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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

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

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

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

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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