Tau (an unofficial brilliant wiki made by me)

τ\large\tau is defined as 2π2\pi and if you don't know what π\pi is, its not the sweet treat, I can assure you.

π\pi is defined as the ratio of the circumference of a circle to its diameter. This means - π=circumferencediameter\pi = \dfrac{\text{circumference}}{\text{diameter}}

\(\pi\) Visualized :) π\pi Visualized :)

Now, back to τ\large\tau, it is 2π2\pi. This means - τ=2×circumferencediameter or circumferenceradius\tau = \dfrac{2 \times \text{circumference}}{\text{diameter}} \ \text{or} \ \dfrac{\text{circumference}}{\text{radius}}

The first 1010 digits of π\pi(sometimes written as 'pi' and pronounced 'pie') go like this - 3.1415926533.141592653 \ldots

And the first 1010 digits of τ\large\tau(sometimes written as 'tau' and pronounced 'tau') are twice that - 6.2831853076.283185307 \ldots

Since τ\large\tau is 2×2 \times an irrational number(π\pi), it is also irrational.

But here comes the big question, why waste another Greek Letter, to define it as 22 times an already existing number?


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More about τ\large\tau

τ\large\tau is the 1919th letter of the Greek Alphabeet and has a value of 300300 in Greek Numerals.

Hi, I'm Tau and this is Capital Tau. I am \(2\pi\) even though I look  like half of \(\pi\). Capital Tau looks like T, but don't tell him that, he has very sensitive feelings :) Hi, I'm Tau and this is Capital Tau. I am 2π2\pi even though I look like half of π\pi. Capital Tau looks like T, but don't tell him that, he has very sensitive feelings :)

The Greek Letter is used in many aspects of Math and Science, some of which are listed below -

  • Used to represent the Golden ratio (1.618...), although ϕ\phi(phi) is more common.

  • Represents Divisor function in number theory, also denoted d\text{d} or σ0\sigma_{0}.

  • Tau is an elementary particle in particle physics.

  • Tau in astronomy is a measure of optical depth, or how much sunlight cannot penetrate the atmosphere.

  • It is the symbol for tortuosity in hydro-geology.

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Advantages of τ\large\tau

π\pi is found in many of the famous formula's and identity. But sometimes, π\pi is preceded by a 22, and we all know by now, that 2π=τ2\pi = \large\tau.

Here are the well known formula's where 2π2\pi can be replaced with τ\large\tau, making the formula simpler and more elegant -

Integral over space in polar coordinates -

  • 2π00f(r,θ) r dr dθ\displaystyle\int_{\color{#D61F06}{2\pi}}^{0} \int_{\infty}^{0} f(r,θ) \ r \ dr \ dθ

Gaussian (normal) Distribution -

  • 12πσe(xμ)22σ2\dfrac{1}{\sqrt{\color{#D61F06}{2\pi}}\sigma} \text{e}^{\dfrac{(x-\mu)^{2}}{2 \sigma^{2}}}

NthN^{th} roots of unity -

  • zn=1z=e2πinz^{n} = 1 ⇒ z = e^{\dfrac{\color{#D61F06}{2\pi} \color{#333333}i}{n}}

Cauchy's Integral Formula -

  • f(a)=12πiγf(z)zadzf(a) = \dfrac{1}{\color{#D61F06}{2\pi} \color{#333333} i} \displaystyle\oint_{\gamma} \dfrac{f(z)}{z-a} dz

Fourier Transform -

  • f(x)=F(k)e2πikxdkf(x) = \displaystyle\int_{- \infty}^{\infty} F(k) e^{\color{#D61F06}{2\pi} \color{#333333} i k x} dk

  • F(k)=f(x)e2πikxdxF(k) = \displaystyle\int_{- \infty}^{\infty} f(x) e^{- \color{#D61F06}{2\pi} \color{#333333} i k x} dx

Riemann Zeta Function -

  • ς(2n)=k=112kn          =B2n2(2n)!(2π)2n, n=1,2,3,......\varsigma (2n) = \displaystyle\sum_{k = 1}^{\infty} \dfrac{1}{2^{kn}} \\ \\ \ \text{ } \ \ \ \ \ \ \ \ = \dfrac{|B_{2n}|}{2(2n)!} (\color{#D61F06}{2π} \color{#333333})^{2n}, \text{ } n = 1,2,3,......

