One big happy family

An enlightening discussion about pi and tau.

One big happy family

by Kodegadulo » Sun Jun 10, 2012 7:04 am

Which constant, [tex]\pi[/tex] or [tex]\tau[/tex] should be THE circle constant? What a silly question! Who says there must be only one circle constant? Why not have an entire family of circle constants, including both [tex]\tau[/tex] and [tex]\pi[/tex]? The real question is, which circle constant should be the fundamental one, from which the rest of the family is derived? Well, [tex]\tau[/tex] is the upper limit of the set [tex][0,\tau)[/tex] which we can define as the set of all "proper angles", i.e., the minimal set which represents every possible direction or orientation. Every other real value is congruent with some value in this set, via the operation [tex](\theta\ mod\ \tau)[/tex]. So there can't be any interesting circle constant greater than [tex]\tau[/tex], because it would just be congruent with some other interesting constant between [tex]0[/tex] and [tex]\tau[/tex]. So if [tex]\tau[/tex] is a candidate for "fundamental circle constant", it must be the largest possible one. That would make any other interesting circle constants some sort of fraction of [tex]\tau[/tex]. It pretty much follows that [tex]\tau[/tex] must be the more fundamental.

I propose the following convention: [tex]\forall n \in \mathbb{N}^*[/tex], let [tex]\tau_{n} = \frac{\tau}{n}[/tex]. Some of the more interesting examples:

  • [tex]\tau_{1} = \tau =[/tex] the full-circle constant
  • [tex]\tau_{2} = \frac{\tau}{2} = \pi =[/tex] the semicircle constant = straight angle = "semitau"
  • [tex]\tau_{3} = \frac{\tau}{3} =[/tex] the tertiant-circle constant = triangle exterior angle = "tertiantau"
  • [tex]\tau_{4} = \frac{\tau}{4} = \eta =[/tex] the quadrant-circle constant = right angle = square exterior angle = "quadrantau"
  • [tex]\tau_{6} = \frac{\tau}{6} =[/tex] the sextant-circle constant = hexagon exterior angle = "sextantau"
  • [tex]\tau_{8} = \frac{\tau}{8} =[/tex] the octant-circle constant = diagonal angle = octagon exterior angle = "octantau"
  • [tex]\tau_{12} = \frac{\tau}{12} =[/tex] the unciacircle constant = dodecagon exterior angle = "unciatau"

This way, [tex]\pi[/tex]-ists should have no complaint about not being able to use [tex]\pi[/tex] as a single constant wherever they think it makes an equation more "elegant" or "efficient". However [tex]\pi[/tex] would appear in the guise of [tex]\tau_{2}[/tex], which [tex]\tau[/tex]-ists can read as [tex]\tau/2[/tex], and proceed to refactor the equation into a more "elegant" and "efficient" [tex]\tau[/tex] form.

In other words, you can all have your have your [tex]\pi[/tex] and [tex]\eta[/tex] too!
Last edited by Kodegadulo on Tue Jul 10, 2012 11:22 pm, edited 4 times in total.
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Re: One big happy family

by xander » Sun Jun 10, 2012 8:49 am

Kodegadulo wrote:In other words, you can all have your have your [tex]\pi[/tex] and [tex]\eta[/tex] too!

Ow...

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Re: One big happy family

by dimension10 » Sun Jul 01, 2012 11:20 pm

Kodegadulo wrote:Which constant, [tex]\pi[/tex] or [tex]\tau[/tex] should be THE circle constant? What a silly question! Who says there must be only one circle constant? Why not have an entire family of circle constants, including both [tex]\tau[/tex] and [tex]\pi[/tex]? The real question is, which circle constant should be the fundamental one, from which the rest of the family is derived? Well, [tex]\tau[/tex] is the upper limit of the set [tex][0,\tau)[/tex] which we can define as the set of all "proper angles", i.e., the minimal set which represents every possible direction or orientation. Every other real value is congruent with some value in this set, via the operation [tex](\theta\ mod\ \tau)[/tex]. So there can't be any interesting circle constant greater than [tex]\tau[/tex], because it would just be congruent with some other interesting constant between [tex]0[/tex] and [tex]\tau[/tex]. So if [tex]\tau[/tex] is a candidate for "fundamental circle constant", it must be the largest possible one. That would make any other interesting circle constants some sort of fraction of [tex]\tau[/tex]. It pretty much follows that [tex]\tau[/tex] must be the more fundamental.

I propose the following convention: [tex]\forall n \in \mathbb{N}^*[/tex], let [tex]\tau_{n} = \frac{\tau}{n}[/tex]. Some of the more interesting examples:

  • [tex]\tau_{1} = \tau=[/tex] the full-circle constant
  • [tex]\tau_{2} = \frac{\tau}{2} = \pi =[/tex] the semicircle constant = straight angle
  • [tex]\tau_{3} = \frac{\tau}{3} =[/tex] the tertiacircle constant = hexagon internal angle
  • [tex]\tau_{4} = \frac{\tau}{4} = \eta =[/tex] the quadrant-circle constant = right angle
  • [tex]\tau_{6} = \frac{\tau}{6} =[/tex] the sextant-circle constant = equilateral triangle angle
  • [tex]\tau_{8} = \frac{\tau}{8} =[/tex] the octant-circle constant = diagonal angle
  • [tex]\tau_{12} = \frac{\tau}{12} =[/tex] the unciacircle constant = equilateral complement angle

This way, [tex]\pi[/tex]-ists should have no complaint about not being able to use [tex]\pi[/tex] as a single constant wherever they think it makes an equation more "elegant" or "efficient". However [tex]\pi[/tex] would appear in the guise of [tex]\tau_{2}[/tex], which [tex]\tau[/tex]-ists can read as [tex]\tau/2[/tex], and proceed to refactor the equation into a more "elegant" and "efficient" [tex]\tau[/tex] form.

In other words, you can all have your have your [tex]\pi[/tex] and [tex]\eta[/tex] too!


The problem is that it will be very confusing for amateurs to suddenly switch from tau to pi.
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NOTE: I'm a high school, student, not kindergartener, unlike what my avatar says, but I can't seem to change it.
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Re: One big happy family

by xander » Mon Jul 02, 2012 9:07 am

dimension10 wrote:The problem is that it will be very confusing for amateurs to suddenly switch from tau to pi.

Who are you putting in the category of "amateurs?" If you are talking about students who have never been introduced to the topic before, there is no reason to assume that they will be confused, since they will never have seen it before. If you are talking about slightly more advanced students, they would not be forced to work with [tex]\tau[/tex], and would continue using [tex]\pi[/tex]. The transition needn't happen overnight, and anyone else can cope.

As I've said before, the best argument in favor of [tex]\tau[/tex] is that it is easier for a beginning student (i.e. a student who has never seen a radian or a trig function before) to understand. In that context [tex]\tau[/tex] is a pedagogical tool, rather than a mathematical tool. Mathematically, the choice of [tex]\tau[/tex] or [tex]\pi[/tex] is irrelevant.

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