Matt Parker: An Attempt to Visualise Minimal Surfaces and Maximum Dimensions

Abstract:
Much of Karen Uhlenbeck's ground-breaking work involved abstract mathematical concepts which are beyond our normal human intuition. And even though there may be practical applications of the results of her work, that does not make minimal surfaces in higher dimensions any less esoteric. Matt Parker will attempt to provide visual demonstrations of both minimal surfaces and higher dimensions (although, probably not at the same time) to allow a small glimpse into the scope of Uhlenbeck's work and achievements.

If you liked this video, you can see more of Matt Parker on his own YouTube channel @standupmaths

This lecture was one of four Abel Lectures held as part of the Abel Prize Week in Oslo.
The University of Oslo, May 22, 2019

Program for the Abel Lecture 2019
1. "Some Thoughts on the Calculus of Variations" by Abel Laureate Karen K. Uhlenbeck, University of Texas at Austin, USA
2. "Solitons in Geometry" by professor Chuu-LianTerng, UC Irvine
3. "Limits, Bubbles, and Singularities: An introduction to the fundamental ideas of Karen Uhlenbeck" by professor Robert Bryant, Duke University
4. "An Attempt to Visualise Minimal Surfaces and Maximum Dimensions" by Matt Parker, @standupmaths

[LeoStaley]

My obsession with Matt Parker finally convinced the YouTube algorithm to show this to me. If that's what it takes.

[thekylehampton]

"I haven't done it. But it should be reasonably easy.." Parker's Last Theorum.

[Lugmillord]

"He was so proud, he named himself after the loop"
I can't not give a thumbs up for jokes like that.

[VulpesScabiosa]

I guess the algorithm just decided to no longer ignore the video

[jurian]

Thank you! The YouTube Algorithm for presenting me this 1 year and 4 months since the talk happened :D

[theexnay]

I tend to use opacity to represent the fourth dimension when i try to manipulate 4D objects in my mind. I just keep in mind that if two pieces have different opacities then they don't touch. I like to define 0 in the opacity dimension as 50% opacity so i have room to go in both directions

Then for rotations it helps to realize what a rotation is. So for 2D you pick a point (a 0 dimensional shape) and everything keeps the same shape and every other point stays the same distance from that point. In 3D you take an infinite line (a 1D shape) and everything keeps the same shape and distance from the line. So in 4D you take a plane (a 2D shape) and do the same thing. I recommend limiting what planes you can use just to keep things simple. So the 3D equivalent is just rotating around the point at (0, 0). In 3D it's the lines corresponding to (X, 0, 0), (0, Y, 0), and (0, 0, Z). In 4D each plane is defined by two coordinates. If one of those coordinates is opacity then it just looks like one of the lines you use in 3D and the rotations look the same, that accounts for 3 of the planes ([o, x, 0, 0], [o, 0, y, 0], and [o, 0, 0, z]). The other three are the planes where you've picked 2 of the normal 3 dimensions (x, y, or z). You've got four dimensions and you're picking 2 of them, that's why there are 6 rotations in 4D, it's 4 choose 2 (look up the choose function to learn more or to easily calculate it for higher dimensions).

To rotate around the latter 3 options the points either stay in the same spot (if they're already touching the plane, like how the earth's pole's don't move when the earth rotates), they move farther away in 3D while their opacity matches the plane more or move closer to the plane while their opacity starts to match it less. Things on one side of the plane should become more opaque if they move closer and things on the other side should become more transparent as they move closer, that way opposite sides of the object maintain the same 4D distance apart even if they are in the same spot in the 3 normal dimensions

I recommend starting by rotating 3D objects in 4D. Like try a 3D sphere that starts centered on (0, 0, 0, 0). All possible to be done just in your mind

[martinwulf]

This man is an absolutely brilliant communicator.
He’s funny, he has insight, he knows what matters, and he can put it all together into a coherent and fluid presentation.

That it what I want to learn from him. How to be this good at presenting on a topic.

[nomasan]

"Now you stand there awkwardly"

"wow.. you actually did that"

[Pika]

From 39:00 in, the number of possible rotations in d dimensional Euclidean space over the reals *R* where d > 1: (d choose (d - 2)) = (d choose 2) = d(d - 1)/2. In other words the triangular numbers 1, 3, 6, 10, 15 and so on. So Matt was right when he said 10 for d = 5.

[brianderocher]

Matt would like you to think the soapy mixture found the minimal surface with a small cube bubble in the large one. I think he was using a square straw.

[tormodguldvog]

I was just going to relax for a few minutes and Matt’s videos are always great fun. 50 minutes later I experience the true relativity of time. Matt is so cool. And what a wonderful way to round off the Abel Lecture.

[EebstertheGreat]

I love that he starts his Abel lecture with Numberphile's video on a non-standard summation method. Very fitting.

[gregorybutcher]

shiiiit. The upmost respect for dealing with a heckler live on stage at the very beginning - that was awesome

[billborrowed]

I mean, gotta love the talks by Matt. It‘s always a pleasure to see a subject, which I understood half-way so far and having it explained with that amount of humorous elements.

[archviceroy]

“A lot of the time mathematicians have the solution of not caring. A lot of problems in math are fixed by, ‘Have you tried not worrying about it?’”

I have never related to a statement so strongly.

[portlyoldman]

A few years on now but I love seeing all these maths profs loving Matt’s show 😁

[simonmultiverse]

For the 3D/4D cube, try reading a short story by Robert Heinlein. The story is called "And He Built A Crooked House". Essentially, he built a house which was an unfolded tesseract, i.e. it was four cubes stacked vertically, with extra cubes glued on each of the four sides on the first floor. This is EXACTLY like the picture at 35:45 or thereabouts.
In the story, he built the house in California, and California has earthquakes, so... one of these earthquakes caused the house to fold itself into a real tesseract. Mayhem ensued.

[sander_bouwhuis]

This video deserves more views and likes. I am going to do the 2 Möbius strips to interlocked hearts with my daughter and her friend that is coming to stay over tonight. I also loved the cube bubble inside the cube.
Thanks!

[PawelKraszewski]

@MattParker, some extension to 1-twist Möbius strip cutting:
* Cut strip in half as you did. Then cut it *again* in half.
* Cut strip in 1/3 of its width, not 1/2.

[mrshinebox]

As much as I love the content of this video, what I really need to know is what bubble mixture did Matt use?