Coxeter-Dynkin diagram

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In geometry, a Coxeter-Dynkin diagram is a graph representing a relational set of mirror (or reflectional hyperplanes) in space for a kaleidoscopic construction. Each node in the graph represents one mirror, and each edge represents the dihedral angle between two mirrors.

As a graph itself, the diagram represents Coxeter groups.

The diagram can also represent polytopes by adding rings (circles) around nodes. The rings express information on whether a generating point is on or off the mirror. Specifically a mirror is active (creates reflections) only when points are off the mirror, so adding a ring means a point is off the mirror and creates a reflection.

Edges are labeled with an integer n (or sometimes more generally a rational number p/q) representing a dihedral angle of 180/n. If an edge is unlabeled, it is assumed to be 3. If n is two the angle is 90 degrees and the mirrors have no interaction, and the edge can be left unmarked. Two parallel mirrors can be marked with an infinity (∞) symbol.

In general n mirrors can be represented in an n-simplex graph where all n*(n-1)/2 edges are drawn. In practice interesting sets of mirrors will have a number of right angles, and such edges can be left undrawn.

Polytopes and tessellations can be generating using these mirrors and a single generator point. Mirror images create new points as reflections. Edges can be created between points and a mirror image. Faces can be constructed by cycles of edges created, etc.

Examples:

A single node represents a single mirror. All points off the mirror are drawn with a ring. Connecting such an off mirror point to its reflection creates a digon or edge perpendicular to the mirror.
Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, or a square if the point is equal distance from both mirrors.
Two nodes attached by an n edge can create an n-gon if the point is on one mirror, and a 2n-gon if the point is off both mirrors.
Three mirrors in a triangle form images seen in a traditional kaleidoscope and be represented by 3 nodes connected in a triangle. Repeating examples will have edges labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn in a line with the 2 edge ignored. These will generate uniform tilings.
Three mirrors with a common point can generate uniform polyhedrons, including rational numbers is the set of Schwarz triangles.
Three mirrors with one perpendicular to the other two can form the uniform prisms.

In general all regular n-polytopes, represented by Schläfli symbol symbol {p,q,r,...} can have their fundamental domains represented by a set of n mirrors and a related in a Coxeter-Dynkin diagram in a line of nodes and edges labeled by p,q,r...

Examples

- {p,q} - a regular polyhedron or tiling
- {p,q,r} - a regular polychoron or honeycomb

[edit] See also

[edit] References

Coxeter Regular Polytopes (1963), Macmillian Company
- Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Section 11.3 Representation by graphs)
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 99-35678 (Chapter 3: Wythoff's Construction for Uniform Polytopes)