abstract algebra

An animated introduction to geometric algebra, presented as a language that can describe most of physics. It was made with manim and covers multivectors and how they generalize familiar tools like cross products and quaternions.

3Blue1Brown introduces group theory and abstract groups, then builds toward classification of finite simple groups and the enormous Monster group with its 196,883 dimensions.

3Blue1Brown's chapter on eigenvectors and eigenvalues in the Essence of Linear Algebra series. It builds a visual intuition for what they mean, why an eigenbasis is useful, and ends with a puzzle.

A 10-minute 3Blue1Brown chapter explaining the determinant as the factor by which a linear transformation scales area or volume. It builds the idea geometrically with animations before any formulas.

Veritasium tells the history of how imaginary numbers arose from the quest for a general solution to the cubic equation, including the duels and secrecy among Renaissance mathematicians. It argues math had to give up the requirement that it mirror reality.

3Blue1Brown, in collaboration with minutephysics, introduces the math of superposition and quantum states using polarized light. It builds up the quantum behavior of photons as a simple counting and probability problem.

Matt Parker talks Numberphile through the different kinds of numbers: algebraic, transcendental, computable, normal, and beyond. It shows how few numbers we can actually name or describe.

Technology Connections explains why the superheterodyne radio has its name and how mixing frequencies lets it tune stations well. It walks through the underlying wave phenomena and algebra and why the design solved a hard tuning problem.

A PBS Space Time episode asking whether the universe's information can ever truly be lost, tying quantum mechanics, determinism and black holes together. It sets up the black hole information paradox.