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Awesum
Приєднався 12 сер 2022
On this channel, we try to highlight hidden gems that we come across while studying mathematics. Most of our videos will heavily rely on visuals, either to prove well known theorems in a more insightful way or to just show some quirky fact for the sake of it.
Counting Tilings (with Linear Algebra)
What does a chessboard have to do with trigonometry, complex numbers and linear algebra?
Well, quite a lot if you want to calculate the number of possibilities to tile said board with two by one tiles!
In this video, we will apply methods from seemingly unrelated fields to arrive at one of the most beautiful results in combinatorics, Kasteleyn's formula.
This video has been produced as part of the #SoME2.
0:00 Introduction
3:04 First Observations
8:56 The Signing
17:07 Linear Algebra Ahead
18:52 Final Sprint
25:59 Holes and Circles
Writing and Animation: Michael Zheng and Levin Kiefer
Narration: Michael Zheng
Editing: Levin Kiefer
Music by glaciære/Stevia Sphere: steviasphere.bandcamp.com/
(licensed under CC BY 3.0)
Tracklist:
Part 0: Bittersweet (Two Months of Moments)
Part 1: Mitosis (Music For Slime Creatures)
Part 2: Synth Gunk (Music For Slime Creatures) & Going to bed on a warm summer night (water slide)
Part 3: Relaxing in the hammock (hammock)
Part 4: Polar bears (hammock) & New Age Website (Reality is not a computer simulation)
Part 5: Ocean waves and Square waves (pool water blue)
Idea: Jiří Matoušek (2010), Thirty-three Miniatures - Mathematical and Algorithmic Applications of Linear Algebra.
Secondary sources:
Andries E. Brouwer, Willem H. Haemers (2012), Spectra of Graphs
Richard Kenyon (2008), An Introduction to the Dimer Model
Linyuan Lu (2009), Spectral Graph Theory - Handout 3: people.math.sc.edu/lu/teaching/2009spring_778S/adjeig.pdf
Image sources:
umdphysics.umd.edu/images/igallery/resized/1101-1200/Fisher_2-1110-1280-720-100.jpg
academictree.org/photo/057/cache.569571.Pieter_Kasteleyn.jpg
www.stfaiths.co.uk/wp-content/uploads/2017/05/Neville-Temperley-e1494584970554-1439x575.jpg
For a video that introduces a method to count the number of matchings for a larger class of graphs, see:
ua-cam.com/video/Y-gDWLQFE4g/v-deo.html
If, on the other hand, you are more fascinated by the Arctic Circle Phenomenon (or want to find the source of the "necklace figure"), see:
ua-cam.com/video/Yy7Q8IWNfHM/v-deo.html
Well, quite a lot if you want to calculate the number of possibilities to tile said board with two by one tiles!
In this video, we will apply methods from seemingly unrelated fields to arrive at one of the most beautiful results in combinatorics, Kasteleyn's formula.
This video has been produced as part of the #SoME2.
0:00 Introduction
3:04 First Observations
8:56 The Signing
17:07 Linear Algebra Ahead
18:52 Final Sprint
25:59 Holes and Circles
Writing and Animation: Michael Zheng and Levin Kiefer
Narration: Michael Zheng
Editing: Levin Kiefer
Music by glaciære/Stevia Sphere: steviasphere.bandcamp.com/
(licensed under CC BY 3.0)
Tracklist:
Part 0: Bittersweet (Two Months of Moments)
Part 1: Mitosis (Music For Slime Creatures)
Part 2: Synth Gunk (Music For Slime Creatures) & Going to bed on a warm summer night (water slide)
Part 3: Relaxing in the hammock (hammock)
Part 4: Polar bears (hammock) & New Age Website (Reality is not a computer simulation)
Part 5: Ocean waves and Square waves (pool water blue)
Idea: Jiří Matoušek (2010), Thirty-three Miniatures - Mathematical and Algorithmic Applications of Linear Algebra.
Secondary sources:
Andries E. Brouwer, Willem H. Haemers (2012), Spectra of Graphs
Richard Kenyon (2008), An Introduction to the Dimer Model
Linyuan Lu (2009), Spectral Graph Theory - Handout 3: people.math.sc.edu/lu/teaching/2009spring_778S/adjeig.pdf
Image sources:
umdphysics.umd.edu/images/igallery/resized/1101-1200/Fisher_2-1110-1280-720-100.jpg
academictree.org/photo/057/cache.569571.Pieter_Kasteleyn.jpg
www.stfaiths.co.uk/wp-content/uploads/2017/05/Neville-Temperley-e1494584970554-1439x575.jpg
For a video that introduces a method to count the number of matchings for a larger class of graphs, see:
ua-cam.com/video/Y-gDWLQFE4g/v-deo.html
If, on the other hand, you are more fascinated by the Arctic Circle Phenomenon (or want to find the source of the "necklace figure"), see:
ua-cam.com/video/Yy7Q8IWNfHM/v-deo.html
Переглядів: 4 548
thank you so much! thing make a lot of sense now!
Just saw this - amazing video! Nice work :)
Thank you for addressing "how do you know you are done?" right at the beginning!
14:44 I didn't get this part, in the example there are 11 red and blue outside edges so the product of weights will be -i and 1 and how does them differ by (-1)^16 = 1?
Nice work ! I did a project on this during my bachelor (mainly explaining the same things you did, but with less visuals). One of the things you did not present was that the eigenvalues of the line graphs are actually (almost) the roots of Tchebyshev polynomials of the second kind. This avoids some calculations in the second half of the video but requires to know those polynomials in the first place. Anyway, thanks for your work !
Great video!
Please make background other color I think that my phone turned off every time
amazing i like counting problems
Nice voice
amazing video!!!! keep it up
I got lost in the middle. I find it funny because I was working with a kinda similar problem but with imperfect matching. I compute by hand the first values but those values are not in the OEIS, at this point I gave up.
Aewsom
More video is needed
I got lost in the middle. But it sounds like a really fascinating calculation. I will definitely go through it one more time more carefully. It is really nicely presented. It's amazing what different pieces enter into the proof.
Great Video. Easy to follow and nice style and animations.
This is sumthing really awesum!
Yu sed it!
SoMEthing wonderful is brewing