continued fraction

International “geek” holiday, Pi Day, started at San Francisco’s Exploratorium. 

Creator Larry Shaw, physicist, tells how.

From San Francisco to New York, in museums, universities, classrooms and in the privacy of one’s own home, people are celebrating Pi. 

It’s the anniversary of the celebration of Pi Day, an international holiday born at San Francisco’s Exploratorium. The number is Pi, 3.1415926535…ad infinitum. It’s today’s date and the number you get when you divide the circumference of a circle by its diameter, and it cannot be expressed as a fraction. It continues forever. 

In an era when math and mathematicians have become sexy again, it’s worth recounting how Pi Day came to be and why it is that people still go to the Exploratorium and gather around the Pi Shrine to perform pi-related rites and eat ritual food – be it apple pie or pizza pie – in honor of this special number. People sing Pi Day songs, bead a pi string (a physical manifestation of the never ending value of pi), and circumnavigate a pi shrine. Pi Day celebrations culminate, appropriately enough, on March 14 at 1:59pm. That’s the third month, the fourteenth day, at 1:59pm, corresponding to the first 6-digits of Pi. And as an added bonus, 3/14 is also Einstein’s birthday.

The original Pi guy is Larry Shaw, a physicist with streaming white hair, a white beard and a transcendent glow. It was 1987, Shaw was thinking a lot about the concept of rotation into another dimension – the sorts of things he was actually paid to do. Pi represents the relationship between one dimension to another in the sense of the linear dimension and the plane; or the relation of the linear dimension and the sphere. So for Shaw, Pi was in the air and definitely on his mind. He and his colleagues were talking about a Pi Shrine or a Pi Day, something to make the concept of rotation noteworthy. And so it all came together. 

For the first Pi Day, they installed a Pi Shrine (a small brass plate engraved with pi to a hundred digits) at the exact center of a circular Exploratorium classroom, a spot that also corresponds to the center-line of the museum’s building. And they walked around the shrine because as Shaw notes, “People go around things to show respect to them in many cultures and religions.” And they ate pie.

It wasn’t until 1989 at the 3rd Pi Day, that the overlap with Einstein’s birthday was uncovered by Shaw’s daughter Sara. She was writing a report on Einstein and told her dad that Pi Day – 3/14 – was also Einstein’s birthday. Voila. With all that mathematical kismet going for it, Pi Day gradually took on an international life of its own.

Today the public comes to circumambulate the Pi shrine approximately 3.14 times. And, bonus! We eat pie.

Continued fractions for the square root of 13

Yesterday we looked the equation x2-13y2=1 and used the continued fraction of √13 to find solutions. We did not specify how to construct such a representation of √13. For certain numbers you can make them, with a bit of perseverance.

Let us consider √13. The first thing we do is note that 3<√13<4 and work with √13-3 rather than √13. Next we note that (√13-3)(√13+3)=4. We can (re)write this equality as

But then you can replace the √13-3 in the denominator by the whole right-hand side, and again, and again, …, this leads to this continued fraction

That is not yesterday’s continued fraction but you can use it too to find approximations of √13. If you add 3 to the convergents of this fraction then you get 11/3, 18/5, 119/33, 393/109, 649/180, …. This is a subsequence of yesterday’s approximations. The solutions of x2-13y2=1 and 13y2-x2=1 can also be found in this sequence.

Can we use this to make yesterday’s continued fraction? Yes, all you need is pencil and paper and the first fraction, 4/(6+(√13-3)) that is. Divide numerator and denominator by 4; the numerator becomes 1 and the denominator will be 6/4+(√13-3)/4; this you can rewrite to 1+(√13-1)/4. Next: divide the numerator and denominator of (√13-1)/4 by √13-1; we get 1 and 4/(√13-1), respectively. But, as above, we observe that (√13-1)(√13+1)=12 and so 12/(√13-1)=(√13+1), or 4/(√13-1)=(√13+1)/3 which we can turn into 1+(√13-2)/3. The results of these two steps are displayed below:

If you keep doing this you will after two steps more arrive at the fraction on the left below. Now we read our initial equality as (√13-3)/4=1/(6+(√13-3)) and this gives of the fraction below on the right

But now we can replace (√13-3) by that fraction on the right, and again, and again, … and this leads to yesterday’s continued fraction.

Exercise Do the same for √2-1 and √3-1 and construct approximations of √2 and √3 in this way, and thus also find solutions to the Pell equations x2-2y2=±1 and x2-3y2=±1.

Math Mistake at Boston Museum of Science Found by a Teenager

15-year-old Jordan Rosenfeld found a mistake in the Boston Museum of Science’s exhibit “Mathematica: A World of Numbers… and Beyond.” For those of you who have been to the museum, you know this exhibit is not new at all and has been around for nearly 35 years. Still, the young man realized there were minus signs rather than plus signs in one of the formulas for the Golden Ratio. After leaving a note about the error, he was contacted by the museum and told they will fix the mistake.

I can’t find an article that says exactly which formula had the mistake, but I’d guess it was either the continued fraction expression

or the continued square root expression

I would be surprised if a high school student recognized a mistake in the infinite series expression, but then again, the error has been there for 35 years…