Lyrics:(to the tune of Rudolph The RedNosed Reindeer)
We know Einstein and Euclid and Sir Isaac Newton Lifelong devotions there ain’t no disputin’ But do you recall… The most tireless one of them all? Ludolph the Mathematician Had a special thing for Pi He made it his life’s mission To help the number specify All of his fellow teachers Never understood his plan To unlock the number’s magic By calculating it by hand! Then in 1599, Ludolph set his goal... I’ll find digit 35 With geometry as my guide! Then how the math world loved him All his hard work helped them see Ludolph the Mathematician You’ll go down in history! Video can be found at http://youtu.be/U2uVoDxZpaQ Lyrics:Look, if you had…one shot…one opportunity…
To recite the digits of the number pi… One moment… Would you capture it… or just let it slip? His palms are sweaty, knees weak, head is heavy The numbers cloud his vision already, keep it steady He’s nervous, but on the surface he looks calm and ready To recite, but it’s not quite offsettin’ That he’s froze now, and the whole crowd knows he vowed To get it perfect, but the numbers won’t come out He’s chokin’, how – everybody’s jokin’ now Three point one five—woops—kablouw! Snap back to reality, oh, it’s a malady Oh, it’s so random, he choked He can’t stand, but he won’t give up That easy, no He won’t have it, he knows – all his fans really hope He can shatter the glob… …al world record of Fo’… …rtytwothousand*, although Ifhelooks down at his notes, he knows that’s when it’s Back to the learnin’ by rote To feel rapture He’d better capture those digits a little faster You better lose yourself in the digits of pi, it’s a high But you got a thousand more to go The numbers do not stop Or drop into pattern, no And memorizin’ ‘em takes most of a lifetime If I’m not mistaken, my memory’s wakin’ This number’s mine for the taking Hear it ring, as I list all the, digits in order And whole numbers are borin’, approximations just keep me snorin’, Pi only grows longer to get more precise A roundedoff fraction just won’t cut the ice So twentytwo sevenths, you take my advice If I catch you round here, you will Pay the price, so give up this façade, Raysheeos are oh so flawed They close the door to those who like infinite Strings of oh’s, 2s, 9s, 3’s, and 6’s, it Shows the woes ofallthe schmoes who chose To forgo Pi’s logic, and I Don’t suppose those bozos applaud it When I throw down the gauntlet, with a “Three point one four”… I got it! You better lose yourself in the digits of pi, it’s a high But you got a thousand more to go The numbers do not stop Or drop into pattern, no And memorizin’ ‘em takes most of a lifetime Pi is 3.141592653589793238462643383279… No more games, I’ma bust right outta my cage, and Get my Pirecitin’ groove on, on this here stage I was playin’ in the beginnin’, the mood all changed I been broken up from soakin’ up this whole page But I kept churnin’ and kept learnin’ to decipher The rhythm of the number that makes me so hyper All the pain and strife amplified by the Fact that I can’t get back to my life, I’m Whacked and I don’t exactly know why, I’m Acting like I am attracted to Pi! So Back to a kid jampacked with random numbers I think this whole process is makin’ me dumber It still shakes me up, man, it makes me sentimental That a number so essential is also transcendental Caught up between bein’ in fashion and bein’ irrational It’s passion flashin’ through my brain Whenever I explain it I just think pi is nice, Would you... like a slice? I’ve gotten to the point, it’s like a vice, I’m not Afraid of gettin' caught, but still it hurts a lot Success is my only mathematic option, failure’s not Pi, I love you, but these people got to know A million tiny circles haunt my every thought So here I go with my shot Brain, fail me not, This may be the only opportunity that I got! You better lose yourself in the digits of pi, it’s a high But you got a thousand more to go The numbers do not stop Or drop into pattern, no And memorizin’ ‘em takes most of a lifetime Pi is… 3.14159265358979323846264338327950288419716939937510582097494459…yeeeah. You can do anything you set your mind to, man. Even memorizin’ pi. I’m out. * We realize that the new world record (though not yet verified) is 100,000 digits. The song was written when the record was 42,815. We stayed with the original number, because frankly, it fits the rhyme better! "Lose Yourself (In The Digits)" lyrics are the property of TeachPi.org. The Pi Manifesto was written by MSC. You can email the author at [email protected]. Last updated July 4th, 2011. **THIS IS NOT OUR OWN WORK** This post requires the use of Javascript. It will use MathJax which when viewed in our RSS feed, it will not show correctly. Please visit the post at http://www.calculushumor.com/3/post/2013/02/thepimanifesto.html to see the correct version. The Pi ManifestoWritten by MSC [email protected]Last updated July 4th, 20111 π versus τ1.1 The Tau MovementThis article is dedicated to defend one of the most important numbers in mathematics: π. Quite recently, a phenomenon known as the Tau Movement has steadily grown and is gaining more and more followers (called Tauists) by the day. This is largely due to three driving forces:
