[math-fun] minimum area convex lattice polygons - results
file systematic-results.txt results from systematic searches (with various programs, as, aq, ai, ao, af, and ae, co and ce) and from the dynamic programming approach (with various programs phalf, pmap, pndx, pvrt) best systematic for even n min min min longest max edge running n area perim edge bound used now later 04 1 4 1 50 06 3 8 2 50 08 7 12 2 50 10 14 18 3 50 12 24 24 3 50 14 40 30 3 50 16 59 36 3 50 18 87 44 4 50 20 121 52 4 50 22 164 62 5 50 24 210 72 5 50 26 274 80 5 50 28 345 90 5 50 30 430 98 5 50 32 523 108 5 40 34 632 118 5 30 36 749 128 5 20 38 890 138 5 18 40 1039 148 5 18 42 1222 162 7 16 44 1412 176 7 16 46 1620 192 7 14 48 1838 208 8 14 50 2088 218 8 14 52 2357 234 8 14 54 2651 244 8 14 56 2953 260 8 14 58 3278 278 9 12 14 60 3612 296 9 12 14 62 4020 308 9 12 14 64 4439 322 9 12 14 66 4902 334 9 12 14 68 5387 348 9 12 14 70 5898 362 9 12 14 72 6418 376 9 12 14 74 6974 390 9 12 14 76 7557 404 9 12 14 78 8182 420 9 12 14 80 8835 434 9 12 14 82 9512 450 9 12 14 84 10218 464 9 13 14 86 10984 482 9 13 14 88 11759 500 9 13 14 90 12635 516 9 13 14 92 13525 532 9 13 14 94 14448 550 9 13 14 96 15399 568 9 13 14 98 16415 588 10 13 14 100 17473 608 10 13 14 best systematic for odd n min min min longest edge bound running n area perim edge max used now later 03 0.5 4 2 50 05 2.5 8 2 50 07 6.5 12 3 50 09 10.5 16 3 50 11 21.5 22 3 50 13 32.5 28 4 50 15 51.5 34 4 40 17 75.5 42 5 30 19 106.5 50 5 20 21 144.5 60 5 18 20 23 193.5 66 5 16 20 25 248.5 76 5 14 20 27 312.5 88 7 14 20 29 391.5 96 7 14 20 31 483.5 106 7 14 20 33 588.5 120 7 14 18 35 704.5 124 6 14 18 37 835.5 136 7 14 18 39 977.5 146 7 14 16 41 1139.5 156 7 14 16 43 1325.5 168 7 14 15 45 1525.5 182 7 14 16 47 1746.5 196 7 14 16 49 1984.5 212 8 14 51 2244.5 228 9 14 53 2522.5 246 9 14 55 2811.5 258 9 14 57 3121.5 276 9 12 14 59 3495.5 282 9 12 14 61 3865.5 302 9 12 14 63 4273.5 314 9 12 14 65 4713.5 326 9 12 14 67 5178.5 340 9 12 14 69 5669.5 350 9 12 14 71 6179.5 368 9 12 14 73 6729.5 382 9 12 14 75 7307.5 394 9 12 14 77 7905.5 408 9 12 14 79 8536.5 426 9 12 14 81 9215.5 442 9 12 14 83 9916.5 458 9 12 14 85 10660.5 474 9 13 14 87 11450.5 492 10 13 14 89 12263.5 510 10 13 14 91 13141.5 528 11 13 14 93 14052.5 546 11 13 14 95 14996.5 562 10 13 14 97 15979.5 578 11 13 14 99 17009.5 602 11 13 14 end of file systematic-results.txt
The last minute of 2005 will contain 61 seconds. http://en.wikipedia.org/wiki/Leap_second Apparently, the Times Square "ball drop" may last a bit longer: http://ny1.com/ny1/NY1ToGo/Story/index.jsp?stid=101&aid=55925 http://www.abs-cbnnews.com/storypage.aspx?StoryId=25934 It probably wouldn't be a good time to be flying, due to inteference with GPS navigation: http://gpsinformation.net/main/gpstime.htm Some radio wags have contended that it would be cheaper to fiddle with the Earth's actual rotation, than to keep screwing around with leap seconds. How much effort would it really require to speed up or slow down the Earth's rotation by one second in one year?
I seem to remember when I was teaching physics that the big winds of el Nino several years ago managed to extend the day by a few milliseconds (big lump of angular momentum in one direction reduced the angular momentum of the solid earth to keep the sum a constant). So, I suspect that it's a non-trivial task. :-) Or, we can just wait a zillion years and let the moon do it naturally. Kerry On 12/30/05, Henry Baker <hbaker1@pipeline.com> wrote:
Some radio wags have contended that it would be cheaper to fiddle with the Earth's actual rotation, than to keep screwing around with leap seconds. How much effort would it really require to speed up or slow down the Earth's rotation by one second in one year?
I suggest that we eliminate bad ideas to narrow the search. Here's one: If everyone on earth faces east and belches at the same time, it will have absolutely no effect.
Some radio wags have contended that it would be cheaper to fiddle with the Earth's actual rotation, than to keep screwing around with leap seconds. How much effort would it really require to speed up or slow down the Earth's rotation by one second in one year?
Some radio wags have contended that it would be cheaper to fiddle with the Earth's actual rotation, than to keep screwing around with leap seconds. How much effort would it really require to speed up or slow down the Earth's rotation by one second in one year?
So let's do the calculation. The angular velocity is to have a relative increase of 1 s/1 yr = 3e-8. Assume the Earth has uniform density, radius R=6.4e6 m, mass M=6.0e24 kg. We will transfer a shell of mass m and thickness d uniform over the Earth's surface to or from the poles. The moment of inertia must have a relative decrease of 3e-8. For the Earth, I=(2/5)MRR, for the shell i=(2/3)mRR. Then i/I=(5/3)(m/M)=3e-8, or m/M=2e-8. This is the same as the volume ratio, v/V=[(4pi)RRd]/[(4pi/3)RRR]=3(d/R)=2e-8, and the depth is d=(2/3)e-8 R = 0.042 m. If we use the ocean for this purpose, then about 70% of the surface is available and the average density of the Earth is 5.5 times that of water. To speed up the Earth and eliminate the leap second, we need to reduce sea level by 0.33 m, about 1 foot, and transfer that water to the South polar ice cap (so it is supported by land). This is easily in the range of an ice age. On the other hand, it's not clear that humans have any influence over global temperatures, for, Al Gore fanboys and Kyoto treaty signers notwithstanding, Earth's climate is controlled by the Sun. Gene __________________________________________ Yahoo! DSL Â Something to write home about. Just $16.99/mo. or less. dsl.yahoo.com
participants (5)
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Chris Landauer -
Eugene Salamin -
Henry Baker -
Kerry Mitchell -
Steve Gray