The Kronecker limit formulas -

  • E(τ,s)=πs1+2π(γlog(2)log(yη(τ)2))+O(s1)E(\tau ,s)={\pi \over s-1}+\color{#D61F06} 2\pi \color{#333333} (\gamma -\log(2)-\log({\sqrt {y}}|\eta (\tau )|^{2}))+O(s-1)

  • Eu,v(τ,1)=2πlogf(uvτ;τ)qv2/2E_{{u,v}}(\tau ,1)=-\color{#D61F06} 2\pi \color{#333333} \log |f(u-v\tau ;\tau )q^{{v^{2}/2}}|

I don't know what this limit is called, please tell me! -

  • 2π=32limn(n+1)k=1nk2(2k+1)2\color{#D61F06}2\pi \color{#333333} = 32 \displaystyle\lim_{n\to\infty} (n+1) \prod_{k=1}^{n} \dfrac{k^{2}}{(2k + 1)^{2}}

I have highlighted the 2π2\pi's in red. If replaced with τ\large\tau, these formula's could be simpler. This is only one of the advantages of τ\large\tau.

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Another advantage that is very useful is in Trigonometry.

When dealing with Unit Circles in Trigonometry, we usually use Radians as our angle measure. As already stated, π\pi radians is a 180180^{\circ} while τ\large\tau radians is a full turn or 360360^{\circ}. Due to this, using τ\large\tau is much easier.

When we use π\pi, 112\dfrac{1}{12} of the unit circle is actually π6\dfrac{\pi}{6} Radians, as shown in the picture below. This is cause for major confusion.

But if we use τ\large\tau all our problems vanish, and everything makes sense again. 112\dfrac{1}{12} of the unit circle is τ12\dfrac{\large\tau}{\normalsize 12} radians, as shown in the picture below.

This really shows that τ\large\tau might actually be better to use than π\pi. Yet, we all have our weaknesses, so let's look at the Disadvantages of τ\large\tau next.

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Disadvantages of τ\large\tau

Coming Soon!\large \text{Coming Soon!}

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τ\large\tau(Tau) vs π\pi(Pi) Conflict

The Pi vs Tau Conflict is a very long conflict between people who love π\pi and people who love τ\large\tau.

It is a battle to find which of the 22 irrational numbers is better and makes our calculations easier.

An illustration of the epic battle :) An illustration of the epic battle :)

The basic arguments given by both teams were as follows -

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Team Tau against π\large \underline{\text{Team Tau against }\pi}

π=CircumferenceDiameter\pi = \dfrac{\text{Circumference}}{\text{Diameter}} while τ=CircumferenceRadius\large\tau = \normalsize \dfrac{\text{Circumference}}{\text{Radius}}

A circle is usually defined by its radius and the radius is something that mathematicians are generally more interested in than the diameter.

The formula for a circle's circumference is also simplified with τ\large\tau

C=τrC = \large\tau\normalsize r

τ\large \tau Radians is equal to 360360^{\circ}, while π\pi Radians is equal to 180180^{\circ}. This value of Pi makes it a bit confusing in dealing with Trigonometry using Unit Circles, but Tau makes it look like easy as pie (pun intended)

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Team Pi against τ\large \underline{\text{Team Pi against }\large \tau}

Pi has been around since a long time. The number Tau also has its flaws. For example - The formula for the area of a circle is made even more complex by using τ\large\tau

A=πr2=τr22A = \pi r^{2} = \dfrac{\large\tau\normalsize r^{2}}{2}

Also, redefining the circle constant differently and replacing it will destroy the beautiful Euler Identity - eiπ+1=0e^{i\pi} + 1 = 0

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This conflict is as long as the amount of digits in Pi and Tau. I will not take sides and either number isn't necessarily better. This is only a fragment of the conflict. This conflict has been around for a few years and hasn't officially ended yet.

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External Links

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Related Wikis on Brilliant

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Try some problems related to π\pi and τ\large \tau

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More Digits

These are the first 100100 digits of Tau, but if you want more, go here -

6.28318530717958647692528676655900576839433879875021164194988918461563281257241799725606965068423413596.2831853071795864769252867665590057683943387987502116419498891846156328125724179972560696506842341359 \ldots

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Next Update

Disadvantages of Tau - Due 21st to 25th Sep\large \text{Disadvantages of Tau - Due 21st to 25th Sep}


Image Credits - Thanks to https://www.wikipedia.org/, for the 'Visualising Pi' image, the 'capital and small tau' image and the 'Pi vs Tau' image and to https://tauday.com/tau-manifesto for the Unit Circle images in 'Advantages of Tau'.

