A browser puzzle, based on "Knot Theory". Not sure I learned anything from playing this, but that was fun:<p><a href="https://brainteaser.top/knot/index.html" rel="nofollow">https://brainteaser.top/knot/index.html</a>
> <i>Every knot is “homeomorphic” to the circle</i><p>Here's an explanation:<p><a href="https://math.stackexchange.com/questions/3791238/introduction-to-topology-knots-and-circles-are-homeomorphic" rel="nofollow">https://math.stackexchange.com/questions/3791238/introductio...</a>
Can anyone explain why they bothered with the fractal at all, instead of using a 3 dimensional grid? Doesn't a grid of the appropriate resolution provide the exact same? Or is it to show that they can do everything within even a subset of a 3D grid limited in this way?
Quanta Magazine consistently explains mathematics/physics for an advanced lay audience in ways that don't terribly oversimplify / still expose you to the true ideas. It's really nice! I don't know of any other sources like this.
I've always wondered if it's possible to harness teen minds to solve significant math problems in high school, if you formulated them well and found the right scope. I think it's possible.
> But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large.<p>I'm struggling to understand what is counterintuitive here. Am I missing something?<p>Also, it's still (always) going to be in the shape of a cube. And if we are going to argue otherwise, we can do that without invoking infinity—technically it's not a cube after even a single iteration.<p>This feels incredibly sloppy to me.