From J.B. Wood: > On 01/02/2013 06:04 AM, firstname.lastname@example.org wrote: > > > The reason why Tau is a fundamental number is because it is the Identity > > Operator for Rotation. Pi is lacking in such fundamental utility. > > > > Tau is as fundamental to Rotation as Zero is to Addition and One is to > > Multiplication. Tonight I watched several YouTube videos about Tau and > > I didn't see anyone make mention of this fact. Bob Palais wrote an > > excellent article back in 2001. But the arguments that I've seen favoring > > Tau over Pi were basically a matter of style and taste. > > > > The reason why Tau is worthy of primacy over Pi is not because of any such > > artistic reasons, but because of this fundamental mathematical reason.
> Hello, and you cite "rotation" as the reason why this number should be > considered a fundamental mathematical constant in the same way that pi, > (and perhaps e and a few others) is viewed? I don't think so. Pi crops > up in so many places in applied mathematics that it would be difficult > not to assign it special status. Likewise for e. Sincerely, > > -- > J. B. Wood e-mail: email@example.com
Howdy. That was my first post to this forum and I'm glad to be here. I will elaborate on what I was saying...
The purpose of my post was in effort toward correcting misconceptions, or more accurately to improve less-than-optimal conceptions. I say it that way because I do not believe in the view that "Pi is wrong", as the current movement seems to like to say. On the contrary, I see Pi to be perfectly accurate. The problem with Pi is that it is not fundamental. And let's be clear that I'm not saying that Tau should be treated "in the same way as pi". My position is that Tau should be used *instead of* Pi. I am not aware of a single use of Pi where it would not be an improvement to replace it with Tau. This is also the position of the "Pi is Wrong" movement, however they go too far when they take the step of calling it "wrong". There is nothing illogical about Pi. It simply is not fundamental, and there is nothing "wrong" with not being fundamental. That would be like saying "Water is Wrong" because you learned that its constituent parts are Hydrogen and Oxygen.
You say that "Pi crops up in so many places", but the position of this movement is that it is not Pi that has cropped up. It is Tau, and we have contorted that by putting it into terms of Pi instead of the fundamental number.
And while were on the quest for improvement, it would be excellent to scrutinize terminology like "mathematical constant". Yes, it is accurate to say that Pi is a mathematical constant. But so is absolutely any other number! Pick a number. Be it complex, rational, whatever - that number will never change. It remains constant, and belonging to the realm of math it is therefore a mathematical constant. What makes Tau special is not that it is a mathematical constant, but rather that it is a Geometric Constant of Proportionality for the Circle.
Now that is a mouthful to say. And it is totally understandable that people will want to abbreviate the label associated with the concept. Boiling it down to "mathematical constant" is to evaporate its essence. A much more sensible abbreviation would be "Constant of Proportionality", or "Geometric Constant". The problem arises when trying to categorize Pi with other special numbers like e and Phi. While Phi is likewise a Geometric Constant of Proportionality, e is not. What makes e special is that when used as an exponential function it becomes Invariant to Differentiation/Antidifferentiation. It is the Eigenvector of Integral Calculus. Clearly it would be inaccurate to lump e in under the label of Constant of Proportionality. A good general term for these special numbers is just that: Special Numbers.
To label them "Mathematical Constants" is to contort the term 'constant'. This is the same essential problem with using Pi instead of Tau. While it is accurate to say that Tau, Phi and e are all mathematical constants, it completely misses the fundamental essential quality of this special set of numbers. And the reason why the term "Special Numbers" can be improved on is because that label does nothing to indicate *why* these particular numbers are special.
Once it is recognized that the reason why this set of numbers is special, then it becomes easy to arrive at terminology that is both accurate and conveys their essential quality. Tau, Phi and e are special because they form invariants under particular operations. And the quality of an Input passing through an Operation forming an Output with an unchanged quality is captured by the word "Eigen". This word has several connotations in English, such as inherent/inborn/particular/specific/proper. And because no single English word fully captures the concept, it was decided to carry this term straight over from the original language.
Tau, Phi and e all have this Eigen Fundamental Characteristic to them. A rotation/antirotation by Tau is returned unchanged. A differentiation/antidifferentiation of exponentiated e returns with its direction(/antidirection) intact. So instead of calling these numbers by the bland-to-the-point-of-meaninglessness Mathematical Constants, they can be collectively known as:
And as stated in my first post, the Special Numbers of Zero and One have this quality as well. Zero has this Eigen Property for the Operation of Addition/Antiaddition, as does One for the Operation of Multiplication/Antimultiplication. Now while it might be tempting to lump 0 & 1 into this group of Eigen Numbers because of this fact, it would be a mistake to do so because 0 & 1 have much greater special qualities than this Eigen Property. It is similar reasoning as to why 1 is not counted as a Prime Number, even though its only factor is One(/Itself). Rational Numbers have an inherent Dimensionality to them, as identified by their Prime Factorization. The Number 4 (2x2) and the Number 6 (2x3) have a Dimensionality of 2. The Number 8 (2x2x2) is the first number with a Dimensionality of 3. The Number 12 (2x2x3) also has a Dimensionality of 3. The Number 16 is the first with a Dimensionality of 4. The reason why Prime Numbers are special is because they are the only numbers with a Dimensionality of 1. They are the "Building Blocks" of other numbers, if you will. The Number 1 is not counted as a member of that group because it has a quality that is *more fundamental* than Primacy. That quality is Unity, and has a Dimensionality of Zero. It is the "Building Block" of all the Prime Numbers. Unity is the fundamental quality that makes it more special than the rest of the numbers that have this Prime Quality.
And using this understanding of why the Number One is more special than all of the other numbers that exhibit a Prime Quality, it sheds light on why Tau is more special than Pi. Both Tau and Pi share the quality of maintaining a Circular Constant of Proportionality. And as there are an infinite set of Prime Numbers, there are also an infinite set of multiples and fractions of Tau that all have this Circular Constant Quality. But from that entire infinite set, it is only the multiples of Tau that share the stronger, more special quality of being Invariant Under Rotation. Tau has this Eigen Property. Pi does not. I see this as an open and shut case. And the remedy is:
Out with Pi. In with Tau.
I stop short of saying "Pi is Wrong", as much as I understand the purpose behind the hyperbole. This problem is as basic as a hypothetical case where conventional wisdom within the math community holds the Number One as included as a Prime Number, and no one recognizes its Unity Quality as giving it a status that is much more special than the Set of Primes. Tau has a fundamental quality that Pi does not. It is a Special Number that happens to be more special than Pi, in a similar way to how 1 is more special than 2, 3, 5 or 7. Tau is the Eigen Number that returns invariance under rotation. It is the Identity Operator for Rotation.
We have now rolled into the year labeled as '2013'. We are approaching a full dozen years since Bob Palais went public with his paper. During that period we went through two years, 2003 and 2011, that are Prime Numbers. The next will be 2017. There is unwavering consensus that the Number 1 is more special than the set of Prime Numbers. Is it possible that by the year 2017, a scant four years from now, that understanding about Tau and Pi will spread to the point where there will then be a solid consensus that Tau is the fundamental number to be used? With the power of today's internet and how knowledge can spread virally, I expect that four years would mark the slow track. We can get there much more quickly. Instead of drawing the veil of our ignorance slowly, pull by gentle pull, it can be lifted in one swift motion of exponential growth in human consciousness for this one particular issue.
This thread is my contribution toward the swell of that wave. Truth has a power that is unstoppable.