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 The true value of PI
 Arthur Kalliokoski Second in Command February 2005 I was reading this page, which is full of tongue-in-cheek jokes about the value of PI, and it occurred to me that the well known value would only apply to Euclidian universes. It seems to me that with the positive curvature of space, then the larger the circle is, the lower the value of PI would be for that circle. Comments?[EDIT]As an example, a circle is actually on the surface of a sphere, and the diameter would also follow the sphere (since it'd be the shortest distance, whereas a chord would not be shortest). Suppose the circle was made into a great circle of the sphere, then the diameter would necessarily be 1/2 of the circumference. “Throughout history, poverty is the normal condition of man. Advances which permit this norm to be exceeded — here and there, now and then — are the work of an extremely small minority, frequently despised, often condemned, and almost always opposed by all right-thinking people. Whenever this tiny minority is kept from creating, or (as sometimes happens) is driven out of a society, the people then slip back into abject poverty. This is known as "bad luck.”― Robert A. Heinlein
 Bruce Perry Member #270 April 2000 π(0,R) = 3.14159265358979323846..., R ≠ 0π(r,R) = R/r·π(0,R)·sin r/R[EDIT]I'm going to use this post to test the maths stuff, since the preview won't seem to render any equation that isn't the specific quadratic formula example from the help page.π(r,R) = \frac{R}{r}·π(0,R)·sin \frac{r}{R}[EDIT 2]Didn't work --Bruce "entheh" Perry [ Web site | DUMB | Set Up Us The Bomb !!! | Balls ]Programming should be fun. That's why I hate C and C++.The brxybrytl has you.
 SiegeLord Member #7,827 October 2006 You don't need to go to different universes to get a different value for PI, just choose a different distance metric. Remember that a unit circle is a set of all points that are distance 1 from the center, while PI is circumference over diameter.E.g with the Manhattan distance circles look like diamonds and PI is 4.As for the PI on a sphere... you'd need a bunch of math to compute it. I don't feel like it, so here's a program that does it for you:#SelectExpand 1#include 2#include 3 4double sq(double a) 5{ 6 return a * a; 7} 8 9double great_circle_distance(double phi_s, double lambda_s, double phi_f, double lambda_f) 10{ 11 double delta_phi = phi_f - phi_s; 12 double delta_lambda = lambda_f - lambda_s; 13 14 double y = sqrt(sq(cos(phi_f) * sin(delta_lambda)) + sq(cos(phi_s) * sin(phi_f) - sin(phi_s) * cos(phi_f) * cos(delta_lambda))); 15 double x = sin(phi_s) * sin(phi_f) + cos(phi_s) * cos(phi_f) * cos(delta_lambda); 16 17 return atan2(y, x); 18} 19 20int main() 21{ 22 double radius = 0.4; 23 24 double num_divisions = 10000; 25 double delta_lambda = (2 * 3.14159265358) / num_divisions; 26 double circumference = num_divisions * great_circle_distance(radius, 0, radius, delta_lambda); 27 28 printf("PI is %f\n", circumference / (2 * radius)); 29 30 return 0; 31} Note that I get that PI is radius dependent, in the way OP described... but I may have made a mistake . "For in much wisdom is much grief: and he that increases knowledge increases sorrow."-Ecclesiastes 1:18[SiegeLord's Abode][Codes]:[DAllegro5]:[RustAllegro]
 Karadoc ~~ Member #2,749 September 2002 Arthur Kalliokoski said:it occurred to me that the well known value would only apply to Euclidian universes. It seems to me that with the positive curvature of space, then the larger the circle is, the lower the value of PI would be for that circle. Comments? Back when I was doing 1st year physics at university, my lecturer told story to make this point.The story when roughly like this. A mathematician tells a physicist about a number called 'pi' and says that one can calculate the exact circumference of a circle using the formula C = 2*pi*r; and the exact value of pi is 3.1415[...], defined by such-and-such formula.The physicist likes the idea, and but being a physicist rather than a mathematician, he sets out to test it in the real world.First, he gets a piece of paper and a string. He pins the string in place and uses it to draw a circle. (The radius of the circle is the length of the string.) He then measures the radius, and measures the circumference of the circle (perhaps using a second piece of string, or whatever).. And he uses the formula to compare them and finds that the formula is exactly right.So, the experiment supports the theory, but the physicist wasn't satisfied yet. He wants to try a bigger circle; so he goes to a football field with a really long string / rope and repeats the experiment. Again, the formula is accurate, and the so the C = 2*pi*r theory is looking pretty sound.But the physicist still isn't satisfied. He decides to do one last experiment, on a truly grand scale. He gets an extremely long rope for this one, and makes a really huge circle - spanning the area of several large countries. The physicist is careful to make sure that there are no hills or anything in the way to disrupt the experiment (perhaps he does it in the ocean or something, who knows... the point is that the circle is really really big.) --- And, to the physicist's surprise, he finds that the formula fails! The circumference is noticeably less than that predicted by the formula.Good thing he tested it!