{"id":1235,"date":"2010-12-08T20:06:09","date_gmt":"2010-12-09T01:06:09","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/?p=1235"},"modified":"2010-12-08T20:06:09","modified_gmt":"2010-12-09T01:06:09","slug":"really-is-wrong","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2010\/12\/08\/really-is-wrong\/","title":{"rendered":"π really is wrong!"},"content":{"rendered":"

I’ve written recently about several different crackpots who insist, for a variety of completely ridiculous reasons, that pi is wrong. But the other day, someone sent me a link to a completely serious site<\/a> that makes a pretty compelling argument that pi really is<\/em> wrong.<\/p>\n

<\/p>\n

The catch is that they aren’t saying that the value<\/em> of pi is wrong. They’re saying that pi is the wrong constant<\/em> for talking about circles.<\/p>\n

There’s a pretty good case for that. The fundamental measure that we use for a circle is the radius – and there are a lot of good reasons for that. But pi is based on the diameter<\/em>: it’s the ratio of the circumference to the diameter. If you use the radius as the fundamental measure, and you go to define a circle constant, the natural choice isn’t pi. It’s 2pi. In the linked article, the author proposes naming this tau.<\/p>\n

It really does make a lot of sense. Look at basic mathematical systems that use pi, and you find an awful lot of 2s – and the argument is that those 2s are all over the place because of the fact that we’re using the wrong damned constant.<\/p>\n

For example, what’s the equation for a fourier transform?<\/p>\n

f(x)=int_{-infty}^{infty} F(k)e^{2pi i k x}dx<\/center>
\n
F(k)=int_{-infty}^{infty} f(x)e^{-2pi i k x}dx<\/center><\/p>\n

Why are those 2s there? Because pi is wrong. We should use tau, which gives us the cleaner equations:<\/p>\n

f(x)=int_{-infty}^{infty} F(k)e^{tau i k x}dx<\/center>
\n
F(k)=int_{-infty}^{infty} f(x)e^{tau i k x}dx<\/center><\/p>\n

It’s no a big deal, but it does actually make those equations make more more sense. As we’ll see below, it actually helps to clarify the meaning of those equations!<\/p>\n

So, suppose we decide to use tau=2pi as the fundamental circle constant. What effect does it really have? A whole lot of things actually make a lot of sense. What’s one full turn around a circle in radians? tau. And that’s quite beautiful and natural. A quarter circle is frac{1}{4}tau radians. That’s lovely.<\/p>\n

What’s the width of a sin curve? tau.<\/p>\n

Even Euler’s equation is more beautiful using tau:<\/p>\n

e^{itau}=1<\/center><\/p>\n

And it even makes sense! What that says is: the complex exponential of turning around a full circle is 1. Which means: if you multiply a complex number by e^{itheta}, that’s basically the same thing as treating it as a vector, and rotating<\/em> it by an angle of theta: so what e^{itau} means is rotating it all the way around the circle: Euler’s equation becomes a very clear statement of the fact that the complex plane is symmetric with respect to that rotation.<\/p>\n

It really does<\/em> make sense. I actually think that he’s right. I think that human inertia is going to make it close to impossible to convince people to change – but I think he’s right, and the correct fundamental circle constant is<\/em> tau.<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve written recently about several different crackpots who insist, for a variety of completely ridiculous reasons, that is wrong. But the other day, someone sent me a link to a completely serious site that makes a pretty compelling argument that really is wrong.<\/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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[22],"tags":[95,96,214,241],"class_list":["post-1235","post","type-post","status-publish","format-standard","hentry","category-good-math","tag-95","tag-96","tag-pi","tag-tau"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-jV","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/1235","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=1235"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/1235\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=1235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=1235"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=1235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}