July 2018 - mathfun@mailman.xmission.com

[math-fun] Number of sums of squares ...
by Colin Wright 19 Jul '18

19 Jul '18

I recently saw the following claim: "On the average, the number of ways of expressing a positive integer n as a sum of two integral squares, x^2 + y^2 = n, is pi" Can anyone confirm or deny this? Thanks. Colin -- Some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way, ... -- Tom Lehrer

4 3

Re: [math-fun] What is the simplex best at being best at?
by Keith F. Lynch 19 Jul '18

19 Jul '18

Dan Asimov <dasimov(a)earthlink.net> wrote: > At this point, I'm pretty sure *what* I think is the number one > distinction between the extremes of the regular simplex and the > sphere, though I haven't yet convinced myself I know exactly *why* > I believe this. > Meanwhile, if anyone else would like to express their opinion, I'd > be curious to learn what others think about this. The simplex is sharper, hence more likely to hurt your hand when you pick it up. The sphere is rounder, hence more likely to roll away and fall off the table when you set it down. I'm not sure I understand your question. :-) The simplex is best at being best at being self-dual. (Except in 2 dimensions where every regular polygon is self-dual.) I think it's the most rigid solid even if the edges are linked by hinges.

1 0

[math-fun] Optimal embeddings of tangles
by James Propp 19 Jul '18

19 Jul '18

Consider the points A=(1,0,0), B=(-1,0,0), C=(0,1,0), and D=(0,-1,0). There are lots of ways to draw two disjoint 3D arcs joining A with B and C with D. What if we want them to be short without getting too close to each other? We might set up some “short-but-not-too-close” objective functional and optimize it using calculus of variations. There are probably many ways to do this, some solvable and some not. Does anyone know of work along these lines? In the spirit of calculus-of-variations-of-variations, we might mandate the optima in advance (each arc is a half-circle on the sphere x^2+y^2+z^2=1, one in the upper hemisphere and one in the lower hemisphere) and then ask, For what objective functional is this pair of arcs locally optimal? What got me wondering along these lines was an object I saw in the halls of the Dartmouth math department, apparently implementing Conway’s rational tangles game. It made me wonder whether there’s a “best” way to embed any given tangle. Jim Propp

1 0

[math-fun] Amazing hit by ries
by Bill Gosper 18 Jul '18

18 Jul '18

Integrate[((2 + 1/Sqrt[3 + t])*ArcSec[t])/(Sqrt[1 + t]*(2 + t)), {t, 1, 2}] == Pi^2/30 + (1/6)*Pi*ArcSin[(1/16)*(7 - 3*Sqrt[5])] The integral {t,1,6} = 2 π^2/15 = Coxeter's still-mysterious orthoscheme integral. --rwg

4 7

[math-fun] On a clear day at Battersea power station
by Fred Lunnon 18 Jul '18

18 Jul '18

On a clear day at Battersea power station ----------------------------------------- Dick Hess enquired << Imagine a power station with towers of negligible width (= 0) built at the four corners of a rectangle on a flat plane. At a certain viewing point V on the plane, the bases of the four towers are equally spaced in viewing by an angle v . V is at a different distance from each corner and the distance from V to the closest tower equals the length t of the long side of the rectangle. For this case v = 90/7 degrees to 15 places of accuracy, but I’m unable to prove equality. >> I wasted an enormous amount of time in a doomed attempt at a relatively simple-minded program generating all possible solutions; but was obliged finally to grasp the nettle, sit down and implement a proper general-purpose multivariant polynomial equation solver, a project which I have been avoiding for several decades. As often happens in such instances, the task proved much simpler and far more satisfactory and effective than my earlier half-cocked efforts. The equations have a number of solutions which are degenerate, but are conveniently filtered by rejecting those including any zero entry. Three-quarters of the remainder are redundant, since congruent to (visually indistinguishable from) some previous one; those are excluded by requiring rectangle side s and angle cosine u to be positive. The remaining 13 solutions are all real and reasonably non-trivial: though distances o,p,q,r are equipped with a sign, its significance remains mysterious. Just 4 solutions satisfy the extra constraints that absolute distances be unequal and that side s < t = o = 1, as follows: x y p q r s u -0.623490 -0.781831 1.801938 2.076521 1.436997 0.512858 0.974928 0.900969 0.433884 0.445042 -0.228243 0.924139 0.639524 0.781831 -0.309017 0.951057 1.618034 1.376382 0.525731 0.525731 0.951057 0.809017 0.587785 -0.618034 0.324920, 0.850651, 0.850651 0.587785 Second & fourth viewpoints are located inside the rectangle! Various sectors in which viewpoint V = (x,y) may lie are distinguished by 12 orders in which towers might appear: for example <r, o, q, p> denotes the SSW sector where first & third viewpoints lie, with angles v = pi/14, pi/7 resp. Here further redundancy occurs: a corner sector and its opposite or reverse have identical equations, reducing the number of distinct cases to 8. Case <r, o, q, p> diagram: (0,s) (t,s) @--------------@ /| t /| / | / | / | s / | s / | / t | / @---/----------@ r / / / / (1,0) / o/ /q / / / / / p /v/ / / //v/v/ /// @ V = (x,y) , u = cos(v) , o = t = 1 ; View source without proportional spacing! A scale diagram these solutions might be appreciated here ... ? Magma program and output, including integer polynomials with individual solution entries as roots, is available online at https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt Fred Lunnon [14/07/18]

