Modern Geometry

1. Modern Geometry © Dr Dragoljub Pokrajac 2006

2. What is Geometry Geometry is the study of figures in a space of a given number of dimensions and of a given type (Wolfram MathWorld). Field of knowledge dealing with spatial relationships (Wikipedia, 8/31/2006, 9:12PM EST) The branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space (Encyclopedia Britannica) “Without geometry, life is pointless”

3. Types of Geometry Planar geometry Solid geometry Spherical geometry Non-Euclidean Geometries Computational geometry Projective geometry Analytic geometry

4. Some Terminology Protractor (device used to measure angles) Types of angles: Acute: between 0 and 90 degrees Right: 900 Obtuse: between 900 and 1800 Straight: 1800 Reflex: between 1800 and 3600 Types of triangles: Equilateral, isoscales, right triangle, scalene (no two sides are equal) Types of quadrilaterals: Parallelograms, rhombuses, rectangles, squares, trapezoids, Types of polygons: Triangle, quadrilateral, pentagon, hexagon, …, n-gon Elements of circles: Chord, tangent, secant, arc

5. Some High School Theorems Sum of angles of n-gon in Euclidean geometry: (n-2)*1800 Area of parallelogram: A=ah Circumference and area of the circle: C=2?r, A= ?r2 Pythagorean theorem: c2=a2+b2 Volume of prism: V=Ah Volume of a pyramid: V=1/3 Ah Area and volume of a sphere: S=4r2 ?, V=4/3 r3 ?

6. Some High School Theorems (Ct’d) Congruence of triangles (SAS, ASA, SSS) Triangle similarity Exterior angle of triangle in Euclidean geometry

7. Some High School Theorems (Ct’d) Angle bisector Segment bisector Midsegment theorem

8. Some High School Theorems (Ct’d) Length of the arc of the circle Inscribed angle theorem

9. Some High School Theorems (Ct’d) Perpendicular bisector of the chord contains the center of the circle

10. Some High School Theorems (Ct’d) Circumscribed circle and circumcenter Orthocenter Inscribed circle and incenter Centroid (baricenter)

11. Origins of Geometry Beginning as applied geometry in Egypt and Mesopotamia Area of rectangle Areas of right and isoscales (and perhaps general) triangles Volume of a right prism… No evidence of proofs

12. Old Greek and Hellenic Mathematicians Thales of Miletus, 6th c. B.C.: Deductive method Pythagoras, 6th c. B.C.: Famous theorem Euclid, 3th c. B.C. The Elements, first axiomatic geometry treatise Archimedes, 3th c. B.C. Measurement of circle (? approximated) Spirals, quadrature of parabola 3D geometry of spheres, cylinders, conoids and spheroids Apollonius, 3th c. B.C. Conic sections Lost work: tangencies, Plane Loci… Heron, Menelaus, Claudius Ptolemy Pappus (5th c. A.D.)

13. Later Developments Descartes Coordinate (analytic) geometry Lobacevski/Rieman Non-Euclidean geometry Hilbert Strict axiomatic of Euclidean geometry

14. Axiomatic Systems A formal mathematical system consists of: unproved statements (axioms) Theorems Theorems are deduced from axioms using deduction (logic) Axioms are statements about: Undefined terms (primitive elements) Relations among elements Operations upon elements

15. Examples Primitive elements Points (in geometry) Real numbers (in algebra) Relations among elements “Point A lies on line l” (geometry) x<y (algebra) Operations upon elements Laying off along the line two line segments forming the third segment (geometry) Addition of two real numbers (algebra)

16. Axioms Axioms are assumptions that are agreed to use Axioms are not self-evident truths Choice of axioms is arbitrary (A can be an axiom and B can follow as a theorem or vice versa) Mathematicians may choose “strong” axioms to make proofs easy, or to try to weak the axioms and see whether the theorems are still provable Choice of primitive elements is also arbitrary (e.g., circle is usually not considered as primitive) Modern mathematicians try to state all the axioms explicitly

17. Properties of Axiomatic Systems Consistency: The axiomatic system that lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. Axioms must form a consistent set (also, if we add axioms to the existing system, the new axioms must be consistent with old ones) Independency: In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent. Completeness: An axiomatic system is complete if for every statement, either itself or its negation is derivable. This is, as shown by the combined works of Godel and Coen, impossible for axiomatic systems involving infinite sets

