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$(\pi+2)$-Day: Buffon's Noodle

Maxim Arnold and Nathan Williams
Department of Mathematical Sciences
University of Texas at Dallas
$\pi$: the ratio of the circumference of a circle to its diameter.

Mathematics: an Experimental Science

In 1773, Georges Louis Leclerc, le Comte de Buffon devised a
mathematical experiment...
Buffon's Needle

To perform this experiment at home, you'll need:
• Needles. (Preferably substitute something less sharp.)
• A grid of equally spaced parallel lines (unit distance apart).
Buffon's Needle

How often does a randomly tossed needle cross a line?
Call this probability $E$ (depends on length $\ell$ of a needle).
Now compute
$\frac{2}{\text{average number of crossings}} \approx \frac{2}{E}$
Now compute
$\frac{2}{\text{average number of crossings}} \approx \frac{2}{E} = \pi?$
Buffon's Needle

By breaking a needle in two, one checks that $E=C \ell,$ where $\ell$ is the length of the needle and $C$ is some constant.

But how can we determine this constant?
Buffon's Noodle

$E=C\ell$ only depends on $\ell$, and doesn't change for a "noodle".
Buffon's Noodle

$E=C\ell$ only depends on $\ell$, and doesn't change for a "noodle".
Since we can approximate a noodle by straight needles.
Buffon's Noodle

$E=C\ell$ only depends on $\ell$, and doesn't change for a "noodle".
Use a circular noodle of unit diameter.

The length $\ell$ of such a noodle is our definition of $\pi$.

Such a noodle will always have exactly two intersections:
Use a circular noodle of unit diameter.

The length $\ell$ of such a noodle is our definition of $\pi$.

Such a noodle will always have exactly two intersections:

Since $E = C \ell$, we can now determine our constant $C$: $2 = C \pi, \text{ or } C = \frac{2}{\pi}.$
We conclude that if $\ell=1$
$\frac{2}{\text{average number of crossings}} \approx \frac{2}{E} = \frac{2}{C\ell} = \frac{2}{\pi/2} = \pi.$
Other values of $\pi$...

Other values of $\pi$...

How to compute (digits of) $\pi$?

Approximating $\pi$ using Buffon's needles is slow. There are more efficient ways....
• Inscribe/circumscribe regular polygons in a circle.

• Infinite series.
Some famous formulas
$\frac{\pi}{4} = \sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$ $\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)$ $\frac{\pi}{2} = \frac{ 2\cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots}{ 1\cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdots}$ $\frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{2+\sqrt{2}}}{2}\cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \cdots$
Thank you!

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