Conway-Thurston

Conway-Thurston symbols
William Thurston showed that all groups acting on a two-dimensional surface, make use of at most four different symmetry operators. John Conway formalised this into a name-scheme for the symmetries on a sphere, the euclidean and hyperbolic planes. From this it was hoped to make a decorated notation, such as the wythoff-dots on the Coxeter-Dynkin symbol. Such attempts have largely eluded researchers, due to the existance of large distance things in the hyperbolic plane.

The basic form is to place a number of **cones** at the front, followed by chains of mirrors, miracles and wanders.

Any number of these symbols, and any number of mirror-chains, represents a valid hyperbolic group, and most often, is the source of uniform tilings in H2, generally.

Hatch Loop
An early notation proposed by Don Hatch was a loop, comprising of mirror-edges and snubs, with most nodes occupied by non-zero elements.

Conway Archiform
The archiform is a description of the edges arriving at a vertex, these are classified as if the edge connects a reflection [], or is a rotated image, or has 'around-symmetry', in other words, a reflection where the edge lies in the mirror <>. A simple mirror-edge might be [1]. This exists where the vertices are reversed, as in the omnitruncates [1] [2] [3]. This has three edges, all different, which connects reversed vertices.

The edges are numbered in the repeated sector, from 1 to n. If an edge returns to a different position on the next vertex, then the second number is also included. For example, a digon, rotated at the centre, is say (1). If this is turned into a polygon, it would become (1,2), being rotated around the centre of the polygon.


 * (1) (2) (3)  represents a tetrahedron, rotations at the centre of each edge.
 * (1,2) (3) (4)   represents an antiprism, (1,2) is the base, and the zigzag is alternately edges 3 and 4.
 * (1,2) (3,4) (5) is the form of the snub cube and snub dodecahedron.  These have two (different) polygons.
 * (1,2) [3]    is a prism with rotary symmetry.
 * <1> is an example of a regular polytope. The edge serves as a mirror between both faces and vertices,
 * <1> [2] is a truncated polytope, with various edges as mirrors, and other ones as mirror-edges (reflecting vertex to vertex)

The non-symmetric triangles in the first three are not presented in the figure, but foreshadow a major problem that exists with CT figures: active regions. These are parts of the symmetry region that do not fall into one of the vertex-objects, but rather are constructed in the symmetry itself. Other issues that arise are things like 'strutt-edges', which are chords of polytons.

The tiling of alternate bands of squares and triangles, is [1] (2,5) (3) (4). Here (2,5) would be a polygon were the numbers consecutive. Instead, there is a division of edges 1 on one side, and 3,4 on the other. These are the unclosed sets of edges that run between the squares and triangles, and represent a wander.

The notation, is capable of describing all the uniform polyhedra in H2. Unfortunately, it describes a lot of other things which do not have a separate realisation. None the less, the notation is versitile enough to see what is going on.