Notes on Euclidean Geometry Kiran Kedlaya Based
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LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
On the Deformation of Inversive Distance Circle Packings, Ii
ON THE DEFORMATION OF INVERSIVE DISTANCE CIRCLE PACKINGS, II HUABIN GE, WENSHUAI JIANG ABSTRACT. We show that the results in [GJ16] are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo’s combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston’s theorem to some extent. 1. INTRODUCTION 1.1. Background. We continue our study about the deformation of inversive distance circle patterns on a surface M with triangulation T. If the triangulated surface is obtained by taking a finite collection of Eu- clidean triangles in E2 and identifying their edges in pairs by isometries, we call it in Euclidean background geometry. While if the triangulated surface is obtained by taking a finite collection of hyperbolic triangles in H2 and identifying their edges in pairs by isometries, we shall call it in hyperbolic background geometry. Consider a closed surface M with a triangulation T = fV; E; Fg, where V; E; F represent the sets of ver- tices, edges and faces respectively. Thurston [Th76] once introduced a metric structure on (M; T) by circle patterns. The idea is taking the triangulation as the nerve of a circle pattern, while the length structure on (M; T) can be constructed from radii of circles and intersection angles between circles in the pattern. More π precisely, let Φi j 2 [0; 2 ] be intersection angles between two circles ci, c j for each nerve fi jg 2 E, and let r 2 (0; +1) be the radius of circle c for each vertex i 2 V, see Figure 1.1. -
Extension of Algorithmic Geometry to Fractal Structures Anton Mishkinis
Extension of algorithmic geometry to fractal structures Anton Mishkinis To cite this version: Anton Mishkinis. Extension of algorithmic geometry to fractal structures. General Mathematics [math.GM]. Université de Bourgogne, 2013. English. NNT : 2013DIJOS049. tel-00991384 HAL Id: tel-00991384 https://tel.archives-ouvertes.fr/tel-00991384 Submitted on 15 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE Extension des méthodes de géométrie algorithmique aux structures fractales ■ ANTON MISHKINIS Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE THÈSE présentée par ANTON MISHKINIS pour obtenir le Grade de Docteur de l’Université de Bourgogne Spécialité : Informatique Extension des méthodes de géométrie algorithmique aux structures fractales Soutenue publiquement le 27 novembre 2013 devant le Jury composé de : MICHAEL BARNSLEY Rapporteur Professeur de l’Université nationale australienne MARC DANIEL Examinateur Professeur de l’école Polytech Marseille CHRISTIAN GENTIL Directeur de thèse HDR, Maître de conférences de l’Université de Bourgogne STEFANIE HAHMANN Examinateur Professeur de l’Université de Grenoble INP SANDRINE LANQUETIN Coencadrant Maître de conférences de l’Université de Bourgogne RONALD GOLDMAN Rapporteur Professeur de l’Université Rice ANDRÉ LIEUTIER Rapporteur “Technology scientific director” chez Dassault Systèmes Contents 1 Introduction 1 1.1 Context . -
On a Combinatorial Curvature for Surfaces with Inversive Distance Circle Packing Metrics
On a combinatorial curvature for surfaces with inversive distance circle packing metrics Huabin Ge, Xu Xu May 30, 2018 Abstract In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton [21]. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to α-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture [6]. We also use the combinatorial Ricci flow to study the corresponding α-Yamabe problem. Mathematics Subject Classification (2010). 52C25, 52C26, 53C44. 1 Introduction This is a continuation of our work on combinatorial curvature in [17]. This paper gen- eralizes our results in [17] to triangulated surfaces with inversive distance circle packing arXiv:1701.01795v2 [math.GT] 29 May 2018 metrics. Circle packing is a powerful tool in the study of differential geometry and geo- metric topology and there are lots of research on this topic. In his work on constructing hyperbolic structure on 3-manifolds, Thurston ([28], Chapter 13) introduced the notion of Euclidean and hyperbolic circle packing metrics on triangulated surfaces with prescribed intersection angles. -
