Tuesday, 13 March 2018

Pi Day 2018 –PrIme Numbers and PI

Pi Day 2018 –PrIme Numbers and PI

Anyone who studies even a bit of advanced mathematics soon discovers the fact that the number pi (3.14159…) shows up all over the place.  Sometimes that seems fairly easy to understand - e.g. the unit circle is fundamental to trigonometry, so it seems natural that pi should show up there.  Other times, it seems more mysterious – for example, number theory is concerned with prime numbers, which are integers, so why does pi show up so often in that subject?  But it does.

Anyway, here’s an example from a book that I own, called “Recreations in the Theory of Numbers”:

If you take any two random numbers and compare them, the probability that they are relatively prime turns out to be = 6/pi**2. 

I won’t bother with a proof of this claim – no doubt you can find a reference on wiki or some other source.  But I did think it would be interesting to test it experimentally, so to speak.

This was a lunch hour exercise at work, so I relied on an Excel hack.  No doubt a more elegant method could work better and would be more robust.  Here’s the method:

  • First, I set up a spreadsheet model, with two random numbers between 1 and 200.
  • Then, I then tested all of the numbers between 1 and 100 with the modulus function, to see whether they divided evenly into the random numbers chosen.
  • If the two numbers had a common divisor, I noted that.  Basically, if the sum of any given modulus of the two numbers was greater than 0, that indicated that they shared a factor, and therefore were not relatively prime.  Otherwise, they were relatively prime
  • Then, I copied the columns that did the job for those numbers, such that I could test several thousand such pairs in the same worksheet (Case 1 tested 1152 pairs of numbers, while Case 2 tested 2304 pairs).
  • I counted up the number of pairs that were relatively prime, and divided that into the number of pairs tested.
  • Then, I compared that to the value of 6/pi**2.
  • With a little algebra, that could be used to estimate the value of pi, via this purely statistical number theory method.
  • Then, I repeated that process one hundred times, recording the results, and graphing them (Case 1).
  • I repeated the process, doubling the number of pairs tested, and recorded those results (Case 2).

·       Yes, I know this model could have been made a lot more efficient (e.g. no point testing all the numbers between 1 and 100), but I just wanted to get some quick results for Pi Day, so I didn’t slave over the details.

Below are the graphs of the results:

Case 1 - In this case, the mean value computed for pi over the 100 runs was 3.135 (to three decimal places), with a standard deviation of .040 and a median of 3.131.  The graphs is “sort-of” normal, but with a lot of deviation from a Gaussian.

Case 2 - In this case, the mean value computed for pi over the 100 runs was 3.129 (to three decimal places), with a standard deviation of .026 and a median of 3.127.  Again, the graph is “sort-of” normal, but with a lot of deviation from a Gaussian.

As you can see, the histogram is tighter in case 2, reflecting the fact that there were twice as many pairwise comparisons for each run.

It is interesting that both runs underestimated the true value of pi, by about 0.3% in one case and about 0.5% in the other case.  I don’t know if that is just a fluke, or if there is some subtle bug, either in my implementation or in Excel’s random number generator.  I have often had my doubts about the latter.

At any rate, it seems clear that this method converges slowly to the true value of pi.  In the days before computers, it would have been quite a challenge to generate the data.  Now, though, one could run a test like this a few million times (e.g. in Python or SPSS) and probably get quite a reasonable estimate for pi.  But it is interesting that pi can be estimated from a method that has no geometrical basis, at all.

And here’s a picture of one of my wife’s (and SF writer) Pi Day pies – pumpkin pie, which is scrumptious with whipped cream and a great favorite of my data warehouse and data science colleagues at the office.

Recreations in the Theory of Numbers, Albert Beiler, Dover

So, now that you have done some math, you should read a science fiction book, or even better, a whole series.  Book 1 of the Witches’ Stones series even includes a reference to pi.:

Kati of Terra

How about trying Kati of Terra, the 3-novel story of a feisty young Earth woman, making her way in that big, bad, beautiful universe out there. 

The Witches’ Stones

Or, you might prefer, the trilogy of the Witches’ Stones (they’re psychic aliens, not actual witches), which follows the interactions of a future Earth confederation, an opposing galactic power, and the Witches of Kordea.  It features Sarah Mackenzie, another feisty young Earth woman (they’re the most interesting type – the novelist who wrote the books is pretty feisty, too).

The Magnetic Anomaly: A Science Fiction Story

“A geophysical crew went into the Canadian north. There were some regrettable accidents among a few ex-military who had become geophysical contractors after their service in the forces. A young man and young woman went temporarily mad from the stress of seeing that. They imagined things, terrible things. But both are known to have vivid imaginations; we have childhood records to verify that. It was all very sad. That’s the official story.” 

The Zoo Hypothesis or The News of the World: A Science Fiction Story

In the field known as Astrobiology, there is a research program called SETI, The Search for Extraterrestrial Intelligence. At the heart of SETI, there is a mystery known as The Great Silence, or The Fermi Paradox, named after the famous physicist Enrico Fermi. Essentially, he asked “If they exist, where are they?”.

Some quite cogent arguments maintain that if there was extraterrestrial intelligence, they should have visited the Earth by now. This story, a bit tongue in cheek, gives a fictional account of one explanation for The Great Silence, known as The Zoo Hypothesis. Are we a protected species, in a Cosmic Zoo? If so, how did this come about? Read on, for one possible solution to The Fermi Paradox.

The short story is about 6300 words, or about half an hour at typical reading speeds. 

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