A method for thinking about high-dimensional spheres, introduced in the context of a classic example involving a high-dimensional sphere inside a high-dimensional box.

What is a change of basis, and how do you do it?

An introduction to the quantum behavior of light, specifically the polarization of light. The emphasis is on how many ideas that seem "quantumly weird" are actually just wave mechanics, applicable in a lot of classical physics.

What exactly are fractals? A common misconception is that they are shapes which look exactly like themselves when you zoom in. In fact, the definition has something to do with the idea of "fractal dimension".

Derivatives are about slope, and integration is about area. These ideas seem completely different, so why are they inverses?

Colors: you see them every day, and you probably have a favorite. Pigments, light, and even noise all color how we experience the world.

An algorithm for solving continuous 2d equations using winding numbers.

A classic problem that Johann Bernoulli posed to famous mathematicians of his time, such as Newton, and how Bernoulli found an incredibly clever solution using properties of light.

What is the dot product? What does it represent? Why does it have the formula that it does? All this is explained visually.

An overview of gradient descent in the context of neural networks. This is a method used widely throughout machine learning for optimizing how a computer performs on certain tasks.

How does bitcoin work? What is a "block chain"? What problem is this system trying to solve, and how does it use the tools of cryptography to do so?

How do you think about a non-square matrix as a transformation?

The cross product is a way to multiple to vectors in 3d. This video shows how to visualize what it means.

Have you ever wondered what color is? In this first installment of a series on light, Colm Kelleher describes the physics behind colors-- why the colors we see are related to the period of motion and the frequency of waves.

When you’re learning about science for the first time, it can be easier to break things down into a simpler form, and you can end up with a few misconceptions about the world. But sometimes this is the first step to understanding that...

Where do glaciers and icebergs get their beautiful blue color? This unique blue might be nature's most brilliant, and the color arises in a very special way thanks to some surprising interactions between light and water molecules. Who...

The animal kingdom boasts an incredible diversity of eyes. Some rotate independently while others have squiggly-shaped pupils. Some have protective lids, others squirt blood. But which creature has the best sight? Which sees best in the...

An information puzzle with an interesting twist

An overview of what a neural network is, introduced in the context of recognizing hand-written digits.

Can you turn your pants inside out without taking your feet off the ground?

The way modern researchers study the sky hasn’t really changed in the last few centuries. For the most part, astronomers still study things by analyzing their light.

How do you defeat a creature that grows two heads for every one head you chop off? You do the math.

What is a vector? Is it an arrow in space? A list of numbers?

It's really difficult for life to create blue pigments, but the color can appear in a handful of compounds that create just the right conditions to reflect blue photons.