(The reason the formula didn't work is because the earth is not a flat plane. It's a sphere...) I thought it was a pretty dumb story at the time, because pi is a mathematical entity. It's value isn't determined experimentally; but rather is defined mathematically. The definition isn't something one tests. It is true by definition (because it's a defined such). The experiments in the story don't show that the maths was wrong, they only show that the formulas were misapplied, because they assume the earth was flat....But that was some years ago, and since then the story has kind of grown on me. I think it highlights a few important points about science and physics and our understanding of the world, and the relationship with all of those things to mathematics. Maths may be absolutely perfect, and never ever wrong, but for it to be of any use in the real world, it has to be used to describe real world things - and it isn't always easy to find the correct mathematical model to describe the things one wants to describe. (That's essentially what physics is all about.) -- As for our understanding of the world; I think the story highlights that although we might be confident that our understanding is sound - it may later turn out that our understanding was actually incomplete - and that we always need to be aware of that possibility.For example, consider Galilean relativity vs Special relativity. Galilean relativity may seem pretty natural, and it seems to be accurate for essentially everything we see and do. But when we start looking at extremely fast things, it turns out that Galilean relativity is completely wrong. It was just an approximation; and the approximation breaks down for really fast stuff.So... despite not liking the story at first, I do think it's helped me understand and become more comfortable with the fact that essentially everything I think I know about the physical world is really 'just an approximation'. -----------
 bamccaig Member #7,536 July 2006 I think the story is dumb for at least two reasons:Firstly, it's completely unreasonable to reliably make a perfect circle with a rope surrounding multiple countries (in the hypothetical situation where the Earth is flat). The infrastructure to actually make this possible would be a huge undertaking and completely impractical for such an experiment. Secondly, it seems pretty biased to believe that a Physicist would not understand that the Earth's surface is not flat (well, sure, maybe many years ago), and that the rope would then not represent a perfect circle. -- I mean the best with what I say. It doesn't always sound that way.
 Karadoc ~~ Member #2,749 September 2002 Sure, I agree with both of those points. The story isn't plausible. But I think the core point of the story is a sound one. (As I said, I thought it was a stupid story when my lecturer told it as well.) -----------
 LennyLen Member #5,313 December 2004 bamccaig said: I think the story is dumb for at least two reasons:Congratulations for taking the ogy out of analogy.
 Johan Halmén Member #1,550 September 2001 bamccaig said: Secondly, it seems pretty biased to believe that a Physicist would not understand that the Earth's surface is not flat (well, sure, maybe many years ago) Eratosthenes measured Earth's circumference in 3rd century B.Chr. Euclid was some 50 years older.I would tell Karadoc's story to my students, if I taught maths or physics. It's dumb, but it still might trigger some thinking. I teach music and music history. To the history part I include some history of ideas, mainly the antique vs. renaissance thing, since it touches art. Karadoc's story can't be true in the same sense as the story about how Eratosthenes measured the circumference of Earth. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.
 Arthur Kalliokoski Second in Command February 2005 I'm pretty sure Karadocs story is more like a parable than actual fact or even apocryphal. A "thought experiment" maybe. “Throughout history, poverty is the normal condition of man. Advances which permit this norm to be exceeded — here and there, now and then — are the work of an extremely small minority, frequently despised, often condemned, and almost always opposed by all right-thinking people. Whenever this tiny minority is kept from creating, or (as sometimes happens) is driven out of a society, the people then slip back into abject poverty. This is known as "bad luck.”― Robert A. Heinlein
 Bob Keane Member #7,342 June 2006 The true value of pi is 3. I will pray for your soul if you don't believe it. By reading this sig, I, the reader, agree to render my soul to Bob Keane. I, the reader, understand this is a legally binding contract and freely render my soul.If we get apple juice from squeezing apples, and we get prunes from drying out plums, where does prune juice come from?Hi Randall Monroe.
