Introduction

1. Introduction • SSCM 4623 (Non-Euclidean Geometry) • Instructor: Dr. Niki Anis bin Ab Karim • H/P: +60127534037 (also on WhatsApp) • E-mail: nikianis@utm.my • Room: C13 310 (make an appointment!)

2. Synopsis • This course is a survey of main concepts of Euclidean geometry with the emphasis on the axiomatic approach, constructions and logic of proof including historical aspects. • A study of axioms of Euclidean geometry, inference rule, some basic theorems of Euclidean geometry and rigorous proofs will be offered. • Non-Euclidean geometry is introduced. • The similarities and differences between Euclidean and non-Euclidean geometries will be discussed

3. Course Objectives • Understand geometry as a postulational system. • Use the basic assumptions, techniques, and constructions of Euclidean geometry. • Understand the consequences of the Euclidean parallel postulate. • Demonstrate familiarity with non-Euclidean geometry in particular Hyperbolic geometry and Elliptic geometry. • Solve problems and prove theorems in non-Euclidean geometry.

4. Syllabus • Foundations and History of Geometry: Sets, logic, deductive proofs, direct and indirect proofs, axioms, definitions and theorems. Early origins of geometry, Thales and Pythagoras, the rise of axiomatic method. Properties of axiomatic systems and Euclid’s axiomatic geometry. • Euclidean Geometry: Angles, lines and parallels. Congruent triangle and Pasch’s axiom. Measurement in Euclidean geometry. Similar triangles and circle geometry. Euclidean Isometries. • Non-Euclidean Geometry: Background and history. Introduction to Hyperbolic geometry and some basic results. Lambert quadriterals and triangles. Area in hyperbolic geometry. Introduction to Elliptic geometry. Basic results in Elliptic geometry. Proof of the consistency of non-Euclidean geometry.

5. References • The Internet (of course) • James R. Smart (1988), Modern Geometries, 5th. Edition. Brook/Cole Publishing Co. • Marvin J. Greenberg (1974), Euclidean and Non-Euclidean Geometries, 2nd. Edition. W. H. Freeman and Co. • Richard L. Faber (1983), Foundations of Euclidean and Non-Euclidean Geometry, Marcel Dekker Inc. • David Gans (1973), Introduction to Non-Euclidean Geometry, Academic Press.

6. Evaluation

7. Elems. of Mathematical Facts

8. Elems. of Mathematical Facts • Definitions: Used to identify the elements in a statement and how they interact with each other. • Axioms: A statement accepted without doubt, usually as starting premise. aka “postulate”, “assumption” • Theorem: A statement proven based on previous elements (theorems, definitions, axioms). • Proof: Process of logical argument based on a consistent deductive system. • Mathematicians as the lawyers of Science?

9. Elems. of Mathematical Facts • Statement: “1+1=2” • Modern Proof: (algebra style) • Let x = 1 and y = 1 • x + y = 1 + 1 = 2 QED (Quod Erat Demonstrandum) • Definitions: “unknowns”, “algebraic addition”, “equality”, etc. • Axioms: “adding two variables is the same as adding the numbers they represent”

10. Elems. of Mathematical Facts • Statement: “1+1=2” • “Classical” Proof: • You have one item in a particular collection. • You add one more item to it. • Count the number of items in the collection and you get two items. • Therefore adding a collection of one item to one item gives you a collection of two items. QED • Definitions: “collection”, “item”, “adding” • Axioms: “adding an item to a collection of items makes them part of the same collection”

11. Elems. of Mathematical Facts • Things to consider: • What does “logically consistent” mean? • Why is set notation used for mathematical proofs? • Everything we do here are playthings of the mind… and yet they do show up in Nature. • Physical Scientists observe the universe, but Mathematicians provide the means to understand & speak about it. (Hence “queen” to the “king”, paraphrasing Einstein.) • Axioms/assumptions can be questioned to give rise to new fields of study. E.g. fuzzy sets/numbers, non-Euclidean geometry.

12. Proof Methods • Direct proof • Proof by mathematical induction • Proof by contraposition • Proof by contradiction • Proof by construction • Proof by exhaustion • Probabilistic proof • Combinatorial proof • Nonconstructive proof • Statistical proofs in pure mathematics • Computer-assisted proofs

13. Proof Methods • Direct proof • Proof by mathematical induction • Proof by contraposition • Proof by contradiction • Proof by construction • Proof by exhaustion • Probabilistic proof • Combinatorial proof • Nonconstructive proof • Statistical proofs in pure mathematics • Computer-assisted proofs

14. Proof Methods • Direct proof: Use axioms, theorems, definitions to show a conclusion based on a premise. • E.g.“Two even numbers added together gives another even number.”

15. Proof Methods • Proof by mathematical induction: when given a statement of the type “P is true for all cases n” • Show case n = 1 is true. • Show case n+1 is true when case n is true. • E.g.“2n–1 is odd for all positive integers n”

16. Proof Methods • Proof by contraposition: When a statement can expressed as “if P is true, then Q is true” then proving “if Q is false, then P is false”(the contrapositive) also proves the statement.I.e.P → Q is equivalent to Q* → P* • E.g. “Given an integer x: if x2 is even, then x is even.”

17. Proof Methods • Proof by contradiction: Given a statement, show that the statement being false leads to a contradiction. (“reductio ad absurdum”) • E.g. “√2 is an irrational number.”

18. Proof Methods • Proof by construction: Show something exists by constructing an explicit example. Alternatively, show something is invalid for all possibilities by making a counterexample. • E.g. Georg Cantor’s diagonal argument defines a set of infinite numbers larger than set of integers.

19. Proof Methods • Proof by exhaustion: Do it until you’re tired. Show a statement is true for all cases by proving it’s true for each possible case. • E.g. Proof for the Four Colour Theorem in 2011 tested for more than 600 possible configurations.

20. Proof Methods (2) • For this course, proofs may also use visual proofs alongside the “classical” worded arguments. • “Ancient” mathematics did not have the benefit of modern notations and algebra, so this was de facto way of proving various statements in geometry.

21. Proof Methods (2) • E.g. Pythagorean theorem • Still need words to explainthe steps!

22. Proof Methods (3) • Proofs are not unique.…but some proofs are better than others. • A “better” proof: • Takes fewer steps. • Uses fewer axioms & theorems. • Easier to understand. • Does more than it’s supposed to (leading to new knowledge).

23. Assignment (i.e. homework) • Write two proofs using different methods for the following statement: • “The result of adding two odd numbers is an even number.” (3%) • Every step should be described, and mention definitions and axioms used. Be rigorous. • Deadline: within Week 2

24. I’ll be out of town early part of Week 3 • Discuss a replacement class forMonday, 29th of February. (2 hours)