Fundamental domains for non-arithmetic lattices
The following are projections of 4-dimensional polytopes that appears in my joint paper with J. Parker and J. Paupert.
The polytopes are invariant under a regular elliptic element (of order 7, 8 or 30 depending on the group), so they are actually a bit bigger than fundamental domains. There are twelve groups, hence twelve polytopes; for each polytope we show two projections of the 1-skeleton, onto each of the invariant complex lines of the regular elliptic element (the projections are displayed on the left for one axis, on the right for the other).
Among the 12 lattices, only two are arithmetic (can you spot them? :D)
Here is a picture of some individual faces.
If you cannot see the picture (or want a better version than the gif) try here.