## Things that equal Pi

by

So you want to make a pie for Pi Day, but you don’t want to decorate it with the traditional symbol $\pi$.  What other expressions could you use that are equivalent?

You could go with the elegant:  a picture of a circle and the ratio of the circumfirence to the diameter

$\frac{C}{d}$

In a similar vein, you could move up a dimension to area

$\frac{A}{r^2}$

or volume $\left(\frac{3V}{4r^3}\right)$, although in this case you’d have to draw a sphere and I can tell you right now that I’d lose points for clarity.

If geometry isn’t your thing, you could decorate your confection with an infinite sum, perhaps the Madhava-Gregory-Leibniz series (discovered by Madhava of Sangamagram, India about 600 years ago, and then rediscovered by James Gregory of Scotland and Gottfried Wilhelm Leibniz of Germany 200 years later)

$\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots$

or the slightly more complicated

$\sqrt{\frac{6}{1}+\frac{6}{4}+\frac{6}{9}+\frac{6}{16}+\cdots}$

found by Leonard Euler of Switzerland in 1735.  Or even the Bailey-Borwein-Plouffe formula (which is, face it, kind of fun to say) that was discovered only 14 years ago(!) by Simon Plouffe of Quebec, Canada:

$\displaystyle\sum_{k=0}^{\infty}\frac{1}{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)$

Incidentally, Simon Plouffe and Neil Sloane are the authors of the Encyclopedia of Integer Sequences, which gave rise to the online version.

But back to $\pi$.  Do you prefer products?  Then maybe you’d want to turn to Wallis’s product, discovered by John Wallis of England in 1655:

$2\cdot\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdot\cdots\right)$

We’ll end on a more radical note:  the Viète formula, which was named after François Viète of France, but actually found by Euler.

$2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdot\cdots$

Tags:

### 10 Responses to “Things that equal Pi”

1. More Fun With Pi « Let’s Play Math! Says:

[…] Things that equal Pi […]

2. PI day 2009 - The Chaotic-Neutron Says:

[…] Things that equal Pi. Btw, the 360 blog has some really interesting posts and you should definitely subscribe to […]

3. Michelle M. Says:

Oh my god – your blog scares me.

4. Pi Day « Every Day’s a Holiday Says:

[…] Because Pi Day coincides with Einstein’s birthday, many treat March 14 as a celebration of science and mathematics in general. I’ve noticed a dearth of these festivals. Religious and political holidays fill each day of the calendar several times over while the scientifically minded find themselves forced to rally around a handful of dates like “Square Root Day” (3/3/09…) and “Mole Day” (June 2 at 10:23). Not to mention our beloved Pi Day. [https://threesixty360.wordpress.com/2009/03/13/things-that-equal-pi/] […]

5. Ξ Says:

Hi Michelle! Here’s the real question: is this post more or less scary than the one that follows?

6. The Carnival of Π « JD2718 Says:

[…] The day before she put up some suggestions for Π day activities, “Things that equal π“ […]

7. Michelle M. Says:

It is National Potato Chip Day as well. Here’s some math: If Michelle eats a whole bag of chips (at on sitting), how much weight will she gain?

8. N Says:

Look up “pi rap battle” on Youtube!

9. Pi Day & Albert Einstein Says:

[…] Because Pi Day coincides with Einstein’s birthday, many treat March 14 as a celebration of science and mathematics in general. I’ve noticed a dearth of these festivals. Religious and political holidays fill each day of the calendar several times over while mathematicians and scientists find themselves forced to rally around a handful of dates like “Square Root Day” (3/3/09…) and “Mole Day” (June 2 at 10:23). Not to mention our beloved Pi Day. [https://threesixty360.wordpress.com/2009/03/13/things-that-equal-pi/] […]

10. Happy Pi Day II | Let's Play Math! Says:

[…] Things that equal Pi […]