Tauists claim that π is the wrong circle constant and believe the true circle constant should be τ=2π. They celebrate Tau Day (June 28th), wear τshirts and spread protau propoganda. But are tauists doing more harm than good? In this article we will explore this very question and provide several reasons why π will prevail in the intriguing π versus τ battle. 1.2 Any publicity is good publicityThe buzz around the blogosphere and on various online news sites is that there is a battle happening in mathematics, namely π versus τ. Headlines in newspapers and on blog articles often declare that π is wrong and tend to mislead the general public:
According to an article published by The Telegraph on Tau day: "Leading mathematicians in India, the UK and the US appeared oblivious to this campaign today and asserted that there has been no debate or even discussion over replacing 2πwith τ in serious mathematical circles." Mathematician Alexandru Ionescu at Princeton University says: "Either one is just fine, it won't make any difference to mathematics." Siddhartha Gadgil, a mathematician at the IISc, says: "The whole notion of replacing π by 2π is silly since we all are very comfortable with π and multiplication by two." In fact, one grad student in mathematics goes on to say: "Of course it had to be a physicist who would want to get rid of the usage of π... Theconcept of π has been around since the time of the ancient Babylonians (the greek letter representing this number was popularized by Euler in the 18th century)... so why change now and trash it? This isn't the first thing that physicists have tried to change in the field of mathematics (notation wise, anyways). I for one believe that the mathematics community will not be lemmings here and go with this idea; I know I'm certainly not going to accept tau as a replacement for pi." It is debatable whether the media coverage of τ is good publicity or bad publicity for mathematics, but regardless, the Tau Movement has definitely sparked an interest. Even those with very little mathematical background are curious about it! I think most mathematicians would agree that anything that generates interest in math is a definite plus. As seen from the quotes above, a lot of mathematicians simply shrug off the Tau Movement as being silly. In this article we attempt to give a serious rebuttal to τ in the defence of π. Any suggestions and reasons why π is better than τ (or τ is better than π) are more than welcome! 1.3 The Tau Manifesto is wrongTauists argue that by using the constant τ=2π a lot of formulas become simpler. Unfortunately, the Tao Manifesto is full of selective bias in order to convince readers of the benefits of τ over π. They pinpoint formulas that contain 2π while ignoring other formulas that do not. We demonstrate below that when making the change to τ, there are lots of formulas that either become worse or have no clear advantage of using τ over π. Tauists also claim that their version of Euler's formula is better than the original, but we will see that it is in fact weaker. The benefits of τ only appear when viewing π from a narrow minded two dimensional geometrical point of view, but these benefits disappear when looking at the bigger picture. We will see how the importance of π shines through as it shows up all over mathematics and not just in elementary geometry. 2 Definitions of π2.1 The Traditional DefinitionThe Tau Manifesto relies on the traditional definition of π, namely, the constant that is equal to the ratio of a circle's circumference to its diameter: \[\pi\equiv\frac{C}{D}\approx 3.14159... \] The manifesto then goes on to suggest that we should be more focused on the ratio of a circle's circumference to its radius: \[\tau\equiv\frac{C}{r}\approx 6.283185...\] In particular, since a circle is defined as the set of points a fixed distance (i.e., the radius) from a given point, a more natural definition for the circle constant uses r in place of D. So why did mathematicians define it using the diameter? Likely because it is easier to measure the diameter of a circular object than it is to measure its radius. In the Tau Manifesto, Hartl says: "I’m surprised that Archimedes, who famously approximated the circle constant, didn't realize that $\displaystyle \frac{C}{R}$ is the more fundamental number. I’m even more surprised that Euler didn't correct the problem when he had the chance."But Dr. Hartl, there is no problem to correct, π is not wrong, and we will soon see that we have been using the right constant all along. There are numerous reasons to define the circle constant using $\displaystyle \frac{C}{D}$ . Some of these reasons include:
