# But what is a Fourier series? From heat flow to circle drawings | DE4

# Solving the heat equation | DE3

# But what is a partial differential equation? | Overview of differential equations, chapter 2

# Overview of differential equations | Chapter 1

# Cramer's rule, explained geometrically | Essence of linear algebra, chapter 12

# More elegant solution to the block collision puzzle

# Solution to the block collision puzzle

# The block collision puzzle

# But WHY is a sphere's surface area four times its shadow?

# Sneaky Topology

# Turbulence

This project was done in collaboration with Dianna Cowern, of Physics Girl fame. We also did a video on her channel, in which we play with vortex rings and look at how their shape is affected by different initial conditions.

The video below is mainly about what turbulence is, and one particular bit of order that we do understand within this famously chaotic and unpredictable phenomenon thanks to Kolmogorov.

Feynman described turbulence as the most important unsolved problem in classical physics, and it stands out as a powerful example of how knowing the fundamental laws of physics is very different from understanding physics fully. With so much focus given to deep laws, and a desire for some “theory of everything”, it can be all too easy to forget just how much remains unknown among the patterns emerging from what foundational laws we do know.

At the heart of this video is one particular method for visualizing air currents, involving a smoke machine and a planar laser. The setup is due to Dan Walsh, although later we realized it had been done before. As to whether this is convergent evolution, or subconscious pattern matching, I can’t say for sure. In either case, the setup is only incidental to the underlying topic here. In watching the footage, I had to continually remind myself that this was the real world, and not computer generated.

Thanks also to Gabe Weymouth for providing a lovely simulation of 2d turbulence.

Thanks to those who caught my speako describing Kolmogorov as a "19th-century" mathematician. Of course, I meant the 1900's. His work is quite recent and remains relevant to a number of active research efforts in a surprising breadth of fields. It's crazy to me that his name is as relevant to fluid dynamicists as to those studying machine learning.

# Explorable videos: Quaternions and 3d rotation

Following the last video, this project gets into how quaternions apply to 3d rotation. This is introduced with a short video on YouTube, but things get much more interesting after that...

The bulk of this lesson is actually hosted off YouTube, where thanks to the fantastic web development wizardly of Ben Eater it is done in the form of "explorable videos" that you can interact with during the narration. To accommodate anyone landing on the page of explorable videos without having watched the ~36 minutes of YouTube content above, some of the explorables are a bit redundant with the previous video. But a little repetition never hurts when we're dealing with a difficult topic.