This application calculates and visualises shadows and co-shadows in Coxeter groups that can be realised as tilings of the euclidean or hyperbolic plane. The article Shadows in the Wild serves as an excellent introduction to shadows. The hyperbolic plane is drawn in the Poincaré disk model.
For a shadow, we need three ingredients: An underlying Coxeter group, a gallery in its Coxeter-complex, and an orientation on the Coxeter-complex.
Using the selection-box in the top right-hand corner, you can select from different affine and hyperbolic Coxeter groups. For technical reasons, we had to omit the tilde-decoration for the names of affine coxeter groups. The notation used for hyperbolic Coxeter groups describes n-gon-groups, i.e. Coxeter groups with defining relations:
We only consider galleries starting in the fundamental chamber. Such galleries are in one-to-one correspondence with decorated words. Unlike in mathematical writing, where a word is written as a sequence of generators, words are written here as a sequence of digits, starting at 0. For example, if a Coxeter group is generated by three elements s0, s1, s2, the word usually written as
There are two ways to input a gallery:
Galleries are visualised as paths starting in the fundamental chamber and ending in their final chamber.
Currently, there are four orientations implemented. The selected orientation can be inverted by checking the "Invert orientation" checkbox.
Orientations can be visualised by checking the "Draw orientation arrows" checkbox. For a chamber c and a panel p of c, this draws a coloured arrow from p to c iff φ(c, p)=+1. If φ(c, p)=-1, nothing is drawn.
Shadows are visualised by colouring the inside of the chambers it comprises. The app allows you to visualise up to three shadows at a time, which use the colours blue, yellow and purple. To show/hide the galleries, orientations and shadows of a colour, check/uncheck the checkbox at the top left of its container. If a chamber lies in multiple shadows at the same time, its colour follows this Venn diagram: