Two Cans & String

{ "thread_id": "11653", "posts": [ { "id": "1186064", "time": "1642604583", "html": "I decided it would be a good use of my time to approximate e in terms of pi and approximate pi in terms of e<br /><br />I challenge you to make a closer approximation than mine:<br /><br /><span style=\"font-weight:bold;\"><span style=\"text-decoration:underline;\">RULES:</span></span><br />Can use any operations<br />Choose 1 irrational number to use<br />Choose 1 irrational number to represent<br />You can only use that one irrational number to approximate the second one you chose<br />Get as close as you can without going over<br />Along with your approximation, provide a percentage in the form of 100(approximation)/(actual number)", "user": "dementedkermit" }, { "id": "1186067", "time": "1642604821", "html": "here are my 2 entries so far<br /><br /><a href=\"https://www.desmos.com/calculator/nhyk1ukxws\">approximation of pi in terms of e</a><br /><a href=\"https://www.desmos.com/calculator/qq1omkbok3\">approximation of e in terms of pi</a><br /><br /><div style=\"font-style:italic;color:#888;\"> also just for fun: <a href=\"https://www.desmos.com/calculator/rg8iw8z1yw\">approximation of 8 in case you need to use it on the fly</a></div>", "user": "dementedkermit" }, { "id": "1188479", "time": "1643124237", "html": "<a href=\"https://www.desmos.com/calculator/qx0jrkwpun\">I improved your approximation of pi in terms of e</a><br /><br />edit: <a href=\"https://www.desmos.com/calculator/4akpwzbgzh\">improved e in terms of pi too</a><br /><br />edit 2: why approximate pi with e when you can find its exact value with e? arccos(e - e - e/e) is exactly equal to pi. Should we limit the set of legal operations?", "user": "aprzn123" }, { "id": "1188533", "time": "1643128163", "html": "<a href=\"https://www.desmos.com/calculator/tr4zdtuzus\">I feel I might have over-approximated</a><br />Accurate to 16 digits", "user": "kylijoy" }, { "id": "1188534", "time": "1643128196", "html": "<a href=\"https://www.desmos.com/calculator/zpsz8zhdn1\">improved on the approximation of e in terms of pi</a><br /><br />working on the pi in terms of e<br /><br />ninjad damn", "user": "dementedkermit" }, { "id": "1188535", "time": "1643128282", "html": "<div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/kylijoy\">KylIjoy</a> said:</div><div style=\"border:1px solid #888; padding:20px;\"><a href=\"https://www.desmos.com/calculator/tr4zdtuzus\">I feel I might have over-approximated</a><br />Accurate to 16 digits</div></div><br />holy shit, mines only accurate to 4", "user": "dementedkermit" }, { "id": "1188536", "time": "1643128332", "html": "(Maclaurin series might be cheating)", "user": "kylijoy" }, { "id": "1188538", "time": "1643128351", "html": "<div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/kylijoy\">KylIjoy</a> said:</div><div style=\"border:1px solid #888; padding:20px;\"><a href=\"https://www.desmos.com/calculator/tr4zdtuzus\">I feel I might have over-approximated</a><br />Accurate to 16 digits</div></div><br />How did you manage this? (ninja'd, nvm)", "user": "aprzn123" }, { "id": "1188539", "time": "1643128378", "html": "Maclaurin series. If you keep adding pi's over the first sigma, it'll keep converging (really fucking slowly, though).", "user": "kylijoy" }, { "id": "1188548", "time": "1643129317", "html": "why did I think <a href=\"https://www.desmos.com/calculator/ynsglyadss\">this</a> was a good idea<br />it didn't even work", "user": "aprzn123" }, { "id": "1188549", "time": "1643129532", "html": "<a href=\"https://www.desmos.com/calculator/nxsgimdzlb\">Pi, accurate to seven digits</a>.