Closure set by Poncelet

Two elliptical conic sections and, for example, two (of an infinite number) pentagons .

The closure set of Poncelet is a set of projective geometry and says: Can a -Eck ( ) at the same time a conic rewrite and another conic enroll, there are infinitely many more -Ecke with this property. ${\displaystyle n}$ ${\displaystyle n>2}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle n}$

Alternative formulation: , are conic sections. lie within . It then starts the following chain of structures: from a point on the tangent is to be drawn that in another point cuts by this point, the second tangent is to drawn so the figure formed by the tangent sections closes again at the point , so says the proposition that there is still an infinite number of other such figures to the conic sections , are. You can start with any other point from and get a closed polygon again. The polygons obtained in this way are also called Poncelet polygons. ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle N}$ ${\displaystyle C}$ ${\displaystyle M}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle D}$

In his Traité des propriétés projectives des figures of 1822, Jean-Victor Poncelet gave a (“synthetic”) geometric proof. Carl Gustav Jacobi (Journal for Pure and Applied Mathematics, Vol. 3, 1828) gave a proof with elliptic functions. A modern proof by Phillip Griffiths makes it clear that the group properties of elliptical curves are behind this theorem. According to Griffiths, the theorem is equivalent to the law of addition of elliptic integrals. Many other famous mathematicians have contributed to the theorem and its generalization, for example Arthur Cayley gave explicit conditions for when conic sections have such Poncelet polygons (Philosophical Magazine Vol. 6, 1852, 99, Phil.Trans.Royal Society Vol. 151 , 1861, p. 225, also in Henri Lebesgue : Les coniques. 1942). This is also shown from the standpoint of the theory of elliptic curves in Griffiths, Harris On Cayley's explicit solution to Poncelet's porism. L'enseignement Mathematique, 24 (1978).

The theorem is the prime example of a class of geometric problems called closure problems .

See also

• Closing phrase from Steiner

literature

• Wolf Barth , Thomas Bauer: Poncelet theorems. Expositiones Mathematicae, Vol. 14, 1996, 125-144.
• Henk Bos , C. Kers, Frans Oort , DW Raven: Poncelet's closure theorem. Expositiones Mathematicae, Vol. 5, 1987, p. 289.
• Kenji Ueno , Koji Shiga, Shigeyuki Morita: A Mathematical Gift II. American Mathematical Society 2000
• Dmitry Fuchs , Serge Tabachnikov A graph of mathematics , Springer Verlag 2011 (Chapter 29)
• Griffiths Complex analysis and algebraic geometry , Bulletin AMS, 1979, p. 607
• Griffiths, Harris: A Poncelet theorem in space. Comm. Math. Helvetici Vol. 53, 1977, p. 145.
• Griffiths: Variations on a Theorem of Abel. Inventiones Mathematicae Vol. 35, 1976.
• Rohn: The closure problem of Poncelet and its extensions. Annual report DMV 1913.
• Friedrich Dingeldey on the closing sentence in: Encyklopädie der Mathematischen Wissenschaften
• Stutz: On the application of the elliptical functions to Poncelet's closure problem. 1909, dissertation
• Hurwitz About infinitely ambiguous geometric problems, especially the closure problems, Mathematische Annalen, Volume 15, 1879, pp. 8-16
• Jacobi: About the application of the elliptical transcendent to a known problem of elementary geometry.
• Ervedoza, Pouchin Le théorème de Poncelet , 2003, French, pdf
• Serge Tabachnikov : Billards , Société Mathématique de France, Panoramas et Synthèses, 1995 (as well as Geometry and Billiards , Online, pdf )
• Prasolov, Tikhomirov Geometry , AMS 2001.
• Vladimir Dragović, Milena Radnović: Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances, Bulletin AMS, Volume 51, 2014, pp. 373–445, online

Web links

• Website of Prof. Dr. Th. Bauer from the University of Marburg
• Animation on the website of the University of Mainz, with state examination thesis by Ulrike Herr

Individual evidence

1. ↑ Simple proof for the special case of circles (pp. 179f). An extension to any conic sections (ellipses) comes from AA Panov (Moscow), see Alexander Shen Mathematical Entertainments , Mathematical Intelligencer 1998, No. 4, p. 31f