We give an exposition of unpublished fragments of Gauss where he discovered (using a work of Jacobi) a remarkable connection between Napier pentagons on the sphere and Poncelet pentagons on the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 = {Vadim Schechtman},
title = {Pentagramma mirificum and elliptic functions {(Napier,} {Gauss,} {Poncelet,} {Jacobi,} ...)},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {353--375},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 22},
number = {2},
year = {2013},
doi = {10.5802/afst.1375},
zbl = {1275.33026},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1375/}
}
TY - JOUR AU - Vadim Schechtman TI - Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...) JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2013 SP - 353 EP - 375 VL - 22 IS - 2 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1375/ DO - 10.5802/afst.1375 LA - en ID - AFST_2013_6_22_2_353_0 ER -
%0 Journal Article %A Vadim Schechtman %T Pentagramma mirificum and elliptic functions (Napier, Gauss, Poncelet, Jacobi, ...) %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2013 %P 353-375 %V 22 %N 2 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1375/ %R 10.5802/afst.1375 %G en %F AFST_2013_6_22_2_353_0
Vadim Schechtman. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1375/
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