If we draw all the fractions in the world as a line instead of a circle, and draw all the curved triangles in the upper half-plane instead of a disk, and subdivide each triangle into 6 smaller ones, we get this picture.
Here the yellow and purple curved triangles are 6 pieces of a single larger triangle.
And yes, two of are very distorted, because they have a vertex at 1/0, which is infinitely far up the page!
@johncarlosbaez Excuse me, thank you for the nice explanation! I have a question, is there any group larger than the SL(2, Z) which still has some nice properties, e.g. notion of modular forms? I wonder how we can distinguish the black and yellow regions in your figure, since they are combined into the fundamental region of SL(2, Z). If we consider some congruence subgroups, the ability to distinguish becomes weaker, but I don't know if we can consider the group containing contains SL(2, Z) and distinguish the yellow and black regions. Thank you!
@todokuro wrote: " I wonder how we can distinguish the black and yellow regions in your figure, since they are combined into the fundamental region of SL(2, Z)."
If you read my article you'll see the answer to this explained in detail:
https://johncarlosbaez.wordpress.com/2023/09/23/the-moduli-space-of-acute-triangles/
In brief the answer is GL(2,Z), a group which is 'twice as big' as SL(2,Z).
@johncarlosbaez Thank you. Are that construction of GL and the fact that GL(2, Z) is not just the SL(2, Z) times Z2, but a semidirect-product of them related in some way?
@todokuro - it must be related somehow. That semidirect product is saying that rotations don't commute with reflections.