NATURAL STRUCTURES in DIFFERENTIAL GEOMETRY Lucas
NATURAL STRUCTURES IN DIFFERENTIAL GEOMETRY Lucas Mason-Brown A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in mathematics Trinity College Dublin March 2015 Declaration I declare that the thesis contained herein is entirely my own work, ex- cept where explicitly stated otherwise, and has not been submitted for a degree at this or any other university. I grant Trinity College Dublin per- mission to lend or copy this thesis upon request, with the understanding that this permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement. Contents 1 Summary 3 2 Acknowledgment 6 I Natural Bundles and Operators 6 3 Jets 6 4 Natural Bundles 9 5 Natural Operators 11 6 Invariant-Theoretic Reduction 16 7 Classical Results 24 8 Natural Operators on Differential Forms 32 II Natural Operators on Alternating Multivector Fields 38 9 Twisted Algebras in VecZ 40 10 Ordered Multigraphs 48 11 Ordered Multigraphs and Natural Operators 52 12 Gerstenhaber Algebras 57 13 Pre-Gerstenhaber Algebras 58 irred 14 A Lie-admissable Structure on Graacyc 63 irred 15 A Co-algebra Structure on Graacyc 67 16 Loday's Rigidity Theorem 75 17 A Useful Consequence of K¨unneth'sTheorem 83 18 Chevalley-Eilenberg Cohomology 87 1 Summary Many of the most important constructions in differential geometry are functorial. Take, for example, the tangent bundle. The tangent bundle can be profitably viewed as a functor T : Manm ! Fib from the category of m-dimensional manifolds (with local diffeomorphisms) to 3 the category of fiber bundles (with locally invertible bundle maps). -
Lectures – Math 128 – Geometry – Spring 2002
Lectures { Math 128 { Geometry { Spring 2002 Day 1 ∼∼∼ ∼∼∼ Introduction 1. Introduce Self 2. Prerequisites { stated prereq is math 52, but might be difficult without math 40 Syllabus Go over syllabus Overview of class Recent flurry of mathematical work trying to discover shape of space • Challenge assumption space is flat, infinite • space picture with Einstein quote • will discuss possible shapes for 2D and 3D spaces • this is topology, but will learn there is intrinsic link between top and geometry • to discover poss shapes, need to talk about poss geometries • Geometry = set + group of transformations • We will discuss geometries, symmetry groups • Quotient or identification geometries give different manifolds / orbifolds • At end, we'll come back to discussing theories for shape of universe • How would you try to discover shape of space you're living in? 2 Dimensional Spaces { A Square 1. give face, 3D person can do surgery 2. red thread, possible shapes, veered? 3. blue thread, never crossed, possible shapes? 4. what about NE direction? 1 List of possibilities: classification (closed) 1. list them 2. coffee cup vs donut How to tell from inside { view inside small torus 1. old bi-plane game, now spaceship 2. view in each direction (from flat torus point of view) 3. tiling pictures 4. fundamental domain { quotient geometry 5. length spectra can tell spaces apart 6. finite area 7. which one is really you? 8. glueing, animation of folding torus 9. representation, with arrows 10. discuss transformations 11. torus tic-tac-toe, chess on Friday Different geometries (can shorten or lengthen this part) 1. describe each of 3 geometries 2. -
Arxiv:1806.04129V2 [Math.GT] 23 Aug 2018 Dense, Although in Certain Cases It Is Locally the Product of a Cantor Set and an Interval
ANGELS' STAIRCASES, STURMIAN SEQUENCES, AND TRAJECTORIES ON HOMOTHETY SURFACES JOSHUA BOWMAN AND SLADE SANDERSON Abstract. A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination. A homothety of the plane is a similarity that preserves directions; in other words, it is a composition of translation and scaling. A homothety surface has an atlas (covering all but a finite set of points) whose transition maps are homotheties. Homothety surfaces are thus generalizations of translation surfaces, which have been actively studied for some time under a variety of guises (measured foliations, abelian differentials, unfolded polygonal billiard tables, etc.). Like a translation surface, a homothety surface is locally flat except at a finite set of singular points|although in general it does not have an accompanying Riemannian metric|and it has a well-defined global notion of direction, or slope, again except at the singular points. -
