Prisms, an iconic physics experiment and album cover symbol,are a result of light slowing down and bending as it enters the glass. This occurs due to the interaction of the light waves with charged pa

The computation of a convolution between two Gaussian functions reveals that the result is another Gaussian distribution. This connection explains why the normal distribution holds a special place in

Submit your own math explainer video, blog post, or interactive game to the Summer of Math Exposition contest by August 22nd. Winners will be featured in a 3b1b video and receive custom gold plushie p

The central limit theorem states that regardless of the initial distribution, as sample sizes increase, the distribution of the sum tends to follow a bell curve. The mean of the distribution represent

The Summer of Math Exposition encourages people to put up math explainers online. Success means more good explanations online. The winners are not the focus, but getting more good explanations is. Thi

When polarized light passes through sugar water, it gets twisted in a unique way, resulting in different colors appearing in diagonal stripes. The twisting rate depends on the frequency of the light,

Join the Summer of Math Exposition, a contest and collaboration event for creating math content online. Subscribe to the mailing list, submit your content by August 15th, and engage with other partici

Matrix exponents are a powerful tool in mathematics and physics that can be used to solve differential equations and describe complex systems. The process involves exponentiating a matrix, which resul

In the Summer of Math Exposition contest on YouTube, smaller math channels gained attention and generated over 7 million views collectively. The peer-review process and shared deadline played a signif

Despite the high accuracy of medical tests, their predictive value can be unexpectedly low due to conditional probabilities.

Neural networks can effortlessly recognize complex patterns, even with different variations, thanks to the power of the brain's visual cortex. In this video, we delve into the structure and function o

Discover the mind-bending world of Bell's inequalities, a quantum mechanical topic that sheds light on the simple math behind quantum weirdness. This video explores the foundations of quantum mechanic

Hamming codes are an elegant and efficient way to detect and correct errors in data transmission. By using a clever combination of parity checks and XOR operations, these codes can locate the position

Convolution is a fundamental operation that combines two lists or functions to create a new one. It can be used to add, multiply, or convolve lists. Convolution has various applications in fields such

The Riemann zeta function is a complex function with an infinite sum that defines values for inputs on the right half of the complex plane. Its extension, known as analytic continuation, unlocks the f

Light passing through sugar water twists its polarization direction, leading to the appearance of colored diagonal stripes. This effect is due to the interaction between light and sugar molecules, whi

Discover three false proofs that lead to incorrect mathematical results, including a fake proof for the surface area of a sphere and a simple argument that tries to prove pi equals 4. These examples h

The Summer of Math Exposition was a contest aimed at encouraging more people to create math explainers. Over 1200 submissions were received, showcasing a wide range of topics and teaching styles. Five

When dealing with probabilities in continuous settings, we can't assign probabilities to individual values, but instead use probability density functions (PDFs) to describe the distribution. The area

When faced with multiple sellers offering the same product at the same price, it can be challenging to decide which one to buy from. This video explores a Bayesian approach to make that decision by an

In this video, we explore the concept of convolution by adding two random variables. We visualize the process through two different methods and discuss the continuous case. We provide an interactive d

In this video, we explore the mathematical beauty of the Gaussian distribution and uncover its surprising connections to circles and population statistics. Through elegant proofs and clever insights,

In this video, we solve a challenging puzzle involving subsets of whole numbers using generating functions. We discover the surprising role of complex numbers and explore the properties of the generat

Bayes' Theorem is a crucial formula in probability that is used in scientific discovery, machine learning, AI, and more. It helps update beliefs based on new evidence and is essential for understandin

Leonard Cohen's famous song Hallelujah was originally inspired by a mathematical pattern. This video explores the relationship between math and music and shares the lesser-known backstory of the song.

Prime numbers exhibit interesting patterns when plotted in polar coordinates. Through polar coordinates, the distribution of prime numbers and their relationship with rational approximations for pi ca

Code bug in Wordle solver slightly affected performance; Bug relates to color assignment for repeated letters; Final analysis reveals optimal opening guess for Wordle game

Dive into the world of vector fields and explore the concepts of divergence and curl. Discover how these concepts can be applied to fluid flow, electricity and magnetism, and even population dynamics.

