Fundamentals of Geometry Math 3181

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1 Fundamentals of Geometry Math 3181 January 10 th, 2012 Instructor: Dr. Franz Rothe Office: Fretwel 345D Phone: Office hours: TWR 11 a.m. - 1 p.m. and by appointment Best book: Manuscript: Robin Hartshorne Geometry: Euclid and Beyond second printing Springer 2000 ISBN Franz Rothe Several Topics from Geometry frothe/3181all.pdf Examinations: two tests Final Exam M, May 7, 2 p.m p.m. in FRET 206 or: T, May 8, 2 p.m p.m. in FRIDY Homework sheets some weekly quizzes Attendance and participation Grading Letter grades are only given for the entire course. They are assigned based on the part of the total possible credit you have gained. I intend to use the following scheme: 1000 points points A 879 points points B 759 points points C 639 points points D otherwise F This table is only intended to be a guideline. It assumes the total of 1000 points. The grading will be adapted to the total actually occurring at the end of the course. Tools You need to have a straightedge, compass, protractor and colored pencils for this class. As simple as these tools are, they are really essential. Please load on your computer the drawing system It is really useful. The files with your drawings can be exported easily in different formats, for example as.png (portable network graphics). During the semester, please print out the relevant parts of my manuscript and make them available to you as hardcopy. 1

2 Course Topics: Most topics are included in the manuscript, or very good available in Hartshorne s book Euclid and beyond. It is the best geometry book I know and rather cheap to get on amazon.com. Too, you can try the website abebooks.com. CORE TOPICS Part I Neutral Geometry Section 1 Incidence and Order 1.1 Logic 1.2 David Hilbert s axiomatization of Euclidean geometry Introduction from Hilbert s Foundations of Geometry Hilbert s axioms Section 3 Congruence of Segments, Angles and Triangles Congruence of segments 3.2 Some elementary triangle congruences 3.3 Congruence of angles 3.4 SSS congruence 3.5 The equivalence relation of angle congruence 3.6 Constructions with Hilbert tools 3.7 The exterior angle theorem and its consequences Section 4 Measurement and Continuity The Archimedean axiom Section 5 Legendre s Theorems The First Legendre Theorem 5.2 The Second Legendre Theorem 5.3 The alternative of two geometries 5.4 What is the natural geometry? Part II Euclidean Geometry Section 9 Some Euclidean Geometry of Circles Thales Theorem 9.2 Rectangles and the converse Thales Theorem 9.3 Construction of tangents to a circle 9.4 A bid of philosophy 9.5 Common tangents of two circles 9.6 Angles in a circle Section 11 Euclidean Geometry and Ordered Fields Ordered fields 11.2 Construction of the field of segment arithmetic 2

3 Commutativity The distributive law The associative law A direct proof of commutativity of segment arithmetic Alternative proof of associativity 11.3 A Hierarchy of Cartesian planes Section 12 Similar Triangles Basic properties from Euclid 12.2 Some exercises 12.3 Secants in a circle 12.4 Trigonometry 12.5 Common tangents of two circles Section 13 Area in Euclidean Geometry Equidecomposable and equicomplementable figures 13.2 The Theorem of Pythagoras and related results 13.3 Dudeney s dissection problem 13.4 Area of rectilinear figures Very Important References [5] H.E. Dudeney, The Canterbury Puzzles, and other curios problems, Nelson, London, [8] Marvin J. Greenberg, Euclidean and Non-Euclidean Geometry, fourth ed., W.H. Freeman and Company, New York, [9] Marvin J. Greenberg, Old and new results in the foundations of elementary plane Euclidean and Non-Euclidean geometries, The American Mathematical Monthly 117 (2010), [10] Robin Hartshorne, Geometry: Euclid and Beyond, second printing, Springer, [12] David Hilbert, Foundations of Geometry, 2nd English ed., Open Court, La Salle, [17] Stefan Mykytiuk and Abe Shenitzer, Four significant axiomatic systems and some of the issues associated with them, The American Mathematical Monthly 102 (1995), 62. [18] David Park, The Grand Contraption the world in myth, number, and chance, second printing, Princeton University Press, [23] John Stillwell, The Four Pillars of Geometry, Springer, Good Additional Topics Part I Neutral Geometry 21 1 Incidence and Order Incidence geometry 1.4 The axioms of order and their consequences 3

4 1.5 Interior and exterior of a triangle 1.6 Left and right, orientation 1.7 The restricted Jordan Curve Theorem 3 Congruence of Segments, Angles and Triangles SSA congruence 3.9 Reflection 4 Measurement and Continuity Axioms related to completeness Cantor s axiom Dedekind s axiom Hilbert s axiom of completeness 6 Neutral Geometry of Circles and Continuity Immediate consequences of neutral geometry 6.2 The tangent is the limiting position of a secant 6.3 Mutual placement of two circles 6.4 Continuity principles for circles 6.5 Continuity principles for circles are independent of Hilbert s axioms 6.6 Derivation of continuity principles from Dedekind s axiom 8 Towards a Natural Axiomatization of Geometry The Uniformity Theorem 8.2 A hierarchy of planes 8.3 Wallis axiom 8.4 Proclus Theorem 8.5 More about Aristole s axiom Part II Euclidean Geometry Pappus, Desargues and Pascal s Theorems Pappus Theorem 10.2 Desargues Theorem 10.3 Pascal s Theorem 13 Area in Euclidean Geometry About the volume of polyhedra 13.6 The role of the Archimedean axiom 13.7 The parallelogram equation 13.8 Heron s formula for the area of a triangle 13.9 Algebraic relations for the pieces of a triangle 14 Standard Euclidean Triangle Geometry The circum-center 4

