Each of these is called a Mapping Class. This set also inherits the group structure of SL(2,Z) (maybe harder to see given the clothing analogy, but remember we are really talking about self-homeomorphisms of the torus, and those can be composed!). This is the Mapping Class Group.
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Matrices in SL(2,Z) come in three types: finite order, diagonalizable, and having a generalized 1-eigenspace. Does this distinction mean anything for mapping classes? Yes! (brb - more after I teach)
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Finite order are somewhat boring (like putting your shirt on backwards, if you take it off and do it again you’re back to a normal front-facing shirt) also, if you look for finite order integer matrices you’ll quickly realize there aren’t many anyway!
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The main group of mapping classes for the torus correspond to the diagonalizable matrices (this is everything with trace greater than 2 in absolute value)- which are called Anosov (yay! We have finally returned to the picture!) we will talk about the remainder in another thread.
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These diagonalizable mapping classes have two eigen-directions: one stretching and one compressing (because determinant is 1) so this corresponds to unzipping the sweater into a square, stretching into a parallelogram, twisting it back around the tours and zipping back up.
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Here’s what that looks like for the mapping class in the original picture (corresponding to the matrix {{2,1},{1,1}}) (image from Wikipedia)pic.twitter.com/X3BYoO5Fyf
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Now a little treat for those of you who have stuck with me so far. Why am I thinking about clothes on a donut? To make cool 3-manifolds to do raytracing in of course! After a little more explanation (sorry not sorry
) I promise to show you some 3D virtual reality renders.1 reply 1 proslijeđeni tweet 6 korisnika označava da im se sviđaPrikaži ovu nit -
Recall to build a closed surface, we can start with a polygon and pair up the sides particular ways. We can do the same for 3 manifolds! Let’s start here with a cubepic.twitter.com/HLpIKb5w8o
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To get started, let’s pair up opposing vertical faces and identify them. This takes each horizontal plane and turns it into a torus - so the result looks kind of like a thickened toruspic.twitter.com/yjTcvbLdgq
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To get a closed 3 manifold we somehow need to still glue together the remaining surfaces - that is we need to specify a map from the inner torus to the outer! If we perturb this map a little bit it won’t affect the topology of the result, so we really need a Mapping Class.
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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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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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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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