18. Definitions Definitions are not considered as components of a mathematical system Definition is just an abbreviation (e.g., a single word “ellipse” instead of “in plane, locus of points the sum of whose distances from two fixed points is constant)

19. Euclid’s Elements 13 books devoted to geometry and number theory 465 prepositions Original work with systematization of earlier work Book I: Axioms, postulates, first propositions Book II: Geometric equivalent of algebraic identities Book III: Geometry of circles, chords, tangents, angles Book IV: Constructions of regular polygons Book V: Theory of proportions Book VI: Theory of similar triangles Books VII-IX: Number theory Book X: Irrationals Books XI-XIII: Solid geometry (with exception of spheres)

20. Euclidean Primitives and their “Definitions” Point A point is that which has not part Line A line is length without breadth The extremities of a line are point A straight line is a line which lies evenly with the points on itself Surface Surface is that which has only length and breadth The extremities of a surface are lines Etc.

21. Euclidean Postulates A straight line can be drawn between any two points A finite line can be extended infinitely in both directions A circle can be drawn with any center and any radius All right angles are equal to each other Given a line and a point not on the line, only one line can be drawn through the point parallel to the line .

22. Euclidean Axioms Things which are equal to the same thing are also equal to each other If equals be added to equals, the wholes are equal If equals be subtracted from equals, the reminders are equal Things which coincide with one another are equal to one another The whole is greater than the part

23. Problems with Euclidean Axiomatic Assumptions implicitly involved in proofs but not stated in the list of axioms and postulates Hence, some of the proofs from the Elements are invalid Some of Euclidean definitions (point, line) are inadequate

24. Major Missing Axioms Continuity Ensures that there is a point between any two points on a line Infiniteness of a straight line Axioms of linear order

25. Hilbert’s Axioms Classification of Axioms Primitives and basic relations Axioms of Connection (Existence and Incidence) Axioms of Order Axioms of Congruence Axiom of Parallels Axioms of Continuity and Completeness

26. Primitives and basic relations Primitives Point, line Primitives are not defined! Basic relations On, between, congruent

27. Axioms of Existence and Incidence 1.There exists at least one line 2.On each line, there exist at least two points 3.Not all the points lie on the same line 4.There is one and only one line passing through two distinct points

28. Some Theorems That Can Be ProvenUsing Axioms of Incidence Through each point, there are at least two distinct lines Not all lines pass through the same point Two distinct lines meet in at most one point HW: Prove these theorems

29. Axioms of Order 5. If B is between A and C, A, B, C are distinct and collinear 6. If B is between A and C, then B is between C and A 7.If points A and C are distinct, there is point B such that B is between A and C and there is point D such that C is between A and D 8. Given three distinct collinear points, one and only one is between two others 10. Given three noncollinear points A, B and C and a line not passing through any of the points, if a point of a segment AB lies on the given line, a point of AC or point of BC also lies on the given line (Pasch’s axiome)

30. Some Theorems That Can Be ProvenUsing Axioms of Order Given four distinct collinear points, it is possible to name them A, B, C, D such that: B is between A and C B is between A and D C is between A and D C is between B and D Note: Sometimes, this theorem is also considered as an axiom(9), which would make the system of axioms non independent

31. Theorems that can be proven-ctd. If point O in on the line l, it divides all points of l into two classes. Points are in the same class if O is not between them. Any line divides the set of all points not on the line into two classes. Two points are in the same class iff the segment they determine does not contain a point lying on the line

32. Useful Definitions Segment: Segment [A,B] is the set of all points between A and B Ray: If O is a point on line l, a ray on line l with terminal point O is the set of all points on the same side of O together with O Angle If h and k are distinct rays with a common terminal points, the pair of rays constitute an angle (h,k). The rays are sides of the angle, and the common terminal of the angle is vertex Straight angle is an angle where h and k are rays on the same line

33. Axioms of Congruence 11. Given a segment [A,B] and a point A’ on a line a, on each ray on a with terminal point A’ there is only one point B’, such that [A,B] [A’,B’]

34. Axioms of Congruence (cont’d) 12. Any interval is congruent with itself: [A,B] [A,B] 13. The relation of interval congruency is symmetric: If [A,B] [A’,B’] then [A’,B’] [A,B] 14. The relation of interval congruency is transitive If [A,B] [A’,B’] and [A’,B’] [A’’,B’’] Then [A,B] [A’’,B’’]

