Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 2, p. 353-375

We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the plane. As a corollary we find a parametrization in elliptic functions of the classical dilogarithm five-term relation.

On présente des fragments non-publiés de Gauss où il a découvert (en utilisant des résultats de Jacobi) une connexion remarquable entre les pentagones de Napier sur le sphère et les pentagones de Poncelet sur le plan. Comme conséquence on trouve une paramétrisation de la relation à cinq termes du dilogarithme en termes de fonctions elliptiques.

@article{AFST_2013_6_22_2_353_0,
     author = {Schechtman, Vadim},
     title = {Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {2},
     year = {2013},
     pages = {353-375},
     doi = {10.5802/afst.1375},
     zbl = {1275.33026},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_2_353_0}
}
Schechtman, Vadim. Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...). Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 2, pp. 353-375. doi : 10.5802/afst.1375. http://www.numdam.org/item/AFST_2013_6_22_2_353_0/

[1] Baxter (R.J.).— Exactly solved models in statistical mechanics, Academic Press (1982). | MR 690578 | Zbl 0723.60120

[2] Bos (H.J.M.), Kers (C.), Oort (F.), Raven (D.W.).— Poncelet’s closure theorem, Expos. Math. 5, p. 289-364 (1987). | MR 917349 | Zbl 0633.51014

[3] Bowditch (B.H.).— A proof of McShane’s identity via Markoff triples, Bull. London Math. Soc. 28, p. 73-78 (1996). | MR 1356829 | Zbl 0854.57009

[4] Coxeter (H.S.M.).— Frieze patterns, Acta Arithm. XVIII, p. 297-304 (1971). | MR 286771 | Zbl 0217.18101

[5] Gauss (C.F.).— Pentagramma Mirificum, Werke, Bd. III, p. 481-490; Bd VIII, p. 106-111.

[6] Flatto (L.).— Poncelet’s theorem, AMS, Providence (RI) (2009). | MR 2465164 | Zbl 1157.51001

[7] Fricke (R.).— Bemerkungen zu [5], ibid. Bd. VIII, p. 112-117.

[8] Gliozzi (F.), Tateo (R.).— ADE functional dilogarithm identities and integrable models, hep-th/9411203.

[9] Greenhill (A.G.).— Applications of elliptic functions, 1892 (Dover, 1959). | MR 111864 | Zbl 0087.08501

[10] Griffiths (P.), Harris (J.).— On Cayleys explicit solution to Poncelet’s porism, Enseign. Math. (2) 24, p. 31-40 (1978). | MR 497281 | Zbl 0384.14009

[11] Hardy (G.H.).— Ramanujan, Cambridge, 1940 (AMS Chelsea, 1991). | Zbl 0025.10505

[12] Jacobi (C.G.J.).— Fundamenta nova theoriae functionum ellipticarum.

[13] Jacobi (C.G.J.).— Über die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie, Crelles J. 3 (1828). | Zbl 003.0129cj

[14] Kirillov (A.).— Dilogarithm identities, hept-th/9408113.

[15] Litttlewood (J.E.).— A mathematical miscellany, Review of Collected Papers of S. Ramanujan, Mathematical Gazette, April 1929, v. XIV, no. 200.

[16] Napier (J.).— Mirifici Logarithmorum canonis descriptio, Lugdini (1619).

[17] Onsager (L.).— Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, p. 117-149 (1944). | MR 10315 | Zbl 0060.46001

[18] Snape (J.).— Applications of elliptic functions in classical and algebraic geometry, Dissertation, Durham.