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Poncelet's Theorem in Finite Projective Planes and Beyond

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Poncelet's Theorem in Finite Projective Planes and Beyond Research Collection Doctoral Thesis Poncelet's Theorem in finite projective planes and beyond Author(s): Kusejko, Katharina Publication Date: 2016 Permanent Link: https://doi.org/10.3929/ethz-a-010682687 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23366 Poncelet's Theorem in finite projective planes and beyond A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Katharina Kusejko MSc Math. ETH Zurich born on 01.10.1989 citizen of Austria accepted on the recommendation of Prof. Dr. Norbert Hungerb¨uhler,advisor Prof. Dr. Alexander Pott, co-advisor 2016 "Perhaps I could best describe my experience of doing mathematics in terms of en- tering a dark mansion. One goes into the first room and it is dark, entirely dark. One stumbles around, bumping into the furniture. Gradually you learn, where each piece of furniture is. And finally, after six months or so, you find the light switch, you turn it on and suddenly, everything is illuminated and you can see exactly where you are." Andrew Wiles Zusammenfassung Die vorliegende Dissertation handelt von Kegelschnitten in endlichen projektiven Ebenen sowie von Kreisketten in endlichen M¨obiusebenen. Die Hauptresultate wur- den bereits in [22], [23], [26] und [24] ver¨offentlicht. Ein besonderer Fokus liegt auf dem Schließungssatz des franz¨osischen Mathematikers Jean-Victor Poncelet, der 1813 fur¨ Kegelschnitte in der reellen projektiven Ebene eine interessante Beobachtung zu Polygonzugen,¨ dessen Kanten tangential zu einem Kegelschnitt sind und dessen Ecken auf einem anderen Kegelschnitt liegen, forma- lisiert und bewiesen hat. Dieses klassische Resultat, welches zu Beginn der These im Detail erkl¨art wird, wurde in den letzten 200 Jahren wiederholt in verschiedenen Teilgebieten der Mathematik - unter anderem in der Analysis, der Geometrie und der Theorie elliptischer Kurven - aufgegriffen. Endliche projektive Ebenen, welche uber¨ einem endlichen K¨orper definiert werden, erfullen¨ eine endliche Version des Poncelet'schen Satzes. Diese Tatsache wird in meh- reren Kapiteln mit Hilfe unterschiedlicher Techniken, darunter kombinatorische, syn- thetische und algebraische Methoden, bewiesen. Im Falle jener Ebenen, welche nicht uber¨ endlichen K¨orpern konstruiert werden, ist die Poncelet'sche Schließungseigen- schaft nicht zwangsl¨aufig gegeben. Dies wird anhand endlicher projektiver Ebenen der Ordnung 9 im Detail erl¨autert. Weiterfuhrende¨ Fragen zum Schließungssatz von Poncelet, wie etwa ein Poncelet'sches Schließungskriterium, werden fur¨ Spezialf¨alle von Kegelschnittpaaren betrachtet. Dies liefert unter anderem einen neuen Zugang zu trigonometrischen Identit¨aten in endlichen projektiven Ebenen, fur¨ welche ein geometrischer Zugang zur Trigonometrie a priori nicht gegeben ist. Im Vordergrund des behandelten Spezialfalles stehen Kegelschnitte in Diagonalform, was in der Folge Fragen zur Diagonalisierbarkeit von Kegelschnittpaaren im Endlichen aufwirft. Mit Hilfe kombinatorischer Uberlegungen¨ werden Bedingungen fur¨ die Diagonalisierbar- keit pr¨asentiert und Unterschiede zum reellen Fall erkl¨art. Eine dem Poncelet'schen Satz verwandte Aussage ist jene des Schweizer Mathema- tikers Jakob Steiner, welche von Ketten sich beruhrender¨ Kreise handelt. Hier wird eine analoge Aussage fur¨ Steiner-Ketten in endlichen M¨obiusebenen, welche uber¨ endlichen K¨orpern konstruiert werden, im Detail studiert. Ahnlich¨ wie im Satz von Poncelet fuhren¨ auch hier die Berechnungen eines Schließungskriteriums fur¨ Kreis- ketten zu einer Interpretation trigonometrischer Identit¨aten in endlichen Ebenen. Durchgehend werden die jeweiligen Eigenschaften reeller und endlicher Ebenen ver- glichen, und vor allem erstaunliche Parallelen und kontraintuitive Unterschiede her- vorgehoben. Abstract This dissertation is mainly about conics in finite projective planes and chains of circles in finite M¨obiusplanes. The main results are already published in [22], [23], [26] and [24]. A particular focus lies on Poncelet's Theorem, a famous result by the French math- ematician Jean-Victor Poncelet. In 1813, he made an interesting observation about two conics in the real projective plane and a polygonal chain with vertices on one conic and the edges tangent to the other conic. This classical result, which is dis- cussed in detail at the beginning of the thesis, was studied repeatedly in the last 200 years in various areas of mathematics - such as analysis, geometry and in the theory of elliptic curves. Finite projective planes defined over a finite field satisfy a finite version of Pon- celet's Theorem. This is proven in several chapters of this thesis by using different mathematical tools, such as combinatorial, synthetic and algebraic methods. In the case of finite projective planes not defined over finite fields, the finite version of Poncelet's