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 Re: Al Zimmermann - Polygonal Areas26.04.2017, 11:43

01/06/12
1015
 Here are some 13 max:(2,1),(1,7),(3,13),(11,12),(9,9),(4,5),(8,8),(5,3),(6,4),(7,6),(10,10),(12,11),(13,2)(9,11),(6,6),(10,9),(5,5),(3,2),(12,1),(13,7),(11,13),(1,12),(2,3),(4,4),(7,8),(8,10)(13,4),(12,13),(3,12),(6,8),(5,9),(7,6),(10,7),(8,5),(9,3),(4,10),(2,11),(1,2),(11,1)

 Re: Al Zimmermann - Polygonal Areas26.04.2017, 13:22

14/12/14
27
 None of your 3 examples has one of the "standard" frames used in Tom's lists of results for hull3, e.g. 13:113, whereas yours are (rotated to support visual comparison):13:11313:11313:113So the task of finding all solutions for the max problems is not so easy. Using Markus Sigg's breadth first search (which has no guarantee of finding all solutions) I've found 27 configs with area 113, which include mirrored solutions.The max 13 might have at least 14 essentially different optimal configs. Definitely not mature enough for inclusion into an OEIS sequence.

 Re: Al Zimmermann - Polygonal Areas26.04.2017, 16:40

01/06/12
1015
 yae9911 в сообщении #1212638 писал(а):Definitely not mature enough for inclusion into an OEIS sequenceYou can have the sequence go up to n=12 and add the "more" keyword. Or you can use "-1" to mean "don't know" and continue it for further n.

 Re: Al Zimmermann - Polygonal Areas26.04.2017, 18:32

24/04/17
5
 Each hull shape has a "quality": how close to the theoretical bound it can come. "hull4" has a quality of 0, but others hulls have quality of -1.5, and so on. When you're searching for an area equal to the theoretical bound, you must have a hull quality of 0, and there are only a few of these. When you're searching for slightly less area, there are more hull shapes to consider. (When you're searching for much less area, you'll need to consider hulls with more than one "cavern" carved into them (probably won't be a factor)).

 Re: Al Zimmermann - Polygonal Areas27.04.2017, 12:54

01/06/12
1015
 TomSirgedas в сообщении #1212659 писал(а):"hull4" has a quality of 0, but others hulls have quality of -1.5, and so on. When you're searching for an area equal to the theoretical bound, you must have a hull quality of 0, and there are only a few of theseSo here is what confuses me: some of your results for hull4 are below optimal (or best found), so how is that possible if hull4 guarantees optimality? Or does it mean that you weren't able to find the best hull4 solution?

 Re: Al Zimmermann - Polygonal Areas27.04.2017, 19:36

24/04/17
5
 hull4 doesn't guarantee optimality. But if you start with this shape: http://antiton.de/PolygonalAreas/index.html?(3,2),(4,4),(2,3),(1,9),(8,10),(10,8),(9,1) and "grow" the polygon by only adding triangles with area .5, you'll have an optimal solution, like this:Sometimes the task is impossible, and triangles with larger area are necessary. And/or a sub-optimal starting point like http://antiton.de/PolygonalAreas/index.html?(3,2),(6,6),(2,3),(1,9),(8,10),(10,8),(9,1) can be used.

 Re: Al Zimmermann - Polygonal Areas01.05.2017, 12:30

01/06/12
1015
 Some optimal max results:11:75.5(4,3),(5,5),(6,6),(3,4),(7,7),(9,10),(2,11),(1,2),(11,1),(10,9),(8,8)(2,9),(5,7),(8,3),(6,6),(7,5),(9,4),(4,8),(3,10),(10,11),(11,2),(1,1)12:94.5(6,6),(7,7),(4,5),(8,8),(10,11),(2,12),(1,3),(3,1),(12,2),(11,10),(9,9),(5,4)14:146(9,10),(7,7),(4,5),(10,12),(2,3),(1,13),(13,14),(14,2),(5,1),(3,4),(11,9),(6,6),(12,11),(8,8)(5,9),(8,4),(4,10),(9,7),(10,6),(6,11),(13,3),(14,13),(3,14),(1,12),(2,1),(12,2),(11,5),(7,8)(1,2),(2,12),(4,10),(9,4),(6,8),(11,5),(7,11),(10,6),(8,7),(5,9),(3,13),(13,14),(14,3),(12,1)15:161.5(5,5),(3,2),(14,1),(15,14),(1,15),(2,3),(4,4),(10,11),(6,6),(7,7),(12,13),(9,9),(13,12),(8,8),(11,10)(13,2),(11,5),(5,10),(10,6),(3,12),(7,9),(8,8),(4,13),(9,7),(6,11),(12,4),(14,3),(15,15),(1,14),(2,1)(4,13),(7,9),(8,8),(3,12),(9,7),(5,10),(12,4),(13,1),(1,2),(2,15),(15,14),(14,3),(11,5),(6,11),(10,6)16:188.5(15,14),(16,2),(3,1),(1,3),(2,16),(14,15),(12,12),(6,7),(11,11),(4,5),(8,8),(9,9),(5,4),(10,10),(7,6),(13,13)

 Re: Al Zimmermann - Polygonal Areas02.05.2017, 10:55

14/12/14
27
 Using some programs from Markus Sigg's polygon optimization toolbox, I've found an improvement of Tom Rokicki's Max 131:16574improved toMax 131:16576which achieves the optimum area.

