Ooh, this is so cool, part of my PhD was on the topology of liquid crystal textures like these! The LC's in this movie are in the "cholesteric nematic" phase: the molecules are rod-shaped and like to line up with each other with a "twist", wiki: https://en.wikipedia.org/wiki/Cholesteric_liquid_crystal … 1/nhttps://twitter.com/vancew/status/1087870532911955968 …
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The molecules aren't too happy in this state though, they'd prefer twisting! In some places they do; there you see bright colors in the movie frame, since the polarization of light rotates as it passes through the sample. (These two sketches are from https://doi.org/10.1364/OE.20.006870 …) 3/npic.twitter.com/f49NhlUbAf
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In the movie, the bright regions form curves and loops, which correspond to structures called "double twist tubes". The molecules in the center of the tubes point along the axis, and they slant more and more as you go radially outwards
(image from https://doi.org/10.1088/1468-6996/16/3/033501 …). 4/npic.twitter.com/IFUZbLKdF1
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The little circular loops are interesting – when a double-twist tube closes on itself, the molecules follow a pattern like this. Math lovers will recognize this as a part of the Hopf fibration! In the movie, these loops grow from / shrink to little pinwheel-patterned disks. 5/npic.twitter.com/ap2dFn9k4P
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The pinwheels arise from structures called "torons", which look like this! The blue dots correspond to "hyperbolic hedgehog" defects, where the orientation of the rods is undefined. Here's a challenge – try to visualize the lines changing from the last picture to this one! 6/npic.twitter.com/hbJ7gv3BoC
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It's quite hard (for me at least) because there's a different direction living at each point in 3D. That's a lot of data to keep in your head at once, and seeing how they all flow is harder still. But what if I told you there's another way?
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Way back in 2013, I published a paper with Paul Ackerman, Gareth Alexander, Randy Kamien (my PhD advisor!) and Ivan Smalyukh that uses the "Pontryagin-Thom construction" to visualize 3D line fields like this as colored surfaces. The Hopf fibration becomes a rainbow donut
: 8/npic.twitter.com/vH1pJpFg9Q
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Very rough idea: draw a surface element at each point in the sample where the lines lie in the xy-plane. The color hue corresponds to the angle in the xy-plane (e.g. red is E/W, blue is N/S). The pattern of colors can be interpreted in terms of the topology of the texture, 9/n
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which makes it less sensitive to noise and thus applicable to the messy real world! Here's a donut computed from data from a double-twist loop, and also two views of a toron. The toron's surface is a rainbow-striped beach ball, and the points at the top and bottom where the 10/npic.twitter.com/mYjEdiGfTj
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rainbow colors collide correspond to the hedgehog defects. Now, to turn a toron into the double-twist loop, open up the defects into holes, and then pull the holes together through the middle and glue into a donut! Much easier to visualize and it's topologically the same. 11/npic.twitter.com/9VhDed3jzs
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To read more on this, see this nice summary of our work by Miha Ravnik: https://physics.aps.org/articles/v6/65 (a link to our paper is there too) and this more recent article on some wild new textures that Paul and Ivan found called "twistions": https://phys.org/news/2017-02-never-before-seen-topological-solitons-experimentally-liquid.html … 12/13
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To summarize, (1) liquid crystals are awesome, (2) topology is awesome, and (3) you should follow
@vancew to see more beautiful pictures and movies! Thanks for reading my first thread! 13/13 oh god it's been years, i'm just a theorist, i hope i haven't misinterpreted the movie...Prikaži ovu nit
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