Math is captivating in 2D and 3D, but becomes teasingly frustrating in higher dimensions. Visualizing analytic facts geometrically is useful, but beyond 3D, it's challenging. Real estate analogy helps

Bitcoin is a fully digital currency with no government or banks, and this video explains the process of inventing it step by step. It starts with keeping track of payments using a communal ledger and

Hamming codes are a clever way to correct errors in data transmission by adding redundancy without sacrificing space. By using carefully chosen parity checks, Hamming codes can identify and fix single

Prime numbers have a surprising connection to natural logarithms, as shown by various mathematical sequences. The sum of reciprocals of prime numbers converges to the natural logarithm of 2, while the

Fractals are not just perfectly self-similar shapes, they are shapes with non-integer dimensions that capture roughness. A shape's dimension can be measured by counting the number of boxes it touches

Understanding the connection between trigonometry and complex numbers strengthens your understanding of both. Exploring trigonometric functions and their graphs leads to non-trivial insights. Squaring

Taylor series are powerful tools for approximating functions using polynomials. By taking derivatives of a function at a specific input, you can construct polynomials that closely resemble the origina

A group of YouTubers attempt to solve the 'impossible' puzzle on a mug using lines that cannot intersect, through creative use of the mug's handle, and recognizing that the mug's shape allows for conn

Discover two distinct problem-solving styles, embodied by Alice and Bob, when solving a puzzle involving the average area of a shadowed cube. Alice prefers a high-level overview before diving into the

Holomorphic dynamics is a field of math that studies functions with complex number inputs and outputs. It explores what happens when these functions are repeatedly applied, leading to behaviors such a

In this video, we delve into the connection between sliding actions in the additive group and stretching actions in the multiplicative group. By understanding this relationship, we can gain new insigh

This video explores the core ideas of calculus, with a focus on finding the area under a curve and calculating derivatives. It introduces the fundamental theorem of calculus and emphasizes the importa

Discover the concept of Fourier transform and how it can decompose frequencies in signals using a mathematical machine. Get a glimpse into its application in sound editing and frequency analysis. A fa

Discover the fascinating relationship between prime numbers, complex numbers, and pi, and how they intertwine in a mesmerizing trio. Uncover a formula for pi through a deep dive into the Riemann zeta

This video explores the classification of finite groups, including simple groups and sporadic groups. Finite groups come in various families and there are 26 sporadic groups that do not fit into the e

The uncertainty principle is a general trade-off between knowledge of position and momentum in various wave phenomena, including sound waves and Doppler radar. It arises from the interplay between fre

Newton's method for finding polynomial roots produces a mesmerizing fractal pattern in the complex plane. The iteration of linear approximations creates chaos at the boundaries between regions, showca

Discover the secrets of refraction and understand why light slows down in different mediums. Explore key insights, FAQ, and timestamped summaries for a thorough understanding.

In the chessboard puzzle, you and your fellow inmate must deduce the location of a hidden key by flipping coins. However, it is impossible to have a universal strategy for all board sizes not a power

In cryptography, guessing a specific string of 256 bits can require an unimaginable number of attempts. Even with the computational power of billions of computers, the chances of finding the correct g

Logarithms are mathematical functions that allow for easy scaling and visualization of data. They can be used to represent exponential growth as a straight line on a logarithmic plot. Logarithms can a

In this video, we explore the math behind complex Fourier series, which involves decomposing functions into rotating vectors. Each vector rotates at a constant frequency, and when their lengths are ad

Learn a quick trick to compute eigenvalues of 2x2 matrices using the mean and product of the matrix's elements. This trick simplifies the process and provides a direct way to find the eigenvalues with

Explore Moser's Circle Problem and how it deceivingly appears to follow a pattern of powers of 2 but falls short by 1. Discover the connection between this problem, Pascal's triangle, and Euler's form

Discover how the game Wurdle and the concept of entropy are connected through information theory. Follow along as we analyze the game and develop an algorithm that maximizes the expected information w

The sequence of integrals involving the sinc function follows a predictable pattern and equals pi until a certain point. This pattern is analogous to a sequence of moving averages. The connection betw

Eigenvectors and eigenvalues are important concepts in linear algebra. This video provides a visual guide to understanding them. By visualizing linear transformations and their effects on vectors, eig

Each year, over 100 countries send their brightest teenagers to compete in the International Math Olympiad (IMO), where they solve challenging math problems over two days. In 2011, Lisa Sauermann from

Vectors are fundamental building blocks in linear algebra. They can be represented as arrows in space, ordered lists of numbers, or generalized mathematical objects. Vector addition and scalar multipl

Exponentials, such as 2 to the x and 7 to the x, are important functions in calculus. However, the most critical exponential function is e to the x. The derivative of e to the x is equal to the functi

GPT stands for Generative Pretrained Transformer, a type of neural network that generates new text. Pretrained means it learned from a massive amount of data. Transformers are the core invention under

The attention mechanism is a key component of the transformer, a powerful tool in AI. It allows words to influence each other's meaning and context. By using query, key, and value vectors, the model c

Learn about the cross product, its geometric representation, and its connection to linear transformations. Explore the standard introduction as well as a deeper understanding of this mathematical conc

Learn the geometric interpretation of the cross product, how to compute it, and its connection to linear transformations and duality.

Explore the geometric interpretation of Cramer's Rule for solving linear systems of equations and understand the significance of determinants in transformations. Learn how to find the coordinates of t

Learn how to translate vectors and coordinate systems between different languages for precise mathematical representation.