1. 3 Main Branches of Modern Mathematics Analysis Algebra Geometry

2. 伍業輝 Ng Ip Fai A Brief History of Geometry M aggie & M ichael ILG131 2008/6/2 16:00

3. What is Geometry? A study of spaces and transformations  Spaces: the geometric spaces  Transformations: translations, rotations, symmetries, etc.

4. Ancient Period The concept of  Pythagorean Theorem Pareto’s Five Regular Polyhedrons Euclid’s Elements Archimedes’s Volume Formula of Sphere Zu Chong-Zi’s ( 祖沖之 ) Principle

5. Middle / Towards Modern Stage Cartesian coordinate system Newton & Leibniz: Calculus Gaussian Elegant Theorem and Gauss-Bonnet’s Theorem Non-Euclidean Geometry and Riemannian Geometry Lagrange: Calculus of Variation Laplace: Celestial Mechanics Euler’s Characteristic & Wave Equation Klein’s Program

6. Modern Time Poincaré’s plane & fundamental groups Hilbert’s Foundations of Geometry Einstein’s General Relativity de Rham’s Cohomology, Hodge’s Theory, Cartan’s Differential Form Chen’s ( 陳省身 ) characteristic class, Chen-Weil and Chen-Simon’s Theories

7. Contemporary Era Rauch’s Theorem Atiyah-Singer’s Index Theorem Yau ( 丘成桐 ): Geometric Analysis Donaldson, Seiberg-Witten’s Theories M. Gromov: Symplectic Geometry Mandelbrot: Fractal Geometry and Chaos Computational Geometry (Wu & Chen 2004: 24-25)

8. Geometry as an Experiential Science Ancient Period I

9. Beginning of Geometry Ancient Civilizations  Chinese, Babylonian, Egyptian  Greek “Geometry” is derived from Greek roots  geo: earth  metry: measurement To measure the areas of lands

10. Pythagorean Theorem Pythagoras (About 6 th Century BC) a 2 + b 2 = c 2 in a right triangle Concept of Directions Trigonometry 5 3 4

11. Geometry as a Logical Science Ancient Period II (3 rd century BC onwards)

12. Euclidean Geometry Euclid (About 300 – 260 BC) The beginning of Axiomatic Geometry  Breakthrough of Mathematics Elements: Words and graphs only; no symbol

13. Common Notions x = a, y = aimply x = y x = y, a = bimply x + a = y + b x = y, a = bimply x – a = y – b CoincideimpliesEqual The whole >The part

14. Postulates I ~ IV A straight line can be drawn from any point to any other point. A finite straight line can be produced continuously in a line. A circle may be described with any center and distance. All right angles are equal to one another.

15. Postulate V: Parallel Postulate If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which are the angles less than two right angles. Given a line and a point not on the line, exactly one line can be drawn through the point parallel to the line.

16. Any Triangle is Isosceles? Let the perpendicular bisector of BC meet the bisector of  A at E. Let EF  AB, EG  AC. BC = EC, EF = EG.  BEF   CEG,  AEF   AEG. BF = CG, AF = AG. AB= AF+ FB = AG+ GC = AC.  ABC is an isosceles triangle. (Li & Zhou 1995: 351-52)

17. Geometry as a Quantitative Science Towards Modern Stage (17 th century onwards)

18. Cartesian coordinate system Descartes (1596-1650)  Points on Plane  Pairs of Numbers  Plane Curves  Equations with 2 Variables Algebra  Geometry E 2  R 2 ; or generally, E n  R n

19. Calculus Newton (1642-1727) & Leibniz (1647-1716) Calculus is NOT geometry; but a tool for it Finding tangent lines and areas Analytical Geometry

20. Geometry as an Invariant in a Transformation Group Modern Time

21. Concept of Group A group is a set G, together with a binary operation * on G which satisfies:  g*h  G,  g, h  G  f *(g*h) = (f *g)*h,  f, g, h  G   !e  G such that g*e = g = e*g,  g  G   g  G  h  G such that g*h = e = h*g Compare it with a vector space yourselves.