 Neil Walker Member #210 April 2000 bamccaig said:The infrastructure to actually make this possible would be a huge undertaking and completely impractical for such an experimentWhat if you they were giants the size of planets from another universe. That string spanning a flat earth would be like us drawing a circle the size of a dustbin lid. wii:0356-1384-6687-2022, kart:3308-4806-6002. XBOX:chucklepie
 van_houtte Member #11,605 January 2010 my apple pie = 22/7 Sometimes you may have to send 3-4 messages
 Johan Halmén Member #1,550 September 2001 Mine is 144029661/45846065. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.
 Arthur Kalliokoski Second in Command February 2005 With all those digits in your fractions to remember, you might as well memorize the decimal value anyway. “Throughout history, poverty is the normal condition of man. Advances which permit this norm to be exceeded — here and there, now and then — are the work of an extremely small minority, frequently despised, often condemned, and almost always opposed by all right-thinking people. Whenever this tiny minority is kept from creating, or (as sometimes happens) is driven out of a society, the people then slip back into abject poverty. This is known as "bad luck.”― Robert A. Heinlein
 Johan Halmén Member #1,550 September 2001 I guess it's more or less the same which ever fraction you use. 22/7 has 3 digits and it gives you only 3.14. 3 / 1 = 3.0000000000000000 13 / 4 = 3.2500000000000000 16 / 5 = 3.2000000000000002 19 / 6 = 3.1666666666666665 22 / 7 = 3.1428571428571428 179 / 57 = 3.1403508771929824 201 / 64 = 3.1406250000000000 223 / 71 = 3.1408450704225350 245 / 78 = 3.1410256410256410 267 / 85 = 3.1411764705882352 289 / 92 = 3.1413043478260869 311 / 99 = 3.1414141414141414 333 / 106 = 3.1415094339622640 355 / 113 = 3.1415929203539825 52163 / 16604 = 3.1415923873765359 52518 / 16717 = 3.1415923909792425 52873 / 16830 = 3.1415923945335709 53228 / 16943 = 3.1415923980404887 53583 / 17056 = 3.1415924015009380 53938 / 17169 = 3.1415924049158366 54293 / 17282 = 3.1415924082860780 54648 / 17395 = 3.1415924116125322 55003 / 17508 = 3.1415924148960475 55358 / 17621 = 3.1415924181374497 55713 / 17734 = 3.1415924213375437 56068 / 17847 = 3.1415924244971145 56423 / 17960 = 3.1415924276169265 56778 / 18073 = 3.1415924306977261 57133 / 18186 = 3.1415924337402399 57488 / 18299 = 3.1415924367451775 57843 / 18412 = 3.1415924397132304 58198 / 18525 = 3.1415924426450741 58553 / 18638 = 3.1415924455413671 58908 / 18751 = 3.1415924484027520 59263 / 18864 = 3.1415924512298559 59618 / 18977 = 3.1415924540232916 59973 / 19090 = 3.1415924567836564 60328 / 19203 = 3.1415924595115348 60683 / 19316 = 3.1415924622074964 61038 / 19429 = 3.1415924648720983 61393 / 19542 = 3.1415924675058848 61748 / 19655 = 3.1415924701093871 62103 / 19768 = 3.1415924726831244 62458 / 19881 = 3.1415924752276041 62813 / 19994 = 3.1415924777433228 63168 / 20107 = 3.1415924802307655 63523 / 20220 = 3.1415924826904056 63878 / 20333 = 3.1415924851227071 64233 / 20446 = 3.1415924875281229 64588 / 20559 = 3.1415924899070968 64943 / 20672 = 3.1415924922600618 65298 / 20785 = 3.1415924945874427 65653 / 20898 = 3.1415924968896545 66008 / 21011 = 3.1415924991671029 66363 / 21124 = 3.1415925014201855 66718 / 21237 = 3.1415925036492913 67073 / 21350 = 3.1415925058548009 67428 / 21463 = 3.1415925080370872 67783 / 21576 = 3.1415925101965145 68138 / 21689 = 3.1415925123334407 68493 / 21802 = 3.1415925144482157 68848 / 21915 = 3.1415925165411820 69203 / 22028 = 3.1415925186126747 69558 / 22141 = 3.1415925206630235 69913 / 22254 = 3.1415925226925498 70268 / 22367 = 3.1415925247015695 70623 / 22480 = 3.1415925266903915 70978 / 22593 = 3.1415925286593192 71333 / 22706 = 3.1415925306086496 71688 / 22819 = 3.1415925325386738 72043 / 22932 = 3.1415925344496771 72398 / 23045 = 3.1415925363419395 