Timeline for ♢ ⧫ ⬠: the fourth kind of Penrose tiling?
Current License: CC BY-SA 3.0
26 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2017 at 10:17 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
A sequence of simple rules is found that makes ♢ ⧫ ⬠ from a P2
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| Jul 18, 2017 at 12:11 | comment | added | Gerry Myerson | What is this strange thing you are doing with the capital version of the letter "i"? | |
| Jul 18, 2017 at 0:46 | comment | added | mkreisel | I don't see it listed at tilings.math.uni-bielefeld.de/glossary/mld-class-penrose either. If you want to know whether it's truly new, I'd suggest sending an email to an expert on tilings, perhaps Professor Goodman-Strauss: comp.uark.edu/~strauss. | |
| Jul 17, 2017 at 20:57 | answer | added | Incnis Mrsi | timeline score: 0 | |
| Jul 11, 2017 at 8:46 | comment | added | მამუკა ჯიბლაძე | The corresponding construction is this: given your tiling, identify each red rhombus angle with the acute angle of a kite that lies on the axis of symmetry. Some green rhombi become covered by these kites. Then identify each remaining green rhombus angle with the acute angle of a dart that lies on the axis of symmetry. | |
| Jul 11, 2017 at 8:42 | comment | added | მამუკა ჯიბლაძე | In fact I now realized - there is also a simple correspondence with the second generation P2: kites $\texttt{<->}$ red rhombi; after that, darts $\texttt{<->}$ remaining green rhombi. | |
| Jul 10, 2017 at 16:27 | comment | added | მამუკა ჯიბლაძე | You said yourself that white rhombi are actually of two different roles. And actually there also are at least two different role pentagons (blue-green and red-green, not counting chirality of red-greens that you also mentioned yourself). In any case I believe one may simply list all permitted vertex figures, no? | |
| Jul 10, 2017 at 12:24 | comment | added | Incnis Mrsi | @მამუკა ჯიბლაძე: We can encode such concept as “a green ♢” locally only sacrificing this very small cardinality (3) of the tiles’ set – the thing Ī̲’m unwilling to accept. Sure, “blue” is synonymous with “outer”; the former was my original terminology that Ī̲ later changed. | |
| Jul 10, 2017 at 11:55 | comment | added | მამუკა ჯიბლაძე | As for outer and inner sides, these are just blue and non-blue sides respectively. | |
| Jul 10, 2017 at 11:54 | comment | added | მამუკა ჯიბლაძე | Not sure if it is redundant or helps with anything but also: a green ♢ cannot share an edge with a ⬠, while a red ♢ can share no more than one edge with a ⬠. | |
| Jul 10, 2017 at 9:14 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
rv incompetent copyediting, including alleged grammar improvements (“of each tile” is nice, whereas “for each tile” isn’t) and messing with the author’s typography
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| S Jul 10, 2017 at 8:50 | history | suggested | Glorfindel | CC BY-SA 3.0 |
grammar corrections
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| Jul 10, 2017 at 7:49 | review | Suggested edits | |||
| S Jul 10, 2017 at 8:50 | |||||
| Jul 10, 2017 at 7:43 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
yet another rule, the last one Ī̲ hope
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| Jul 9, 2017 at 21:49 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
yet another condition on ♢ — the sufficiency likely would fail without anything of the sort
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| Jul 9, 2017 at 20:54 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
terminology tweaks
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| Jul 9, 2017 at 20:39 | history | edited | Incnis Mrsi | CC BY-SA 3.0 |
Update: rules of the tiling necessary (and possibly sufficient) for the Penrose-equivalence
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| Jul 9, 2017 at 17:39 | answer | added | მამუკა ჯიბლაძე | timeline score: 8 | |
| Jul 9, 2017 at 17:29 | comment | added | მამუკა ჯიბლაძე | OK let me make one more try - in an answer, maybe this time it is less wrong :D | |
| Jul 9, 2017 at 17:28 | comment | added | მამუკა ჯიბლაძე | In fact I now see I missed blue edges completely, they are not covered at all, along with quite big rhombi around them! | |
| Jul 9, 2017 at 17:17 | comment | added | Incnis Mrsi | @მამუკა ჯიბლაძე: First: do you remember Wikipedia’s “originality” policy? Second: it might be true, but the rule about dart is tricky. Each red edge that separates neighboring ⬠s struts to ⧫ both ends, and we have to know where ⬠’s green edge is to decide about orientation of the dart; that is, which of two red endpoints becomes its reflex (216°) angle. So even♢ ⧫ ⬠ ⇒ P2 isn’t a fairly trivial deal. Even harder is the reverse conversion; you can look at SVG source code to find that the algorithm requires position data spanning three generations. | |
| Jul 9, 2017 at 16:48 | comment | added | მამუკა ჯიბლაძე | However it looks that it is derivable from P2, as follows: turn every red edge into the mid-line of a dart; observe that every green edge sticks into the middle of a blue rhombus, so every (green edge+its continuation into the blue rhombus) can be made into the mid-line of a kite. I believe these darts and kites then cover everything without overlaps, and form a P2, no? | |
| Jul 9, 2017 at 16:27 | comment | added | მამუკა ჯიბლაძე | In any case it is worth adding to the Wikipedia List of aperiodic sets of tiles | |
| Jul 9, 2017 at 15:15 | comment | added | Incnis Mrsi | @Sylvain JULIEN: They certainly knew decagonal patterns since medieval times, but Ī̲ never saw such long (really endless) chains of ⬠s in Islamic art. | |
| Jul 9, 2017 at 15:12 | comment | added | Sylvain JULIEN | It may have been used in Muslim architecture already. Marcus du Sautoy in his book "symmetry" describes the different kinds of symmetry used in Alhambra in Granada, he may be able to tell you more. | |
| Jul 9, 2017 at 14:55 | history | asked | Incnis Mrsi | CC BY-SA 3.0 |