Theorem is proven to be wrong. This is shown by examining the finite projective planes of order 9. Further questions related to Poncelet's Theorem, such as a criterion for a polygonal chain to close up, are discussed for special cases of conics. This leads to a new approach of trigonometric identities in finite projective planes, for which a geometric interpretation of trigonometry is a priori not given. A particular focus lies on conics in diagonal form and therefore questions about the diagonalisation of conics in finite projective planes are discussed. Conditions on the simultaneous diagonalisation of conic pairs are presented by using combinatorial methods. These conditions are then compared to the real case. A result related to Poncelet's Theorem deals with chains of touching circles and is due to the Swiss mathematician Jacob Steiner. In this dissertation, we discuss a similar statement for Steiner chains in finite M¨obiusplanes constructed over finite fields. Again, by investigating a criterion for the chain of circles to close, we get an interpretation of trigonometric identities in finite M¨obiusplanes. Throughout, the properties of real and finite planes are compared, with an emphasis on surprising parallels and counter-intuitive differences. Contents 1 Introduction 1 2 Preliminaries 7 2.1 Finite projective planes . .7 2.2 Finite Fields . .9 2.3 The Desarguesian Plane . 10 2.4 Conics . 12 2.5 Pencil of conics . 13 3 Poncelet's Theorem 15 3.1 The classical version . 15 3.2 Interpretation in finite planes . 17 3.3 Degenerate Cases . 19 3.4 Examples . 21 4 Poncelet's Theorem in finite projective planes of order 9 25 4.1 Poncelet's Theorem in PG(2; 9)..................... 25 4.2 Poncelet's Theorem in the finite projective planes over S ....... 33 4.2.1 The miniquaternion near-field S ................. 33 4.2.2 The plane Ω . 35 4.2.3 The plane ΩD ........................... 38 4.2.4 The plane Ψ . 40 5 Some proofs for Poncelet's Theorem 45 5.1 A synthetic proof . 45 5.2 A combinatorial proof . 53 5.2.1 Preliminaries and preparation . 53 5.2.2 Poncelet's Theorem for triangles . 55 5.2.3 Proof of Poncelet's Theorem . 57 Contents 6 A Poncelet Criterion 63 6.1 Construction and properties . 63 6.2 Poncelet's Theorem for conics Ok .................... 64 6.3 Relations for pairs of conics . 66 6.4 A Poncelet Criterion . 72 6.4.1 Poncelet coefficients . 72 6.4.2 Poncelet polynomials . 80 6.5 Comparison to other methods . 82 6.5.1 Comparison to the Euclidean Plane . 82 6.5.2 Comparison to Cayley's Criterion . 85 7 Simultaneous Diagonalization 87 7.1 The disjoint case . 87 7.2 The nondisjoint case . 94 7.3 Summary . 98 8 Steiner's Theorem in the finite M¨obiusplane M(q) 99 8.1 Miquelian M¨obius planes . 100 8.2 Steiner's Theorem in M(q)........................ 100 8.3 The plane M(5) .............................. 103 8.4 Existence and length of Steiner chains . 105 8.5 Generalization . 110 8.5.1 A M¨obiusinvariant for pairs of circles . 110 8.5.2 Transformation of non-intersecting circles into concentric circles111 8.5.3 General criterion for Steiner chains . 113 8.6 Comparison to the Euclidean plane . 113 Bibliography 117 Dank 121 Curriculum Vitae 123 Chapter 1 Introduction This thesis is about a branch of geometry called finite projective geometry. In a nutshell, projective geometry is like Euclidean geometry with additional infinite elements. Imagine standing on a train track and looking at the horizon. Although the tracks are parallel, they seem to meet at infinity. 1 This is exactly what projective geometry deals with. In particular, there are no parallel lines in a projective plane. This explains the idea of the term projective. Now for the finiteness. In our continuous world there are infinitely many points on every line. But one can describe a concept of geometry also by taking only finitely many points. In particular, a line will be a collection of finitely many points. This branch of geometry is relatively new and has for example applications in coding theory and cryptography. In this thesis, we are mainly interested in one particular property of a finite projective plane, namely the property of being a Poncelet plane. Jean-Victor Poncelet (1788 - 1867) [36] was the first mathematician who formalized projective geometry, and it was him, in 1813, to state one of the most beautiful results in this field. One version reads as follows. 1Picture taken from http://www.thoughtsupport.net/how-to-get-back-on-track/ 1 Introduction Theorem (Poncelet's Theorem). Let C and C0 be two conics generally situated in the real projective plane. If it is possible to find an m-sided polygon, m 3, such that the vertices lie on C0 and the sides are tangent to C, then there are≥ infinitely many other such m-sided polygons. Moreover, one cannot find such an m0-sided polygon for m = m0 for the same pair of conics C and C0. 6 Figure 1.1: Example of Poncelet's Theorem In Chapter 2, we will start with some preliminaries about finite projective planes.
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