Math 141, Winter 2020

Math Diaries

Textbook Reading (Jan 6): Sections 1.1 and 1.2.

Wednesday Jan 8: Isometries of Euclidean Plane I.

Definition of Distance-Preserving Transformation. Examples and Non-Examples.
How to show that a map is or is not distance-preserving. Three Types of Isometries.
Rotations Centered at Origin. Translations along a vector. Example of Reflection.

Textbook Reading (Jan 8): Section 1.2.

Friday Jan 10: Isometries of Euclidean Plane II.

Reflection along an axis. Composition of Isometries and conjugation.
Rotations with arbitrary center. Reflections along arbitrary line.
The group generated by Translations. The group generated by Rotations.
The composition of reflections is not a reflection.

Textbook Reading (Jan 10): Section 1.3.

Monday Jan 13: The group generated by reflections.

The notion of a group. Inverses under composition of isometries.
The product of two reflections: Parallel Lines and Intersecting Lines.
Theorem: An isometry is two reflections iff it is a translation or rotation.
Isometries and their Effect on Triangles.

Textbook Reading (Jan 13): Sections 1.3 and 1.4.

Wednesday Jan 15: Classification of Isometries in Euclidean Plane.

Sets defined by distances to points. Two Examples: Circle and lines.
Third Example: The set of points equidistant to three non-colinear points is a point.
Isometry in Euclidean plane are uniquely determined by images of three non-colinear points.
Theorem: Every isometry of the Euclidean plane is the product of 1,2 or 3 reflections.

Textbook Reading (Jan 15): Sections 1.3 and 1.4.

Friday Jan 17: Classification of Isometries in Euclidean Plane II.

Proof of the Classification Theorem for Isometries in the Euclidean Plane.
Isometries form a group. Orientation-Preserving and Orientation-Reversing Isometries.
The products of three reflections. Glide Reflections: Definition and Examples of gliding.
Theorem: Every isometry of the Euclidean plane is a translation, a rotation or a glide reflection.

Textbook Reading (Jan 17): Sections 1.4 and 1.5.

Monday Jan 20: Martin Luther King Jr. Day.

Wednesday Jan 22: The Euclidean Cylinder.

Euclidean Surfaces. Quotient construction and main properties.
Formal definition of the Cylinder: quotienting by isometries of the Euclidean Plane.
Distance in the Cylinder: Definition and Properties. Example of Distances.

Textbook Reading (Jan 22): Sections 2.1 and 2.2.

Friday Jan 24: The Twisted Cylinder and the Euclidean Torus.

Definition of the Twisted Cylinder. Definition of the Euclidean Torus.
Distances in Quotients of Euclidean Plane by Isometries. Examples of Distances.
Practical Demonstration about Distances in Euclidean Surfaces: Rolling Papers.

Textbook Reading (Jan 24): Sections 2.3 and 2.4.

Monday Jan 27: Quotient Surfaces.

Definition of Discontinuous Subgroup. Review of Limit Points. Fixed point free actions.
Theorem: Discontinuous and Fixed-Point Free iff neighborhood with unique orbit representative.
Examples of Discontinuous Subgroups. Example of Non-Discontinuous Subgroup.

Textbook Reading (Jan 27): Section 2.5 and 2.6.

Wednesday Jan 29: Classification of Discontinuous and Fixed-Point Free Groups I.

Theorem: Discontinuous Fixed-Point Free Group generated by translations has 1 or 2 generators.
Example of the Euclidean Cylinder and the Euclidean Torus: Many translationsa as 1 or 2 translations.
Case of Glide Reflections. Example of the Twisted Cylinder and the Klein Bottle.

Textbook Reading (Jan 29): Section 2.5.

Friday Jan 31: Classification of Discontinuous and Fixed-Point Free Groups II.

Proof of the Classification of Quotient Surfaces through Isometry Subgroups.
The Four Quotients: Euclidean Cylinder, Twisted Cylinder, Euclidean Torus and Klein Bottle.

Textbook Reading (Jan 31): Section 2.5.

Monday Feb 3: Euclidean Surfaces.

Definition of a surface. Definition of a Euclidean surface.
Examples of Euclidean Surfaces. Covering of a Surface by the Plane.
The Pencil Map. Sketch of the Proof for Classification of Euclidean Surfaces.

Textbook Reading (Feb 3): Sections 2.7 and 2.8.

Wednesday Feb 5: Review Session for Midterm Exam

Friday Feb 7: Midterm Exam

Monday Feb 10: The Distance in Spherical Geometry.

Definition of the 2-sphere. Distance in Euclidean 3-space and Distance in the 2-sphere.
Properties of the Distance in the 2-sphere. Equidistant Points in the sphere.
Computations and Examples for distances between points.

