In the last years, I read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this "Pentagramma Mirificum" (latin for "the miraculous pentagram"). Unfortunately, there is no wikipedia article about this, and I didn't find lower-level and accesible discussions of it, but only a few articles about it by H.S.M Coxeter and others. This object is related to spherical pentagrams (planar pentagrams are star like figures with 5 sides), but since I didn't find enough good sources on it, I wasn't able to extrapolate if it's a phenomenon of algebraic nature (some references say it was a special case of "cluster algebras"), geometric nature, or perhaps both. I've read that the geometer Coxeter invented something called "frieze paterns", which gave a modern interpretation of Gauss's calculations about this phenomenon.
So, first of all I want to understand what the "Pentagramma Mirificum" is about. Since I know almost nothing about it, I simply listed a few articles that might serve as an aid in an exposition of the subject. I think it will be good not only for me, but for stack exchange, if there will be at least one post about this - I didn't find a single post about it in stack exchange. I'll be glad if someone will also be able to explain a little bit of Gauss's investigations about it (which go quite deeply; for example, in his later fragments he applies Mobius's barycentric calculus to a chain of pentagons infinite in both directions). So here is a list of articles:
- PENTAGRAMMA MIRIFICUM AND ELLIPTIC FUNCTIONS - this is an exposition article for some of Gauss's fragments. It also gives some background, so I think it's the most suitable article to use in order to answer my question.
- Frieze patterns - an article by H.S.M Coxeter.
- There is also the reference "Draft Translation of Gauss’ Fragmentary Notes on the Pentagramma Mirificum" (just google these words and you will find it), which is an English translation of Gauss's original fragments.