Things that equal Pi

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piSo 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

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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:

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  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 […]

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