Textbook Reading (Feb 10): Section 3.1.

Wednesday Feb 12: Rotations in the 2-sphere.

Definition of Rotation along the z-axis. Conjugation and Rotation along any oriented axis.
Explicit formula for rotation along the x-axis. Interlude reviewing Linear Algebra.
Isometries of the 2-sphere changing the axis of rotation. Explicit Examples.

Textbook Reading (Feb 12): Sections 3.1 and 3.2.

Friday Feb 14: Lines and Reflections in the 2-sphere.

Definition of a Line in the 2-sphere. Examples and Non-Examples of Lines.
Characterization of Lines in the 2-sphere as Intersections with linear 2-planes.
Reflection along the equator in the 2-sphere. Reflection along any Line in the 2-sphere.

Textbook Reading (Feb 14): Sections 3.1 and 3.2.

Monday Feb 17: Presidents' Day 2020.

Wednesday Feb 19: Classification of Isometries in 2-sphere.

Properties of Reflections in the 2-sphere. Isometries are determined by 3 non-collinear points.
Theorem: Every isometry is a composition of 1,2 or 3 reflections. Corollaries.
Exploration of each type of Isometries in the 2-sphere. The group of rotations.

Textbook Reading (Feb 19): Sections 3.1 and 3.2.

Friday Feb 21: Problem Session on Triangles in the 2-sphere.

Definition of Triangles in the 2-sphere. Polygons in the 2-sphere.
Problem 1: Interior Angle Sum for Triangles in the 2-plane.
Problem 2: Interior Angle Sum for Triangles in the 2-sphere.
Problem 3: Interior Angle Sum for n-gons in the 2-plane and the 2-sphere.

Textbook Reading (Feb 21): Sections 3.8.

Monday Feb 24: Triangles in the 2-sphere.

Triangles in the 2-sphere. Interior Angles in the 2-sphere and Their Sum.
Excess of a spherical triangle. Examples of Triangles and Interior Angle Sums.
Proof of Harriot's Theorem: Sum of Interior Angles in Spherical Triangles is Pi plus Area.
Introduction to the Stereographic Projection: Definition and First Constructions.

Textbook Reading (Feb 24): Sections 3.8 and Section 3.3.

Wednesday Feb 26: Stereographic Projection.

Definition of the Stereographic Projection from the North Pole. First Examples of Images of Points.
Formula for the Stereographic Projection and its Inverse. Properties of the Stereographic Projection.
Proof that the stereographic projection is not an isometry. Images of lower and upper hemispheres.

Textbook Reading (Feb 26): Section 3.3.

Friday Feb 28: Problem Session on Complements of Lines.

Problem 1: Count the number of regions in the complement of n generic lines in the 2-plane.
Problem 2: Count the number of regions in the complement of n generic lines in the 2-sphere.
Problem 3(a): Show that the Euler characteristic of a connected planar graph is one.
Problem 3(b): Show that the Euler characteristic of a connected spherical graph is two.
Problem 4: Show that there are only five Platonic solids and classify them.
Problem 5: Show that a Game of Brussel Sprouts always finished in eight rounds.

Textbook Reading (Feb 28): Sections 3.1 and 3.2.

Monday Mar 2: Euclid's Axioms and Hyperbolic Geometry.

The three geometries: Euclidean, Spherical and Hyperbolic. Similarities and differences.
Euclid's Four Axioms and the Fifth Parallel Axiom. The Resolution by Bolyai-Lobachevski-Gauss.
Hyperbolic Geometry via Examples in Real Life and Science. Saddle Surfaces.
Pizza, Pringles, Butterfly wings, Mantis Shrimp Kick (see also this article).

Textbook Reading (Mar 2): Section 4.1.

Wednesday Mar 4: The hyperbolic distance.

M.C. Escher's Hyperbolic Fish and Hyperbolic Art: All fish have the same size.
The two main questions: Computing Hyperbolic Distances and Hyperbolic Lengths.
The hyperbolic distance: definition, and main formula. Examples and Computations.
Definition of hyperbolic length and its formula. Final Example and relation to length.

Textbook Reading (Mar 4): Sections 4.2 and 4.3.

Friday Mar 6: Hyperbolic Lines.

The main question: What are lines in Hyperbolic Upper-Half Plane ?
First Theorem: Euclidean Vertical Lines are Hyperbolic Lines. Proof and Examples.
Second Theorem: Semi-Circles centered at the x-axis are Hyperbolic Lines. Isometries and Proof.
The first Four of Euclid's Axioms hold in Hyperbolic Geometry. Failure of the Parallel Axiom.

Textbook Reading (Mar 6): Section 4.2.
Additional Sources: Caroline Series on Hyperbolic Geometry (Formulas on Page 22 are useful).

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