Tau- twice the Pi

June 28th, 2016 Sections: Calendar, Maths-Stats
Tau = 2pi

τ  (Tau)  = 2pi = 6.28…

I have never understood why, but the Americans write dates in the format day/month/year so the tragic events of September 11th, 2001, are universally known as 9/11 … and everyone understands what is being referred to.

For Mathematicians this means that 14th March is often known as Pi Day, because the date is written as 3/14 and the first few figures of the Mathematical constant π are 3.14… it actually goes on and on – ad infinitum, without end and without repeating itself.  It is possible to find on the internet the first 10,000 digits of π. the first 100,000 digits, the first million digits … the first four million digits … the first 50 million digits … the first four billion digits … even, the first five trillion digits ….

OK – I can hear a lot of people asking: So  what?  Who on earth would want to to know the value of Pi to five trillion digits? … and I have some sympathy with that question … but there are some people who do … and pi is actually a fundamental concept in Mathematics which crops up all over the place … and although it is defined as the ration between the Circumference of a circle and it’s diameter … it crops up all over the place – even where circles are nowhere to be found.

In 2015, there was more attention than usual given to Pi Day … because the next few digits in the value of Pi are 15, (… π=3.1415926535897932… ), and the date was 3/14/15 … some people even went so far as to define a Pi-moment: 9:26am and 53 seconds on March 14, 2015 …that is:  3/14/15 – 9:26:53.

Now, in case anyone wants to take me to task … I am well aware that in ‘normal’ “mathspeak”, 3/14 means three fourteens. or three divided by 14 … or 0.21428571463… it also goes on and on – ad infinitum, without end, but this number does enter into a regular cycle of repeatng digits.  Different people use a variety of separators between month, day and year – there is no reason why uit has to be a backslash (/) and could just as easily be a dot – or, as the Americans like to call it, a period (.) so, the fourteenth of March could easily be 3.14

Of course – that gives us a different problem with our Pi-moment … because there are two decimal points … and a couple of colons … but, hey, what’s a couple of decimal points between friends?

I remind myself that it doesn’t really matter … it’s only a convention.  There’s nothing special about March 14th; no metaphysical or mystical connection between pi and that date … like there is between the diameter of a circle and its circumference.

If we were catoon characters, and only had four fingers on each hand, then we would probably use Octal (base 8) as a counting system … using just the digits 0, 1, 2, 3, 4, 5, 6, and 7 …

… and, yes, there I am using the Oxford comma – another convention … but not one that is generally accepted.

Pi would still exist … but it wouldn’t be 3.14159265… it would be 3.1103755242… and Pi-day, (Octal Pi-Day, that is), would be March 11th.

If anyone wants to question my sums .. don’t worry – it’s been done already … for example here and here.

Mathematicians have other ways of describing pi …

Histotically, before decimals were invented, mathematicians had other ways of expressing it … for example as common – or vulgar – fractions:

\begin{equation} \frac31,\frac{22}7,\frac{333}{106},\frac{355}{113},\frac{103993}{33102}, \frac{104348}{33215},\frac{208341}{66317},\frac{312689}{99532} \end{equation}

None of them actually equal pi … but, then, neither does 3.1415926… or 3.1103755242… – they are all approximations … and the further along the sequence the close the approximation gets.  The fourth term in the sequence was known to the ancient Chinese and is actually a better approximation than 3.14159265.  They worked it out using one of my favourite approximations comes in the form of a ‘continued fraction’:

\pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\ddots }}}}}}}}}}}}}}

… but it’s much more satisfying in these formats:

  \pi =\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\ddots }}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\ddots }}}}}}}}}}=\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\ddots }}}}}}}}}}

 

Here’s another one named after the great German Mathematician Gottfried Leibnitz:

\pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots

… or, how about this one one from the 15th century:

\pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots

… but that’s just scratching the surface …

 

Although most people have heard of π … not everone has heard of Tau (τ) which is a similar concept: the ratio between the circumference and the radius of the circle (as opposed to the diameter).  It’s supposed to be more intuitive, easier to understand, make the formulas easier and ties in conveniently with the Mathmatical concept of Circular Measure – or radians. … but, personally, I have my doubts … and I think it has as much chance at catching on as the draft law proposed in Indiana in 1897 … to define the value of pi as 3.  (Yes, there really was an attempt to pass such a law … which, fortunately, never made it to the statute book … three might be a good enough value for a rule of thumb aimed at getting a quick estimate when performing a calculation involving pi … but introduces an error which can very quickly lead to serious discrepancies between the ‘estimate’ and reality.)

For the record: τ works out to be tice the size of π – 6.283185 … which makes the 28th June, (or 6/28 in the American convention for writing dates), Tau Day.  So:

Happy Tau Day!
28th June

 

For the record: 28th June is also interesting because it’s Perfect Day.  Both 6 and 28 are perfect numbers – their factors add up to the number itself … in the case of 6, for example, the factors are 1, 2, and 3, (we don’t count the special case of the number itself) and whem they are added together …. the result is equal to thr number itself – 6 … and the same is true for 28, (the factors:  1, 2, 4, 7, and 14 – add up to 28), but there are very few ‘perfect numbers’.  The next two are 496 and 8128 … They appear to be more and more scarce as

 

Is it half full? ... or half empty?

Is it half full? … or half empty?

Although today’s postcard is not specifically about Kyrgystan, Tau Day reminds us that the way we see things, the way we understand things all depends on our viewpoint, on how we look at things … like in the old adage about the optimist and the pessimist looking at a glass of water … is it half empty … or half full?

…  when faced with a situation, is it a problem? … or an opportunity?

We need to remember that we all see things in different ways … but that doesn’t mean to say that one person’s view is right and another’s is wrong.  Oh, it could well be that one is mistaken … and the other not … but, all to often it’s a case that both have a valid point of view … and we need to try and understand that point of view … why they have it … and where it it leads.

When Max, for example, takes a photograph, then it can make a difference to the final result which lens he uses … two photographs of the same scene can look quite different … he achieves different effects … but that doesn’t necessarily mean that one is better than the other.

It’s a lesson that we often forget …

 

 

 

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There is one comment. to “Tau- twice the Pi”

  1. ian
    June 30th, 2016 at 07:41
    1

    Just one of several posts to commemorate Tau Day and explain why some Mathematicians think it is a better choice than Pi:

    http://www.livescience.com/55209-tau-is-better-than-pi.html

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