Cite as : Tau, Brilliant.org. Retrieved on some date and some time, from https://brilliant.org/discussions/thread/tau/

Note by Percy Jackson
1 month, 1 week ago

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EXAMPLE\hspace{-50px}\color{grey}\\[-22px]\tiny\textsf{EXAMPLE}
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Páll Márton (no activity) - 1 month, 1 week ago

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How? Please comment this in Latex Club @Páll Márton

Percy Jackson - 1 month, 1 week ago

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so tau is double the value of pi even though the symbol is half of pi interesting

Nathan Soliman - 1 month, 1 week ago

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Yes, a fun fact @Nathan Soliman :)

Percy Jackson - 1 month, 1 week ago

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what do we use tau to calculate

Nathan Soliman - 1 month, 1 week ago

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@Nathan Soliman Trigonometry graphs and functions that require unit circle method, because tau radians is 360 degrees :)

Percy Jackson - 1 month, 1 week ago

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Can you please update?

Lin Le - 1 week, 3 days ago

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I've completed Advantages of Tau. Only Disadvantages left. It'll be done by next week :)

Percy Jackson - 1 week, 1 day ago

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@Frisk Dreemurr - I'm making a wiki for fun :)

How is it?

Percy Jackson - 1 month, 1 week ago

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@Frisk Dreemurr

Percy Jackson - 1 month, 1 week ago

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Nice @Percy Jackson, I never noticed any note on the debate of Tau and Pi until today :)

Frisk Dreemurr - 1 month, 1 week ago

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@Frisk Dreemurr Thanks, the debate is actually a pretty big thing :)

Percy Jackson - 1 month, 1 week ago

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@Percy Jackson @Frisk Dreemurr - I added a rating system LOL :)

Percy Jackson - 1 month, 1 week ago

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@Percy Jackson does it have a number count

Nathan Soliman - 1 month, 1 week ago

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@Nathan Soliman What? @Nathan Soliman

Percy Jackson - 1 month, 1 week ago

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@Percy Jackson the rating system

Nathan Soliman - 1 month, 1 week ago

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@Nathan Soliman No, its just for fun :)

Percy Jackson - 1 month, 1 week ago

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@Percy Jackson do you want to hear something stanley hudson said?

Nathan Soliman - 1 month, 1 week ago

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@Páll Márton - I was looking at your formatting guide while making this, its really useful, thanks :)

Percy Jackson - 1 month, 1 week ago

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BTW go to my feed to see my latest note on buttons. You can add that to your guide @Páll Márton

Percy Jackson - 1 month, 1 week ago

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@Yajat Shamji - Look at this :)

Percy Jackson - 1 month, 1 week ago

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I really like it. However, I am impartial to both π\pi and τ\tau so I am neutral in this war/conflict.

Yajat Shamji - 1 month, 1 week ago

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I wrote that I'm neutral but I'm actually a Tauist LOL. Glad you like it :)

Percy Jackson - 1 month, 1 week ago

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@Yajat Shamji - There is an astronomy point in 'More About Tau' :)

Percy Jackson - 1 month, 1 week ago

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Thanks.

P.S. 55 stars!

P.S.S. I challenge you to find geometric sums and limits involving π\pi that result in τ\tau and geometric sums and limits involving τ\tau that result in π\pi.

Yajat Shamji - 1 month, 1 week ago

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@Yajat Shamji I'll try :) @Yajat Shamji

P.S Thanks you for 5 stars!

Percy Jackson - 1 month, 1 week ago

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@Yajat Shamji Done @Yajat Shamji, I added a few integral formula's that I could think of. I still don't have a limit though, I'll work on that :)

Percy Jackson - 1 week, 1 day ago

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@Percy Jackson You know I'll make one...

Yajat Shamji - 1 week, 1 day ago

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@Percy Jackson I found some (but the result's π\pi so you'll need to multiply the entire limit by 22):

Yajat Shamji - 1 week, 1 day ago

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@Yajat Shamji I need something where I don't need to multiply actually. There are many Pi limits - Pi limits

Percy Jackson - 1 week, 1 day ago

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@Yajat Shamji @Yajat Shamji - I added a limit formula that doesn't need me to multiply the limit by 2 and do some crazy math, but I don't know its name.

Percy Jackson - 1 week, 1 day ago

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no, the beautiful Euler identity would not disappear, since e^i tau - 1 = 0 (but I am a pie fan)

I Love Brilliant - 1 week, 2 days ago

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Its what they argue, its the truth. It may not be correct, but its an arguement

Percy Jackson - 1 week, 2 days ago

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