What are the modern connections of the Pentagramma Mirificum studied by Gauss?

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 miracalous 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 goemeter 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 ecxhange. 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 barychentric 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 backgroud, 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.
• hey!! i don't know who voted to close my question, but if it's unclear i'll explain it better: i just want to understand what is the pentagramma mirificum phenomenon. Since it seems to be very advanced (so i know almost nothing about it), i wasn't able to define my question exactly. But don't make a mistake - it's a very important question! so, meanwhile, just view my question as a demand for clear exposition/description of the phenomenon and it's related results. Aug 12 '18 at 16:04
• @user2554: You yourself mentioned two excellent sources on the subject. Just read these sources if you want to learn what is it. It is still unclear what is your question. Aug 12 '18 at 22:02
• I word of warning: before you spend time on reading the paper recommended by @Nick R, I advise you to learn where it comes from by typing "LaRouche movement" on Wikipedia. Aug 12 '18 at 22:06
• @AlexandreEremenko Oops. Does it! Maybe I should have read more of it. I'll delete the link.
– NWR
Aug 12 '18 at 22:14
• See the Coxeter reference, page 298, for a picture and an explanation. matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf Aug 13 '18 at 12:36

FWIW, I have written a Polish Wikipedia article on pentagramma mirificum. For the time being, you can try to make sense of it with Google Translate.

Edit (December 28, 2018): Lo and behold, an article in English Wikipedia.

Edit (January 6, 2019): Also, a short video about the construction of pentagramma mirificum.

• Wow!! thanks sincerely! It will be of great help for me. I'll use google translate to English and read it. I guess that thanks to you perhaps we'll soon see also an english article on it (someone will use your article in the polish wikipedia). Thanks again! Dec 26 '18 at 10:56
• @user2554 You can now enjoy the English version of the article. :) Dec 28 '18 at 11:32
• Thanks! by writing this article you made a very essential contribution to wikipedia , because the pentagramma mirificum isn't a well known topic. By the way, I read that Gauss found a connection between the gnomonic projection of those spherical pentagons and elliptic functions. Can you tell something about this too? Anyway, thanks again. Dec 28 '18 at 11:47
• @user2554 Unfortunately, elliptic functions are beyond my dilettantish knowledge of maths. I prefer to understand what I write. Maybe someone more knowledgeable than I will add the information. Dec 28 '18 at 16:09
• @user2554 I also made a 3Blue1Brownish video on pentagramma mirificum. Some day, I may cover more of its properties. Jan 6 '19 at 7:50

The Pentagramma Mirificum is a spherical figure formed by a series of five great circle arcs, each orthogonal to the next, and it probably does deserve a Wikipedia entry. However some elementary information about it is available online. A good start is Math Central where we read that

"The story begins in 1602, when Nathaniel Torporley (1564-1632) began to investigate the five "parts" a, A, b, B, c of a right-angled spherical triangle (right-angled at C). According to De Morgan, Torporley anticipated by a dozen years the famous rules of Napier which Gauss embodied in his pentagramma mirificum... The "core" of the pentagram is a pentagon whose vertices are (obviously) poles of these five arcs; it is thus a self-polar pentagon. [This means that if a vertex of the inside pentagon is placed at the north pole of the sphere, the opposite side lies along the equator.] The whole figure can be derived from the right-angled triangle ABC (appearing at the top) by extending the sides and drawing also the polar great circles of the vertices A and B."

Wikipedia does have an article on Napier's rules, which are a mnemonic for ten identities relating every triple chosen from the set a, b, c, A, B. The identities are algebraic when written in terms of their square tangents. If we call the arc lengths $L_i$ instead, and set $a_i=\tan^2 (L_i)$ then $a_ia_{i+1}=1+a_{i+3}$, adding the indices mod $5$.

Gauss noticed that the first three of these identities imply the other two, incidentally this proves the horizontal $5$-periodicity of the Coxeter's frieze pattern of width $2$. Coxeter's frieze patterns are arrengements of numbers into horizontal rows so that $2\times2$ determinants formed by neighboring four numbers are $1$. For an elementary exposition see the first lecture of Tabachnikov's Three Lectures on Frieze Patterns. For a more extensive survey see Coxeter's Frieze patterns at the Crossroads of Algebra, Geometry and Combinatorics by Morier-Genoud, where the pentagramma mirificum is also discussed. Frieze patterns lead into less elementary precincts. For those interested, the space of classical Coxeter’s frieze patterns can be presented as a discrete version of a coadjoint orbit of the Virasoro algebra, and the canonical cluster pre-symplectic form on it is a discretization of the Kirillov symplectic form, see Coxeter's Frieze Patterns and Discretization of the Virasoro Orbit by Ovsienko and Tabachnikov.

Another modern relation is to cluster algebras of Fomin and Zelevinsky, already covered by Wikipedia. The algebras are generated by subsets of fixed size called clusters, whose elements satisfy relations similar to $a_ia_{i+1}=1+a_{i+3}$. Fomin is a great enthusiast of their connection to the pentagramma, see his Bielefeld page, where we read:

"It would be a stretch to say that Gauss discovered cluster algebras: the pentagon equation, in its algebraic form, was found by W. Spence in 1809. In the context of Pentagramma Mirificum, the realization that the 5-cycle represents a nontrivial algebraic identity seems to appear first in print by Arthur Cayley; see [A. Cayley, On Gauss's Pentagramma Mirificum, Philosophical Magazine, vol. XLIL (1871), 311-312; The collected mathematical papers of Arthur Cayley. Vol. 7]."

The "nontrivial algebraic identity", the pentagon equation, expresses the fact that the map $F:(a,b)\mapsto(b,\frac{b+1}a)$ satisfies $F^5=$id. This can be depicted as a pentagon of Laurent polynomials, and leads to quivers and a different view of cluster algebras, see Lampe's lecture notes.