Here are the angle ratios, single spaced. Note 105 | 81 | 31 -> {2.747569809613456, 0.09098949883831027, 4} Will we see a 5?? --rwg On Sat, Jul 14, 2018 at 2:27 PM Bill Gosper <billgosper@gmail.com> wrote:
Whoa, I missed a bunch--It's not always the shortest!
10 | 8 | 3 -> {2.883374909936108, 0.1156052694308558, 3}, 12 | 9 | 7 -> {1.360101615395144, 0.3676195913161512, 2}, 16 | 15 | 9 -> {2, 0.4230193607355411, 1.181978997676621}, 20 | 16 | 9 -> {1.970233608424166, 0.2537770129705129, 2}, 30 | 25 | 11 -> {2.577918191949451, 0.1939549523182869, 2}, 33 | 28 | 16 -> {2, 0.3108932957857233, 1.608268839430409}, 35 | 25 | 24 -> {1.052955878126168, 0.4748537050667273, 2}, 42 | 36 | 13 -> {3.184489452982539, 0.1570110397230892, 2}, 56 | 45 | 25 -> {2, 0.2521798885164634, 1.982711638669623}, 56 | 49 | 15 -> {3.790459140112012, 0.1319101410984276, 2}, 63 | 49 | 32 -> {1.665627030397474, 0.3001872513324205, 2}, 72 | 64 | 17 -> {4.396066569535012, 0.1137380410626691, 2}, 85 | 66 | 36 -> {2, 0.2154608388247404, 2.320607321160153}, 88 | 64 | 57 -> {1.155588726469411, 0.43267988735717, 2}, 90 | 81 | 19 -> {5.001438829466385, 0.09997123168921095, 2}, 95 | 84 | 49 -> {2, 0.3563865094455654, 1.402971175249748}, 99 | 81 | 40 -> {2.274267025448232, 0.2198510528469961, 2}, 105 | 81 | 31 -> {2.747569809613456, 0.09098949883831027, 4}, 105 | 104 | 64 -> {2, 0.488328556341499, 1.023900801022046}, 110 | 100 | 21 -> {5.606649726655717, 0.08917981760530679, 2}, 120 | 91 | 49 -> {2, 0.1900458759172064, 2.630943700234912}, 130 | 100 | 69 -> {1.563911529448709, 0.3197111796830694, 2}, 132 | 121 | 23 -> {6.21174508248556, 0.08049267852439472, 2}, 143 | 121 | 48 -> {2.881301422635132, 0.1735326946608448, 2}, 154 | 121 | 75 -> {1.767243027729367, 0.2829265653645964, 2}, 156 | 144 | 25 -> {6.816754857592237, 0.07334868429999643, 2}, 161 | 120 | 64 -> {2, 0.1712601680894115, 2.919534679768387}, 165 | 121 | 104 -> {1.206800306873896, 0.4143187544385066, 2}, 175 | 144 | 81 -> {2, 0.2776530159912519, 1.800808819651913}, 182 | 169 | 27 -> {7.421699471930491, 0.06737001436005906, 2}, 189 | 170 | 100 -> {2, 0.3755931654327534, 1.331227631429094}, 195 | 125 | 112 -> {1.133746416703989, 0.294010484551206, 3}, 195 | 169 | 56 -> {3.487530812195983, 0.1433679089662771, 2}, 203 | 198 | 121 -> {2, 0.4697763467834818, 1.064336260059615}, 208 | 153 | 81 -> {2, 0.1567202506526263, 3.190398164358864}, 208 | 169 | 87 -> {2.172972392505991, 0.2300995639541341, 2}, 210 | 196 | 29 -> {8.026593324341455, 0.06229292799520558, 2}, 221 | 169 | 120 -> {1.513010766797108, 0.3304669146925171, 2}, 234 | 169 | 155 -> {1.114571612093318, 0.4486028484620488, 2}, 238 | 196 | 93 -> {2.375519777211096, 0.2104802514365969, 2}, 240 | 209 | 121 -> {2, 0.3394475959969932, 1.472981414204592}, 240 | 225 | 31 -> {8.63144686044193, 0.05792771572185756, 2}, 255 | 225 | 64 -> {4.093298500957322, 0.1221508765810904, 2}, 261 | 190 | 100 -> {2, 0.1450768789752975, 3.446448555631915}, 266 | 196 | 165 -> {1.237498204650487, 0.4040409902180163, 2}, 272 | 256 | 33 -> {9.236267842722823, 0.05413441971520409, 2}, 279 | 220 | 121 -> {2, 0.2319574293602586, 2.155567947873048}
On 2018-07-14 12:49, Tomas Rokicki wrote:
There's some information on Wikipedia:
https://en.wikipedia.org/wiki/Integer_triangle#Integer_triangles_with_one_an...
