Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)

Vadim Schechtman[1]

  • [1] Institut des Mathématiques de Toulouse, UPS, 118 route de Narbonne, 31062 Toulouse, France

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 2, page 353-375
  • ISSN: 0240-2963

Abstract

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We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the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.

How to cite

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Schechtman, Vadim. "Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)." Annales de la faculté des sciences de Toulouse Mathématiques 22.2 (2013): 353-375. <https://eudml.org/doc/275393>.

@article{Schechtman2013,
abstract = {We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the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.},
affiliation = {Institut des Mathématiques de Toulouse, UPS, 118 route de Narbonne, 31062 Toulouse, France},
author = {Schechtman, Vadim},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {2},
pages = {353-375},
publisher = {Université Paul Sabatier, Toulouse},
title = {Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)},
url = {https://eudml.org/doc/275393},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Schechtman, Vadim
TI - Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...)
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 2
SP - 353
EP - 375
AB - We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the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.
LA - eng
UR - https://eudml.org/doc/275393
ER -

References

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  10. Griffiths (P.), Harris (J.).— On Cayleys explicit solution to Poncelet’s porism, Enseign. Math. (2) 24, p. 31-40 (1978). Zbl0384.14009MR497281
  11. Hardy (G.H.).— Ramanujan, Cambridge, 1940 (AMS Chelsea, 1991). Zbl0025.10505
  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). 
  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). Zbl0060.46001MR10315
  18. Snape (J.).— Applications of elliptic functions in classical and algebraic geometry, Dissertation, Durham. 

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