I know that 3Blue1Brown made a video on the hairy ball theorem, but there is a more elegant proof by Heinz Hopf in 1946 that generalises to all other surfaces!
Watch now: youtu.be/TLHbOMNKtzw
How do we visualise differential forms? What are differential forms? We will start with an alternative view of integration.
Watch now: youtu.be/j79ihqK0-gE
Starting a new video series on differential forms! The story of how we got here, why we got this definition, and why differential forms are important.
Watch here: youtu.be/dNT8OLNMsQ0
There is a hidden 4D rotational symmetry in Newtonian gravity. But where is the 4D? Here is a teaser.
Watch now: youtu.be/PpsPSr49HGM
Apart from energy and angular momentum, another quantity known as LRL vector is also conserved in Newtonian gravity. We'll rediscover it using a method first documented in 2008 - turns out, LRL vector is just "complex" energy!
Watch now: youtu.be/dMu_e3K4Y1g
Black background with white text "conformally dual forces" on top. There are two pictures side by side on the bottom with a white double-arrow on top.
Every power-law force is "conformally dual" to some other forces, e.g. Hooke's law is conformally dual to inverse square law, and the inverse 7th power law is also conformally dual to inverse 4th power law. What other pairs can we have?
Watch now: youtu.be/8XmktRwSGYQ
There are two sides: one side with the text "Newtonian gravity" with a figure of a planet orbiting the Sun in an ellipse; the other side with the text "Hooke's law" with a figure of a planet attaching to the Sun by a spring.
Newtonian gravity and Hooke's law (2D or 3D) both produce elliptical orbits. Is this a coincidence? Of course not, and Newton found out a connection between the two seemingly distinct forces.
Watch now: youtu.be/1ubTR3HzAI4
Thumbnail of a video split into two sides: the left with "Newtonian gravity" written and a planetary orbit as a representative picture; and the right with "Quantum mechanics" written and the hydrogen spectral lines as a representative picture.
New video series on things you don't know about Newtonian gravity. This intro video tells you why you need to care: some obscure facts about Newtonian gravity were useful to quantum mechanics!
Watch now: youtu.be/C8woZdYisEc
A particular quantity in Newtonian gravity was so obscure that it was rediscovered many times in history, as shown in the GIF here. Can you guess what that quantity is?
A new video series on the way, and the intro is already posted on Patreon: www.patreon.com/posts/new-in...
I can mathematically prove that action principle works, but something about it feels deeply unsatisfying, until I found a paper that explains what action is and why we minimise it.
Watch now: youtu.be/Ohrl3S2wcBU
Arguably the most important theorem in differential geometry, Gauss-Bonnet theorem links geometry and topology, local and global, continuous and discrete. Can we prove it without technical calculations?
Watch now: https://buff.ly/3Dp6zsv
The 4-colour theorem was proved in 1976 with the help of a computer, but the corresponding theorem on the torus was proved way earlier in 1890. Why is it easier on the torus?
Watch now: https://buff.ly/3BrIkJx
While working on the next video, I needed to use V - E + F ≥ 2 - 2g, but this is one of those theorems where you've heard of, but don't know how to prove (and most textbooks treat it as exercise). So this is my thought process.
Watch now: https://buff.ly/3DtPj52
If the inequality (instead of the much more common equality) is mentioned, it is usually an exercise. So I tried to do it, and later on I Googled a bit more to see there is a StackExchange answer roughly along the lines of what I'm thinking, and it became this video.
A bit about my next video (preview here: https://buff.ly/4fVMlVy while working on another video, I wanted to intuitively understand V - E + F ≥ 2 - 2g. This is different from the usual = sign, because I'm considering all graphs, not just ones where all faces are homeomorphic to R2.🧵(1/2) #mathsky
I guess I should introduce myself: I'm Trevor, and run the YouTube channel Mathemaniac, which showcases underrated pieces of maths and physics. Btw, I used to not have the @mathemaniac.bsky.social handle on Twitter, because Simon from Numberphile already took it, but over here, I am THE mathemaniac!
A picture that has three pictures of different shapes of soap films, and a title "One formula describes them all!".
New post here!
I said on YouTube that I'm not making the third video in the series of minimal surfaces, just typing this up instead:
www.mathemaniac.co.uk/s/All__shape...
Mainly due to too many algebraic manipulations and fewer visuals than I envisioned.
Next video(s) coming in December!