35. Axioms of Congruence (cont’d) 15. If [A,B] and [B,C] are segments on a line a with no common point, if [A’,B’] and [B’,C’] are segments on line a’ with no common points, and if [A,B] [A’,B’] and [B,C] [B’,C’] then [A,C] [A’,C’]

36. Axioms of Congruence (cont’d) 16. Given an angle (h,k) that is not a straight angle and given a ray h’ on a line a, on each side of a there is one and only one ray k’ such that (h,k) (h’,k’). A straight angle is congruent only to straight angles

37. Axioms of Congruence (cont’d) 17. Any angle is congruent with itself: (h,k) (h,k) 18. The relation of angle congruency is symmetric: If (h,k) (h’,k’) then (h’,k’) (h,k) 19. The relation of angle congruency is transitive If (h,k) (h’,k’) and (h’,k’) (h’’,k’’) Then (h,k) (h’’,k’’)

38. 20. If in triangles ABC and A’B’C’, [A,B] [A’,B’], [A,C] [A’,C’] and ?A ?A’, then ?B ?B’ Axioms of Congruence (cont’d)

39. Triangles ABC and A’B’C’ are congruent if: [A,B] [A’,B’], [A,C] [A’,C’] [B,C] [B’,C’] ?A ?A’ ?B ?B’ ?C ?C’ Useful Definition

40. SAS congruence: Triangles are congruent if [A,B] [A’,B’], [A,C] [A’,C’] ?A ?A’ ASA congruence: Triangles are congruent if [A,B] [A’,B’], ?A ?A’ ?B ?B’ SSS congruence Theorems That Can Be Proven

41. Axiom of Parallels 21. Given a line l and a point A not on line, there is one and only one line l’ containing the given point A and having no common points with a given line l (Playfair axiom)

42. This is the most controversial axiom of Euclidean geometry There were numerous attempts to derive it from other axioms Alternative geometries not containing this axiom exist Note on Axiom of Parallels

43. Useful Definition Two lines are parallel if they have no common points

44. Axioms of Continuity and Completeness 22.Given points A and B and points A1, A2,… such that A1 is between A and A2, A2 is between A1 and A3,…such that [A,A1] [A1,A2] [A2,A3] … then there is a positive integer n such that B is between A and An (Archimedes’ axiom)

45. Note This axiom makes possible to measure segment [A,B] using segment [A,A1]

46. Axioms of Continuity and Completeness (Cont’d) 23. No additional points or lines can be added to the system without violating one of the preceding axioms (Postulate of completeness) Note: Axiom 23. is necessary for establishing mapping between the points on line and real number and hence for analytic geometry Note: For solid geometry, Hilbert axioms involve 6 more axioms

47. Basic Geometric Constructions Tools Straightedge and compass What is solvable and what is not

48. Tools for Constructions Straightedge (ruler with no parallel sides and with no graduation) Compass (draws circles) Modern compass can translate the distance from one place to another Euclidean compass cannot translate distances (collapses) “Rusty” compass Cannot change the distance, which is fixed

49. Classic Constructions Classic constructions, that could be accomplished with straightedge and Euclidean compass can also be accomplished with: Straightedge and modern compass Straightedge and rusty compass These three pairs of tools are equivalent We will do “constructions” with straightedge and modern compass

50. Basic Constructions Construct the perpendicular bisector of a line segment, or construct the midpoint of a line segment. Given a point on a line, construct a perpendicular line through the given point. Given a point not on the line, construct a perpendicular line through the given point. Construct the bisector of an angle. Construct an angle congruent to a given angle. Construct a line through a given point, parallel to a given line. Construct an equilateral triangle, or construct a 60º angle. Divide a line segment into n congruent line segments.C Construct a line through a given point, tangent to a given circle. Construct the center point of a given circle.C Construct a circle through three given points. Circumscribe a circle about a given triangle. Inscribe a circle in a given triangle.

51. Steps in Geometric Construction Analysis Construction Proof Discussion

52. Feasibility of Geometric Constructions Some tasks cannot be performed using only straightedge and compass Doubling the cube Quadrature of the circle Trisection of arbitrary angle PROJECT!