3Blue1Brown

After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math, so to speak.

What fractal dimension is, and how this is the core concept defining what fractals themselves are.

The Borsuk-Ulam theorem from topology solving a counting puzzle. Truly unexpected, truly beautiful.

How e to the pi i can be made more intuitive with some perspectives from group theory, and why exactly e^(pi i) = -1.

I want you to feel that you could have invented calculus for yourself, and in this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.

Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?

A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.

A visual explanation of what the chain rule and product rule are, and why they are true.

A visual explanation of what the chain rule and product rule are, and why they are true.

Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).

Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.

What is an integral? How do you think about it?

Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.

A very quick primer on the second derivative, third derivative, etc.

Taylor polynomials are an incredibly powerful for approximations, and Taylor series can give new ways to express functions.

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.

Can we describe all right triangles with whole number side lengths using a nice pattern?

The Bitcoin protocol and blockchains explained from the viewpoint of stumbling into inventing your own cryptocurrency.

Supplement to the cryptocurrency video: How hard is it to find a 256-bit hash just by guessing and checking? What kind of computer would that take?

Space filling curves, turning visual information into audio information, and the connection between infinite and finite math.

How do you think about a sphere in four dimensions? What about ten dimensions?

What's actually happening to a neural network as it learns?

This one is a bit more symbol heavy, and that's actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of the series, hopefully providing some connection between that video and other texts/code that you come across later.

A difficult geometry puzzle with an elegant solution.

Original Title: Rediscovering Euler's formula with a mug (not that Euler's formula) A mug with some unexpectedly interesting math.