The late Fields medalist Maryam Mirzakhani posed a wonderful puzzle that @emilyriehl answered for the square; @math3ma wrote up the answer in a terrific exposition: https://www.math3ma.com/blog/is-the-square-a-secure-polygon …
Now I’ve generalised their answer to a triangle and a hexagon:
https://plus.google.com/113086553300459368002/posts/gFGPmCXHKUs …pic.twitter.com/kbWn55tfPu
the theta series (when suitably unrolled) for this is Iain Banks bonkers... http://owen.maresh.info/hyperbolic.html http://mathworld.wolfram.com/ThetaSeries.html …