Mathemaniac | How physics solves a math problem (and a 3D graphics problem) @mathemaniac | Uploaded 1 month ago | Updated 2 hours ago
Should’ve been titled “accidentally stumbling onto an area of active research way out of my depth”.
The Plateau's problem asks for the minimal surface (with minimum surface area) with a given boundary. Because nature seems to solve this problem easily, as demonstrated by soap films, we turn to physics for inspiration. This is simulated by (hyperbolic) mean curvature flow, which turns out to be a hot topic for research in both maths and physics, and somehow also used in 3D graphics.
Files for download:
Go to mathemaniac.co.uk/download and enter the following password: meancurvatureflow
**Further reading**:
/Mathematical/ side of things:
1) General overview of Plateau's problem:
arxiv.org/pdf/1509.03797
2) General overview of the field of geometry in 2000 (by Shing-Tung Yau): intlpress.com/site/pub/files/_fulltext/journals/ajm/2000/0004/0001/AJM-2000-0004-0001-a016.pdf
3) The relevant Wiki articles:
en.wikipedia.org/wiki/Minimal_surface
en.wikipedia.org/wiki/Plateau%27s_problem
4) An undergrad/masters project on mean curvature flow:
math.mcgill.ca/gantumur/math581w12/downloads/OlivierMercier.pdf
/Physical/ side of things:
1) Original Brakke's paper on mean curvature flow in annealing metal (1978): kenbrakke.com/papers/downloads/motionbook.pdf
2) What got me started for this video: https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node41.html
3) Implementation on how to simulate soap films:
cs.uwaterloo.ca/~thachisu/hgf.pdf
4) Research into how instability arises:
http://www.damtp.cam.ac.uk/user/gold/pdfs/CatenoidCollapse.pdf
5) The YouTube video on @cambridgeuniversity for soap film collapse:
youtube.com/watch?v=3QgHxnDsrGQ
6) Relevant Wiki articles (on Young-Laplace equation):
en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation
7) An undergrad/masters thesis on physics of soap films:
https://www.princeton.edu/~stonelab/Teaching/FredBraszFinalPaper.pdf
/Computer graphics/ side of things:
1) How to smooth 3D contours with mean curvature flow:
mathematica.stackexchange.com/questions/172517/smoothing-3d-contours-as-post-processing/172603#172603
2) How to improve on the mean curvature flow procedure to solve Plateau's problem:
mathematica.stackexchange.com/questions/72203/can-mathematica-solve-plateaus-problem-finding-a-minimal-surface-with-specifie
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
Should’ve been titled “accidentally stumbling onto an area of active research way out of my depth”.
The Plateau's problem asks for the minimal surface (with minimum surface area) with a given boundary. Because nature seems to solve this problem easily, as demonstrated by soap films, we turn to physics for inspiration. This is simulated by (hyperbolic) mean curvature flow, which turns out to be a hot topic for research in both maths and physics, and somehow also used in 3D graphics.
Files for download:
Go to mathemaniac.co.uk/download and enter the following password: meancurvatureflow
**Further reading**:
/Mathematical/ side of things:
1) General overview of Plateau's problem:
arxiv.org/pdf/1509.03797
2) General overview of the field of geometry in 2000 (by Shing-Tung Yau): intlpress.com/site/pub/files/_fulltext/journals/ajm/2000/0004/0001/AJM-2000-0004-0001-a016.pdf
3) The relevant Wiki articles:
en.wikipedia.org/wiki/Minimal_surface
en.wikipedia.org/wiki/Plateau%27s_problem
4) An undergrad/masters project on mean curvature flow:
math.mcgill.ca/gantumur/math581w12/downloads/OlivierMercier.pdf
/Physical/ side of things:
1) Original Brakke's paper on mean curvature flow in annealing metal (1978): kenbrakke.com/papers/downloads/motionbook.pdf
2) What got me started for this video: https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node41.html
3) Implementation on how to simulate soap films:
cs.uwaterloo.ca/~thachisu/hgf.pdf
4) Research into how instability arises:
http://www.damtp.cam.ac.uk/user/gold/pdfs/CatenoidCollapse.pdf
5) The YouTube video on @cambridgeuniversity for soap film collapse:
youtube.com/watch?v=3QgHxnDsrGQ
6) Relevant Wiki articles (on Young-Laplace equation):
en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation
7) An undergrad/masters thesis on physics of soap films:
https://www.princeton.edu/~stonelab/Teaching/FredBraszFinalPaper.pdf
/Computer graphics/ side of things:
1) How to smooth 3D contours with mean curvature flow:
mathematica.stackexchange.com/questions/172517/smoothing-3d-contours-as-post-processing/172603#172603
2) How to improve on the mean curvature flow procedure to solve Plateau's problem:
mathematica.stackexchange.com/questions/72203/can-mathematica-solve-plateaus-problem-finding-a-minimal-surface-with-specifie
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
