3Blue1Brown Explainers

A collection of notes on several standalone "explainer" videos from 3Blue1Brown, covering a range of beautiful mathematical topics.

The Fourier Transform

The Fourier Transform is a mathematical tool that allows us to decompose a signal into its constituent frequencies. It's like taking a musical chord and figuring out which individual notes are being played.

The Core Idea: Winding a Graph

Imagine you have a signal, which is a function of time, like a sound wave. The core idea of the Fourier Transform is to take this signal and "wind" it around a circle.
* You choose a frequency for how fast you wind the graph.
* If the chosen frequency is present in the original signal, the wound-up graph will be lopsided, and its "center of mass" will be far from the origin.
* If the chosen frequency is not present, the graph will be more evenly distributed around the circle, and its center of mass will be close to the origin.

The Fourier Transform is a new function that, for each frequency, gives you the position of the center of mass of the wound-up graph. The magnitude of this complex number tells you the amplitude of that frequency in the signal, and its angle tells you the phase.

The Formula

$$
\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx
$$
* $f(x)$ is the original signal.
* $e^{-2\pi i x \xi}$ is the mathematical representation of the "winding" process. It's a point on the unit circle in the complex plane that rotates as $x$ changes. The frequency of rotation is given by $\xi$.
* The integral sums up the contributions of the entire signal to find the "center of mass" for that frequency.

Quaternions and 3D Rotation

Quaternions are a number system that extends the complex numbers. They are particularly useful for representing rotations in 3D space, avoiding problems like gimbal lock that can occur with other representations like Euler angles.

From Complex Numbers to Quaternions

• 2D Rotation: Multiplying a complex number by another complex number of magnitude 1, $e^{i\theta}$, rotates it by the angle $\theta$ in the 2D complex plane.
• Quaternions: A quaternion is a number of the form $q = a + bi + cj + dk$, where $a, b, c, d$ are real numbers, and $i, j, k$ are imaginary units that satisfy the relations:
  $$
  i^2 = j^2 = k^2 = ijk = -1
  $$
  This means that multiplying by $i, j,$ or $k$ can be thought of as rotating between different imaginary axes.

Using Quaternions for 3D Rotation

A rotation in 3D space by an angle $\theta$ around a unit vector axis $(x, y, z)$ can be represented by the quaternion:
$$
q = \cos(\theta/2) + (x i + y j + z k) \sin(\theta/2)
$$
To rotate a vector $\vec{v}$ (represented as a quaternion with a zero real part), you "sandwich" it between the rotation quaternion $q$ and its conjugate $q^*:
$$
\vec{v}_{\text{rotated}} = q \vec{v} q^
$$

The Mandelbrot Set

The Mandelbrot set is one of the most famous fractals in mathematics, generated by a very simple rule.

The Rule

The set is defined by an iterative process in the complex plane. For any complex number $c$, you generate a sequence starting with $z_0 = 0$:
$$
z_{n+1} = z_n^2 + c
$$

The Question

For a given starting complex number $c$, does the resulting sequence of $z_n$ values fly off to infinity, or does it remain bounded (staying within a certain distance of the origin)?

The Mandelbrot set is the set of all complex numbers $c$ for which this sequence remains bounded. The intricate, infinitely complex fractal shape emerges from this simple rule. The colors often seen in images of the Mandelbrot set represent how quickly the sequence for points outside the set goes to infinity.

Euler's Number (e)

Euler's number, $e \approx 2.71828$, is a fundamental constant that appears throughout mathematics and science. It is the base of the natural logarithm.

The Intuition: Continuous Growth

The number $e$ arises naturally from the idea of continuous compound growth.
* Imagine you have $1 that grows at a rate of 100% per year. After one year, you have $2.
* What if you compound the interest twice a year? You get 50% interest twice. This is $(1 + 1/2)^2 = 2.25$.
* What if you compound it $n$ times a year? You get $(1 + 1/n)^n$.
* The limit as $n$ approaches infinity (continuous compounding) is $e$:
$$
e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n
$$

The Calculus of e

The function $f(x) = e^x$ has the unique property that it is its own derivative:
$$
\frac{d}{dx}e^x = e^x
$$
This means that the rate of growth of the function at any point is equal to its value at that point. This is the mathematical essence of continuous, exponential growth.

The Riemann Hypothesis

The Riemann Hypothesis is one of the most famous unsolved problems in mathematics. It is a conjecture about the zeros of the Riemann zeta function.

The Riemann Zeta Function

The zeta function is defined as the infinite sum:
$$
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \dots
$$
This function is deeply connected to prime numbers through the Euler product formula.

The Hypothesis

The zeta function can be extended to the complex plane. It has some "trivial" zeros at the negative even integers. The Riemann Hypothesis is concerned with the "non-trivial" zeros. It states:

All non-trivial zeros of the Riemann zeta function have a real part of 1/2.

This means they all lie on a specific vertical line in the complex plane, known as the "critical line".

Why It Matters

The locations of these zeros are intimately related to the distribution of prime numbers. If the hypothesis is true, it would provide a much more precise understanding of how the primes are distributed among the integers.

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• Explainers: 3Blue1Brown