Timeline for Distribution over Penrose Tilings?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2021 at 10:28 | comment | added | Dan Rust | @NickS You're right. I've done that, and also added some detail on how the frequencies of patches are calculated. | |
| Jun 20, 2021 at 10:28 | answer | added | Dan Rust | timeline score: 7 | |
| Jun 18, 2021 at 22:32 | comment | added | Nick S | @DanRust I think you should make this comment an answer. | |
| May 21, 2021 at 2:50 | history | became hot network question | |||
| May 21, 2021 at 0:06 | comment | added | Dan Rust | By Birkhoff's ergodic theorem then, the unique measure is the one which assigns to a set of tilings with a patch $P$ at the origin, the frequency of that patch $P$ in a penrose tiling (which is independant of the choice of tiling) | |
| May 21, 2021 at 0:04 | vote | accept | Bill Bradley | ||
| May 21, 2021 at 0:03 | comment | added | Dan Rust | The tiling space of the penrose tilings is uniquely ergodic with respect to the translation action. As is any reasonable notion of the 'canonical transversal' which is essentially the one you describe as the set of tilings with a vertex at the origin (the action on this set is a little more difficult to describe though and really needs groupoids). This is all in 'Aperiodic Order, Vol. 1' by Baake and Grimm. | |
| May 21, 2021 at 0:02 | history | edited | Bill Bradley | CC BY-SA 4.0 |
Clarifying/fixing definition of fragment containment.
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| May 20, 2021 at 23:33 | comment | added | Mateusz Kwaśnicki | @StevenStadnicki: The wording of the question is perhaps misleading, but if one changes it to "...contains that fragment attached at the origin", I think it makes perfect sense. | |
| May 20, 2021 at 18:30 | answer | added | RavenclawPrefect | timeline score: 8 | |
| May 20, 2021 at 16:33 | comment | added | Bill Bradley | Thank you, @StevenStadnicki ; it sounds like perhaps my condition is misguided. Is there a better notion of "this measure covers all Penrose tilings" that might work better? | |
| May 20, 2021 at 16:02 | comment | added | Steven Stadnicki | Any arbitrary large finite subset of tiles selected from any Penrose tiling appears in every Penrose tiling infinitely often, so your measure is essentially trivial. This is in fact the point of the tilings' appearance in Connes' book, IIRC — traditional measures (in both senses) are insufficient because there is no finitary way to distinguish any two of the uncountably many tilings. | |
| May 20, 2021 at 15:55 | comment | added | Mateusz Kwaśnicki | My guess is that the (compact - with the usual topology where "being close enough" means being equal on a large centred ball) space of such tiling has a unique ergodic measure with respect to translations, but at this moment I have nothing to back this up. "Inflation-deflation" approach would then give a way to sample from this distribution. | |
| May 20, 2021 at 15:27 | comment | added | Malkoun | I am not knowledgeable in that topic, but I do remember Alain Connes's Noncommutative Geometry book mentioning Penrose tilings as one of the examples. Maybe this helps? | |
| May 20, 2021 at 14:58 | history | asked | Bill Bradley | CC BY-SA 4.0 |