A moduli space of polygons
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- Опубліковано 6 жов 2024
- Polygons in three dimensions might seem simple, but their geometry connects many areas of math and has surprising applications to chemistry and signal processing. In this video, I give a brief introduction to the moduli space of polygons in Euclidean space. This is a submission for the Summer of Math Exposition 2.
Apologies for any audio issues. The recording had an echo which I couldn't fully eliminate. Next time I'll use a lavalier mic.
References (in order of their appearance):
Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames
Jameson Cahill, Dustin G. Mixon & Nate Strawn
arxiv.org/abs/...
Symplectic geometry and connectivity of spaces of frames
Tom Needham & Clayton Shonkwiler
arxiv.org/abs/...
The symplectic geometry of polygons in Euclidean space
Michael Kapovich & John J. Millson
www.math.ucdav...
The Bring sextic of equilateral pentagons
Lyle Ramshaw
arxiv.org/abs/...
Complex Differential Geometry
Fangyang Zheng
A Note on the Moduli Space of Polygons
Alessia Mandini
www.math.tecni...
The symplectic geometry of closed equilateral random walks in 3-space
Jason Cantarella & Clayton Shonkwiler
arxiv.org/abs/...
Determination of the absolute handedness of knots and catenanes of DNA
Mark A. Krasnow, Andrzej Stasiak, Sylvia J. Spengler, Frank Dean, Theo Koller & Nicholas R. Cozzarelli
The Moduli Space of Spatial Polygons
Alex Wright
www-personal.um...
Geogebra files:
The moduli space of equilateral quadrilaterals:
www.geogebra.o...
In this activity, you can manipulate the shape of the quadrilateral, which changes the associated point in the hemisphere. You can also change the value of the s parameter to identify the boundary of the moduli space. Here, the cusps indicate the singular quadrilaterals, but the picture isn't intended to be an isometric depiction of the moduli space.
Equilateral hexagons and their moment polytope:
www.geogebra.o...
Knotted link:
www.geogebra.o...
This gives a polygonal trefoil knot which approximates the knotted DNA strand from the paper by Krasnow et. al. Of all the graphics, this one was the most unexpected painful one to make.
If you are interested in the fugue excerpt at the end of the video, you can download a rough version of the sheet music here.
differentialge...
#SoME2 #polygons #geometry
At 1:36 I finally understood moduli space, seriously thank you! Can't wait for more videos.
Thanks! I'm glad you enjoyed the video.
omg the reverb in the audio unbelievable
Sorry about that. I've since changed my microphone set-up so future videos should have better sound quality.
Hi, I came here from the SoME2 judging, and you have a very interesting video. I found it fascinating when you visually showed how the 4-sided quadrilateral could be compared to a topological sphere. Definitely earned my subscription! 😁
Thanks! I'm glad you enjoyed it. The fact that the space turns out to be a sphere was fascinating for me also when I learned it.
I can't wait to see the followup about symplectic geometry!! Brilliant video that now has me puzzling over the topic!
The space of quadrilaterals is a lot easier to deal with if your specified edge lengths aren't all the same, or more precisely, if you can't achieve a+b=c+d with the four lengths. Then it's just a smooth 2-sphere, without these three nasty points. (Half my 1996 thesis was about polygon spaces.)
Thanks for the comment and that's a great point! The magic fact that Lyle found is that for equilateral quadrilaterals, the moduli space has constant curvature away from the three cone points and it is possible to construct the orbifold covering from a round sphere explicitly. For more general side lengths, the metric and curvature are quite a bit more complicated (this is what makes the Mathematica computation so hairy).
As such, even though the topology of the smooth moduli spaces is simpler than the one I discussed, the geodesics, Riemannian metric, etc. are much more complicated.
@@GabeKhan The topology is very simple for n-gons with lengths (1,1,1,...,1,n-1-epsilon). I bet the geometry becomes quite simple as epsilon -> 0 (assuming one scales up the metric, whose diameter is also heading to zero). I should answer Lyle's email...
We have investigated the Gromov-Hausdorff limits for quadrilaterals, and there are some nice phenomena (e.g., limits in the smooth regime becoming round). However, the metric gets really difficult to compute as you increase the number of sides (we still haven't even figured out equilateral pentagons), so it would be really nice if there is some way to understand the metric for those n-gons without having to brute force the computations.
Absolutely amazing video! Thanks for showing the book; I'm currently going through it. Looks really good so far. Subscribed.
Thanks! It is definitely a great book
Good explanation❤
Thanks!
Great content. The audio is obviously pretty bad, but the subtitles make up for that flaw.
Thanks for the feedback. Going forward I'm using a different option to capture the sound, and hopefully that will work better.
4:59 why are first two polygones are distinct?
Great question! If you label the vertices 1,2,3 and 4, the first picture corresponds to the case where the first and third vertex coincides, whereas the second corresponds to when the second and fourth vertex coincides. To see why these are genuinely different, it's helpful to imagine what would happen if the edge lengths were (1.01,1.01,1,1) instead of being exactly the same. In this case, if I tried to make the first line segment, I would end up with a triangle where the long edge has length 2 (composed of the last two edges) and the other sides have length 1.01. On the other hand, I could still make a segment where the second and fourth vertex coincides, although one of the edges would be slightly longer than the other.
Great animations.
Thanks!
Hey, the audio is pretty rough to listen to, some words I can barely understand with this reverbing room...
Next time consider placing a friend's phone on a table near you, start recording, do a little clap to synchronize video to audio, it'll be much better.
Thanks for the advice and the feedback! I'll definitely need to figure some other way to do the audio for future videos. In the meantime, I've found that this one sounds a lot better with headphones and I edited the closed captioning to make sure it is correct as well.
@@GabeKhan oh I totally forgot that cc was an option. Great idea!
Gabe, you need a lapel microphone, my man.
For sure. Thanks for the advice. I used a different setup for the next video which is hopefully a bit better and I have a lapel mic on the way.
cant hear shit