classical geometry

Welcome to the math228 course webpage! Here, you'll find info about the class and links to some helpful ressources. The page will be updated every week and it will serve as the main communication tool. If any link is broken, please let me know.

schedule

mwf 10:30-11:30

Trottier, room 2120

Please note that the courses take place in-person. Courses at McGill start offically on Wednesday September 1st and end on Monday December 6th (see key academic dates here).

course outline

The course is designed to (re)introduce the method and results from Euclid's classical geometry coming from both a historical and a mathematical point of view. You can find the up-to-date course outline here.

structure of the course

• I. Reconnecting with geometry
• II. A review of Euclid's Elements
• II.1 The Books
• II.2 A review of Book 1
• II.3 A pot-pourri of Euclid's work
• III. Visiting geometry
• III.1 Centers of triangles
• III.2 Relating quantities in a triangle
• III.3 Conic sections
• III.4 Hilbert's axiomatization of geometry
• III.5 Solid geometry

exercises

Here is a concise list of the suggested exercises.

• [Har13] 1.1.(4,5,6,7,8,9,10), 1.5.(8,9,10,13,14,19), 2.6.(1,3,5,6,8,10), 2.7.(1,2,4,6)
• [CG67] 1.1.3, 1.3.(1,3,4)
• homemade exercises
• practice midterm and the solutions
• practice final

references

Our main references will be [CG67] and [Har13].

books

• [AZ07] Akopyan, A. & Zaslavsky, A. (2007). Geometry of conics. American Mathematical Society.
  Links: McGill's library and pdf
• [CG67] Coxeter, H. S. M. & Greitzer, S. L. (1967). Geometry revisited. Maa.
  Links: McGill's library and pdf
• [Eve72] Eves, H. (1972). A survey of geometry. Allyn & Bacon.
  Links: pdf
• [Eve97] Eves, H. (1997). Foundations and fundamental concepts of mathematics. Courier Corporation.
  Links: McGill's library
• [Fit07] Fitzpatrick, R. (2007). Euclid’s elements of geometry. Euclidis Elementa.
  Links: Fitzpatrick's webpage
• [HS21] Hall, H. S. & Stevens F. H. (1921). A school geometry. Macmillan and co.
  Links: pdf
• [Har13] Hartshorne, R. (2013). Geometry: Euclid and beyond. Springer Science & Business Media.
  Links: McGill's library and pdf
• [Hea56] Heath, T. L. (Ed.). (1956). The thirteen books of Euclid's Elements. Courier Corporation.
  Links: McGill's library, pdf books I-II, pdf books III-IX and pdf books X-XIII

articles, course notes and more

• [Eve58] Eves, H. (1958). Pappus's extension of the Pythagorean Theorem. The Mathematics Teacher, 51(7), 544-546.
  Links: pdf
• [Mad89] Mader, A. (1989). A Euclidean model for Euclidean geometry. The American Mathematical Monthly, 96(1), 43-49.
  Links: pdf
• [Wol21] Wolfram, S. (2021). The Empirical Metamathematics of Euclid and Beyond. arXiv preprint arXiv:2107.07337.
  Links: Wolfram's blog, arXiv

drawings, scans and proofs

• drawings
• the Euler line and the nine-point circle
• the relationship between H,N and O
• the Dandelin spheres
• the 5 Platonic solids
• some non-Platonic solids
• scans
• the Euler line
• the nine-point circle
• the excircle
• the power of the incenter
• proofs
• prop 6, Book I
• prop 20, Book I

course calendar

The following calendar can give you a good impression of what's to come! Don't forget to scroll down to get some more precise info about quizzes and exams. If you enjoy zooming in a lot, you can also download the pdf version.

evaluations

Evaluations will be in three formats : 5 quizzes (3% each), 1 midterm (25%), a final (40%) and an oral exam (20%). They will all be closed book.

