Geometry
Elementary Geometry MTH312 @ IISER Pune
Instructor : Dr. Anupam Kumar Singh
Tutor : Ayesha Fatima
Schedule : August - November 2012, 3 hours (2 lectures + 1 tutorial) every week
Evaluation : Test I - 25%
Test II (mid sem exam) - 25%
Test III - 25%
Test IV (end sem exam) - 25%
All tests will be of 3 hours duration.
Prerequisite : High School geometry and Linear Algebra
Goal of the Course : To learn basic geometry of different kinds such as Euclidean, Spherical and Hyperbolic geometries. History of mathematics is closely related to the development of geometries of various kind. Much of the modern Mathematics, Physics and Engineering is based on geometry and skills (thinking) obtained that way. The topics such as Engineering Design, Perspective Drawings, Theoretical Physics specially String Theory, Differential Geometry and Algebraic Topology are based on understanding the basic geometries. The fact that Euclid (300 BC) thought of axiomatizing geometry and Perlman (2002) solved Geometrization Conjecture (famously known as Poincare conjecture) says a lot about importance of Geometry throughout.
Proposed Course Content : Euclidean geometry in 2 and 3 dimension, Complex Numbers, Quaternions, Projective geometry, Spherical geometry and Hyperbolic geometry.
Fun Things to Look Forward : Classical problems in straightedge and compass construction, Trisecting an angle in Origami, Perspective drawings (1 and 2 point perspectives), Symmetries of regular polygons, Tiling, Playing Carrom in Hyperbolic plane.
Text Books :
Geometry - Fenn
The Poincare Half Plane - Stahl
Hyperbolic Geometry - Anderson
Reference Books :
Geometry - Audin
Curved Spaces - Wilson
Geometry : Euclid and beyond - Hartshorne
The Four Pillars of Geometry - Stillwell
Introduction to Geometry - Coxeter
Course in Modern Geometries - Cederberg
The Poincare Half Plane : A Gateway to Modern Geometry - Stahl
Geometry of Surfaces - Stillwell
02/08/2012 Overview of the course. Euclidean, Spherical and Hyperbolic Geometry in 2D, Euclid and axiomatization in Mathematics.
06/08/2012 Axioms of Euclidean Geometry in the Plane, Models, Coordinate Geometry. The 5th postulate of Euclid!!!
07/08/2012 Straightedge and compass constructions, 5 basic constructions and corresponding algebraic equations, Constructible Numbers. How constructible numbers form a field using construction?
09/08/2012 Tutorial Assignment I
Links to Enjoy :
Teaching geometry according to Euclid by Hartshorne
Straightedge and Compass constructions
13/08/12 Distance in R^2, Norm and Bilinear form, explicit formula for translations, rotations and reflections; Isometries of R^2; Composition of reflections.
14/08/12 Orthogonal Transformations; explicit description of O(2) and SO(2); rotations vs reflections.
16/08/12 Tutorial Assignment II
20/08/12 Holiday
21/08/12 Classification of isometries in 2D, i.e., any isometry of R^2 is of the form f(x)=Ax+b where A is in O(2) and b is a vector in R^2. Dilations, Congruent triangles and Similar triangles from Isometry point of view.
23/08/12 Tutorial Assignment III
27/08/12 Origami and Trisection of an angle; Huzita axioms of origami; demonstration how to trisect an angle and proof. In Origami one can solve cubic and quartic equations too!!
The Power of Origami - an article in Plus magazine
Origami and Geometric Constructions - Robert Lang
28/08/12 Tutorial
30/08/12 Tutorial
Kali Software for Tessellations
01/09/12 Test I
03/09/12 Geometry of Complex Numbers; representing C as 2x2 real matrices; the circle group S^1; Equation of lines and circles in complex variable; Basic transformations of the complex plane, dilation, translation, rotation, reflection and division.
Resonance article by A. R. Shastri on Complex Numbers and Plane Geometry
04/09/12 Mobius Transformations and extended complex plane (Riemann Sphere).
06/09/12 Tutorials
YouTube video on Mobius Transformations
Douglas Arnold's page on Mobius Transformation
10/09/12 Riemann Sphere and Stereographic Projection; division and inversion map on C^+, circles get mapped to circles under basic Mobius transformation (which generate all Mobius transformations).
11/09/12 Cross Ratio and Mobius Transformations.
13/09/12 Tutorials Assignment IV
17/09/12 3 dimensional geometry (Solid Geometry), Sphere S^2, spherical coordinate system, inner product, cross product (which makes R^3 into Lie algebra), scalar triple product and vector product.
18/09/12 Isometries, Hamilton's Quaternions; How composition of two isometries can be achieved by multiplying vectors in quaternions (analog of 2d geometry and complex numbers).
20/09/12 Quaternions and 3 dimensional geometry, the map from H^1 to SO(3).
24/09/12 Tutorials Assignment V
25/09/12 No class
26/09/12 Mid Sem Exam at 2:30
Flatland by Abbott a novel where world is two dimensional
08/10/12 Projective plane, how to think of RP^2 in different ways (as line in R^3, antipodal point identified on sphere S^2, R^2 U {as many infinites as many directions in R^2}), RP^2 as 2-dimensional manifold, lines in projective plane.
09/10/12 Answer-sheet of mid sem exam
11/10/12 Perspective Drawing; how to draw cubes in one-point perspective and two point perspective, draw a house, draw tiles on a floor, draw sky-scrapers, drawing circles, cubes, pyramid and other geometric pictures in perspective. Demonstration of how to find perspective point(s) on photographs.
There are a lots of nice videos on YouTube, search for "one point perspective" and "two point perspective"
15/10/12 Incidence and duality, Desargues' Theorem
16/10/12 Projective Transformations
18/10/12 Tutorial Assignment VI
22/10/12 Projective Completions of curves, Affine classification of conics, projective classifications of conics.
23/10/12 Spherical Geometry , geodesics, triangles, distance function, Area of a triangle, sum of interior angles in a geodesic triangle > pi.
25/10/12 Tutorial Assignment VII
29/10/12 Isometries of sphere is O(3), Euler's theorem for polygon (V-E+F=2).
30/10/12 Hyperbolic plane (upper half plane model), geodesics, lines passing through two points, parallel lines, Mobius transformations as maps.
31/10/12 Test III at 2:30 PM
01/10/12 Tutorial
05/11/12 Hyperbolic length of a curve, classification of geodesics as length minimizing curves, hyperbolic distance between two points, triangles.
06/11/12 Classification of circles in the hyperbolic plane, Euclidean circles are Hyperbolic circles and vice-versa, finding center and radius formula, Euclidean vs Hyperbolic geometry (comparing axioms and birth of non-Euclidean geometry). Introduction to Disc model.
08/11/12 Tutorials Assignment VIII
Escher and Hyperbolic geometry
12/11/12 Hyperbolic triangles, sum of interior angles is less than pi, area of a hyperbolic triangle.
13/11/12 Holiday on the occasion of Diwali.
15/11/12 Tutorial.
End sem exam on 29/11/12 at 2PM.