NoShade.Vision

Summarized Videos

3Blue1Brown

But what is a Fourier series? From heat flow to drawing with circles | DE4

This video discusses the mathematics behind a complex Fourier series, which involves a collection of rotating arrows that can draw anything. Fourier discovered that the heat equation can be solved using an infinite family of solutions, leading to the creation of solutions for tailor-made initial conditions despite having different distributions than cosine waves. The focus shifts to breaking down functions into rotating vectors for a more generalized approach. The speaker explains how Fourier series can be used for drawing animations using rotating vectors, and relates this back to the simpler one-dimensional example of a step function. The video concludes that this foray into Fourier series is a glimpse at a deeper idea, as exponential functions play an important role in differential equations.

Oh, wait, actually the best Wordle opener is not “crane”…

In this video, the speaker admits a minor error in the code used to solve the game Wordle, which had little effect on the results, and provides information on how color is assigned to letters when there are repeated characters in the guess. The best possible opening guess for the game is revealed to be "soare" after weighting all possible patterns, followed by "slane." However, as humans play the game differently, this information is only useful if someone has memorized the word list and will not largely impact gameplay. The speaker discusses the joy of writing algorithms for word games like Wordle and how that contributes to improving skills needed to create more meaningful algorithms for other purposes.

But what is the Central Limit Theorem?

The Central Limit Theorem (CLT) is a fundamental concept in probability theory that states that as the sum of events gets larger, the distribution of those events will increasingly resemble a bell curve, regardless of the distribution of the individual events. This section introduces multiple approaches to understanding the CLT, including simulations with different dice distributions, finding the range of values with 95% certainty, and explaining the third assumption of finite variance. The CLT realigns all the distributions so that their means line up together and rescales them so that all the standard deviations equal one, resulting in a universal normal distribution function. The video emphasizes caution when assuming variables follow a normal distribution without justification and unravels the mystery of why the standard normal distribution has a pi in it, tying it back to circles.

But what is a convolution?

The "But what is a convolution?" video explains that convolutions involve combining two lists of numbers or functions, which cannot be achieved by simple arithmetic operations. Convolution is used in diverse applications such as probability, image processing, and differential equations. The video presents various examples of discrete convolution, including rolling dice, creating a moving average, and blurring images. The speaker notes that convolutions are equivalent to multiplying two different polynomials, and introduces the discrete Fourier transform as a more efficient algorithm for convolution. The video concludes by hinting at the next topic: the continuous case and probability distributions. The fast Fourier transform is also introduced as an efficient algorithm reducing computational complexity.

Researchers thought this was a bug (Borwein integrals)

The Borwein integrals, a sequence of integrals that appear to be unrelated, are shown to be connected through Fourier transforms. The video explains the use of stretched and multiplied versions of the sinc function to demonstrate how integrals generate a stable area equal to pi until they fall just short of ideal value, with a similar factor in front of pi. By evaluating Fourier transforms of functions, the video shows how Borwein integrals can be computed based on the Fourier transform of a function, and how convolutions can be understood using moving averages. Overall, the video highlights the power of Fourier transforms and the convolution theorem to solve tricky problems.

Why π is in the normal distribution (beyond integral tricks)

The video discusses why pi appears in the formula for a normal distribution beyond integral tricks. The speaker goes beyond the classic proof that explains the presence of pi and connects it to the Central Limit Theorem. They explain how to find the volume underneath a bell surface and relate it to the two-dimensional graph of a bell curve. The speaker introduces John Hershel's theory of using the e^-x^2 function to describe a probability distribution in two-dimensional space. They simplify the problem of deriving the normal distribution of numbers using Herschel's two properties and a helper function that satisfies a property of addition to multiplication. The video also explores how the Herschel/Maxwell derivation converges with the Gaussian distribution's defining property of circular symmetry and offers insight into how a convolution theorem and Fourier transforms help explain the e^(-x^2) function in the Central Limit Theorem.

How to lie using visual proofs

The video discusses how visual proofs without rigorous logic can be misleading in mathematics, taking the examples of fake proof of surface area of a sphere and the incorrect visual proof of all triangles being isosceles. The presenter warns against relying solely on intuition and visual aids, emphasizing on the need for proof and logical rigor in mathematics. The video further explains the difference between the geometry of curved surfaces and flat spaces, emphasizing the danger of making assumptions without explicit confirmation and the subtleties of limiting arguments.

Olympiad level counting

The video introduces the problem of finding the number of subsets of a set whose sum is divisible by five. The presenter explains the concept of generating functions and demonstrates how to use them to solve the problem efficiently. The video also discusses the use of complex numbers and roots of unity to manipulate the coefficients of the generating function, allowing for the deduction of information without expanding the expression. The video concludes with a reflection on the unexpected techniques used to solve the problem and suggests further study of generating functions in the book by Herbert Wilf.

We ran a contest for math explainers, here are the results (SoME2)

The creators of the YouTube channel, 3Blue1Brown, discuss the results of their math exposition contest that invited contestants to create math-related videos or articles. The winning submissions were judged based on motivation, clarity, novelty, and memorability. The rise in views for the submissions garnered over 7 million views since the submission deadline for the contest. It is discussed how the benefit of having a well-motivated audience is particularly important when it comes to educational content like math and the importance of clarity in building a viewer's intuition for understanding. The different ways in which motivation can present itself and examples of well-motivated videos and articles are given. The contest winners are also announced, and it is said that the project’s success can be attributed to sponsors and support from the growing community.

Alice, Bob, and the average area of a cube's shadow

This video explores the problem-solving process for finding the average area of a cube's shadow and introduces two different problem-solving styles embodied by Alice and Bob. Alice discovers three insights, including the relationship between the area of a cube's shadow and its individual faces, the fact that the average of the sum of face shadows is the same as the sum of the average of the face shadows, and the general correlation between a convex solid's shadow and surface area. Bob, on the other hand, takes a more detailed approach and uses calculus to find the average area for the shadow of a square. The video emphasizes the importance of both approaches and explains how they contribute to understanding mathematical concepts like convexity and abstraction. The video ends with a challenge to reflect on how Alice and Bob answer the question of what it means to choose a random orientation.