I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I am focussing on an article about pentagonal tilings in the euclidian plane, since this project is mainly focussed on math I am trying to give a decent proof of why these pentagons can tile the plane. I have been trying to read Karl Reinhardt's paper but since its in german and its written in 1916, I am a little lost.
I have discussed this with both my partners and my teachers but we can't seem to find any decent proof. Could you give me a hand? Maybe you know some papers which discuss the subject?
Your help is very welcome!
Thank you,
Roy
I would suggest "The symmetries of things" by J. Conway, C. Goodman-Strauss & H. Burgiel, for learning the very basics of symmetries of tilings. Also in "Tilings and Patterns" by Grumbaum & Shephard you should find whatever you need, though the first option is IMO much more friendly.
I also suggest you to look at this note, pentaplexity by Penrose and Martin Gardner's column.