Is there any hope of a mortal actually understanding the counterexample in R^8? On Fri, Aug 21, 2020 at 4:18 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Hmm... the abstract on arXiv says:
"This result implies that every unit cube tiling of R^7 contains a facesharing pair of cubes."
which suggests that the conjecture is indeed resolved.
https://arxiv.org/abs/1910.03740
Sent: Friday, August 21, 2020 at 8:32 PM From: "Mike Beeler" <mikebeeler2@gmail.com> To: "Dan Asimov" <dasimov@earthlink.net>, "math-fun" < math-fun@mailman.xmission.com> Subject: Re: [math-fun] Correction re: Keller's conjecture now settled in all dimensions
Nice. But I am still confused about whether the tiling version has been proved true in dimension 7. The Wikipedia article says, "However, the translation from cube tilings to graph theory can change the dimension of the problem, so this result doesn't settle the geometric version of the conjecture in seven dimensions.” The article's “Talk” tab mentions this ambiguity, but does not seem to resolve it. The Quanta article implies the computer result is based on the graph version, so the question for the tiling version would still be open. ???
— Mike
On Aug 21, 2020, at 12:36 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Oops! I said the exact opposite of what happened. Sorry about that.
In 7 dimensions it was proved ***true***, not false, contrary to what I wrote.
—Dan
----- 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>. ^^^^^^^^^^^^^^ ***FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE***
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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