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

Question

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.

Answer 1

[I do not have enough reputation to add comments so I have to make this an answer]

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.

Answer 2

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.

Answer 3

Regarding the last fragments of Gauss on Pentagramma mirificum, which are dated to 1843 and are numbered as [9]-[11] in the translation of his fragments, I believe one way to put it into historic context is to remark that Gauss simply discovered here Clebsch's theorem on the pentagram map. Clebsch's theorem (proved by Alfred Clebsch in 1871) states that for any rectilinear pentagon, if one draws all its diagonals to form an inner pentagon, than the smaller pentagon is projectively equivalent to the original pentagon. Two planar figures are projectively equivalent if there is a projective mapping that sends one into the other, when a projective mapping is defined as any composition of perspective projections centered at a series of points $O_1,O_2,O_3$, etc.

In his comments on Gauss's pentagramma mirificum, Robert Fricke states that:

Gauss constructed here first by continuing the diagonal and the lengthening of sides, a chain endless in both directions, of pentagons and recognizes, that this grid (net) is transformed into one another circumscribing pentagons through collineation:$$(5) x' = (\frac{2G-1}{2G'-1})x, y' = (\frac{2G-1}{2G''-1})y$$ For making things clear, compare the figure provided; Gauss himself made ready drawings of this kind, in which the net of pentagons are carried through still much further. The coefficients which stand on the right hand, in (5), are the “exponents” or “coefficients of the rejuvenations” which in [9] and [10] come into action. It lies very close by, to conceive the here-presented relationship of things, in the sense of modern theory of discontinuous substitution groups. The zero point, $O$, and is one of the three boundary points of the form (5) originating cyclical collineation group.

The term collineation refers to one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of any 3 collinear points are themselves collinear. Therefore, if I understand Fricke's remarks correctly, than his description of Gauss's ideas correspond to Clebsch theorem.