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
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Introduction to Fractals: Fractals and how to build a Sierpinski Tetrahedron (Henry Segerman)
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Fun YouTube videos:
- Space-filling curves (3Blue1Brown)
- Henry Segerman on space-filling curves (Numberphile)
- Doodling in Math Class: Infinity Elephants (Vihart)
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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/
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“The Fractal Geometry of Nature” by Benoît B. Mandelbrot (1982)
Probability and Statistics
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Videos:
- The Monty Hall Problem (Numberphile)
- Rock, Paper, Scissors (Numberphile)
- Markov Chains (PBS Infinite Series)
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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)
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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)
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“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.
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“The Shape of Space” by Jeffrey Weeks: a great introduction into the topology of 3-dimensional spaces
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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
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Playlist of 13 short & great videos by Welch Labs explaining Complex Numbers, including their history
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Euler’s Formula with Introductory Group Theory (3Blue1Brown)
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Möbius Transformations Revealed (Douglas Arnold and Jonathan Rogness)
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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)
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Math and Movies - Animation at Pixar (Numberphile): amongst other topics discusses B-spline surfaces
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Any of the videos on these great YouTube channels:
