User:Playitsmart/Tau: The True Circle Constant

Do read http://tauday.com/ for a clearer argument.

As stated by Bob Palais, π is wrong. The number that deserves the fame is known by most as "2π". [img]http://tauday.com/images/figures/pi-angles.png[/img][3]. [img]http://tauday.com/images/figures/tau-angles.png[/img][3]. See one of the fundamental advantages of τ (Tau, 2π) over π? For a new student trying to learn radians, it is much simpler to learn that 1/4 of a circle is τ/4, rather than π/2. Bob Palais compared using π to defining "hour" as "30minutes" but leaving clocks the way they are.[1]

By definition, a circle is "a plane curve everywhere equidistant from a given fixed point, the center"[4]. The distance is know as radius. However, the circle constant π is Circumference / Diameter. This proposed circle constant is defined as Circumference / Radius.

There are many cases, 2π appears throughout mathematics. Michael Hartl and Bob Palais give many examples of the reoccurce.

In the formula, A= πr^2, one might believe that this is more correct than (1/2)τr^2. However, this table shows other examples of a factor of 1/2. "Quantity	Symbol	Expression Distance fallen	y	1/2gt^2 Spring energy	U	1/2kx^2 Kinetic energy	K	1/2mv^2 Circular area	A	1/2τr^2 Table 3: Some common quadratic forms."[3] Using π is equivalent having and using a constant for (1/2) gravity, (1/2) kinetic energy, and (1/2) mass. COME ON! Seriously? Would I go around saying I weigh 2(90) Lbs (4.2kg)? No. I would simply state that i weigh 180 Lbs (8.3kg).

Suggested plan of action for τists: "How can we switch from π to τ? The next time you write something that uses the circle constant, simply say “For convenience, we set τ=2π”, and then proceed as usual. (Of course, this might just prompt the question, “Why would you want to do that?”, and I admit it would be nice to have a place to point them to. If only someone would write, say, a manifesto on the subject…) The way to get people to start using τ is to start using it yourself."[3] "But what about puns? We come now to the final objection. I know, I know, “π in the sky” is so very clever. And yet, τ itself is pregnant with possibilities. τism tells us: it is not τ that is a piece of π, but π that is a piece of τ—one-half τ, to be exact. The identity eiτ=1 says: “Be 1 with the τ.” And though the observation that “A rotation by one turn is 1” may sound like a τ-tology, it is the true nature of the τ. As we contemplate this nature to seek the way of the τ, we must remember that τism is based on reason, not on faith: τists are never πous."[3]

τ to 1000 decimal places
τ=6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234135964296173026564613294187689219101164463450718816256862234900568205403977042211119289245897909860763928857621951331866892256951296467573566330542403818291297133846920697220908653296426787214520498282547449174013212631176349763041841925658508183430728735785180720022661061097640933042768293903883023218866115540731519183906184362234763865223586210237096148924759925499134703771505449782455876366023898259667346724881313286172042789892790449474381404359721887405541078434352586353504669349636935338810264001136254290527121655571542685515579218347274557442936881802449906860293099170742101584559398517847084039912224258043921728068836319627259549542619921037414422699999996745956099902119463465632193627190048918910693816605285044616506689370070523862376342020006275677505773175066416762841234355338294607196506980857510937462319125727764706575187503915563715561064342453613226003855753222918184328403978... '''

In Hex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

Good points not in "pi is wrong" and the "tau manifesto"
''' By Michael Hartl "This gives us reason to suspect that the volume of an n-sphere might have natural factors of 2 for n2 as well.

You're right that my formulas aren't simpler than the one using the gamma function, but that wasn't the claim. I claimed that they are more specific because they use the fact that n is a nonnegative integer. Indeed, if you look at a standard reference, such as Wolfram MathWorld's entry on the hypersphere (http://mathworld.wolfram.com/Hypersphere.html), you'll see formulas along the same lines as the ones I used: Vn(r)=           (2n)!n2rn n!!2(n+1)2(n−1)2rn if n is even; if n is odd

The first thing we notice is that MathWorld writes the volume piecewise, as I did; the integral nature of n results in a piecewise expression for the volume whether we want it to or not. We next notice that they use the double factorial function, also as I did—but, somewhat mysteriously, they use it only in the odd case. (This is a hint of things to come.)

Let's look at the odd case first. I don't know about you, but to me 2(n+1)2(n−1)2

looks an awful lot like 2(2)(n−1)2

And at this point we immediately recover the formula I cited in my original comment.

Now let's look at the even case. Perhaps is more natural in the odd case, but is more natural in the even case, right? It's certainly possible, but one might reasonably ask why MathWorld chose to use a double factorial in one case and only a single factorial in the other. My suspicion is that (like most of the world) their pattern recognition is overly biased toward, so they simply didn't notice that they could unify their formulas using the following identity: 2n!=n!!2n2 Substituting this into the formula for even n then yields n!!2n2n2rn which bears a striking resemblance to n!!(2)n2rn and again we recover the expression in the original comment.

We see from this discussion that using in place of does give additional insight into the relationship between the volume of an n-sphere and the circle constant. I'm guessing from your tone that there's no convincing you on this point, but I hope you'll reserve the right to change your mind about. (I certainly reserve the right to change my mind about . After all, I've changed my mind once already.)"[5].

On other wikipedia pages (and other sites too)

"The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≍ 2π. The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2613-2589 BC and later at the pyramid of Abysir c.2453-2422. "[6] wrong... the ratio was TAU!

Law of Periods T= (τ^2 *r^3)/GM instead of (4pi^2 *r^3)/GM

Problem(s) with Tau
τ is also used for Newton*meter and for some other lesser known uses. However, pi is not used only as (1/2)τ. For example, there are pi bonds.

Some complain about the change in euler's identity when using τ/2 (even though its showing that it is after 1/2 a turn where pi loses that) and think that the zero in (e^iτ= 1 + 0) is unnesesary (even though it natural comes out since e^iθ = cosθ + isinθ), but dont see anything wrong with forcing a zero into (e^iτ/2= -1 + 0 / e^iτ/2 +1=0).

Other
τist click here

http://www.yourworldoftext.com/tau