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by virtuous_signal
2346 days ago
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For anyone wondering about the significance of this, the following is my loose explanation: you probably know that a torus is the surface of a doughnut. Better yet, it can be thought of as a square with edges identified (like the videogame screen when you play asteroids: when you reach the top edge you come out the bottom edge at the corresponding place; same for left and right). The latter representation is "better" than the doughnut in the sense that it's symmetric: the horizontal and vertical directions are treated the same. But it has the downside that it requires "identifications"; corresponding points at the top and the bottom represent the "same" point while points in the middle of the screen are unique. The doughnut representation satisfies uniqueness everywhere, but isn't symmetric: the circles in the vertical direction are all the same size; but if you consider circles in the other direction, they depend on the angle; for instance circles at the top of the doughnut are bigger than the circle encircling the doughnut hole, and smaller than the circle encircling the entire doughnut. You can visualize how this asymmetry happens by turning the square into a doughnut: start with the square above; make it into a roll so that the left and right sides are identified, and then now bend the roll to identify the top and bottom edges: if you bent towards you, then the backside gets stretched a lot more than the front side on the way to forming the full doughnut. Clifford torus is a way to satisfy both properties (symmetry and uniqueness) at the same time and it's a theorem that one needs to be in 4 dimensions for this to happen. |
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