Math Beyond Your Courses

Here are some references for interesting topics which go beyond your usual CEGEP math courses, which you could use for some vacation reading/watching… Most of these are free and available online, some others are actual textbooks, which you should be able to find in our - or a public - library.

Even if the topic titles don’t mean anything to you (yet), or if they don’t sound that interesting, I highly recommend you just watch some of the videos, I think you’ll be pleasantly surprised. Also, don’t be discouraged if you don’t understand everything shown - the goal should be firstly to just get a rough overview of all the different kinds of “Math” which are out there, and see if you find a topic interesting.

Have fun (and let me know if you have questions or want even more recommendations)!

Gabriel Indurskis

Fractals

• Introduction to Fractals: Fractals and how to build a Sierpinski Tetrahedron (Henry Segerman)

• Fun YouTube videos:

  • Space-filling curves (3Blue1Brown)

  • Henry Segerman on space-filling curves (Numberphile)

  • Doodling in Math Class: Infinity Elephants (Vihart)

• The following website, make your way through the listed topics one by one as you like (this was co-edited by the late Benoît Mandelbrot, who is usually credited with “inventing” mathematical fractals):

  http://users.math.yale.edu/public_html/People/frame/Fractals/

• “The Fractal Geometry of Nature” by Benoît B. Mandelbrot (1982)

Probability and Statistics

• Videos:

  • The Monty Hall Problem (Numberphile)
  • Rock, Paper, Scissors (Numberphile)
  • Markov Chains (PBS Infinite Series)

• Download this free, opensource textbook, and read Chapters 1 and 2 (or more if you like!): https://www.openintro.org/stat/textbook.php?stat_book=os

Knot Theory

Knot Theory is a subfield of Topology which is also related to Graph Theory.

• Videos (all of these are on the same playlist of fantastic videos about knots, but I in particular recommend the following):

  • What are mathematical knots? (Numberphile)
  • Knots in Biology, e.g. DNA (Numberphile)
  • The Borromean Rings (Numberphile)

• “The Knot Book” by Colin Adams

• Video: Introduction to Knots & Invariants (Andrews University)

Topology (including Graph Theory)

• Videos:

  • Topology Riddles (Kelsey Houston-Edwards, on PBS Infinite Series)
  • Too Many Triangles (Henry Segerman, on Numberphile)
  • The Bridges of Königsberg (Cliff Stoll, on Numberphile)
  • Sewing and cutting Klein bottles (Cliff Stoll, on Numberphile)
  • Doodling in Math Class: Snakes & Graphs (Vihart)
  • Network Mathematics and Rival Factions (PBS Infinite Series): using graph/network theory to analyze friendship/rivalry relationships (with some references [and mild spoilers] for Game of Thrones)
  • Who cares about topology? The Inscribed rectangle problem (3Blue1Brown)

• “Flatland: A Romance of Many Dimensions”, by Edwin A. Abbott (satirical novella, first published in 1884, now available for free): Note that women are described in a non-favorable way, but this is in fact meant as a social critique of society. Please read the section about this on this Wikipedia page before misunderstanding this aspect of the book.

• “The Shape of Space” by Jeffrey Weeks: a great introduction into the topology of 3-dimensional spaces

• also, this related (old, but great) 8min video about The Shape of Space

• also related, this 3min video by PhDcomics about The Shape of Space from a physics point of view

Complex Numbers, Quaternions, and more

• Playlist of 13 short & great videos by Welch Labs explaining Complex Numbers, including their history

• Euler’s Formula with Introductory Group Theory (3Blue1Brown)

• Möbius Transformations Revealed (Douglas Arnold and Jonathan Rogness)

• Visualizing Quaternions: An explorabale video series by Grant Sanderson (3Blue1Brown) and Ben Eater

Group Theory

• Group theory and why I love 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (3Blue1Brown)

Error-correcting Codes

• Hamming codes, Part 1 and Part 2, by Grant Sanderson (3Blue1Brown)

Useful free/open-source Mathematics Websites/Software

• CoCalc (Collaborative Calculation in the Cloud): This includes the computer algebra system “Sage”, amongst many other things. Highly recommended. Watch the short video on the main page for an introduction.
• Desmos (easy to use online graphing calculator)
• Geogebra (great for geometric constructions)

Mathematical Typesetting (LaTeX)

• Introduction to LaTeX (GVSUmath)
• Intro to writing with LaTeX on CoCalc.com (previously called SageMathCloud) (Vincent King)

Programming

• Python from Scratch: Online introduction to Python by the University of Waterloo
• Pythonchallenge.com: An online programming riddle (can be done with other programming languages)
• CodeCombat.com: Learn programming by playing a game
• Code Academy: More serious (but great) online programming courses
• For even more computer science games & learning resources (some aimed at younger children, but playable at any age), see the list at Computer Science Fun for Kids

General

• What’s the fastest way to alphabetize your bookshelf? (Chand John)
• The mathematical secrets of Pascal’s triangle (Wajdi Mohamed Ratemi)
• Floating Point Numbers and the limitations of computers (Computerphile)

• Math and Movies - Animation at Pixar (Numberphile): amongst other topics discusses B-spline surfaces

• Any of the videos on these great YouTube channels:

  • Numberphile
  • 3Blue1Brown
  • PBS Infinite Series

Silly

• Mr Bean: The Exam

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