Math 465 and Math 584: Foundations of Geometry

Math 465: Foundations of Geometry

Math 584: Euclidean and Non-Euclidean Geometry

Information Page, Fall 2012

The course is worth 4 credits and will meet MW 4:40-6:40.

Texts: Geometry: Euclid and Beyond, by Robin Hartshorne

Elements

Here is an interesting eddition of the "Elements" by Oliver Byrne (from 1847). He attempts to present the proofs using colored pictures and as little text as possible.

Office hours: T 1.20-2.20, W 3:20-4:20, and by appointment.

I will be in my office on Monday, Dec. 17, 1-2 pm.
I finished grading. I will be in my office on Friday, 1-2pm. You may also e-mail me about the results.

HERE is additional information about the organization of the course (date of test, grading policy, etc.).

A short biography of Euclid and Euclid

Here is a short note about Euclid and his "Elements".

Here is a discussion of the Wallace-Gerwien-Bolyai Theorem. Here is a discussion of an analogous problem in three dimensions (often called Hilbert's third problem) - it is a challenging read.

Here are numerous proofs of the Pythagorean Theorem.

Here you can read about power of points with respect to a circle and its application to challenging high-school problems.

Nine point circle

Here are some properties of the nine-point circle.

Here and here you will find a discussion of the orthic traingle and its properties..

Here is a "proof" that all triangles are equilateral.

Here is a proof of the Simson's line theorem, and here is a discussion of the Simson line.

Here is a nice visualization of Miquel point. Here is a proof of Miguel point theorem (Pivot theorem) and also several properties of the Simson line.

Here is a discussion of Miquel point and other results due to Miquel.

Here is a nice article about finite geometries.

Here is a short discussion of affine and projective planes.

Here is an overview of Hilbert's axioms.

Here is a nice discussion of inversion. See various links at the bottom which all show various applications of inversions. In particular, read the one about Apollonian circles theorem, which is related to problem 39.10.

Here is another place where many interesting results related to inversion are discussed.

Here is a short discussion of inversion with many references.

Here is a very nice applet showing the Poincare model of the Hyperbolic plane. Play with it!

Here is another nice applet.

Here is a more substantial discussion and visualization of hyperbolic geometry.

You should be able to state and discuss the five postulates from the Elements and the main definitions of Books 1-4. You should be able to state, prove, and use the following propositions from the Elements.

Book 1: Propositions 8, 9, 10, 11, 12, 23, 26, 27, 29, 31, 32, 47.

Book 2: Propositions 11, 12, 13, 14 (squaring a rectangle).

Book 3: Propositions 16, 17, 18, 19, 20, 21, 22, 23, 31, 32, 35, 36, 37.

Book 4: Propositions 4, 5, 10, 11.

Exam Solutions

Assignments

Assigments must be written carefully. You are allowed (even encouraged) to discuss the assigments with other students but you have to write the solutions on your own. Identical solutions will not be graded. If your solutions are not written clearly, they will not be graded and you will receive no credit.

HW 1: pdf. HW 2: pdf. HW 3: pdf. HW 4: pdf. HW 5: pdf. HW 6: pdf.