72753 / 23158 = 3.1415925382157353 73108 / 23271 = 3.1415925400713336 73463 / 23384 = 3.1415925419089974 73818 / 23497 = 3.1415925437289869 74173 / 23610 = 3.1415925455315543 74528 / 23723 = 3.1415925473169497 74883 / 23836 = 3.1415925490854169 75238 / 23949 = 3.1415925508371956 75593 / 24062 = 3.1415925525725208 75948 / 24175 = 3.1415925542916234 76303 / 24288 = 3.1415925559947300 76658 / 24401 = 3.1415925576820620 77013 / 24514 = 3.1415925593538385 77368 / 24627 = 3.1415925610102731 77723 / 24740 = 3.1415925626515766 78078 / 24853 = 3.1415925642779543 78433 / 24966 = 3.1415925658896100 78788 / 25079 = 3.1415925674867418 79143 / 25192 = 3.1415925690695459 79498 / 25305 = 3.1415925706382137 79853 / 25418 = 3.1415925721929341 80208 / 25531 = 3.1415925737338921 80563 / 25644 = 3.1415925752612699 80918 / 25757 = 3.1415925767752455 81273 / 25870 = 3.1415925782759953 81628 / 25983 = 3.1415925797636914 81983 / 26096 = 3.1415925812385042 82338 / 26209 = 3.1415925827005990 82693 / 26322 = 3.1415925841501404 83048 / 26435 = 3.1415925855872895 83403 / 26548 = 3.1415925870122043 83758 / 26661 = 3.1415925884250404 84113 / 26774 = 3.1415925898259505 84468 / 26887 = 3.1415925912150855 84823 / 27000 = 3.1415925925925925 85178 / 27113 = 3.1415925939586176 85533 / 27226 = 3.1415925953133033 85888 / 27339 = 3.1415925966567908 86243 / 27452 = 3.1415925979892174 86598 / 27565 = 3.1415925993107203 86953 / 27678 = 3.1415926006214323 87308 / 27791 = 3.1415926019214853 87663 / 27904 = 3.1415926032110093 88018 / 28017 = 3.1415926044901310 88373 / 28130 = 3.1415926057589760 88728 / 28243 = 3.1415926070176683 89083 / 28356 = 3.1415926082663281 89438 / 28469 = 3.1415926095050759 89793 / 28582 = 3.1415926107340284 90148 / 28695 = 3.1415926119533020 90503 / 28808 = 3.1415926131630103 90858 / 28921 = 3.1415926143632653 91213 / 29034 = 3.1415926155541780 91568 / 29147 = 3.1415926167358563 91923 / 29260 = 3.1415926179084073 92278 / 29373 = 3.1415926190719370 92633 / 29486 = 3.1415926202265481 92988 / 29599 = 3.1415926213723435 93343 / 29712 = 3.1415926225094237 93698 / 29825 = 3.1415926236378877 94053 / 29938 = 3.1415926247578327 94408 / 30051 = 3.1415926258693556 94763 / 30164 = 3.1415926269725500 95118 / 30277 = 3.1415926280675102 95473 / 30390 = 3.1415926291543270 95828 / 30503 = 3.1415926302330917 96183 / 30616 = 3.1415926313038933 96538 / 30729 = 3.1415926323668195 96893 / 30842 = 3.1415926334219573 97248 / 30955 = 3.1415926344693910 97603 / 31068 = 3.1415926355092054 97958 / 31181 = 3.1415926365414837 98313 / 31294 = 3.1415926375663066 98668 / 31407 = 3.1415926385837554 99023 / 31520 = 3.1415926395939087 99378 / 31633 = 3.1415926405968451 99733 / 31746 = 3.1415926415926414 100088 / 31859 = 3.1415926425813741 100443 / 31972 = 3.1415926435631176 100798 / 32085 = 3.1415926445379463 101153 / 32198 = 3.1415926455059320 101508 / 32311 = 3.1415926464671475 101863 / 32424 = 3.1415926474216631 102218 / 32537 = 3.1415926483695484 102573 / 32650 = 3.1415926493108728 102928 / 32763 = 3.1415926502457041 103283 / 32876 = 3.1415926511741086 103638 / 32989 = 3.1415926520961532 103993 / 33102 = 3.1415926530119025 104348 / 33215 = 3.1415926539214212 208341 / 66317 = 3.1415926534674368 312689 / 99532 = 3.1415926536189365 833719 / 265381 = 3.1415926535810779 1146408 / 364913 = 3.1415926535914038 3126535 / 995207 = 3.1415926535886505 4272943 / 1360120 = 3.1415926535893890 5419351 / 1725033 = 3.1415926535898153 42208400 / 13435351 = 3.1415926535897722 47627751 / 15160384 = 3.1415926535897771 53047102 / 16885417 = 3.1415926535897811 58466453 / 18610450 = 3.1415926535897842 63885804 / 20335483 = 3.1415926535897869 69305155 / 22060516 = 3.1415926535897891 74724506 / 23785549 = 3.1415926535897909 80143857 / 25510582 = 3.1415926535897927 245850922 / 78256779 = 3.1415926535897931  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.
