Symbol of Schläfli

In Mathematical, the symbol of Schläfli is a simple notation which gives a summary of certain important properties of a rectangular Polytope particular.

The symbol of Schläfli was named thus in the honor of the Mathématicien of the 19th sièle Ludwig Schläfli which made important contributions in Géométrie and in other fields.

Regular polygons (plan)

The symbol of Schläfli for a Polygon with N east coasts { N }.

For example, a regular pentagon is represented by {5}.

See the regular polygons and the spangled polygons.

For example, {5/2} is the Pentagramme.

Regular polyhedrons (3-space)

The symbol of Schläfli of a Polyèdre is { p , Q } if its faces is p - gones, and each top is surrounded by Q faces (the Figure of top is a Q - gone).

For example {5,3} is the regular Dodécaèdre. It has pentagonal faces, and three pentagons around each top.

See the 5 solid of Plato, 4 solid of Kepler-Poinsot.

The symbols of Schläfli can also be defined for the Pavage S regular of Euclidean spaces or hyperbolic in a similar way.

For example, the hexagonal Pavage is represented by {6,3}.

Regular polychores (4-space)

The symbol of Schläfli for a regular Polychore is form { p , Q , R }. It has { p } regular polygonal faces, { p , Q } cells, { Q , R } regular polyhedric figures of top and { R } figures of regular polygonal edges.

See the six polychores regular convex and ten not-convex.

For example, the 120-cell is represented by {5,3,3}. It is built by dodecahedral cells {5,3}, and has 3 cells around each edges.

There exists also a regular paving of Euclidean 3-space: the cubic Honeycomb, with a symbol of Schläfli of {4,3,4}, made cubic cells, and 4 cubes around each top.

There exist also 4 hyperbolic regular pavings including {5,3,4}, the Petit hyperbolic dodecahedral honeycomb, which fills space with dodecahedral cells .

Higher dimensions

For the Polytope S of higher size, the symbol of Schläfli is defined by recurrence like: ${p_1, p_2,…, p_ {n-1}} \,$ if the facets have a symbol of Schläfli ${p_1, p_2,…, p_ {N2}} \,$ and the figures of top: ${p_2, p_3,…, p_ {n-1}} \,$ .

There exist only 3 polytopes regular in 5 dimensions and above: the Simplex, {3,3,3,…, 3}; the cross Polytope, {3,3,…, 3,4}; and the Hypercube, {4,3,3,…, 3}. There do not exist not-convex regular polytopes above 4 dimensions.

Duaux polytopes

For the dimension 2 or above, each polytope has a dual .

If a polytope has a symbol of Schläfli ${p_1, p_2,…, p_ {n-1}} \,$ then its dual has a symbol of Schläfli ${p_ {n-1},…, p_2, p_1} \,$ .

If the continuation is the same one towards the left and the right-hand side, the polytope is car-dual . Each polytope regular in 2 dimensions (polygon) is car-dual.

Prismatic forms

The prismatic polytopes can be defined and named like a Cartesian Produit polytopes of lower size:

a prism p - gonal, with a figure of top p.4.4 like $\ begin {Bmatrix} \ \ end {Bmatrix} \ times \ begin {Bmatrix} p \ end {Bmatrix}$ .
a uniform Hyperprisme {p, Q} - edric like $\ begin {Bmatrix} \ \ end {Bmatrix} \ times \ begin {Bmatrix} p, Q \ end {Bmatrix}$ .
a uniform Duoprisme p-q like $\ begin {Bmatrix} p \ end {Bmatrix} \ times \ begin {Bmatrix} Q \ end {Bmatrix}$ .

A prism can also be represented like the truncation of a Hosoèdre like $t \ begin {Bmatrix} 2, p \ end {Bmatrix}$ .

Symbols of Schläfli extended for the uniform polytopes

The uniform polytopes, built starting from a Construction of Wythoff, are represented by a notation of truncation extended starting from a regular form {p, Q,…}. There exists a quantity of parallel forms below symbolic systems which refer the elements of the symbol of Schläfli , discussed by dimensions.

Uniform polyhedrons and pavings

For the polyhedrons, a symbol of wide Schläfli is used in the article of 1954 by Coxeter enumerating the article entitled polyhedral uniforms .

Each regular polyhedron or paving {p, Q} has 7 forms, including the regular form and its dual, corresponding to the positions in the fundamental right-angled triangle. An eighth special form, the softened , corresponds to a alternation of the omnitronquée form.

For example, T {3,3} means simply Tétraèdre truncated.

Second notation, more general, such a used by Coxeter, applies to all dimensions, and is specified by a T followed of a list of indices corresponding to the mirrors of Construction of Wythoff (they also correspond to the nodes ringed in a Diagramme of Coxeter-Dynkin).

For example, the Cube truncated can be represented by t_0,1 {4,3} and it can be looked like halfway between the Cube, t₀ {4,3} and the Cuboctaèdre, t₁ {4,3}.

In each one, a name indicating the operation of the construction of Wythoff is given initially. In second place, some have an alternative terminology (given between brackets) applying only for a given dimension. Precisely, the omnitroncature and the development, the dual relations applying differently in each dimension.

Uniform polychores and honeycombs

There exists with more the 15 forms truncated for the polychores and the honeycombs based on each regular form {p, Q, R}.

See the articles Polychore and convex uniform Honeycomb.

The notation with T in index is parallel to the graphic Diagramme of Coxeter-Dynkin, whose each graphic node represents the 4 hyperplanes of the reflections mirrors in the fundamental field.

References

The Beauty off Geometry: Twelve Essays (1999), Dover Publications ISBN 99-35678 (Chapter 3: Wythoff' S construction for uniform polytopes, p41-53)
Johnson, N.W Uniform Polytopes , Manuscript (1991)
Johnson, N.W The Theory off Uniform Polytopes and Honeycombs , Ph.D. Essay, University off Toronto, 1966
Coxeter, H.S.M.; Regular Polytopes , (Methuen and Co., 1948). (pp. 14,69,149)
Coxeter, Longuet-Higgins, Miller, Uniform will polyhedra , Phil. Trans. 1954,246 has, 401-50. (Extended Schläfli notation defined: Count 1: p 403)

External bonds

Symbols of Wythoff and generalized symbols of Schläfli
polyhedric Names and notations

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