On Inversions in Central Conics
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 9 (2020), No. 1, 40 - 51 ON INVERSIONS IN CENTRAL CONICS ROOSEVELT BESSONI AND GUY GREBOT Abstract. We use plain euclidean geometry to analyse the structure of an inversion in an ellipse or in a hyperbola. By this formulation, results on inversions in circles are directly extended to results on inversions in ellipses. 1. Introduction Jacob Steiner is quoted in [5] as the first mathematician to formalize the bases of inversive geometry in a text dated from 1824 and published after his death by B¨utzberger. In 1965, the mathematician Noel Childress [1] extended the concept of circular inversion to central conics, ellipse or hyperbola. In 2014, Ramirez [6][7] showed other properties concerning inversion in ellipses and Neas [4], in 2017, looked at anallagmatic curves under inversion in hyperbolae. Inversion in circles is a geometrical topic which is usually developped without analytic geometry [8]. Strangely enough, we could not find any work concerning inversion in central conics using plain euclidean geometry arguments. In the studied publications the basic framework is analytical geometry. Also, as far as we could reach, the structure of an inversion in a central conic is not mentioned in the cited works and it seems to us that the analytical framework tends to hide this property. In this communication, we use plain euclidean geometry to analyse the structure of an inversion in an ellipse or in a hyperbola. We show that an inversion in a central conic is given by the composition of compressions and an inversion in a circle or in an equilateral hyperbola; in this sense, an inversion in a central conic is just an affine deformation of an inversion in a circle or in an equilateral hyperbola. -
Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings
Transformation Groups c The Author(s) (2020) Vol. 26, No. 1, 2021, pp. 31–68 CROSS RATIOS ON BOUNDARIES OF SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS J. BEYRER∗ Department of Mathematics Universit¨atHeidelberg Im Neuenheimer Feld 205, 69120 Heidelberg, Germany [email protected] Abstract. We generalize the natural cross ratio on the ideal boundary of a rank one symmetric space, or even CAT(−1) space, to higher rank symmetric spaces and (non- locally compact) Euclidean buildings. We obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties of those cross ratios; for example that (under some restrictions) periods of hyperbolic isometries give back the translation vector. In addition, we show that cross ratio preserving maps on the chamber set are induced by isometries and vice versa, | motivating that the cross ratios bring the geometry of the symmetric space/Euclidean building to the boundary. Introduction Cross ratios on boundaries are a crucial tool in hyperbolic geometry and more general negatively curved spaces. In this paper we show that we can generalize these cross ratios to (the non-positively curved) symmetric spaces of higher rank and thick Euclidean buildings with many of the properties of the cross ratio still valid. 2 2 On the boundary @1H of the hyperbolic plane H there is naturally a multi- plicative cross ratio defined by z1 − z2 z3 − z4 crH2 (z1; z2; z3; z4) = z1 − z4 z3 − z2 2 2 when considering H in the upper half space model, i.e., @1H = R [ f1g. This cross ratio plays an essential role in hyperbolic geometry. -
Dan Pedoe, on Geometrical Matters, P 67-74Mathschron009-008.Pdf
ON GEOMETRICAL MATTERS Dan Pedoe Dedicated to H.G. Forder on his 90th birthday (received 29 June, 1979) Geometers are sadly aware of the present depressed state of their art. The "new" math, swept elementary geometry aside, in a frantic effort to keep up with the Russians, who had convinced the Western world of their technological superiority by launching Sputnik. The "educators" who led American mathematics disregarded, or were sublimely unconscious of the fact that Russian high schools, then and now, study geometry intensely. Yaglom's new book [16]reveals the extent to which the best high schools in Russia plumb non-Euclidean geometry. In America, publishers demand