Symbol of Wythoff

In Geometry, a symbol of Wythoff is a short notation, created by the Mathématicien Willem Abraham Wythoff, to name the regular and semi-regular polyhedrons using a kaleidoscopic construction, by representing them like pavings on the surface of a Sphère, on a Euclidean Plan or a hyperbolic Plan.

The symbol of Wythoff gives 3 numbers p, Q, R and a positional vertical bar (|) who separates the numbers before and after it. Each number represents the order of the mirrors at a top of the fundamental triangle.

Each symbol represents a Polyèdre uniform or a Pavage, although the same polyhedron/paving can have symbols of different Wythoff starting from different symmetrical generators. For example, the regular Cube can be represented by 3 | 4 2 with a O_h symmetry and 2 4 | 2 like a square prism with two colors and a D_4h symmetry, as much as 2 2 2 | with 3 colors and a D_2h symmetry.

Table of summary

There exist 7 pitch points with each whole of p, Q, R: (and some particular forms)

There exist three particular cases:

p Q (R S) | - It is a mixture of p Q R | and p Q S | .
| p Q R - the softened forms (alternate) give this other unusual symbol.
| p Q R S - a single form softened for the U75 which is not constructible within the meaning of Wythoff.

Description

The letters p, Q, R represent the shape of the fundamental triangle for symmetry, more precisely each number is the number of reflexive mirrors which exist at each top. On the sphere, there exist three principal types of symmetries: (3 3 2), (4 3 2), (5 3 2) and an infinite family (p 2 2), for p=2,3,… unspecified (all the simple families have a right angle, therefore r=2)

The position of the vertical bar in the symbol is used to indicate specific forms (a position of category of the pitch point) in the fundamental triangle. The pitch point can be is on or beside each mirror, activated or not. This distinction generates 8 (2 ³) possible forms, neglecting where the pitch point is on all the mirrors.

In this notation, the mirrors are labelled Pr the order of reflection of the opposite top. The values p, Q, R are listed before the bar if the mirror corresponding is active.

The impossible symbol | p Q R , which implies the pitch point is on all the mirrors which is the only possible one if the triangle is generated in a point. This uncommon symbol is réassigné to mean quelquechose the different one. These symbols represent the case where all the mirrors are active, but the enumerated considered images in an odd way are ignored. This generates results of rotational symmetry.

This symbol is functionally similar to the Diagramme of more general Coxeter-Dynkin which shows a triangle marked p, Q, R on the edges, and of the circles on the nodes, representing the mirrors to imply if the pitch point touched this mirror (the diagram of Coxeter-Dynkin is shown as a linear graph when r=2 since there are no reflections interacting through a right angle).

Triangles of symmetry

There exist 4 classes of symmetry of reflections on the Sphère, and 2 for the Euclidean Plan and infinitely much for the hyperbolic Plan, the first:

(p 2 2) diedric Symmetry p=2,3,4… (Order 4p )
(3 3 2) tetrahedral Symmetry (Order 24)
(4 3 2) octahedral Symétrie (Order 48)
(5 3 2) icosahedral Symétrie (Order 120)
(4 4 2) - symmetry *442 - Triangle 45-45-90 (includes the square field (2 2 2 2))
(3 3 3) - symmetry *333 - Triangle 60-60-60
(6 3 2) - symmetry *632 - Triangle 30-60-90
(7 3 2) - symmetry *732 (plane hyperbolic)

The groups of symmetry above include only the whole solutions on the sphere. The list of the triangles of Schwarz include rational numbers, and determine the whole whole of solutions of the polyhedral uniforms.

In pavings above, each triangle is a fundamental field, coloured by even and odd reflections.

Summary of spherical and plane pavings

a selection of the pavings created by the construction of Wythoff are given below.

Spherical pavings (r2)

Planar pavings (r2)

a representative hyperbolic paving is given, and shown like a projection of disc of Poincaré.

Planar pavings (r>2)

The Diagramme of Coxeter-Dynkin is given in a linear form, although it is a triangle, with the segment R trailing connected to the first node.

Coverings of spherical pavings (r2)

pavings are shown as Polyèdre S. Some of these forms are degenerated, given by accodances of the figures of top, with the edges or the tops of covering.

See too

regular Polytope
Polyhedral regular
List of uniform pavings
List of the uniform polyhedrons
Polyhedral uniform

References

Coxeter Regular Polytopes , Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff' S construction)
Coxeter The Beauty off Geometry: Twelve Essays , Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff' S Construction for Uniform Polytopes)
Coxeter, Longuet-Higgins, Miller, Uniform will polyhedra , Phil. Trans. 1954,246 has, 401-50.
pp. 9-10.

External bonds

the symbol of Wythoff
Wythoff symbol
Poster of the uniform polyhedrons by using the method of construction of Wythoff
Description of constructions of Wythoff
KaleidoTile 3 free Software of education for Windows by Jeffrey Weeks which generated many images on the page.

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