<br />There was a trick I'm blanking on that you could use to make the pi series converge faster. Something to do with tangent?<br /><div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/aprzn123\">aprzn123</a> said:</div><div style=\"border:1px solid #888; padding:20px;\">why did I think <a href=\"https://www.desmos.com/calculator/ynsglyadss\">this</a> was a good idea</div></div><br />Me when I look at how long that equation scrolls for:<br /><img src=\"https://i.imgur.com/zdILJXS.png\" alt=\"\" />", "user": "kylijoy" }, { "id": "1188568", "time": "1643131574", "html": "I see your Taylor Series and raise you a <a href=\"https://www.wolframalpha.com/input/?i=%28%28pi%2Bpi%29%2Fpi%29%2Bcontinuedfractionk%28pi%2Fpi%2B%28mod%28k%2C%28pi%2Bpi%2Bpi%29%2Fpi%29*%28pi%2Bpi%29%2Fpi-mod%28k%2C%28pi%2Bpi%2Bpi%29%2Fpi%29%5E%28%28pi%2Bpi%29%2Fpi%29%29*%28%28pi%2Bpi%29k%2Bpi%29%2F%28pi%2Bpi%2Bpi%29%2C+%28k%2C+pi-pi%2C+floor%28pi%5Epi%29%29%29\">continued fraction</a>. It's already accurate enough, but you can make the pi^pi a proper power tower (pi^pi^pi^\u2026) to make it converge real fast (and also <span style=\"text-decoration:line-through;\">very difficult</span> impossible to actually compute, because it's kinda overkill).", "user": "justabitjaded" }, { "id": "1188574", "time": "1643131851", "html": "This has now passed above my understanding", "user": "dementedkermit" }, { "id": "1188577", "time": "1643132361", "html": "I can do my best to explain further it if you'd like, but the gist is that continued fractions are great and e has a decently simple continued fraction.", "user": "justabitjaded" }, { "id": "1188578", "time": "1643132442", "html": "Can you make a continued fraction for pi?", "user": "dementedkermit" }, { "id": "1188579", "time": "1643132641", "html": "I see your continued fraction and raise you by <a href=\"https://imgur.com/a/JIckAek\">basically cheating</a> (screenshot b/c I'm out for lunch and on my phone).<br /><br />That's it, we've done it. It literally cannot be approximated any further.", "user": "kyiijoy" }, { "id": "1188581", "time": "1643132975", "html": "That's not an approximation to pi, that's just pi. Although I suppose you could just make it a Riemann Sum without too much effort.<br /><br />And I did do a <a href=\"https://www.wolframalpha.com/input/?i=ceil%28e%29%2Bcontinuedfractionk%28%28floor%28e%29*k-e%2Fe%29%5Efloor%28e%29%2C+ceil%28e%29%2Bceil%28e%29%2C+%28k%2C+e%2Fe%2C+ceil%28e*e%5Ee%29%29%29\">continued fraction version for pi</a>, but it does not converge quickly at all.", "user": "justabitjaded" }, { "id": "1188583", "time": "1643133085", "html": "Technically, an integral is an already-converged rsum if you thing about it.", "user": "kyiijoy" }, { "id": "1188585", "time": "1643133182", "html": "I just mean that if you view getting it exact as cheating then it would be trivial to get arbitrarily close.", "user": "justabitjaded" }, { "id": "1188589", "time": "1643133489", "html": "<div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/kylijoy\">KylIjoy</a> said:</div><div style=\"border:1px solid #888; padding:20px;\"><div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/aprzn123\">aprzn123</a> said:</div><div style=\"border:1px solid #888; padding:20px;\">why did I think <a href=\"https://www.desmos.com/calculator/ynsglyadss\">this</a> was a good idea</div></div><br />Me when I look at how long that equation scrolls for:<br /><img src=\"https://i.imgur.com/zdILJXS.png\" alt=\"\" /></div></div><br />I was trying to implement Ramanujan's approximation but it had a four digit prime