Mathematics 3210

Euclidean and Non-Euclidean Geometry

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Date Topics covered Reading Homework Assignments
Monday January 12
Chapter 1

Wednesday January 14
Euclid's Geometry.
Origins of geometry, the axiomatic method, undefined terms.

Friday January 16
Euclid's first four postulates.

HW1: Due Friday January 16
.pdf, .tex

Monday January 19

Wednesday January 21
The parallel postulate.
Chapter 2
Friday January 23
Introduction to LaTex.
(See the bottom of this webpage)
HW 2: Due Friday February 6
Chapter 1, Exercises: 1, 2, 3, 6, 9, 10, 12
Monday January 26
Misleading diagrams, straight edge and compass constructions.

Wednesday January 28
Introduction to logic.

Friday January 30

Theorems and proofs, proof by contradiction.

HW 3: Due Friday February 6
Chapter 1, Exercises: 14, 15.
Chapter 2, Exercises: 1, 2, 3.
.pdf, .tex
Monday February 2

Basic logic, cont.

Wednesday February 4
Incidence geometry.

Friday February 6
HW 4: Due Friday February 6
Chapter 2, Exercises: 4, 7, 8, 10 (a)(b), 11.
Monday February 9
Homework review

Wednesday February 11
Sample Midterm I .pdf
Friday February 13

Monday February 16
Affine planes

Wednesday February 18
Projective planes

Friday February 20
Hilbert's Axioms
Flaws in Euclid.
Chapter 3
HW 5: Due Friday February 20
Chapter 2, Exercise: 14.
Chapter 2, ``Major Exercises'': 1-5.
Monday February 23
Axioms of betweenness.

Wednesday February 25
Axioms of congruence.

Friday February 27
Axioms of continuity.

HW 6: Due Friday March 6
Chapter 3 Exercises on Betweenness: 1-5
Monday March 2
Axiom of parallelism.

Wednesday March 4
Neutral Geometry
Geometry without the parallel axiom.  Alternate interior angle theorem.
Chapter 4

Friday March 6
Exterior angle theorem, measure of angles and segments.
HW 7: Due Friday March 13
Chapter 3 Exercises on Congruence: 21-24
Monday March 9
Sachheri--Legendre theorem.

Wednesday March 11
Equivalence of the parallel postulates.

Friday March 13
Angle sum of a triangle.
HW 8: Due Friday March 13
Begin presentation writing assignment
Monday March 16
Homework review

Wednesday March 18
Sample Midterm II .pdf
Friday March 20

Week of March 23-March 27
Monday March 30 Non-Euclidean Hilbert planes, and the defect Chapter 5, Chapter 6
Wednesday April 1
Similar triangles, parallels admitting a common perpendicular.

Friday April 3
Limiting parallel rays, hyperbolic planes.

HW 8: Due Friday April 10
Presentation writing assignment
Monday April 6
History of the Parallel Postulate
Proclus, Equidistance, Wallis, Saccheri.

Wednesday April 8
Clairout's Axiom and Proclus' Theoreom, Legendre, Lambert and Taurinus, F. Bolyai

Friday April 10
Non-Euclidean Geometry
J. Bolyai, Gauss, Lobachevsky

HW 9: Due Friday April 17
Chapter 4, Exercises 1, 2, 6, 10, 14
Monday April 13
Classification of parallels.

Wednesday April 15
Independence of the Parallel Postulate
Consistency of hyperbolic geometry.

Friday April 17 The Beltrami--Klein model

 HW 10: Due Friday April 24
Chapter 5, Exercises 10, 11, 13
Monday April 20
The Poincare models.

Wednesday April 22
Models, cont.

Friday April 24
Further topics

HW 11: Due Friday May 1
Study for final exam
Monday April 27
Further topics

Wednesday April 29
Homework review

Friday May 1

Sample Final .pdf
Monday May 4
FINAL EXAM 7:30 -10:00 PM ECCR 1B51
(Lecture Room)

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I strongly encourage everyone to use LaTex for typing their homework.  If you have a mac, one possible easy way to get started is with texshop.  If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc.  If you are using windows, you're on your own, but I'm sure there's something online.  Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  It is probably easiest to just look at the .tex file, and start to experiment.  There is also the (Not so) short introduction to latex, which will answer most of your questions (although typing your question into your favorite search engine will probably work well, too).  You may also want to try or for a cloud version.