[seqfan] Re: How many squares can you make from n points in the plane?
Kurz, Sascha
Sascha.Kurz at uni-bayreuth.de
Tue Oct 5 15:05:25 CEST 2021
> Sascha: what are you run-times ? How far can you compute these
> configurations ? Can you compute one around n = 140 ? This could
> probably tell circles and octagons appart (my bet is now on circles).
Prescribing a (12,3)-octagon
120 points generate 1286 squares; contained in a 12x12-grid.
___XXXXXX___
__XXXXXXXX__
_XXXXXXXXXX_
XXXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXXXX
_XXXXXXXXXX_
__XXXXXXXX__
___XXXXXX___
it took 4.32 seconds to compute the optimal extension by 20 points on the
surrounding 14x14-grid. (I stopped the extension to an 16x16 grid after
30 minutes, where the LP bounds where between 1752 and 1789.)
140 points generate 1752 squares.
contained in a 13x13-grid.
___XXXXXXX___
__XXXXXXXXX__
_XXXXXXXXXXX_
_XXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXXXX
_XXXXXXXXXXX_
_XXXXXXXXXXX_
__XXXXXXXXX__
____XXXXX____
Prescribing a circle with radius 6.0828
121 points generate 1298 squares; contained in a 13x13-grid.
_____XXX_____
___XXXXXXX___
__XXXXXXXXX__
_XXXXXXXXXXX_
_XXXXXXXXXXX_
XXXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXXXX
_XXXXXXXXXXX_
_XXXXXXXXXXX_
__XXXXXXXXX__
___XXXXXXX___
_____XXX_____
it took 155.42 seconds to compute the optimal extension by 19 points on the
surrounding 15x15-grid. The very same solution was computed
My bet is now on circles too. However, for small numbers of points (k,l)-octagons perfom pretty well:
k l n m n(n-1)/m comment
2 0 4 1 12.000 optimal
3 0 9 6 12.000 optimal
4 1 12 11 12.000 optimal
4 0 16 20 12.000 best known
5 1 21 37 11.351 best known
6 1 32 88 11.273 best known
7 2 37 117 11.385 best known
7 1 45 175 11.314 best known
8 2 52 237 11.190 best known
9 2 69 421 11.145 best known
10 3 76 507 11.243 best known
10 2 88 686 11.160
11 3 97 836 11.139
12 3 120 1286 11.104
13 3 145 1880 11.106
14 4 156 2175 11.117
14 3 172 2643 11.128
15 4 185 3070 11.088
16 5 196 3425 11.159
15 3 201 3602 11.160
16 4 216 4190 11.084
Circles seem to perform better if the number of points gets larger
radius n m n(n-1)/m comment
1.4143 9 6 12.000 optimal
2.0001 13 11 14.182 2 off opimal
2.2361 21 37 11.351 best known
2.8285 25 50 12.000 1 off best known
3.0000 29 67 12.119
3.1623 37 117 11.385 best known
3.6056 45 175 11.314 best known
4.0000 49 204 11.529
4.1232 57 278 11.482
4.2427 61 323 11.331
4.4722 69 421 11.145 best known
5.0000 81 568 11.408
5.0991 89 698 11.221
5.3852 97 836 11.139 octagon
5.6569 101 905 11.160
5.8310 109 1051 11.201 octagon
6.0001 113 1128 11.220
6.0828 121 1298 11.186
6.3246 129 1476 11.187 octagon
6.4032 137 1678 11.104
6.7083 145 1880 11.106 octagon
7.0001 149 1977 11.154
7.0711 161 2304 11.181
7.2112 169 2546 11.152
7.2802 177 2812 11.078
7.6158 185 3070 11.088 octagon
7.8103 193 3336 11.108
8.0001 197 3469 11.131
8.0623 213 4065 11.108
8.2463 221 4387 11.083
8.4853 225 4544 11.092
8.5441 233 4874 11.091
8.6024 241 5228 11.064
8.9443 249 5566 11.095 octagon
9.0001 253 5743 11.102
9.0554 261 6113 11.101
9.2196 277 6901 11.078 better than octagon
9.4340 285 7311 11.071
9.4869 293 7745 11.047
9.8489 301 8155 11.073 octagon
9.8995 305 8368 11.080
10.0001 317 9031 11.092
10.0499 325 9505 11.078
10.1981 333 9987 11.070
10.2957 341 10485 11.058
10.4404 349 10991 11.050
10.6302 357 11497 11.054 better than octagon
However, we will really need quite some points to see the asymptotical fraction 11.008 (and to see differences between
circles and octagons).
I have no glue whether choosing a different grid in a small circle inside another circle, as in Theorem 4 in the paper on
isoclees right triangles, may get us below 11.008.
Is it worth to collect "best known" lower bounds for all numbers of points? Up to which n?
Best regards,
Sascha
More information about the SeqFan
mailing list