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Pi day (3/14)

Today is the Pi day, March 14th, i.e. 3/14 in the American notation. As a Mathematician and Statistician, it is difficult for me not to marvel about the ways in which special numbers like \pi appear in many different, apparently unrelated places in Maths, Stats and real life.

\pi is of course a ratio of a circle circumference and diameter.

The Wolfram’s Mathworld lists many different formulas, either infinite series or products, that produce the number \pi. I found some of these formulas amazing and some frankly bizzare. There is a part of my scientific background that is deeply rooted in Mathematics (thanks to my lecturers at the Jagiellonian University where I did my undergraduate and PhD studies) that causes me to stop and think, why:

\frac{2}{\pi}=\sqrt{\frac{1}{2}}\times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}} \times\ldots

for example?

To me, there is an intrinsic beauty in such mathematical formulas, similar to art, e.g. Horse riders by Escher, or nature:

There is another, much more chaotic, side of \pi – the apparent unpredictability of its digits. \pi is supposedly a normal number in that if in its sequence of digits, all digits and all subsequences of digits are equally represented. Although it has not been proven that \pi is indeed normal, this property seems to be well supported. In other words, if I tell you that the fifth digit is 5, each of the 10 digits (0, 1, 2, …, 9) can come next and so I cannot really conclude that sixth digit is more likely to be 9 (3.14159…).

If we consider a ‘random’ walk governed by digits of \pi, i.e. at each time step a particle moving either to the left or to the right by how far the current digit is from the mean of 5:

  • if the digit is 3, move by 2 to the left;
  • if the digit is 1, move by 4 to the left;
  • if the digit is 4, move by 1 to the left;
  • if the digit is 5, move by 1 to the right;
  • if the digit is 9, move by 4 to the right;

we get the following graph:

which looks random!

I find it really amazing that there is so much to a humble number \pi and that it shows so much simplicity and at the same time so much complication, so much structure, and at the same time so much randomness and unpredictability.

If you want to see more information on the randomness aspects of \pi, please see the slides from my public lecture given at the University of Stirling in 2013. You can find it at SlideShare.

I better stop now, as it is time for lunch:

Broccoli picture by Jon Sullivan - http://pdphoto.org/PictureDetail.php?mat=pdef&pg=8232, Public Domain, https://commons.wikimedia.org/w/index.php?curid=95997

Winter

We have just had a spell of very snowy weather in the UK and this has brought back memories of the 2010-11 winter:

We have not had such a winter (at least in most of the UK) in the seven years interval between 2010-11 and 2017-18. Six years without snow, followed by a winter with a lot of snow…

Those of us who read and enjoyed The Long Winter by Laura Ingalls Wilder remember the scene in chapter 7 when Mr Harthorn’s store is visited by an old Indian:

“You white men,” he said. “I tell-um you.”

He showed seven fingers again. “Big snow.” again, seven fingers. “Big snow.” Again seven fingers. “Heap big snow, many moons.”

(…)

The Indian meant that every seventh winter was a hard winter and at the end of three times seven years came the hardest winter of all. He had come to tell the white men that this coming winter was a twenty-first winter, that there would be seven months of blizzards

Indeed, Laura Ingalls Wilder described in her book the famous 1880-1881 winter which is considered the most severe winter in the United States ever. So, given that in recent years we in the UK had six years without much snow followed by a snowy and cold winter, is there any evidence for the seven years cycle?

Unfortunately, this does not appear to be the case. The MetOffice provides a wealth of historic records, for example, air temperature, air frost and rainfall data for a number of weather stations in the UK. Choosing Braemar in the Cairngorms in the heart of Scottish Highlands, we get the following graphs for the minimum yearly temperature:

and for the maximum number of air frost days (not necessarily consecutive):

Note that 2017-18 winter is really not included, as the data include December 2017 but exclude January 2018. Green lines show the 7 years interval. In addition, the red lines show very snowy winters in the UK following the Snowfall catalogue by Bonacina and O’Hara. More about blue lines below.

It is very clear that although 2010-11 winter had some snow and cold weather, it is 2009-10 winter that stands out. Moving back 7 years, 2002-03 and 2003-04 were quite mild and we need to go back to 1984-85, 1978-79 and 1962-63 to get similar temperature and frost records and snowfall. As shown by blue lines in the plots above, this might suggest a 16 years cycle. This cycle would include 2000-01 winter that was actually quite cold, although only with the average snowfall in the UK; it also points to 2010-11 rather than 2009-10 and predicts 2026-27 as the next ‘big snow’.

A statistician would point to many problems with the above analysis, as it reflects our ability – and probably a need – to find patterns in random data.

Instead, we should be looking for more scientific explanations. There are many hypotheses why 2009-10 was such a cold winter and whether this reflected any cycles. Solar activity (22 years), the North Atlantic Oscillations (no fixed pattern, but 12 years cycles suggested). It is too early to say whether 2017-18 will also prove a really cold winter and why it has so far produced a lot of snowfall in the UK.

In addition to trying to satisfy our scientific interests, we can always simply enjoy the cold and snowy weather: