Can you believe it? Grey Matters is 8 years old today, March 14th! How long is 8 years? When I started this blog, YouTube had been formed as a company, but it would be another month before they would publicly unveil their website.
Besides being this blog's 8th blogiversary, it's also Pi Day (3/14) and Albert Einstein's birthday, so let's have a little fun, shall we?
Mental_Floss.com helps gets the party started by sharing 11 unserious photos of Einstein. Yes, of course the famous tongue picture is there, but there are more with which you may not be familiar.
For a Pi Day party, we need food, and what better food than pies? Matt Parker shows us how to calculate Pi using pies:
If you're concerned about food being used in this way, Matt Parker assures:
Your next concern might be about the accuracy of the measurement, which Wolfram|Alpha gives here to 10 places. At first glance, 3.138 doesn't seem as impressive as it should be.
For the record: we were given the pies, they were no longer fit for consumption (thrown away after) and we donated £314 to a food charity.— Matt Parker (@standupmaths) March 11, 2013
However, if you remember last month's post on bringing pi digits to life, you'll recall that it only takes 38 digits to measure a universe-sized circle with an accuracy to the nearest hydrogen atom. Considering that, measuring a circle in terms of pies to 3.138 is less surprising, and is a considerably good result.
When I was doing research for my continued fractions post, I was thrilled to discover L. J. Lange's continued fraction of Pi, which he developed in his May 1999 paper An Elegant Continued Fraction for π:
As much as you hear about the randomness and unpredictability of Pi, this continued fraction has an astonishingly simple and regular pattern. The denominators, of course, are all 6. The numerators are the squares of all the odd numbers starting with one. In fact the numerator at any level n can be calculated with the formula (2n - 1)2. For example, the 6th numerator is calculated as (2 × 6 - 1)2 = (12 - 1)2 = 112 = 121. Using Gauss' Kettenbruch notation, we can then write this formula for Pi as:
How fast does this get us to the 38 digits for our universe-sized circle which measured to the nearest hydrogen atom?
We can use Wolfram|Alpha to get an idea. The first 10 levels of this fraction give us Pi to 4 digits (the integer part plus 3 decimal places). We get Pi accurate to 7 digits by the 100th level, and to 10 digits by the 1000th level.
Assuming this logarithmic rate of 3 digits for every order of magnitude continues, we would need to go tens of trillions of levels deep to get universe-level accuracy!
Alas, it seems the beauty of this formula's pattern is at the cost of slow convergence to Pi. Since the original formula takes the process to infinite levels, however, at least it gives Pi in the long run. If you're wondering how someone like Archimedes worked out Pi 2200 years ago, without textbooks, calculators, or even calculus, it's actually due to this ingenious approach described at BetterExplained.com. Note that after taking his own approach 96 levels deep, Archimedes also calculated Pi accurate to only 4 digits.
Naturally, many others are celebrating Pi Day today. Check out Ben Vitale's Some Musings on Pi, both part 1 and part 2. The Math Dude podcast also took some time to celebrate the world's best known mathematical constant. One of the more amusing moments in Pi history was the time that the Indiana almost legislated the value of Pi to be exactly 3.2, and James Grime tells the story well.
Thanks to all my readers for reading Grey Matters and keeping this blog going for 8 wonderful years! Now, it's time for you to keep an eye out for what I have in store for my 9th year.