Talk:Coxeter diagram

Edge lengths
Unmarked nodes are zero length edges, or 'o'. Marked edges are x, x marks the spot. The polygon which makes the horizon, is of U sides, of length R. The angle 2pi/U is the smallest resolvable angle, and R is the largest distance one is prepared to traverse the points of. The short chord of U is u which gives 2.

The length r is two points on either side of a mirror, as o|o, allows for figures of zeroish length. So r4o3o is a cube of edge near zero, while o4o3o is a point.

m is used for the dual of x, so o3x4o is a CO, and o3m4o is a rhombic dodecanedron. A special wythof mirror margin construction makes these. --Wendy.krieger (talk) 04:16, 25 July 2020 (UTC)

Two other special lengths are used. In a lacing-tower, the z-length is such to produce a zero-height. Literally, &#z means & 'add a new axis of symmetry', #'destroy the symmetry in this axis', and 'z' loop back on itself = remove axis. So qo3oo4ox&#z is a zero hight (perpendicular to the three dimensions of the cube/octahedron), lacing that has an edge such to make the height zero. The effect is to produce a thatch or 'tegum-sum', that covers the compoumd in question.

The second special length is used in some other constructions, but not mirror-edge. It is %, its role is to make an edge into a chord of a polygon. In some of the workings for the decoration of the Conway-Thurston orbifolds, this is used to make edges into chords. For example, in 3/ & /2% which is the truncated-cube in pyritohedral symmetry, the octagon-faces are formed by rectangles /2%, and by trapezia x%&#tx. The effect is to make /%%/&xt. In 3*3, the result joins three rhombus x%&#tx to the sides of a truncated triangle x3%. In both cases, the length of the edge corresponds to the chord it is to become --Wendy.krieger (talk) 11:37, 26 July 2020 (UTC)