<\/a>. Or it’s not a number, it’s an acceleration<\/em><\/a>.<\/p>\n\n\n\nIt’s amazing. I constantly get mail – mostly from fans of either Jain (the author of the φ-based π mentioned above), or from followers of Miles Mathis (he of “π isn’t a ratio, it’s an acceleration” fame), insisting that I’m part of the great mathematical conspiracy to deny the true factual value of π.<\/p>\n\n\n\n
And yet… It’s so simple to demonstrate how wrong that is.<\/p>\n\n\n\n
My favorite version is a simple program.<\/p>\n\n\n\n
Here’s the idea, followed by the code.<\/p>\n\n\n\n
- Take the unit square – the region of the graph from (0, 0) to (1, 1), and inside of it, an arc of the circle of radius 1 around (0,0).
![\"\"](\"https:\/\/i0.wp.com\/www.goodmath.org\/blog\/wp-content\/uploads\/2020\/08\/arc.jpg?resize=150%2C165\")
<\/li> - Pick a random point, (x, y), anywhere inside of that square.<\/li>
- If the distance from the origin (
) is less than one, then the point is inside the circle. If it isn’t, then it’s outside of the circle.<\/li>
- The probability,
, of any given random point being inside that circle is equal to the ratio of the area of the circle to the area of the square. The area of that region of the circle is:
, and the area of the the square is
. So the probability is
, or
.<\/li>
- So take a ton of random points, and count how many are inside the circle.<\/li>
- The ratio of points inside the circle to total random points is
We can turn that into a simple Python program:<\/p>\n\n\n\n
from random import random\n\ndef computePi(points):\n inside = 0\n for i in range(points):\n x = random()\n y = random()\n if (x*x + y*y) < 1.0:\n inside = inside + 1\n return (inside*1.0)\/points * 4.0\n\n\nfor i in range(30):\n pi = computePi(2**i)\n print(f\"Pi at 2**{i} iterations = {pi}\")\n<\/pre>\n\n\n\nThe exact value that you’ll get when you run this depends on the random number generator, and the initial seed value. If you don’t specify a seed, most random number libraries will use something like last 32 digits of the current system time in nanoseconds, so you’ll get slightly different results each time you run it. I just ran it, and got:<\/p>\n\n\n\n
Pi at 2**0 iterations = 4.0\nPi at 2**1 iterations = 4.0\nPi at 2**2 iterations = 3.0\nPi at 2**3 iterations = 2.0\nPi at 2**4 iterations = 3.5\nPi at 2**5 iterations = 2.75\nPi at 2**6 iterations = 3.0625\nPi at 2**7 iterations = 3.125\nPi at 2**8 iterations = 3.109375\nPi at 2**9 iterations = 3.1875\nPi at 2**10 iterations = 3.171875\nPi at 2**11 iterations = 3.126953125\nPi at 2**12 iterations = 3.12109375\nPi at 2**13 iterations = 3.14013671875\nPi at 2**14 iterations = 3.169677734375\nPi at 2**15 iterations = 3.1324462890625\nPi at 2**16 iterations = 3.14453125\nPi at 2**17 iterations = 3.147247314453125\nPi at 2**18 iterations = 3.138519287109375\nPi at 2**19 iterations = 3.1364669799804688\nPi at 2**20 iterations = 3.1443214416503906\nPi at 2**21 iterations = 3.141223907470703\nPi at 2**22 iterations = 3.141301155090332\nPi at 2**23 iterations = 3.1419320106506348\nPi at 2**24 iterations = 3.1415367126464844\nPi at 2**25 iterations = 3.1421539783477783\nPi at 2**26 iterations = 3.1420511603355408\nPi at 2**27 iterations = 3.1415300369262695\nPi at 2**28 iterations = 3.141532242298126\nPi at 2**29 iterations = 3.1415965482592583\n<\/pre>\n\n\n\n
I suspect that I could do a lot better using a special number library to reduce or eliminate the floating point roundoff errors, but I don’t really think it’s worth the time. Just this much, using a really simple, obvious, intuitive method produces a better result than any<\/em> of the numbers pushed by the crackpots.<\/p>\n\n\n\n
To support that previous statement: the best crackpot value for \u03c0 is the one based on the golden ratio. That version insists that the true value of \u03c0 is 3.14460551103. But you can see – by using the simple metric of counting points inside and outside the circle – that the actual value is quite different from that.<\/p>\n\n\n\n
That’s what makes this breed of denialism so stupid. \u03c0 isn’t complicated: it’s a simple ratio. And it’s easy to test using simple concepts. Pi relates the diameter (or radius) of a circle to the circumference or area of that circle. So any test that works with circles can easily show you what \u03c0 is. There’s nothing mysterious or counterintuitive or debatable about it. It is what it is, and you can test it yourself.<\/p>\n","protected":false},"excerpt":{"rendered":"
To me, the strangest crackpots I’ve encountered through this blog are the \u03c0 denialists. When people have trouble with Cantor and differently sized infinities, I get it. It defies our intuitions. It doesn’t seem to make sense. When you look at G\u00f6del incompleteness theorem – it’s really hard to wrap your head around. It doesn’t […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[],"class_list":["post-3875","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-10v","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3875","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=3875"}],"version-history":[{"count":1,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3875\/revisions"}],"predecessor-version":[{"id":3882,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3875\/revisions\/3882"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=3875"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=3875"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=3875"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}