What would truly feel like a “different” way of wearing a sweater (for humans, say) would be to shove your head through an arm hole, and then figure out the rest accordingly.
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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.Prikaži 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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