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.

Hatch loops provided varying degrees of cover of the tilings that Marek &Ccaron;trn&aacute;ct was producing on the TYLER java applet. As with CD, a zero edge on two consecutive nodes would poke the vertex there. You therefore can not have more than two 'o' nodes. x and s work as in Coxeter-Dynkin groups, but a notation for multiple classes of snub was ever found.

Still, it's the precursor of Conway's notation.

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.

Krieger decorated orbifold
Wendy Krieger's idea here is to construct various kinds of edges in the correct place of the CT diagram. An edge that shows is written in the usual style, as o or x. For a strutt, the form is to use a % sign.