Introducing FOSSEE Animations: Lecture notes and Animations for Undergraduate Mathematics
Geometric intuition is the ability to solve and/or reason out non-geometric problems in geometric ways. Now, there have been numerous derby matches on rigour vs intuition (especially in mathematics). We’ll try to stray away from recapitulation, but instead, stand on the shoulder of giants.
The famous British mathematician Ian Stewart had said, “Intuitive feeling invoked by a picture can, with a little care, be turned into a logically satisfactory mathematical proof that the theorem is true - and because it appeals to intuition, it’s a very convincing proof”. But what if we could somehow extend it beyond just a static picture? Things do become delightfully interesting, to say the least.
One good thing about visualizing mathematics is that it simplifies the complexity. For instance, we can break down an abstract topic into a series of steps - reducing the reliance on prerequisites. The intention here is not to forgo the rigour completely, but instead, make it a bit more rewarding with some visual treats - to look at analysis from the eyes of geometry.
This approach was inspired by Grant Sanderson (creator of 3b1b and the python package manim) - in fact, most of the animations created under this project are made with manim.
“FOSSEE Animations | Math” is a FOSSEE project based in IIT Bombay. The project consists of a repository of lecture notes on selected math topics created for students, by students. (i.e all the notes and animations are created by the students as a part of an internship/fellowship) -- BUT that’s not the end of it - apart from being typical lecture notes, mathematically cogent animations are also present to complement the notes (lecture notes on steroids?).
These lecture notes closely follow a specific structure:
- Definition / Theorem
- Motivation
- Bird’s Eye View - a simplified overview of the topic, essentially explains the main points of the topic in a simplified (non-rigorous) manner.
- Context of the Definition - expands on the topic and, more importantly, explains the what and the why of the theorem/definition. Typically involves breaking down the theorem/definition into smaller pieces and explaining the pieces individually.
For example, why is differentiability of a single-variable function defined on an open set and not a closed set? - Applications
- Pause and Ponder - offers “food for thought” on the given topic
- History / References / Further Reading
Over the course of this summer (2020), we curated a repository of around many such lecture notes ranging from Transformations to Multivariable calculus. These notes were created by the amazing fellows from the FOSSEE Summer Fellowship 2020. Again, the idea behind this concept is that the animations will be complementary to the notes, giving a well-formed geometrical intuition about the mathematical theory.
The process of creating such notes is rather exhaustive, with multiple revisions and reviews - around 3-5 reviews with the mentors and one round of discussion with Prof. Ramachandran. Moreover, before finalizing the notes, they also go through another round of review with a few other professors from IIT Bombay. The fellows have also shown appreciation to this process as they improved their pedagogical writing which in turn would be helpful for the anonymous reader.
In terms of numbers, during the Fellowship, we curated 50+ lecture notes with 200+ animations. These notes are primarily focused on first/second year undergraduate mathematics curriculum, specifically Linear Algebra, Calculus of Several Variables, and Series & Transformations.
And, since these notes are created by students, we certainly have some more opportunities in the bag. While the Summer Fellowship takes place once a year, last year we also gave an internship on Real Analysis - with more planned in the future. The internships follow the same model as the fellowship. The outreach programs also include workshops on Animating Math, where libraries like Manim and Mayavi are introduced along with the general principles of curating explanatory animations are discussed.
Tl;dr: The FOSSEE Animations project works on simplifying the complex topics through animations. This is done by numerous students who take part in the internship and/or fellowship. The lecture notes are available at math.animations.fossee.in and attempts to present the reader with a blend of both the algebraic and geometric approach.