an assured sale of at least 50,000 copies for any book on elementary mathematics before they will handle it, and only calculus books and such make the grade. Any book on geometry, inadvertently published, only sells to a limited extent, and if the publisher has "inventory problems", (that is, if he suffers from a lack of warehouse space, and who does not?), books are soon pulped. Fortunately there are still University Presses, and the excellent Chelsea and Dover publishers who keep books in print, and Forder's remarkable Calculus of Extension [4] is still obtainable, although some of his early books on geometry have vanished. One of my pulped books, Circles, has been reprinted, with added problems and solutions, by Dover Books [10]. Those who defend the present state of mathematics are quick to point out that a lot of geometry is still being studied and taught, under chapter headings such as "combinatorics", "tesselations", and so on. -
Pst-Euclide a Pstricks Package for Drawing Geometric Pictures; V.1.75
pst-euclide A PSTricks package for drawing geometric pictures; v.1.75 Dominique Rodriguez Herbert Voß Liao Xiongfei September 29, 2020 The pst-eucl package allow the drawing of Euclidean geometric figures using LATEX macros for specifying mathematical constraints. It is thus possible to build point using common transformations or intersections. The use of coordinates is limited to points which controlled the figure. I would like to thanks the following persons for the help they gave me for development of this package: • Denis Girou pour ses critiques pertinentes et ses encouragement lors de la décou- verte de l’embryon initial et pour sa relecture du présent manuel; • Michael Vulis for his fast testing of the documentation using VTEX which leads to the correction of a bug in the PostScript code; • Manuel Luque and Olivier Reboux for their remarks and their examples. • Alain Delplanque for its modification theorems on automatic placing of points name and the ability of giving a list of points in \pstGeonode. Thanks to: Manuel Luque; Thomas Söll. 1 Contents 2 Contents I. The package 5 1. Special specifications 5 1.1. PSTricks Options ........................................ 5 1.2. Conventions .................................... ....... 5 2. Basic Objects 5 2.1. Points......................................... ...... 5 2.2. Segmentmark.................................... ...... 7 2.3. Segmentlabels .................................. ....... 8 2.4. Triangles...................................... ....... 9 2.5. Angles ........................................ -
Download Inversive Geometry Free Ebook
INVERSIVE GEOMETRY DOWNLOAD FREE BOOK Frank Morley, F. V. Morley | 288 pages | 15 Jan 2014 | Dover Publications Inc. | 9780486493398 | English | New York, United States Inversive Geometry Fukagawa and D. MR 48, However, inversive geometry is the larger study since it includes the raw inversion in a circle not yet made, with conjugation, into reciprocation. MR 47 MR 19 There may be zero, one, or two points. Selecting the appropriate Inversive Geometry in the Manipulatethe radii are adjusted according to the position of the slanted Inversive Geometry for this third blue circle to exist. It provides an exact solution to the important problem of converting between linear and circular motion. MR 38and 50 Two pairs of inverse points are either collinear with or concyclic on a circle Inversive Geometry to. Help Learn to edit Community portal Recent changes Upload file. Die Inversion und ihre Anwendungen. Magnus, W. Two intersecting circles are orthogonal if they have perpendicular tangents at either point of intersection. New York: Wiley, pp. Courant, R. Has it been superseded by other geometries, i. Webb, C. Home Questions Tags Users Unanswered. In addition, the curve to which a given curve is transformed under inversion is called its inverse curve or more simply, its "inverse". Hence, the angle between two curves in the Inversive Geometry is the same as the angle between two curves in the hyperbolic space. The complex analytic inverse Inversive Geometry is conformal and its conjugate, circle inversion, is anticonformal. Coolidge, J. The two orange disks indicate adjustable tangency points and of the circles and the parallel lines.