I just thought of a kind of hypercube tiling that is even wilder than the unKeller tilings. My intuition says it's impossible, but after hearing about unKeller tilings, I wouldn't be TOO surprised to hear than such a wild tiling had been found in, say, R^24 or R^196882. A wild hypercube tiling is one in which no two facet-adjacent tiles have the same orientation. Two n-cubes which partially meet in an (n-1) facet are forced to line up along one axis, but their mutual orientation still has n-2 degrees of freedom. As I said, this feels impossible -- but I don't see any obvious impossibility proof. On Fri, Aug 21, 2020 at 12:21 PM Dan Asimov <dasimov@earthlink.net> wrote:

Keller's conjecture <https://en.wikipedia.org/wiki/Keller's_conjecture> proposed that if Euclidean space R^n were tiled by n-dimensional cubes any which way, then there must be some pair of n-cubes that share a common (n-1)-dimensional face.

This is intuitively true in dimensions 2 and 3, but was proved true in dimensions up through 6 by Oskar Perron in 1940.

To the surprise of many people, Lagarias & Shor (1994) found a counterexample in dimension 10. This immediately leads to counterexamples in all higher dimensions as well.

Soon after a counterexample was found in dimension 8, showing it was also false in dimension 9 and leaving the only dimension where it was open being dimension 7.

A team of four people — Joshua Brakensiek, Marijn Heule, John Mackey, and David Narváez — used a computer program to search for counterexamples in dimension 7, and in October 2019 they found one: < https://arxiv.org/abs/1910.03740>.

There a nice article about it in Quanta magazine: < https://www.quantamagazine.org/computer-search-settles-90-year-old-math-prob...

...

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—Dan

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