Timeline for Is this an instance of any existing convex pentagonal tilings?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 25, 2015 at 1:58 | vote | accept | Jacky | ||
| Aug 24, 2015 at 17:56 | comment | added | Noam D. Elkies | Each of those still has two parallel sides (this follows from D+E=180), so has a Type I tiling. I don't know whether your alternative tilings with these pentagons are known. That's a "Q2" matter not addressed by the Wikipedia page, though I wouldn't be surprised if some of the people working on "Q1" have also run across your pentagons that tile the plane by translates of a 4-tile region. | |
| Aug 24, 2015 at 10:18 | comment | added | Jacky | Sorry for the bad notations... Do you think any of the two generalizations (below) is instance of a currently discovered type of convex pentagonal tiling? | |
| Aug 24, 2015 at 4:25 | comment | added | Noam D. Elkies | Yes, the argument you give proves that your tiling is not the same as the generic Type 4 tiling. But unless you can generalize your pentagon to one that's not already known to tile the plane in some other way, it's only a "Q2" distinction. | |
| Aug 24, 2015 at 4:09 | comment | added | Jacky | Thanks again Noam. Do you think there's fundamental difference between the pattern I posted and type 4 on Wikipedia (that in type 4, each side of all pentagons overlaps with one side of another pentagon, while in my pattern, there are occasions when a side of a pentagon overlaps with 2 shorter sides of 2 pentagons)? | |
| Aug 24, 2015 at 4:00 | comment | added | Noam D. Elkies | Your tiling is "4-tile" (the pentagon appears in 4 orientations). But your pentagon also has the Type-1 tiling with a "2-tile lattice" that's shared by all convex pentagons with two parallel sides. It looks like this pentagon also fits into Type 4, giving yet another 4-tiling. | |
| Aug 24, 2015 at 3:52 | comment | added | Jacky | Thanks for the answer. May I further enquire that is it a 2-tile lattice or a 4-tile lattice? If it's an example of a 4-tile lattice, can it belongs to type 1? | |
| Aug 24, 2015 at 3:46 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |