Greg Egan on Nostr: In 1926, Menger proved that you can embed any compact 1-dimensional space in his ...

In 1926, Menger proved that you can embed any compact 1-dimensional space in his famous fractal sponge, including a loop, but he didn’t say anything about the knottedness of the embedded loop.

Now three teenagers and their mentor have proved that you can actually embed any *knot* in the Menger sponge: you can deform any knot without the string having to pass through itself so that it ends up as a subset of the Menger sponge.

https://www.quantamagazine.org/teen-mathematicians-tie-knots-through-a-mind-blowing-fractal-20241126/

https://arxiv.org/abs/2409.03639