**Application With Uniform Polytopes**

Coxeter–Dynkin diagrams can explicitly enumerate nearly all classes of uniform polytope and uniform tessellations. Every uniform polytope with pure reflective symmetry (all but a few special cases have pure reflectional symmetry) can be represented by a Coxeter–Dynkin diagram with permutations of *markups*. Each uniform polytope can be generated using such mirrors and a single generator point: mirror images create new points as reflections, then polytope edges can be defined between points and a mirror image point. Faces can be constructed by cycles of edges created, etc. To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is *not* on the mirror(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant from them.) A mirror is *active* (creates reflections) only with respect to points not on it. A diagram needs at least one active node to represent a polytope.

All regular polytopes, represented by Schläfli symbol symbol {*p*, *q*, *r*, ...}, can have their fundamental domains represented by a set of *n* mirrors with a related Coxeter–Dynkin diagram of a line of nodes and branches labeled by *p*, *q*, *r*, ..., with the first node ringed.

Uniform polytopes with one ring correspond to generator points at the corners of the fundamental domain simplex. Two rings correspond to the edges of simplex and have a degree of freedom, with only the midpoint as the uniform solution for equal edge lengths. In general k-rings generators are on k-faces of the simplex, and if all the nodes are ringed, the generator point is in the interior of the simplex.

A secondary markup conveys a special case nonreflectional symmetry uniform polytopes. These cases exist as alternations of reflective symmetry polytopes. This markup removes the central dot of a ringed node, called a *hole* (circles with nodes removed), to imply alternate nodes deleted. The resulting polytope will have a subsymmetry of the original Coxeter group. If all the nodes are *holes*, the figure is considered a *snub*.

- A single node represents a single mirror. This is called group A
_{1}. If ringed this creates a line segment perpendicular to the mirror, represented as {}. - Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, or a square if the point is at equal distance from both mirrors.
- Two nodes attached by an order-
*n*branch can create an*n*-gon if the point is on one mirror, and a 2*n*-gon if the point is off both mirrors. This forms the I_{1}(n) group. - Two parallel mirrors can represent an infinite polygon I
_{1}(∞) group, also called Ĩ_{1}. - Three mirrors in a triangle form images seen in a traditional kaleidoscope and can be represented by three nodes connected in a triangle. Repeating examples will have branches labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn as a line (with the
*2*branches ignored). These will generate uniform tilings. - Three mirrors can generate uniform polyhedra; including rational numbers gives the set of Schwarz triangles.
- Three mirrors with one perpendicular to the other two can form the uniform prisms.

The duals of the uniform polytopes are sometimes marked up with a perpendicular slash replacing ringed nodes, and a slash-hole for hole nodes of the snubs. For example represents a rectangle (as two active orthogonal mirrors), and represents its dual polygon, the rhombus.

Read more about this topic: Coxeter–Dynkin Diagram

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