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Projective Configuration Theorems: Old Wine into New Wineskins

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Geometry in History

Abstract

We survey some recent results concerning projective configuration theorems in the spirit of the classical theorems of Pappus, Desargues, Pascal, …We hope that this modern take on the old theorems makes this evergreen topic fresh again. We connect configuration theorems to completely integrable systems, identities in Lie algebras of motion, modular group, and other subject of contemporary interest.

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Notes

  1. 1.

    See also a recent paper [50] for an extension of Kasner’s theorem from pentagons to Poncelet polygons.

  2. 2.

    The existence of caustics for strictly convex and sufficiently smooth billiard curves is proved in the framework of the KAM theory.

  3. 3.

    See [44] for a curious property of the centroids of Poncelet polygons.

  4. 4.

    See also [25] for another theorem of Ivory and its relation to billiards in conics.

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Acknowledgements

I am grateful to Schwartz for numerous stimulating discussions and to Hooper for an explanation of his work. I was supported by NSF grant DMS-1510055. This article was written during my stay at ICERM; it is a pleasure to thank the Institute for its inspiring and friendly atmosphere.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, USA

    Serge Tabachnikov

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  1. Serge Tabachnikov
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Correspondence to Serge Tabachnikov .

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Editors and Affiliations

  1. University of Mumbai & Department of Atomic Energy, Centre for Excellence in Basic Sciences, Mumbai, Maharashtra, India

    S. G. Dani

  2. Institut de Recherche Mathématique, University of Strasbourg, CNRS, Strasbourg Cedex, France

    Athanase Papadopoulos

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Tabachnikov, S. (2019). Projective Configuration Theorems: Old Wine into New Wineskins. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_9

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