An easy (horizontal) walk through fake octagons

A fake octagon is a genus two translation surface with only one singular point and the same periods as the octagon. Existence of infinitely many fake octagons was established first by McMullen (2007), and more generally follows from dynamical properties of the isoperiodic foliation. The purpose of this note is to describe an infinite family of fakes constructed by means of elementary methods. We describe an easy cut-and-paste surgery and show that the nth iterate of that surgery is a fake octagon Octn. Moreover we show that Octn is different from Octm for n different from m , and that any Octn can be approximated arbitrarily well by some other Octm. This note is intended to be elementary and fully accessible to non-expert readers.