It's not as general as what Bill observes.
I think this is fascinating!
On Sat, Jul 14, 2018 at 12:40 PM Dan Asimov <dasimov@earthlink.net>
wrote:
This is a fascinating pattern! It may have very deep roots. Maybe ask John Conway if he's heard of this.
—Dan
rwg wrote: ----- This is probably old news, but triangles with two angles in ratio n seem always to have their shortest side a perfect nth power:
Sides -> Angle ratios 4 | 5 | 6 -> {0.673407904146284, 2, 0.7424920273751127}, 40 | 39 | 25 -> {1.882031453653618, 1/2, 1.062681495634607}, 48 | 35 | 27 -> {1.363957995353242, 1/3, 2.199481223190493}, 70 | 51 | 49 -> {1.052656745727889, 1/2, 1.899954575047201}, 77 | 72 | 49 -> {1.710412883254859, 1/2, 1.169308311215516}, 117 | 88 | 81 -> {1.113181718197647, 1/2, 1.796651855941546}, 126 | 115 | 81 -> {1.62220636751334, 1/2, 1.232888761906277}, 132 | 119 | 64 -> {2.216537678860817, 1/3, 1.353462216596219}, 176 | 135 | 121 -> {1.153038717449732, 1/2, 1.734547131620664}, 187 | 168 | 121 -> {1.56846197132432, 1/2, 1.275134518123709}, 204 | 145 | 144 -> {1.008861547409698, 1/2, 1.982432579708385}, 228 | 217 | 144 -> {1.779802212058679, 1/2, 1.123720369853131}, 247 | 192 | 169 -> {1.181274044306326, 1/2, 1.693087230384759}, 260 | 231 | 169 -> {1.532276923765853, 1/2, 1.305247092728272}, 273 | 272 | 169 -> {1.982157307044563, 1/2, 1.009001653346091}, 280 | 279 | 125 -> {2.965423277339246, 1/3, 1.011659961977428}, 330 | 259 | 225 -> {1.202325336238882, 1/2, 1.663443279217926}, 345 | 304 | 225 -> {1.506251839762349, 1/2, 1.327799208076356}, 368 | 273 | 256 -> {1.086393099567274, 1/2, 1.840954255689428}, 400 | 369 | 256 -> {1.660214339091134, 1/2, 1.204663731006491}, 425 | 336 | 289 -> {1.21862553465025, 1/2, 1.641193248567542}, 442 | 387 | 289 -> {1.486633544904333, 1/2, 1.345321452522924}, 459 | 440 | 289 -> {1.809233016517494, 1/2, 1.105440803777561}, 476 | 305 | 256 -> {1.216537678860817, 1/4, 3.288019820105906}
In[234]:= Times @@ {0.673407904146284`, 2, 0.7424920273751127`} Out[234]= 1. . . . In[233]:= Times @@ {1.2165376788608173`, 1/4, 3.288019820105906`} Out[233]= 1.
These 2, 1/2, 1/3, etc, were Rationalize'd after FullSimplify failed utterly on all of them. -----