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You can change coordinates to a model where the rotation symmetries act like normal rotations of R3. This makes the geodesics look more like actual helices, but the trade off is now the metric tensor looks complicated. Here’s a rendering of some geodesics in these coordinates.pic.twitter.com/P5Q53sqpym
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Whoops the day got away from me! We will talk about why the earth doesn’t shrink in size as it recedes into the distance (quite unlike what happens in Euclidean space) another day. Here’s a hint
pic.twitter.com/z3QZEYlTZL
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More rings appear as this process repeats: in this figure light reaches the earth along three kinds of paths: headin straight there(green), traversing one large spiral (yellow), or two smaller spirals (blue). This results in a central image surrounded by 1 large and 1 small ringpic.twitter.com/pQkCNOZC3k
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As the earth continues to recede, this gap grows - and we see the ring shaped mirage expanding away from the earth with a wider and wider black annulus in between!pic.twitter.com/EfeZxideDT
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As the earth recedes, the radius of spiraling geodesics which crash back inwards and reach it grows - and eventually becomes equal to the radius of the earth itself. At this point there’s a circle of geodesics which skim Earth tangentially, before escaping into space.pic.twitter.com/8eQ92lRSig
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1) is explained by rotational symmetry: once we have passed a conjugate point to the earth (see previous thread) there are multiple geodesics connecting our eye to the point “directly ahead” on earth. But as soon as there is a second, there is a circles worth by symmetry.pic.twitter.com/3ML3GcgzpN
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Since we are seeing earth in the video by looking along geodesics headed generally "upwards", it helps to draw some of these. First sign of strange behavior: they do not spread out like straight lines in Euclidean space, but rather spiral around the z-axis.pic.twitter.com/DFSJJpOIQk
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A quick crash-course on Nil: the geometry is defined starting with the real Heisenberg group. We then build a metric on this group for which the Heisenberg group is isometries by starting translating around the Euclidean metric at the origin, which gives dx^2+dy^2+(dz-xdy)^2.pic.twitter.com/sW888tHpzs
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Today I'll continue sharing some of the figures from our paper https://arxiv.org/abs/2002.00513 on Nil geometry. Here's what it looks like when you back away from a sphere (flying along the z-axis, in coordinates coming from the Heisenberg group)pic.twitter.com/jPfprMRUS3
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The actual story here is a bit more complicated (if I remember right there is a 'focal line' not just a point, and we are at one of the points on this line. Here's a schematic of the situationpic.twitter.com/p9EFdZtw0l
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And so as we embark on a journey to try and understand new 3-dimensional geometries visually, its no surprise that their curvature causes similar effects. That 'geometric lensing' may warp our view of mathematical realities as well.pic.twitter.com/mTWb6Booxs
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And here's another gravitational lens: not only is the background galaxy smeared into a ring; but the four bright points are actually four distinct images of the same thing! Seen along four geodesics bending around the foreground galaxy and reaching our eyes.pic.twitter.com/AUA22J2JzL
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When gravity does this we call the result a gravitational lens - and see them throughout the cosmos! Here is a picture of one massive foreground galaxy (yellow) bending the light of a background galaxy (blue) into a ring!pic.twitter.com/AKBnNUKCLL
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Since we are viewing the earth in Nil by following Nil-geodesics, seeing the center of our vision imaged again as a ring means that an entire circle of geodesics leaving our eye come back together and converge after traveling to the earth. I've simulated this here:pic.twitter.com/OlAseecTnK
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With this in mind - watch the video above a couple more times, and try to imagine what could be happening. Here's a screen shot from when we are facing the earth, looking directly at South Americapic.twitter.com/rQfdkMsk2r
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So to model sight in a curved space, we can imagine shooting light rays out from our screen into the space along geodesics, and seeing what they hit. (This figure is not from our paper! Just me scribbling in keynote between teaching :-) . )pic.twitter.com/ulCQ62TpKm
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Our research team from
@ICERM (me,@henryseg,@Sabetta_ and@LaMiReMiMath) just finished expository papers on Nil & Sol geometry. I'll post the Arxiv links soon, but first will take you on a tour of the illustrations. Here's a view of the earth rotating in Nil geometry.pic.twitter.com/KmOnMfILftPrikaži ovu nit -
Some color coding may help here. Imagine standing in this manifold and looking down - we see the torus coming from the bottom of the cube (Orange). If we look up we see the torus coming from the top (Blue).pic.twitter.com/iqJzncHnzc
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Here’s a view from inside the resulting manifold, where we are rendering the edges of the fundamental domain by raytracing along geodesics: this is what we would actually see if we were placed inside the identified cube given the Sol metricpic.twitter.com/Tp5u0Ritgx
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As a crazy fact - the type of mapping class we choose determines the geometry that this 3 manifold can have! If we pick an Anosov mapping class, the resulting 3 manifold has Sol geometry.pic.twitter.com/TEsdo95IdT
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