number", "user": "aprzn123" }, { "id": "1188599", "time": "1643135536", "html": "This thread when it started: <span style=\"font-style:italic;\">Fun ways of trying different combinations to come within a few hundredths of irrational numbers</span><br /><br />This thread now: <span style=\"font-weight:bold;\">Nerdy arms race of applying progressively more convoluted methods in order to create impressively accurate approximations</span>", "user": "kylijoy" }, { "id": "1188602", "time": "1643137413", "html": "Maybe we should pick some numbers that are a little bit less well-researched than the two most famous transcendental numbers. Like, maybe we try approximating the <a href=\"https://en.wikipedia.org/wiki/Dottie_number\">cosine fixed point</a> in terms of the <a href=\"https://en.wikipedia.org/wiki/Euler%27s_constant\">Euler-Mascheroni constant</a>.", "user": "justabitjaded" }, { "id": "1188605", "time": "1643138604", "html": "<div style=\"margin:20px; background-image:url(/images/light.png);\"><div style=\"border:1px solid #888; padding:5px;\"><a href=\"/users/justabitjaded\">justabitjaded</a> said:</div><div style=\"border:1px solid #888; padding:20px;\">Maybe we should pick some numbers that are a little bit less well-researched than the two most famous transcendental numbers. Like, maybe we try approximating the <a href=\"https://en.wikipedia.org/wiki/Dottie_number\">cosine fixed point</a> in terms of the <a href=\"https://en.wikipedia.org/wiki/Euler%27s_constant\">Euler-Mascheroni constant</a>.</div></div><br />I like that idea<br /><br />here is my final submissions for approximating e<br /><a href=\"https://www.desmos.com/calculator/pmmgrmukgp\">https://www.desmos.com/calculator/pmmgrmukgp</a>", "user": "dementedkermit" }, { "id": "1188615", "time": "1643141920", "html": "<a href=\"https://www.desmos.com/calculator/x7yaft2bha\">it feels like cheating to use a number less than 1: approximating dottie number using sqrt(i^i)</a>", "user": "dementedkermit" }, { "id": "1209378", "time": "1649551337", "html": "Here is an extremely simple one<br /><br />\u03c0 = ³\u221a31<br /><br />99.993% accuracy<br /><br />I challenge you all to find an approximation as simple with a better accuracy<br /><br />Edit: I have just realized that this forum is like 3 months old and I don't even care. I demand that the abyss answer my call.", "user": "kormagogthedestroyer" } ], "users": { "dementedkermit": { "name": "dementedkermit", "key": "dementedkermit", "url": "/users/dementedkermit", "avatar": "/uploads/drawn/20469.png", "rankClass": "civilian", "rankText": "", "posts": "6194" }, "aprzn123": { "name": "aprzn123", "key": "aprzn123", "url": "/users/aprzn123", "avatar": "/uploads/drawn/31595.png", "rankClass": "civilian", "rankText": "", "posts": "6633" }, "kylijoy": { "name": "KylIjoy", "key": "kylijoy", "url": "/users/kylijoy", "avatar": "/uploads/drawn/30497.png", "rankClass": "civilian", "rankText": "", "posts": "112" }, "justabitjaded": { "name": "justabitjaded", "key": "justabitjaded", "url": "/users/justabitjaded", "avatar": "/uploads/drawn/29113.png", "rankClass": "civilian", "rankText": "", "posts": "4268" }, "kyiijoy": { "name": "KyIIjoy", "key": "kyiijoy", "url": "/users/kyiijoy", "avatar": "/uploads/drawn/30496.png", "rankClass": "civilian", "rankText": "", "posts": "141" }, "kormagogthedestroyer": { "name": "KormagogTheDestroyer", "key": "kormagogthedestroyer", "url": "/users/kormagogthedestroyer", "avatar": "/uploads/drawn/33694.png", "rankClass": "civilian", "rankText": "", "posts": "237" } }, "page_num": 1, "locked": 0, "total_pages": 2 }

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