Timeline for Lattice n-gons with ordered side lengths 1,2,3,...,n
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2017 at 13:05 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Updated links.
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| Mar 10, 2017 at 9:42 | history | edited | CommunityBot |
replaced http://www.gap-system.org/ with https://www.gap-system.org/
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| Jun 5, 2016 at 20:43 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
As reported by Bernardo Recamán Santos today, 2 of the 1482 polygons referred to in the last paragraph had self-intersections. Therefore the count for n = 15 is 584 rather than 586. The linked data files have been corrected as well.
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| May 16, 2016 at 13:58 | comment | added | Bernardo Recamán Santos | The number of all such serial polygons is now sequence A273089 in the OEIS: oeis.org/A273089. | |
| May 4, 2016 at 22:16 | comment | added | Stefan Kohl♦ | @BernardoRecamánSantos: Done! | |
| May 4, 2016 at 22:15 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added data as requested in the comments.
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| May 4, 2016 at 20:32 | comment | added | Bernardo Recamán Santos | Great @StefanKohl: Could you please provide the file with the new polygons. It might now be worth adding the related sequence to the OEIS with a link to this page. | |
| May 4, 2016 at 20:03 | comment | added | Stefan Kohl♦ |
@BernardoRecamánSantos: Certainly more polygons turn up: we have then e.g. $3$ instead of just one for $n=8$, and $5$ instead of one for $n = 11$ and $6$ instead of none for $n = 12$. -- To find this, you only need to change 3 lines in the GAP function I gave (remove the line which checks for nonnegative coordinates, add a check polygon[Length(polygon)-1][2] <> 0 in the first line of the function search and put directions[2] := [[0,2]]; to exclude rotations and symmetrical versions).
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| May 4, 2016 at 12:10 | comment | added | Bernardo Recamán Santos | The question arises: If we do not restrict ourselves to polygons entirely in the first quadrant, do many more serial lattice polygons completely different to those above (i.e. exclude rotations and symmetrical versions) appear? To start with, golygons would turn up, but most likely many others. | |
| May 3, 2016 at 9:42 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added link to zip file with drawings of all 249 polygons for n <= 19.
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| Apr 30, 2016 at 22:31 | comment | added | Stefan Kohl♦ | @JosephO'Rourke: Of course! -- I have added a paragraph on this. | |
| Apr 30, 2016 at 22:30 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added information, in response to Joseph O'Rourke's comment.
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| Apr 30, 2016 at 21:25 | comment | added | Joseph O'Rourke | Could you say something about how you searched for these (delightful) polygons? | |
| Apr 30, 2016 at 20:41 | comment | added | Stefan Kohl♦ | @BernardoRecamánSantos: It seems likely that your conjecture is true (e.g. further search found $225$ such polygons for $n = 19$). But Gerhard Paseman is right that $n$ must be congruent to $0$ or $3$ modulo $4$. -- Consider a checkerboard coloring of the integer points in the Cartesian plane. Then the ends of a side have the same color if its length is even, and they have different color if its length is odd. This holds also for diagonal sides, as you can check. So a closed path must contain an even number of sides of odd length. | |
| Apr 30, 2016 at 17:41 | comment | added | Bernardo Recamán Santos | My conjecture is that there are infinitely many such polygons. Indeed, it could be that there is at least one with n sides for any sufficiently large n. | |
| Apr 30, 2016 at 17:38 | vote | accept | Bernardo Recamán Santos | ||
| Apr 30, 2016 at 13:53 | history | answered | Stefan Kohl♦ | CC BY-SA 3.0 |