Transcript Art of Problem Solving

3 Main Branches of
Modern Mathematics
Analysis
 Algebra
 Geometry

A Brief History of
Geometry
Maggie & Michael
ILG131 2008/6/2 16:00
伍業輝
Ng Ip Fai
What is Geometry?

A study of spaces and transformations
 Spaces:
the geometric spaces
 Transformations:
translations, rotations, symmetries, etc.
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

Middle / Towards Modern Stage

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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
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

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)
Geometry as
an Experiential
Science
Ancient Period I
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
Pythagorean Theorem
Pythagoras (About 6th Century BC)
2
2
2
 a + b = c in a right triangle
 Concept of Directions
 Trigonometry

5
3
4
Geometry as
a Logical Science
Ancient Period II
(3rd century BC onwards)
Euclidean Geometry
Euclid (About 300 – 260 BC)
 The beginning of Axiomatic Geometry

 Breakthrough

of Mathematics
Elements: Words and graphs only;
no symbol
Common Notions
x = a, y = a
 x = y, a = b
 x = y, a = b
 Coincide
 The whole

imply
imply
imply
implies
>
x=y
x+a=y+b
x–a=y–b
Equal
The part
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.

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.
Any Triangle is Isosceles?

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Let the perpendicular bisector of BC meet the
bisector of A at E.
A
Let EF  AB, EG  AC.
BC = EC, EF = EG.
BEF  CEG, AEF  AEG.
F
BF = CG, AF = AG.
G
E
AB = AF + FB
= AG + GC = AC.
B
C
M
ABC is an isosceles triangle.
(Li & Zhou 1995: 351-52)
Geometry as
a Quantitative Science
Towards Modern Stage
(17th century onwards)
Cartesian coordinate system

Descartes (1596-1650)
on Plane  Pairs of Numbers
 Plane Curves  Equations with 2 Variables
 Points
Algebra  Geometry
2
2
n
n
 E  R ; or generally, E  R

Calculus
Newton (1642-1727) & Leibniz (1647-1716)
 Calculus is NOT geometry; but a tool for it
 Finding tangent lines and areas
 Analytical Geometry

Geometry as an
Invariant in a
Transformation Group
Modern Time
Concept of Group

A group is a set G, together with a binary
operation * on G which satisfies:
 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
 g*h
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Compare it with a vector space yourselves.
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.
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
To Classify geometry
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.

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
Gaussian Elegant Theorem
Gauss Theorema Egregium
 Gauss Curvature
 Gauss Curvature is intrinsic.

Global Inner Geometry
Cartan (1869-1951) &
Chen Xing-Shen (1911-2004)
 Localization vs. Globalization
 Viewpoint of Dual

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
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.
Calendar of Geometry
(Wu & Chen 2004:32)
“Date”
Events
Year
01/01
Dawn of Ancient Greek Civilization
1650 B.C.
04/15
Pythagorean Theorem
600 B.C.
05/15
Euclid’s Elements
300 B.C.
05/20
Archimedes’s Volume of sphere
250 B.C.
06/15
(Year 0)
0 AD
11/25
Cartesian coordinate system
1630
11/29
Newton & Leibniz: Calculus
1670
12/14-17 Gauss & Riemann: non-Euclidean & inner geometry
1820-1850
12/22
Poincaré: Fundamental Groups; Ricci: Tensor analysis
1900
12/24
Einstein’s General Relativity
1910-1920
12/27
Cartan & Chen: Global Inner Geometry
1950
12/29
Mandelbrot: Fractal Geometry
1970
12/31
(Year 2000)
2000
Error in the “Proof”
Suppose AB > AC.
 Let the bisector of A
meet BC at D.
 BD/DC = AB/AC > 1.
 BD > DC.
B
 M is on [BD].
A

M
D
E
(Li & Zhou 1995: 352)
C
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, 6th 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
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

Wikipedia: History of Geometry (and others).
http://en.wikipedia.org/wiki/History_of_geometry

The History of Geometry.
http://math.rice.edu/~lanius/Geom/his.html
The End

Thank you for your attention!

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