2.2 Other definitions of πAnother definition for π is to define it to be twice the smallest positive x for which cos(x)=0 [4], or the smallest positive x for which sin(x)=0. With this definition neither π nor τ is simpler than the other. Tauists may claim that τ can be defined as the period of cos(x) or sin(x) but whether this is better is up for debate (in the same way, π can be defined as the period of tan(x)). Another common geometric definition for π is in terms of areas rather than lengths. Take r to be the radius of a circle. Define π to be the ratio of the circle's area to the area of a square whose side length is equal to r, that is, \[\pi\equiv\frac{A}{r^2}.\] In terms of τ, this definition is messy and includes a factor of 2. In particular, define τ to be the twice the ratio of a circle's area to the area of a square whose side length is equal to r, that is, \[\tau\equiv 2\left(\frac{A}{r^2}\right).\] Clearly, this definition favors π over τ and also involves the important radius of a circle. Like the traditional definition, this definition of π depends on results of Euclidean geometry and comes naturally when looking at areas. 2.3 Why stop at redefining π?Mixing things up a bit, in terms of diameter we can define a constant (call it $\displaystyle \frac{\pi}{4}$ ) as follows: \[\frac{\pi}{4}\equiv\frac{A}{D^2}.\] This suggests that perhaps both π and τ are wrong, and $\displaystyle \frac{\pi}{4}$ is the correct circle constant. Others have also suggested similar numbers as the circle constant. In 1958, Eagle suggests that $\displaystyle \frac{\pi}{2}$ is the correct circle constant [1]. In fact, the $\displaystyle \frac{\pi}{2}$ Manifesto is coming soon to a website near you! (Just kidding, I hope). But why stop at redefining π? Terry Tao says: "It may be that 2πi is an even more fundamental constant than 2π or π. It is, after all, the generator of log(1). The fact that so many formulae involving $\displaystyle \pi^n$ depend on the parity of n is another clue in this regard." Clearly, each of π, 2π, $\displaystyle \frac{\pi}{2}$ , $\displaystyle \frac{\pi}{4}$ and 2πi have their benefits, but should we seriously isolate 2π and attempt to redefine it as τ? Sure τ is better in a few instances, but that is because it is a multiple of π. This is no reason to introduce a new constant and encourage mathematics to adopt it. 3 Silly arguments3.1 A silly argument for τThe main argument for τ is its simplicity to calculate the number of radians in a fraction of a circle. I think we all would agree that τ makes this trivial task a bit more trivial. A tauist would ask you: Quick, how many radians in an eighth of a circle? In terms of turns, τ has a slight advantage. Just look at the following two figures that appeared in the Tau Manifesto and tell me you aren't convinced by the power of τ! But this is not a reason to switch to τ. The context is highly relevant in this regard and similar questions which favor π can be posed. Let me demonstrate with an example by using areas rather than angles. Note that the area of a unit circle is π. Now quick, what is the area of an eigth of a unit circle? Tau may have its benefits when looking at turns, but when looking at areas π takes the cake (or rather, pie). Just like the Tau Manifesto, I too can create convincing looking pictures: Looking at Figure 2, it seems that τ is off by a factor of two. This demonstrates that in some situations τ may be better, and in other situations π is better. The reason the Tau Manifesto is so convincing is because of selection bias. They only demonstrate the situations where τ is either better than π or comparable and ignore situations where it is worse. 3.2 A silly argument for πAs demonstrated in Section 3.1, when dealing with areas π outperforms τ. One of the most important problems proposed by ancient geometers is that of squaring the circle. The problem is stated as: Can a square with the same area as a circle be constructed by using only a finite number of steps with compass and straightedge? Reinterpretting this problem in terms of τ is a disaster, which provides more motivation why π is the true circle constant. To conclude this section we reiterate a very important fact: The area of a unit circle is π. This result is so beautiful that it would be a crime to rewrite it using τ. 4 Probability and statistics  A win for πSome formulas with 2π may seem like a win for τ, but just because there is a 2 in the formula does not mean it belongs with the π. Let me demonstrate with an example that appeared in the Tau Manifesto, the Gaussian (normal) distribution. 4.1 The normal distributionThe Gaussian integral is the integral of the Gaussian function $\displaystyle e^{x^2}$ over the entire real line: \[\int_{\infty}^\infty e^{x^2}\,dx = \sqrt{\pi}.