Math 4060 -- Foundations of Geometry

Spring 2017

Class Meets:   MWF 9-9:50 in GAB 317.

Instructor: William Cherry

Office Location: General Academic Building (GAB) 405.
Office Hours:  Mondays & Wednesdays 10-Noon


All students should have successfully completed Math 3000 (Real Analysis I) before taking this course. The reason for this is that this course is a proof intensive course and students should already be familiar with basic proof writing techniques and strategies. Prior or concurrent enrollment in either Math 3610 (Real Analysis II) or Math 3510 (abstract algebra) is also strongly recommended. This is meant to ensure familiarity with proof and proof technique. Nothing from real analysis or abstract algbra will actually be needed. Students who have not yet taken Math 3610 or Math 3510, but who are confident in their proof-writing ability, are welcome in the course.

Course Description

Early interest in geometry was almost certainly motivated by a desire to improve building techniques and to be able to create interesting shapes for temples, altars, toys, and machines. Scholars in ancient Greece "abstracted" the study of shapes into an idealized form. The most widely read textbook in history is a geometry text by Euclid known as the Elements. Thousands of years later, much of what we learn in high school geometry is based on the ideas in Euclid's text. Euclid's goal was to create a logical foundation for idealized geometry. He wanted to provide proofs for geometric facts based on as few axioms as possible. The course will begin with a careful study of parts of Euclid and an exploration of what he was trying to do and why. The course will then move on to the contributions of 19th and 20th century scholars who built on Euclid's work. The second half of the course will look at alternative geometries where the so-called parallel postulate need not hold.

The course will emphasize the role of proof in geometry, its historical development, and the philosophical implications that proof had on the development of scientific thought. Students will begin by reading Euclid and continually develop their own proof writing skills and geometric intuition.


Robin Hartshorne, Geometry: Euclid and Beyond, Springer Verlag, 2000. ISBN 9780387986500
Euclid's Elements, The Thomas L. Heath translation edited by Dana Densmore, Green Lion Press, 2002. ISBN 9781888009194.
Euclid is also available online at

Course Requirements

Grading in the course will be based on weekly homework assignments, in class midterm, and an in class final exam.
Daily Quizzes5%
Gometry Art Project10%
Midterms 28% (14% each)
Final Exam27%
Class attendance will be strictly monitored.

Important Dates

Midterms: Monday, Mar. 6 and Monday, Apr. 10
Geometry Art Project Due: Monday, May 1
Final Exam: Wednesday, May 10, 8:00 - 10:00 a.m.

Course Handouts

Course Syllabus
Article on Rusty Compass Constructions in Islamic Art
Notes on Euclid III.35
Information about the First Midterm Exam
Geogebra illustration of the Euler Line
Pictures Illustrating the 9-Point Circle Proof
Geogebra illustration of the 9-Pont Circle
Geometry Art Project Assignment (due May 1)
Hilbert Axioms
If you missed class, ask for the solutions to some of the homework from section 7 (not available on the web).
Information about the Second Exam
Geogebra file with a tool for drawing P-lines
Geogebra file illustrating P-diameters of P-circles
Geogebra file with tools to drop and erect perpendiculars in the Poincare plane
Handout on the proof of Theorem 34.5
The Boomerang Project website (tries to determine if the geometry of the universe is Euclidean or non-Euclidean); see also NASA's Universe 101 site.
An article about how isometries of the Poincare plane can be used in viewing brain scans.
Articles by Jeffrey Weeks and Neil Cornish on the Poincare Dodecahedral Space and the shape of the univerise: 2004 and 1998.
Kepler's Mysterium Cosmographicum
Information about the Final Exam

Computer Software

Geogebra: Select "Construction Protocol" from the "View" menu to see the steps of your construction or to print the steps.
Geometer's Sketchpad is available in the Arts & Sciences computer labs in GAB, Wooten, and Terrill. Click here for how to print construction steps from Sketchpad.

Homework Assignments

Homework #1 due beginning Jan 23 (the back page is not available on the web; see me if you missed class)
Homework #2 and #3 due Jan 27 and 30.
Homework #4 and #5 due Feb 1 and 3 (due dates postponed to Feb 3 for assignment #4 and Feb 6 for assignment #5).
Homework #6 and #7 due Feb 8 and 10
Homework #8 due by Mar. 3 (the back page is not available on the web; see me if you missed class)
Homework #9 and #10 due Feb 13 and 15
Homework #11 due Feb 17
Homework #12-14 due Feb 20, 22 and 24 (due date postponed to Feb 27 for assignment #14).
Homework #15 and #16 due Mar 1 and 3
Homework #17 and #18 due Mar 8 and 10
Homework #19-21 due Mar 20, 22, and 24
Homework #22-24 due April 3, 5, and 7
Homework #25 due April 12
Homework #26-27 due April 14 and 17 (Homework #27 postponed to Wednesday, April 19)
Optional Extra Credit Homework #28 due April 21.
Homework #29 due April 24.
Homework #30 due April 26.
Homework #31 due April 28.
Homework #32 due May 3.

Trouble reading or printing links on this page?

The homework assignments and supplemental materials above are in Adobe PDF (or Acrobat) format. If you are having trouble reading or printing these hand-outs, click here.

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