1292: Pi vs. Tau

2014-08-22T00:37:52Z

Junmtl: /* Explanation */

{{comic
| number = 1292
| date = November 18, 2013
| title = Pi vs. Tau
| image = pi vs tau.png
| titletext = Conveniently approximated as e+2, Pau is commonly known as the Devil's Ratio (because in the octal expansion, '666' appears four times in the first 200 digits while no other run of 3+ digits appears more than once.)
}}

==Explanation==

This is yet another of [[Randall]]'s [[:Category:Compromise|compromise comics]]. A few mathematicians argue as to whether to use pi, which is the ratio between a circle's circumference and its diameter, or tau, which is the ratio between a circle's circumference and its radius.

Some consider pi to be the wrong convention and are in favor of using tau as ''the'' circle constant (see the [http://tauday.com/tau-manifesto Tau Manifesto], which was inspired by the article "[http://www.math.utah.edu/~palais/pi.html Pi is wrong!]" by mathematician Robert Palais). Others consider proponents of tau to be foolish and remain loyal to pi (see the [http://www.thepimanifesto.com Pi Manifesto]). Of course, regardless of which convention is used, the change is merely in notation — the underlying mathematics remains unaltered. Still, the choice of pi vs. tau can affect the clarity of equations, analogies between different equations, and how easy various subjects are to teach.

Most people know π (pi) by the approximation 3.14, but do not know τ (tau) which, by definition, is twice as large as pi. Randall is suggesting using "pau", which is a portmanteau of "pi" and "tau", as a number situated, appropriately enough, halfway between pi and tau, i.e. 1.5 pi (or 0.75 tau). But of course his number would be inconvenient, as this value does not naturally turn up when working with circles or other mathematical constructs, so there are no commonly used formulas that would use pau.

The title text claims that pau can be approximated by e+2, as both values are roughly 4.7 — a similarity that holds little since it requires another irrational constant, {{w|E_(mathematical_constant)|e}}. It also attributes the nickname "Devil's Ratio" to pau, due to the sequence {{w|Number of the Beast|666}} supposedly appearing four times in the first 200 digits of pau when expressed in the {{w|octal}} base. However, this is not the case, and was likely due to an error in the computer system used by WolframAlpha; for more details see below.

==Transcript==
:[On the left is a "forbidden"-style slashed circle with the π symbol, captioned "Pi". On the right is a "forbidden"-style slashed circle with 2π, captioned "Tau". In the middle it reads 1.5π, captioned "Pau".]
:A compromise solution to the Pi Tau dispute

==Math details==
Possibly, [[Randall]] used [http://www.wolframalpha.com/ Wolfram|Alpha] to calculate the result (he uses it a lot, for example [http://what-if.xkcd.com/70/ What-if 70: The Constant Groundskeeper] or [http://what-if.xkcd.com/62/ What-if 62: Falling With Helium]).
However, when the comic was published, there was (and still is, as of April 29, 2014) a bug in Wolfram|Alpha so that, when getting 200 octal digits from "pau", it just calculates the decimal value rounded to 15 significant digits (this is 4.71238898038469) and expands that as octal digits as far as needed.

This gives a periodically repeating number. In the first 200 digits of the octal expansion, the sequences 666 and 6666 do occur, but each only once. There are 4 occurrences, however, in the first 300 digits:
<pre>
4.554574376314416445676661714336617116240444076666510533533077631151350452060436452476274022621206136310000177621674175071262255702044274154476005744176002676623042402346036604733130522524127534777714554305412763636566643022106616734723661726160312772574551366370203115523402704104015532221722772357666</pre>
Expansion that long indeed does contain 666 (the {{w|Number of the beast|number of the beast}}) four times (with one instance as 6666). It also contains 0000, 222, 444, and 7777, but they only appear once in a run.

{{w|Mathematical coincidence|Coincidentally}}, e+2 is also very similar to 1.5 pi, although only to a few digits.
<pre>
1.5π = 4.71238898038...
e+2 = 4.71828182845...
</pre>

==Trivia==
*For Pi the sequence '666' occurs for the first time at position 2440. Many more occurrences can be found here: [http://www.angio.net/pi/ The Pi-Search Page].
* Note that pau is Catalan for peace, which is a good solution for the pi/tau dispute.
* In the discussion it has been theorized that Randall used [[356: Nerd Sniping|Nerd Sniping]]. In which case he was aware of the mistake in Wolfram!
*For an entertaining introduction to the concept of tau, see this [https://www.khanacademy.org/math/recreational-math/vi-hart/pi-tau/v/pi-is--still--wrong Vi Hart video].

{{comic discussion}}
[[Category:Comics with color]]
[[Category:Math]]
[[Category:Compromise]]

explain xkcd - User contributions [en]

1292: Pi vs. Tau