\] This integral is important and has many applications in mathematics. Notice the integral does not have a 2π, beautiful!! This is when tauists will claim there is a similar formula with 2π in it, but then we end up with a nasty fraction of $\displaystyle \frac{1}{2}$ in the power of e, ew! The only thing worse than multiplying by 2 is dividing by 2: \[\int_{\infty}^\infty e^{x^2/2}\,dx = \sqrt{\tau}.\] Comparing these two integrals most mathematicians would agree that not only is the first one nicer, it is much more natural! When the Gaussian integral is normalized so that its value is 1, it is the density function of the normal distribution: \[f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\, e^{\frac{(x\mu)^2}{2\sigma^2}}.\] However, by grouping the 2 with the $\displaystyle \sigma^2$ rather than with the π, it can easily be written in the form \[f(x)=\frac{1}{\sqrt\pi(\sqrt 2\sigma)}e^{\frac{(x\mu)^2}{(\sqrt 2\sigma)^2}}.\]The Tau Manifesto groups the 2 with the π and gives this formula as an example where τ wins over π. But in fact, the 2 does not belong with the π and this becomes even more apparent when looking at alternate suggestions for the "standard" normal distribution. The distribution with μ=0 and $\displaystyle \sigma^2=1$ is called the standard normal, that is, \[\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{ \frac{1}{2} x^2}.\] Various mathematicians debate on what we should call the standard normal distribution. Note that above by setting $\displaystyle \sigma^2=1$ and grouping the 2 with π rather than the $\displaystyle \sigma^2$ it (falsely) appears to be a win for τ. Gauss suggests that the standard normal should be \[f(x) = \frac{1}{\sqrt\pi}\,e^{x^2}\] and Stigler insists the standard normal to be \[f(x) = e^{\pi x^2}.\] Neither of these suggestions has a 2π because the 2 does not belong with the π in the first place. Unfortunately, ϕ(x) has been adopted as the standard normal, but this does not make it a win for τ. 4.2 Other distributionsWhen analyzing other distributions we see that 2π is not as common in statistics as the Tau Manifesto would lead you to believe. The Cauchy distribution has the probability density function $$f(x)= { 1 \over \pi } \left[ { \gamma \over (x  x_0)^2 + \gamma^2 } \right],$$ and the standard Cauchy distribution has probability density function $$f(x)=\frac{1}{\pi(1+x^2)}.$$ The student's tdistribution has the probability density function $$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{\frac{1}{2}(\nu+1)}.$$ Neither of these have a 2π appearing, but the student's tdistribution does have multiples of πoccuring. In fact, multiples of π show up throughout mathematics, so it is no surprise that 2π shows up in some formulas. 5 Polygons and triangles  Another win for πConsider a triangle with interior angles α, β and γ. Let me ask you, what is the sum of these three angles? Is it τ? That would be nice if it were, but in fact, the answer is the almighty π! $$\alpha+\beta+\gamma=\pi.$$ By looking at polygons we see that π is a clear winner over τ. Take any polygon with ksides and interior angles $\theta_i$ (for i=1,2,…,k). Then the sum of the angles is equal to $$\sum_{i=1}^k \theta_i=(k2)\pi.$$ Once we look beyond specific angles inside of circles, π really does show who's boss! In fact, multiples of π are very important in mathematics, including τ=2π. The importance of τ comes from the fact that it is a multiple of π, but other multiples of π are just as important. We have demonstrated that angles of arcs in circles are a win for τ, interior angles in polygons are a win for π, areas in circles are a win for π, but what about areas of polygons? It is well known that the area of a regular ngon inscribed in a unit circle is: $$A=n\sin\frac{\pi}{n}\cos\frac{\pi}{n}.$$ Clearly, another win for π. 6 Trigonometric functionsWe just can't stress this enough. The reason τ shows up a lot is because it is a multiple of π. We saw the multiple νπ appear in Section 4.2 and the multiple (k−2)π in Section 5. Looking at trigonometric functions we should expect multiples of π to again show up (and indeed they do). The following table shows the domain and period of common trig functions: $$\begin{array}{ccc}\mbox{Function} & \mbox{Domain} & \mbox{Period}\\\hline\sin\theta & \mathbb{R} & 2\pi\\\cos\theta & \mathbb{R} & 2\pi\\\tan\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & \pi\\\csc\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & 2\pi\\\sec\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & 2\pi\\\cot\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & \pi\\\end{array}$$ Notice that π shows up, along with 2π and nπ. By converting the table to τ we would get even more nasty fractions than are already there. 7 Other formulasThe Tau Manifesto had an extremely small list of formulas containing 2π, but what about other well known formulas and functions? The error function: $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{t^2} dt.