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An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

2023 · arXiv

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From the abstract

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

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3 selected excerpts · 3 total mentions

benreesman
Neutral mention · rule
We've arrived at a point in the conversation where your assertions are (for the most part) correct in at least a narrow way. But there were several epistemological sleights of hand involved in getting from point A to point B that you as an eminent philosopher can trivially identify at least as easily as I can, at least in hindsight. First, you've elided the unambiguous context of the OP's essay and my response to it: I said it's dangerous for people who miss getting into and through an elite university education without burdensome and difficult-to-discharge debt to go for broke anyways: the typical outcome there is the typical outcome of modern university education in…
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general, which is a pile of debt for a mediocre education with a lift on employment outcomes asymptotically approaching zero and trivially not justifying the cost of the debt and the debt service, and a network of principally personal social utility. When the cheese at the end of the maze is "getting rich", that's a bad play. Another important piece of context you've used as a wedge (because I didn't dissect his essay via cherry-picked quotes, assuming that everyone who cared had read it and internalized the broader context) is that pg is clearly addressing people who he hopes will become high-achieving technologists, at a point in life where the decision is around a STEM undergraduate education with large class sizes and and a decaying level of interaction with faculty or other eminent thinkers affiliated with the university. This is nothing like a philosophy degree from an eminent university that involves freewheeling face-to-face interaction with some of the sharpest and most disruptive thinkers in a field, and furthermore in a field with very subjective measures on short time horizons about who has or has not made lasting contributions. We argue to this day about the philosophical and epistemological impact of work done millennia ago: right off the top of my head people debate whether or not Augustine was a key figure in the development of a modern sense of personal identity, and the debate about who was worth listening in these areas was very old then [1]. There is no evidence that's ever been presented to me that we can identify worthwhile contributions to philosophy on a time scale measured in years or even decades: people who devote their lives to such pursuits are optimizing for posterity, a very worthy thing to optimize for in my opinion but rarely one a person ever knows the outcome on in their lifetime. Furthermore, the exact immersive scenario you're describing requires nothing like the nepotism, barriers to entry, or cost structure of a modern elite university: we've erected barriers around class dynamics, wealth dynamics, and admissions that had far better solutions with the cost structure of a coffee shop in recent history [2]. We've eradicated that institution intentionally or unintentionally in favor of something defense-procurement markup expensive, with admissions criteria constantly plagued by scandal and lawsuit [3] after scandal [4], which welcomes false positives on admissions when attached to wealth and power [5], and sees false negatives on groundbreaking contribution unambiguous in contribution [6] when not attached to wealth and power, at least until forced to by mathematical rigor. And in your conclusion you've linked learning, broadly construed, with learning how to operate in circles where people face no consequences, continue to act unilaterally on behalf of the commons, and fuck up constantly [6] [7] [8] [9] [10] to put any question about their innate capability at anything other than learning how to work the socio-economic, professional, and academic class systems and maintain those distinctions by stepping on people who were both right and doing the right thing [11] via thuggery [12]. Almost all destructive forces with actual power [13] [14] [15] trace clear intellectual, academic, and philosophical origins to the work of academics or other noted intellectuals on the wrong side of history [16] [17] [18] [19]. So the question for you here isn't whether or not the world currently works they way you're describing, for the elite few it does. The question is what team you're playing for from a post of extremely influential privilege. Things aren't going well for the non-elites, and I don't think pg was trying to send 1-3% of his audience into a decaying elite subsidized by leading the other 97% to the slaughter in the form of useless degrees with mountains of debt. He's backed the wrong horse on a few things, but he's a fundamentally good guy. [1] https://academic.oup.com/book/4734/chapter-abstract/14697047... [2] https://en.wikipedia.org/wiki/Salon_(gathering) [3] https://en.wikipedia.org/wiki/Students_for_Fair_Admissions_v... [4] https://www.theatlantic.com/ideas/archive/2024/01/claudine-g... [4] https://en.wikipedia.org/wiki/Legacy_preferences [5] https://arxiv.org/pdf/2303.10798.pdf [6] https://en.wikipedia.org/wiki/Lawrence_Summers [7] https://en.wikipedia.org/wiki/Long-Term_Capital_Management [8] https://www.npr.org/2008/10/24/96070766/greenspan-admits-fre... [10] https://edition.cnn.com/2023/11/22/tech/larry-summers-openai... [11] https://en.wikipedia.org/wiki/Brooksley_Born [12] https://www.pbs.org/wgbh/frontline/documentary/warning/ [13] https://en.wikipedia.org/wiki/Totalitarianism [14] https://en.wikipedia.org/wiki/Populism [15] https://en.wikipedia.org/wiki/Cancel_culture [16] https://en.wikipedia.org/wiki/Karl_Marx [17] https://en.wikipedia.org/wiki/Milton_Friedman [18] https://en.wikipedia.org/wiki/Ayn_Rand [19] https://en.wikipedia.org/wiki/Michel_Foucault
l0b0
Neutral mention · rule
No, I mean mathematically. I had a skim through the original paper[1], but unfortunately it's not easy to figure out what the coordinates are. If any of them are related as a factor of an irrational number, I don't think it's possible. [1] https://arxiv.org/abs/2303.10798
bazzargh
Neutral mention · rule
The original monotile paper https://arxiv.org/abs/2303.10798 mentions that (at the time that was written) the minimum number of sides had to be 5 <= n <= 13, but also that a smaller tile can't be a polykite like the hat; exhaustive search had shown that there were no other aperiodic monotiles (than the hat and turtle) with < 21 kites But a correction: the basic shape of the spectre is 14 sides with 2 of them colinear. It's only truly aperiodic when you modify the 14 edges, making it clear it is 'really' 14 sided not 13.

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