• Quizzes happen on Monday, every even numbered week (except weeks 2 and 8, details below). Time alloted will be 40 minutes. They will consist of one problem and/or stating definitions. Hints will be given in class the week before.
• Written exams will consist of one midterm and one final. The midterm will be in two parts on week 8, happening on Monday 18 and Wednesday 20. The final exam will take place on December 17th from 2pm to 5pm.
• During the last three weeks of class, students will be asked to discuss geometry one-on-one during a 30 minutes interview.

exams

• quizzes
• quiz 1, solutions
• quiz 2, solutions
• quiz 3, solutions
• quiz 4, solutions
• quiz 5, solutions
• midterm
• part 1
• part 2
• solutions
• final, solutions
• oral. The following sheet details some selected topics.

important info

Labour day on Monday September 6th. Courses only on Wednesday and Friday.

lectures

• 08.09
  I.Reconnecting with geometry (continued and finished)
  Recap of the classification of Pythagorean triples.
  II. A review of Euclid's Elements. II.1 The Books
  A first look a the structure of Euclid's elements and the 5 postulates.
• 10.09
  II.2 A review of Book 1
  Propositions 1,2,3,4,5 from Book 1 and their proofs.

important info

Quiz 1 on Monday.

Starting Wednesday, we will assume some basic results of geometry. You can take a look at [Har13]'s appendix "Brief Euclid" for a review of what we will assume. I highlighted some of them here. You should read the proof of the highlighted results.

lectures

• 20.09
  Quiz 1
  Pdf available here and solutions available here.
• 22.09
  III.Visiting geometry. III.1 Centers of triangles
  Defining what we mean by the center of a triangle. The circumcenter of a triangle.
• 24.09
  III.1 Centers of triangles (continued)
  The centroid and orthocenter of a triangle.

important info

Quiz 2 on Monday. Exercise 10 of homemade problems is a good preparation for the quiz.

lectures

• 04.10
  Quiz 2
  pdf available here and solutions available here.
  
• 06.10
  III.2 Relating quantities in a triangle
  Formulas for relating area, circumradius, inradius and the exradii. Construction of the excircles. The displayed drawing is available here.
  
• 08.10
  Recap!

important info

Midterm week. On Monday and Wednesday, problems will be distributed in class.

lectures

• 18.10
  Midterm part 1
  pdf available here.
• 20.10
  Midterm part 2
  pdf available here and solutions for midterms 1 and 2 available here.
• 22.10
  III.2 Relating quantities in a triangle (continued)
  The power of a point with respect to a given circle.

important info

Quiz 3 on Monday. A good preparation for quiz 3 is ex11 on the homemade exercises. Bonus midterm questions should be submitted by Monday November 1st on midnight.

lectures

• 01.11
  Quiz 3
  pdf available here and solutions available here.
• 03.11
  III.3 Conic sections (continued)
  The Dandelin spheres. Here is a handy sheet explaning the construction for all three families of conics. Included is also a drawing explaning the eccentricity of an ellipse.
  Watch on Youtube an excellent expository video made by Grant Sanderson called "Why slicing a cone gives an ellipse". It does a good job of showing the Dandelin spheres in action.
• 05.11
  III.3 Conic sections (continued)
  Properties of the excribed parabola. The focus point lies on the circumcircle.

important info

Quiz 4 on Monday. A good preparation for quiz 4 is reading (carefully) section 1.4 of [AZ07]. If you plan on doing the bonus question on the quiz, you might want to take a look at ex12 of the homemade exercises.

lectures

• 15.11
  Quiz 4
  pdf available here and solutions available here.
• 17.11
  III.4 Hilbert's axiomatization of geometry (continued)
  Axioms of betweeness
• 19.11
  III.4 Hilbert's axiomatization of geometry (continued)
  The plane separation theorem

important info

Quiz 5 on Monday. A good preparation for quiz 5 is to work on the exercises from section 2.6 and 2.7 in [Har13]. More precisely, you might want to try the exercises 1,3,5,6,8,10 in section 6 and 1,2,4 in section 7.
Warning: In 6.1 and 6.3, you can substitute "up to isomorphism" by "up to labeling of the points" (or go learn the definition of an isomorphism at p. 68 of [Har13]). In 6.5c, you can substitute "if F is any field" by "if F is either R (real numbers) or Q (rational numbers)" (or go learn the definition of a field on p.128 of [Har13]).

lectures

• 29.11
  Quiz 5
  pdf available here and solutions available here.
• 01.12
  III.5 Solid geometry
  The Platonic solids. A handy sheet with all of them is available here. Two examples of non-platonic solids are available here.
• 03.12
  III.5 Solid geometry (continued)
  Proof of the Euler formula and classification of all platonic solids.

The exam will take place the 17th of December at 2pm in BURN 1205. It will be a 3h exam, closed book.

The final is available here and the solutions are available here.