1 1

Re: [math-fun] 4-5-6
by Tomas Rokicki 16 Jul '18

16 Jul '18

Here's a python script that generates integer triangles with angles in any rational ratio (pass the numerator and denominator on the command line). I'm sure someone skilled in Python can write this much more simply. Given this, the only thing we need to show is that these are the only such integer triangles. Sample usage: abacus:inttri rokicki$ python rattri.py 1 6 | head 1771561 2271060 566839 46656 72930 30421 4826809 8761896 4632263 117649 240240 146329 11390625 25591020 17151679 262144 637560 458081 # # Given two integers h and k, find integer triangles with angles in the # ratio h:k. Based on the generic parameterization given here: # # https://en.wikipedia.org/wiki/Integer_triangle#Integer_triangles_with_one_a… # import sys from fractions import gcd import math # h = int(sys.argv[1]) ; k = int(sys.argv[2]) ; q = 1 coscrit = math.cos(math.pi/(h+k)) def choose(a, b): if b < 0 or b > a: return 0 if b + b > a: b = a - b r = 1 for i in range(1, b+1): r = r * (a - i + 1) / i return r ; while True: for p in range(1, q): if q*coscrit >= p: continue if gcd(p, q) > 1: continue a = 0 b = 0 c = 0 for i in range(0, (h+1)/2): a += ((-1)**i)*choose(h, 2*i+1)*(p**(h-2*i-1))*((q*q-p*p)**i) for i in range(0, (k+1)/2): b += ((-1)**i)*choose(k, 2*i+1)*(p**(k-2*i-1))*((q*q-p*p)**i) for i in range(0, (h+k+1)/2): c += ((-1)**i)*choose(h+k, 2*i+1)*(p**(h+k-2*i-1))*((q*q-p*p)**i) a *= q**k b *= q**h g = gcd(gcd(a, b), c) a /= g b /= g c /= g print a, b, c q = q + 1 On Sun, Jul 15, 2018 at 2:55 PM Bill Gosper <billgosper(a)gmail.com> wrote: > Wow. I haven't even digested WDS's thing. --Bill > Maybe it's time for MathOverflow, e.g.? > > On Sun, Jul 15, 2018 at 1:36 PM Tomas Rokicki <rokicki(a)gmail.com> wrote: > >> Any more progress on this? >> >> I have a large number of additional examples if you are interested, >> such as this triple for 6: >> >> 117649 146329 240240 >> >> and these for 5: >> >> 231 1024 1220 >> >> 3024 3125 5555 >> >> 7776 12155 17214 >> >> 16807 34320 42833 >> >> 36400 59049 89001 >> >> 32768 79695 92168 >> >> 59049 162656 178695 >> >> 100000 302499 320210 >> >> 161051 206460 323851 >> >> I have not yet tried to find recurrences for these . . . >> >> It's abundantly clear empirically that your hypothesis works both >> ways (with suitable caveats). There should be a way to prove >> this. >> >> -tom >> >> On Sat, Jul 14, 2018 at 4:14 PM Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> OMG! Nick Baxter somehow found >>> 1024 | 1220 | 231 -> {23.09066960680706, 0.2165376788608164, 1/5}, >>> 46656 | 72930 | 30421 -> {9.357241092137794, 0.6412146423203051, 1/6} >>> >>> 23.1--a toothpick! Even more toothpicular is >>> 357 | 343 | 20 -> {18.99091913441869, 0.01755224857596323, 3}, >>> because it's 1/19 of a smaller angle. >>> >>> The parametric solutions in >>> >>> >>> https://en.wikipedia.org/wiki/Integer_triangle#Integer_triangles_with_one_a… >>> et seq. are surprisingly absent from OEIS. --rwg >>> >>> On Sat, Jul 14, 2018 at 3:21 PM Bill Gosper <billgosper(a)gmail.com> >>> wrote: >>> >>> > 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(a)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_a… >>> >> > >>> >> > 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(a)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. >>> >> >> ----- >>> >> >>> >> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> >> >> -- >> -- http://cube20.org/ -- http://golly.sf.net/ -- >> > -- -- http://cube20.org/ -- http://golly.sf.net/ --