 Re: Al Zimmermann - Polygonal Areas02.05.2017, 11:07

25/04/17
3
 Wow, incredible!

 Re: Al Zimmermann - Polygonal Areas02.05.2017, 12:12

14/12/14
27
 This was an easy prey:Tomas Rokicki's Max149:21532.5improved to the optimalMax149:21535

 Re: Al Zimmermann - Polygonal Areas02.05.2017, 13:21

01/06/12
1015
 Great work!

 Re: Al Zimmermann - Polygonal Areas03.05.2017, 17:52

03/05/17
3
 I have done exhaustive searches up to N = 11 for minimum and maximumThe minimum results follow. I have listed the solutions where the #ofconfigs differs from yae9911 list3 1.5 # 14 1.0 # 15 1.5 # 2(1,1), (2,2), (4,5), (3,3), (5,4), (1,2), (3,3), (4,4), (2,1), (5,5), 6 4.0 # 27 4.5 # 38 3.5 # 49 4.5 # 110 4.0 # 3(1,3), (5,6), (8,8), (6,5), (3,1), (10,7), (4,2), (9,9), (2,4), (7,10), (1,3), (5,6), (8,8), (3,1), (10,7), (4,2), (6,5), (9,9), (2,4), (7,10), (1,3), (8,8), (3,1), (10,7), (4,2), (6,5), (9,9), (5,6), (2,4), (7,10),11 5.5 # 2(1,1), (3,4), (7,11), (2,2), (10,9), (6,6), (4,5), (11,10), (5,3), (9,8), (8,7), (1,2), (5,8), (2,3), (9,11), (3,4), (6,6), (11,10), (4,1), (7,5), (8,7), (10,9), The above results just confirm TomSirgedas minimumsProcessing Tom's minimums upto 20 to remove symmetries the #ofconfigs are12 6.0 # 1213 6.0 # 614 6.5 # 515 7.0 # 1116 7.0 # 417 7.5 # 1118 8.0 # 1919 8.5 # 1720 9.0 # 55The maximum results are3 1.5 # 14 4.0 # 15 9.0 # 16 14.0 # 27 22.0 # 18 32.5 # 19 44.0 # 2(1,2), (2,7), (4,6), (5,4), (6,3), (3,8), (8,9), (9,5), (7,1), (1,2), (2,8), (5,4), (4,6), (6,5), (3,7), (8,9), (9,3), (7,1), 10 59.0 # 311 75.5 # 5(1,1), (2,9), (4,8), (8,3), (7,5), (6,6), (9,4), (5,7), (3,10), (10,11), (11,2), (1,1), (2,9), (5,7), (8,3), (6,6), (7,5), (9,4), (4,8), (3,10), (10,11), (11,2), (1,2), (2,9), (4,8), (8,3), (7,5), (6,6), (9,4), (5,7), (3,10), (10,11), (11,1), (1,2), (2,9), (5,7), (6,6), (8,3), (7,5), (9,4), (4,8), (3,11), (11,10), (10,1), (1,3), (4,4), (8,9), (7,7), (6,6), (9,8), (5,5), (3,2), (10,1), (11,10), (2,11), I am running 12 for minimums and maximums. I am sure the minimum will just confirm Tom's resultsI expect to have results in about three weeks.

 Re: Al Zimmermann - Polygonal Areas07.06.2017, 15:30

14/12/14
27
 All, I have started to edit drafts of 4 new OEIS sequences to document the results for small n. If you could have a look at the drafts I'd be grateful for suggestions and feedback. Links to examples and some visualizations will be added.Values of Minimum AreaNumber of Configurations with Minimum AreaValues of Maximum AreaNumber of Configurations with Maximum AreaIf you are a registered OEIS user you can even log in and edit the sequence drafts.BTW, we have found some further improvements of the Contest and Post Contest results since my last post. The data will be provided later in a separate post.Hugo Pfoertner

 Re: Al Zimmermann - Polygonal Areas08.06.2017, 13:56

14/12/14
27
 Последний раз редактировалось yae9911 08.06.2017, 14:30, всего редактировалось 2 раз(а). List of improvements continued:Min 331: 169.0Min 257: 131.5Min 293: 149.5Min 419: 215.5Min 191: 96.0Max 257: 64892.5Continued in next post.

 Re: Al Zimmermann - Polygonal Areas08.06.2017, 15:25

14/12/14
27
 Continued list of improvements found by "polygon engineering" using Markus Sigg's Tabu-enhanced breadth first search programs:Max 467: 215981.0Max 373: 137451.5Min 373: 188.0Max 167: 27140.5Max 521: 269092.0One more continuation follows.

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