3,4,3 \ end {Bmatrix}}}$ and chart diagram of Coxeter . It also has a lower symmetry construction with index 6 as $s\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}$ ${\ displaystyle s \ left \ {{\ begin {array} {l} 3 \\ 3 \\ 3 \ end {array}} \ right \}}$ or s {3 ^1,1,1 } and , and symmetry with index 3 as $s{\begin{Bmatrix}3\\3,4\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3 \\ 3.4 \ end {Bmatrix}}}$ or sr {3,3,4}, or .

Connected $s{\begin{Bmatrix}3,4,3,3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3,4,3,3 \ end {Bmatrix}}}$ or s {3,4,3,3}, body with lower symmetry as $s{\begin{Bmatrix}3\\3,4,3\end{Bmatrix}}$ ${\ displaystyle s {\ begin {Bmatrix} 3 \\ 3,4,3 \ end {Bmatrix}}}$ or sr {3,3,4,3} ( or ), and with the least symmetry as $s\left\{{\begin{array}{l}3\\3\\3\\3\end{array}}\right\}$ ${\ displaystyle s \ left \ {{\ begin {array} {l} 3 \\ 3 \\ 3 \\ 3 \ end {array}} \ right \}}$ or s {3 ^1,1,1,1 } ( )

Euclidean honeycombs are , s {2,6,3} ( ) or sr {2,3,6} ( ) or sr {2,3 ^[3] } ( )

Other Euclidean (equilateral) cells are s {2,4,4} (and ) or sr {2,4 ^1,1 } ( ):

The only homogeneous planar hyperbolic honeycombs are planar hexagonal mosaic honeycombs , s {3,6,3} and which can also be constructed as , h {6,3,3}, . It is also constructed as s {3 ^[3,3] } and .

Other hyperbolic (isosceles) cells are , s {3,4,4} and .

Notes

↑ Kepler , Harmonices Mundi , 1619
↑ Conway, 2008 , p. 287.
↑ Conway, 2008 , p. 401.

Literature

HSM Coxeter, MS Longuet-Higgins, JCP Miller. Uniform polyhedra // Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. - The Royal Society, 1954. - T. 246 , no. 916 . - S. 401-450 . - ISSN 0080-4614 . - DOI : 10.1098 / rsta . 1954.0003 .
Coxeter, HSM 8.6 Partial truncation, or alternation // Regular Polytopes . - 3rd. - 1973. - S. 154–156. - ISBN 0-486-61480-8 .
Coxeter . Tables I and II: Regular polytopes and honeycombs // . - 3rd. ed .. - Dover Publications, 1973. - S. 154–156. - ISBN 0-486-61480-8 .
HSM Coxeter . Kaleidoscopes: Selected Writings of HSM Coxeter / F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss. - Wiley-Interscience Publication, 1995. - ISBN 978-0-471-01003-6 .
- (Paper 17) Coxeter , The Evolution of Coxeter – Dynkin diagrams , [Nieuw Archief voor Wiskunde 9 (1991) 233–248]
- (Paper 22) HSM Coxeter, Regular and Semi Regular Polytopes I , [Math. Zeit. 46 (1940) 380–407, MR 2.10]
- (Paper 23) HSM Coxeter, Regular and Semi-Regular Polytopes II , [Math. Zeit. 188 (1985) 559–591]
- (Paper 24) HSM Coxeter, Regular and Semi-Regular Polytopes III , [Math. Zeit. 200 (1988) 3–45]
HSM Coxeter . Chapter 3: Wythoff's Construction for Uniform Polytopes // The Beauty of Geometry: Twelve Essays. - Dover Publications, 1999 .-- ISBN 0-486-40919-8 .
Uniform Polytopes. - 1991. - (Manuscript).
- The Theory of Uniform Polytopes and Honeycombs. - University of Toronto, 1966. - (Ph.D. Dissertation).
John H. Conway , Heidi Burgiel, Chaim Goodman-Strass. The Symmetries of Things. - 2008. - ISBN 978-1-56881-220-5 .
Weisstein, Eric W. Snubification on Wolfram MathWorld .
Richard Klitzing. Snubs, alternated facetings, and Stott – Coxeter – Dynkin diagrams // Symmetry: Culture and Science. - 2010 .-- T. 21 , no. 4 . - S. 329–344 .

[1] Kepler , Harmonices Mundi , 1619

[_632b4181e993020d-2] Conway, 2008 , p. 287.

[_63324581e9992785-3] Conway, 2008 , p. 401.

The foundation	Truncation	Full truncation		Dual nost	Sprain


t ₀ {p, q} {p, q}	t {p, q}	t ₁ {p, q} r {p, q}	2t {p, q}	t ₂ {p, q} 2r {p, q}	rr {p, q}	tr {p, q}	h {q, p}	ht ₁₂ {p, q} s {q, p}	ht ₀₁₂ {p, q} sr {p, q}

Operation Snub

Operation Conway's snub

Coxeter snub operations, right and quasi-correct

Examples

Inhomogeneous flat-bottomed polyhedrons

Homogeneous flat-stellated stellate polyhedrons of Coxeter

Flat-faced polyhedra and Coxeter honeycombs in high-dimensional spaces

Examples

See also

Notes

Literature

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