1 0

Re: [math-fun] What is the simplex best at being best at?
by Dan Asimov 16 Jul '18

16 Jul '18

Fair enough. Each function F : C_n —> R depending only on shape* that takes its global maxima (or minima) exactly on all spheres, and the other extrema exactly on all regular simplices . . . is a *candidate* for characterizing the distinction between spheres and simplices as opposite types of convex bodies. I am guessing that maybe there is just one geometric objective function like F that is somehow *at the root* of the gradient between spheres and simplices in C_n. This may be wrong, but it feels right to me. —Dan ————— * something I omitted mention of below Fred Lunnon wrote: ----- << And why is one better than another? >> The question seems ill-posed without some information about the purposes for which such a flow might be "better" ?! WFL Dan Asimov wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.) (Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function F : C_n —> R there exists a global maximum and a global minimum on C_n.) For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.) But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? Say n = 3. We could use the isoperimetric inequality, and look at the boundary-area to volume ratio, or better yet the dimensionless ratio F(c) = A(c)^3 / V(c)^2 for c in C_n. (By a theorem in analysis, the boundary of a convex body is rectifiable, so A makes sense.) Or we could take the ratio of the radii of the inscribed and circumscribed spheres: F(c) = inradius(c) / circumradius(c), c in C_n. Question: Is one of these better than the other, or is something else even better for characterizing the gradient between the sphere and the simplex? And why is one better than another? -----

3 2

[math-fun] What is the simplex best at being best at?
by Dan Asimov 16 Jul '18

16 Jul '18

Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.) (Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function F : C_n —> R there exists a global maximum and a global minimum on C_n. For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.) But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? Say n = 3. We could use the isoperimetric inequality, and look at the boundary-area to volume ratio, or better yet the dimensionless ratio F(c) = A(c)^3 / V(c)^2 for c in C_n. (By a theorem in analysis, the boundary of a convex body is rectifiable, so A makes sense.) Or we could take the ratio of the radii of the inscribed and circumscribed spheres: F(c) = inradius(c) / circumradius(c), c in C_n. Question: Is one of these better than the other, or is something else even better for characterizing the gradient between the sphere and the simplex? And why is one better than another? —Dan

3 2

Re: [math-fun] 4-5-6
by Bill Gosper 14 Jul '18

14 Jul '18

Whoa, I missed a bunch--It's not always the shortest! 10|8|3 12|9|7 16|15|9 20|16|9 30|25|11 33|28|16 35|25|24 42|36|13 56|45|25 56|49|15 63|49|32 72|64|17 85|66|36 88|64|57 90|81|19 95|84|49 99|81|40 105|81|31 105|104|64 110|100|21 120|91|49 130|100|69 132|121|23 143|121|48 154|121|75 156|144|25 161|120|64 165|121|104 (Rushed. Sorry for the double spacing.) 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_a… > > 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(a)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. >> -----

1 2

Re: [math-fun] 4-5-6
by Dan Asimov 14 Jul '18

14 Jul '18

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. -----

2 1