Vebyast
Nascent Transhuman
Knot theory demands the embedding an N-dimensional sphere (note: this is the number of dimensions that can uniquely describe a point on the sphere, so the "sphere" we all know about is a 2-sphere: latitude and longitude) in a space of dimension N+2. For example, the knots we're used to IRL are 1-spheres (that is, circles, where your point on the surface of the 1-sphere can be described with a single value: angle) embedded in 3-space. If an N-sphere is embedded in a space with more than N+2 dimensions, knots are not possible; the extra degree of freedom permits any crossing to be undone. For an illustration, imagine embedding a 0-sphere (2 points on a number line) entangled with a circle in 2-dimensional space versus in 3-dimensional space. In two dimensions, the situation looks like this:
You'll see that there's no way to unknot these: no matter how you slide your 0-sphere or 1-sphere around, you can't unlink them. But if you have a third dimension, it's now trivial to unknot things; you have a ring and a pair of dots, and it's trivial to lift the dots so they're not in the ring's plane, move them away from each other, and then bring them back into the ring's plane with both ends of the 0-sphere outside the 1-sphere:
Now let's extend this to the phenomenon of spontaneous knotting of agitated strings, aka the reason your earbuds always come out of your pocket tangled into a horrible mess. In short, what happens here is that knots can be made easily, by passing the ends of strings through loops, but once a knot is formed the presence of friction makes those knots more difficult to undo. Furthermore, if you get into the topology of knots, thinking of unknotting a string as it passing back through the same loop twice, the probability of an agitated string unknotting itself is somewhat similar to the probability of a random walk stopping, turning around, and exactly retracing its steps. Clearly this is low-probability.
So what happens when add higher dimensions? Well, knots become significantly easier to undo! There are more available movements that do not result in a knot, and even many that undo a knot. Indeed, as you add more and more dimensions, it becomes effectively impossible for knots to be a relevant mathematical concept. As before, recall the difficulty of "tying together" two dots and a circle.
Which brings me to my conclusion: Homura's hair actually is that fantastic because she's lifting parts of it into a higher dimension. The added dimension directly prevents the formations of knots for reasons rooted in the fundamental mathematics of geometry and topology.
You'll see that there's no way to unknot these: no matter how you slide your 0-sphere or 1-sphere around, you can't unlink them. But if you have a third dimension, it's now trivial to unknot things; you have a ring and a pair of dots, and it's trivial to lift the dots so they're not in the ring's plane, move them away from each other, and then bring them back into the ring's plane with both ends of the 0-sphere outside the 1-sphere:
Now let's extend this to the phenomenon of spontaneous knotting of agitated strings, aka the reason your earbuds always come out of your pocket tangled into a horrible mess. In short, what happens here is that knots can be made easily, by passing the ends of strings through loops, but once a knot is formed the presence of friction makes those knots more difficult to undo. Furthermore, if you get into the topology of knots, thinking of unknotting a string as it passing back through the same loop twice, the probability of an agitated string unknotting itself is somewhat similar to the probability of a random walk stopping, turning around, and exactly retracing its steps. Clearly this is low-probability.
So what happens when add higher dimensions? Well, knots become significantly easier to undo! There are more available movements that do not result in a knot, and even many that undo a knot. Indeed, as you add more and more dimensions, it becomes effectively impossible for knots to be a relevant mathematical concept. As before, recall the difficulty of "tying together" two dots and a circle.
Which brings me to my conclusion: Homura's hair actually is that fantastic because she's lifting parts of it into a higher dimension. The added dimension directly prevents the formations of knots for reasons rooted in the fundamental mathematics of geometry and topology.
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