Buffon’s Needle · Transport-First Worked Example

Scope: Worked Example
Mode: Measure Theory, Geometry, Transport
Primary Result: $P = \dfrac{2\ell}{\pi d}$ emerges from angular closure

1. Problem Definition (Formal)

Consider an infinite plane marked with parallel lines separated by distance $d$. A rigid needle of length $\ell \le d$ is placed on the plane with arbitrary position and orientation.

Define:

$x \in [0, d/2]$ — perpendicular distance from the needle midpoint to the nearest line
$\theta \in [0, \pi/2]$ — acute angle between the needle and the parallel lines

The needle intersects a line if and only if:

$$ x \le \frac{\ell}{2} \sin\theta $$

2. Measure-Theoretic Framing

The experiment samples the configuration space:

$$ \Omega = [0, d/2] \times [0, \pi/2] $$

with uniform measure $d\mu = dx\,d\theta$. No stochastic dynamics are assumed — this is a purely geometric sampling of transport degrees of freedom.

3. Crossing Region as Non-Closure Set

The set of configurations that fail to close between two adjacent lines is:

$$ \Omega_c = \left\{ (x,\theta) : 0 \le x \le \frac{\ell}{2} \sin\theta \right\} $$

This region represents angular orientations for which transported endpoints exceed the available transverse gap.

4. Exact Integral Derivation

The measure of the crossing set is:

$$ \mu(\Omega_c) = \int_{0}^{\pi/2} \int_{0}^{(\ell/2)\sin\theta} dx\,d\theta = \frac{\ell}{2} \int_{0}^{\pi/2} \sin\theta\, d\theta $$

Evaluating the angular integral:

$$ \int_{0}^{\pi/2} \sin\theta\, d\theta = 1 $$

Thus:

$$ \mu(\Omega_c) = \frac{\ell}{2} $$

5. Normalization and Emergence of π

The total configuration space has measure:

$$ \mu(\Omega) = \frac{d}{2} \cdot \frac{\pi}{2} = \frac{\pi d}{4} $$

The ratio of non-closing configurations to all configurations is therefore:

$$ P = \frac{\mu(\Omega_c)}{\mu(\Omega)} = \frac{\ell/2}{\pi d /4} = \frac{2\ell}{\pi d} $$

6. Interpretation Without Probability

The quantity $P$ is traditionally labeled a probability. In this framework it is more precisely:

a normalized measure of angular non-closure
a ratio of transport orientations violating spatial constraints
an average over uniform frame-transported rotation

No random variable evolves. No stochastic force acts. The result depends only on angular measure.

7. Why π Is Inevitable Here

π enters because angular transport closes only after $2\pi$ radians. Any problem that:

samples orientation uniformly
imposes linear spatial constraints
counts closure failure

will necessarily normalize by π. This is the same mechanism seen in:

circle circumference
random chord paradoxes
mean projection lengths

8. Transport-First Summary

Buffon’s Needle is not a statement about chance.

It is a precise geometric accounting of how often transported orientation fails to close within imposed linear boundaries.

π appears because closure in angle space costs π.