22. Erlangen’s Guiding Principle Given a set V and a transformation group G on the elements in V. Then V is called a space, its elements is called points, and the subspaces of V is called graphs. And then the study of graphs about the invariants in the group G is called the geometry of V corresponding to G. To Classify geometry

23. Non-Euclidean Geometry Gauss, Bolyai, and Lobatchewsky The Parallel Postulate can’t be proved by the first 4 postulates. Both the geometries with and without the Parallel Postulate are self-consistent.

24. Differential Geometry Gauss (1777-1855), Riemann (1826-1866) Riemann’s “On the Hypotheses which lie at the Bases of Geometry”  Über die Hypothesen, welche der Geometrie zu Grunde liegen Riemannian Geometry  Einstein’s General Theory of Relativity

25. Gaussian Elegant Theorem Gauss Theorema Egregium Gauss Curvature Gauss Curvature is intrinsic.

26. Global Inner Geometry Cartan (1869-1951) & Chen Xing-Shen (1911-2004) Localization vs. Globalization Viewpoint of Dual

27. A Chinese Poem about Chen 天衣豈無縫, 匠心剪接成. 渾然歸一體, 廣邃妙絕倫. 造化愛幾何, 四力纖維能. 千萬寸心事, 歐高黎嘉陳. ---- Yang Zhen-Ning 歐 : Euclid, 高 : Gauss, 黎 : Riemann, 嘉 : Cartan, 陳 : Chen

28. Algebraic Geometry Use algebraic methods to study geometry Deep and Difficult  Not only the background of geometry and analysis are required, but a deep understanding of algebra is also needed.

29. Calendar of Geometry (Wu & Chen 2004:32) “Date”EventsYear 01/01Dawn of Ancient Greek Civilization1650 B.C. 04/15Pythagorean Theorem600 B.C. 05/15Euclid’s Elements300 B.C. 05/20Archimedes’s Volume of sphere250 B.C. 06/15(Year 0)0 AD 11/25Cartesian coordinate system1630 11/29Newton & Leibniz: Calculus1670 12/14-17Gauss & Riemann: non-Euclidean & inner geometry1820-1850 12/22Poincaré: Fundamental Groups; Ricci: Tensor analysis1900 12/24Einstein’s General Relativity1910-1920 12/27Cartan & Chen: Global Inner Geometry1950 12/29Mandelbrot: Fractal Geometry1970 12/31(Year 2000)2000

30. Error in the “Proof” Suppose AB > AC. Let the bisector of  A meet BC at D. BD / DC = AB / AC > 1. BD > DC. M is on [BD]. (Li & Zhou 1995: 352)

31. Bibliography Li, C.-M. & Zhou H.-S. (1995). Research of Elementary Mathematics. Beijing: Higher Education Press. Burton, D. (2007). The History of Mathematics: An Introduction, 6 th ed. New York: McGraw-Hill. Au, K.-K., Cheung K.-L. & Cheung L.-F. (2007). Towards Differential Geometry: Lecture notes of EPYMT 2007. Hong Kong: The Chinese University of Hong Kong. Beardon, A. (2005). Algebra and Geometry. Cambridge: Cambridge University Press. Garding, L. (1977). Encounter with Mathematics. New York: Springer-Verlag

32. On-line References Wu, Z.-Y. & Chen, W.-H. (2004). “A Short History of Development of Geometry”. Mathmedia 28(1). http://www.math.sinica.edu.tw/math_media/d281/ 28103.pdf http://www.math.sinica.edu.tw/math_media/d281/ 28103.pdf Wikipedia: History of Geometry (and others). http://en.wikipedia.org/wiki/History_of_geometry http://en.wikipedia.org/wiki/History_of_geometry The History of Geometry. http://math.rice.edu/~lanius/Geom/his.html http://math.rice.edu/~lanius/Geom/his.html

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