$$ The sinc function: $$\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}.$$ The Gamma function: $$\Gamma(1/2)=\sqrt\pi$$$$\Gamma(3/2)=\sqrt\pi/2$$$$\Gamma(5/2)=3\sqrt\pi/4$$ Euler's Reflection Formula: $$\Gamma(z)\Gamma(1z)=\frac{\pi}{\sin(\pi z)}.$$ Volume of unit nball: $$V=\frac{\sqrt\pi^n}{\Gamma(1+\frac{n}{2})}.$$ Area of an ellipse: $$A=\pi ab$$ Integral of hyperbolic secant: $$\int_{\infty}^\infty \operatorname{sech}(x)\,dx = \pi.$$ Integral of $\displaystyle \frac{1}{\sqrt{1x^2}}$: $$\int_{1}^1 \frac{1}{\sqrt{1x^2}}\,dx = \pi.$$ Integral of $\displaystyle \frac{1\cos x}{x^2}$: $$\int_{\infty}^\infty \frac{1\cos x}{x^2}\,dx = \pi.$$ Integral of $\displaystyle \frac{\sin x}{x}$: $$\int_{\infty}^\infty \frac{\sin x}{x}\,dx = \pi.$$ Integral of $\displaystyle \frac{\sin^2 x}{x^2}$: $$\int_{\infty}^\infty \frac{\sin^2 x}{x^2}\,dx = \pi.$$ Integral of $\displaystyle \frac{1}{1+x^2}$: \[\int_{\infty}^\infty \frac{1}{1+x^2}\,dx = \pi.\] Where are you τ? Ah, it must be hiding in shame. 8 Euler's IdentityOne reason the Tau Manifesto is able to convert so many readers is because of their version of Euler's identity.They claim that $$e^{i\tau}=1$$ is more elegant than the formula $$e^{i\pi}+1=0,$$ but any mathematician can see this is total nonsense. Sure, there may be a nice formula that uses τ, but that is because τ is a multiple of π. In reality, there is also a nice formula for the multiple 3π, but that doesn't mean we should start worshipping 3π. The fact is, their version of the formula may look nice but it is much weaker than the original. Consider the function eix. We ask the following important question: What is the smallest positive solution x so that $\displaystyle e^{ix}$ is an integer? The answer comes as no surprise, it is π. That is, π is the smallest number that brings imaginary powers of e back to the real line. This is why π is more important than τ. Furthermore, the equation $\displaystyle e^{i\pi}=1$ is a much stronger result than $\displaystyle e^{i\tau}=1$ and the τ equation comes trivially from the first equation by squaring both sides: $$\left(e^{i\pi}\right)^2=\left(1\right)^2\quad\implies\quad e^{i\tau}=1.$$ When it comes to Euler's identity, τ just can't compete with the powers of the almighty π. 9 Conclusion and remarks9.1 Engineers are against τI do not have a background in engineering but it is important to consider the applications of π. Regarding the introduction of τ in the Tau Manifesto, Gareth Boyd writes: Dr Hartl's theoretical background would seem to be on show here. He has forgotten about the practical application of mathematics  engineering. Tau is already one of the most important symbols in mechanical engineering as it denotes shear stress. Additionally the ratio of diameter to circumference is very important when we work with bars of material or pipes. We tend not to purchase these by the radius. Perhaps a little more thought and debate are required in this matter before we start a revolution. It should be mentioned that the Tau Manifesto does give a good argument for using the symbol τ. However, we question whether the constant τ=2π is actually needed in mathematics. 9.2 Quadratic FormsThe connection between the formula $$A=\frac{1}{2}\tau r^2$$ and some quadratic forms in physics is certainly interesting, but the traditional formula $$A=\pi r^2$$ is already a quadratic form preferred by mathematicians. The Tau Manifesto would lead you to believe there is a $\displaystyle \frac{1}{2}$ missing by comparing it to a few physics formulas, but then you would be forgetting about the connection the formula has to circles. One strong fact that was mentioned in Section 3.2 is that the area of a unit circle is π. This fact is what makes π more important than τ as a large number of of problems in geometry deal with areas. In my opinion, the only benefit τ seems to have is that it makes computing angles of arcs in a circle a bit more trivial. When looking at areas, π certainly shines. Even when looking at areas where no circle seems to be apparent, π shines through. In particular, the last six formulas in Section 7 show that the area under the specified functions is equal to π  this is truly amazing. 9.3 From the authorI hope you enjoyed reading the Pi Manifesto. This article is meant to be a fun discussion of the importance of π and why π is the right circle constant afterall. It is a first draft and any additional arguments, mathematical facts, or strengthening of the above points in the defence of π are more than appreciated! Please feel free to contact me if you have questions or comments. AcknowledgmentsNone yet. If you have any improvements to the Pi Manifesto let me/[them] know! Copyright and licenseThe Pi Manifesto. Copyright © 2011 by MSC. Please feel free to share The Pi Manifesto, which is available under the Creative Commons Attribution 3.0 Unported License.
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