 Tobias Dammers Member #2,604 August 2002 ---Me make music: Triofobie---"We need Tobias and his awesome trombone, too." - Johan Halmén
 Arthur Kalliokoski Second in Command February 2005 Johan Halmén said: I guess it's more or less the same which ever fraction you use. 22/7 has 3 digits and it gives you only 3.14.Pi is approximately equal to 314159/100000ths so there! It's about 0.999999155336 of the true value. “Throughout history, poverty is the normal condition of man. Advances which permit this norm to be exceeded — here and there, now and then — are the work of an extremely small minority, frequently despised, often condemned, and almost always opposed by all right-thinking people. Whenever this tiny minority is kept from creating, or (as sometimes happens) is driven out of a society, the people then slip back into abject poverty. This is known as "bad luck.”― Robert A. Heinlein
 bamccaig Member #7,536 July 2006 Arthur wins the thread, IMO. -- I mean the best with what I say. It doesn't always sound that way.
 Arthur Kalliokoski Second in Command February 2005 Quote: Arthur wins the thread, IMO. Only for circles that are relatively small compared to the universe. “Throughout history, poverty is the normal condition of man. Advances which permit this norm to be exceeded — here and there, now and then — are the work of an extremely small minority, frequently despised, often condemned, and almost always opposed by all right-thinking people. Whenever this tiny minority is kept from creating, or (as sometimes happens) is driven out of a society, the people then slip back into abject poverty. This is known as "bad luck.”― Robert A. Heinlein
 Tobias Dammers Member #2,604 August 2002 van_houtte said: my apple pie = 22/7 Mine is 3.1415. If I need more digits, I look it up, derive it, or use the constant that is built into every sane programming language (and also a few insane / mentally unstable ones). acos(-1) works well, too. ---Me make music: Triofobie---"We need Tobias and his awesome trombone, too." - Johan Halmén
 Johan Halmén Member #1,550 September 2001 3.1416 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Years of thorough research have revealed that the red "x" that closes a window, really isn't red, but white on red background.Years of thorough research have revealed that what people find beautiful about the Mandelbrot set is not the set itself, but all the rest.
 Bruce Perry Member #270 April 2000 3.142536475869708...Wait, maybe I'm thinking of this --Bruce "entheh" Perry [ Web site | DUMB | Set Up Us The Bomb !!! | Balls ]Programming should be fun. That's why I hate C and C++.The brxybrytl has you.
 NiteHackr Member #2,229 April 2002 {"name":"606279","src":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/a\/2\/a2627110353a0597b6c9f641f342400d.jpg","w":259,"h":194,"tn":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/a\/2\/a2627110353a0597b6c9f641f342400d"}MMmmmmm.... PI....... --Deluxe Pacman 1 & 2 (free) with source code availablehttps://nitehackr.github.io/games_index.html
 Thomas Fjellstrom Member #476 June 2000 {"name":"raspberrypi.jpg","src":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/7\/8\/7873cb7954f4a66d40631d5fa84cd745.jpg","w":652,"h":456,"tn":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/7\/8\/7873cb7954f4a66d40631d5fa84cd745"}MMmmmmm.... PI....... -- Thomas Fjellstrom - [website] - [email] - [Allegro Wiki] - [Allegro TODO]"If you can't think of a better solution, don't try to make a better solution." -- weapon_S"The less evidence we have for what we believe is certain, the more violently we defend beliefs against those who don't agree" -- https://twitter.com/neiltyson/status/592870205409353730
 Bruce Perry Member #270 April 2000 {"name":"raspberry-pie-slice.jpg","src":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/e\/b\/ebdc9bc0ce799a6ee3a2cc2a531cfde4.jpg","w":511,"h":342,"tn":"\/\/djungxnpq2nug.cloudfront.net\/image\/cache\/e\/b\/ebdc9bc0ce799a6ee3a2cc2a531cfde4"}<3 --Bruce "entheh" Perry [ Web site | DUMB | Set Up Us The Bomb !!! | Balls ]Programming should be fun. That's why I hate C and C++.The brxybrytl has you.
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