5 14.2 Double and half size triangles 14.3 The centroid 14.4 The orthocenter 14.5 The in-circle and the three ex-circles 14.6 The road to the orthocenter via the orthic triangle 14.7 The Euler line 15 Harmonic Points The Theorems and Menelaus and Ceva 15.2 The circle of Apollonius 15.3 An application to electrostatics 15.4 The perspective view 16 Advanced Euclidean Geometry A Euclidean egg 16.2 The egg built from inside 16.3 A Euclidean trisection of a segment 16.4 Some triangle constructions 16.5 Archimedes Theorem of the broken chord 16.6 The Theorem of Collignon 16.7 Vectors and special quadrilaterals 16.8 The Theorem of Ptolemy 16.9 The quadrilateral of Hjelmslev 17 The Regular Pentagon The Euclidean construction with the Golden Ratio 17.2 Relation between the sides of pentagon and 10-gon 17.3 The construction with Hilbert tools 17.4 Variants of the Euclidean construction 17.5 A false pentagon 18 Inversion by a Circle Definition and construction of the inverted point 18.2 The gear of Peaucollier 18.3 Invariance properties of inversion 19 A Glimpse at Elliptic Geometry Elliptic geometry is derived from spherical geometry 19.2 The conformal model 19.3 Falsehood of the exterior angle theorem 19.4 Area of a spherical triangle 19.5 Does Pythagoras imply the parallel postulate? Main results obtained by calculus Examples for the lunes of Pythagoras 5

6 19.6 The stereographic projection The stereographic projection is an inversion Many congruent angles 22 Trisection of an Angle and the Delian Problem Trisection by Archimedes 22.2 Trisection by Nicomedes 22.3 Trisection by origami 22.4 Construction of the cubic root by two-marked ruler 22.5 Definition and equations of the concoid 22.6 Nicomedes construction of the cube root and the concoid 22.7 Duplication of the cube by two-marked ruler 22.8 Duplication of the cube and the curve of Agnesi 22.9 A close look at Nicomedes trisection Trisection and the concoid Archimedes trisection yields more solutions Archimedes trisection and the concoid 23 The Heptagon The construction 23.2 Trigonometric calculations 23.3 Using complex numbers 23.4 Geometric proof 6

7 Attendance The general attendance rules of UNCC do apply. I expect you to be in the class regularly. I am willing to disregard 2 absences. For every unexcused absence more, 5 points are deducted from the course credit. Please make sure that you actually get the list of attendance that I pass around. If your name is missing, please add it to the list. Whatever the reason for your absence, you have to get all material you missed. Absence is no excuse for missing an exam, not knowing or not being prepared. Discipline Be aware that withdrawal from a course after the deadline to drop is nearly impossible. The instructor has no authority to assign a W grade after the drop date. Therefore make up your mind early in the first week of the semester, which course to take and check whether you can manage your workload. Please check relevant drop dates. It is necessary to create an atmosphere of politeness and concentration. I intend to speak loud enough, but within natural limits. English is not my native language. If you a have a question, please speak up, such that everybody can hear you. All cheating in the tests is clearly forbidden. There are things that are not tolerable, as for example regular use of cell phone in class. Not allowed are talking in class, coming late or leaving early, using cell phones during class time (at least regularly, there may be exceptions to people with sick relative, babies at home, and so on.) Further reminders Since nobody can control what material is available to you during tests, I announce this guideline, which I believe is fair: Your answers have to be not only formally correct, but even more important: they have to deal with the problem in question. I consider to deduct points, if you answer not only the present but some other problems without making clear what you do. Your work has to have the quality to be understandable by yourself or some friend also some time, say a year later. Final The course has a final prepared by the instructor. I shall adapt the exact content of the final to the selection of the material that I have actually stressed during the course. As you can see from the grading scheme, it is essential to get a decent grade on that final. To this end, you need to have a working knowledge of the entire course material. Otherwise you can easily loose a letter grade, or more, because of the common final. Do not fool yourself thinking that having heard about a procedure means actually mastering it. We shall learn to appreciate careful definitions and correct statement of questions, methods, theorems and results. We have to develop the mathematical abilities required for successful problem-solving which include: recognizing similar problems in different disguises, adjusting the notation to the present case, preparing appropriate drawings, breaking up a problem in several steps, going, if appropriate, to a more general or a more special case, transferring a known method to a different situation, 7

8 giving answers in precise formulation. Goals of the Course 1. To deliver a sound foundation of geometry, keeping up with the standard of rigor established by Hilbert in his ground breaking work Foundations of Geometry of To learn Euclidean geometry thoroughly, especially congruence, similarity, area, constructions. 3. To discuss the theory of segment field, proportions and area in Euclidean geometry, as done by Hilbert in detail. 4. To discuss the history of the parallel postulate, the attempts to prove it, and finally the discovery of its independence. 5. To understand different axiomatic systems of geometry, the relations of their parts, the importance of models on one side, and general theorems on the other side. 8

Fundamentals of Geometry Math 3181

Fundamentals of Geometry Math 3181 January 10 th, 2013 Instructor: Dr. Franz Rothe Office: Fretwel 345D Phone: 687 4908 Office hours: TR 4 p.m. - 5 p.m. and W 11 a.m. - 1 p.m. and by appointment Best book: