Borsuk-Ulam Theorem (1D)

Theorem: For any continuous function $f$ on a circle, there exist two antipodal points with equal values: $f(\theta) = f(\theta + \pi)$.

How this works:

1. Draw a curve below — think of it as sketching a function.
2. The x-axis becomes angle $\theta \in [0, 2\pi]$, the y-axis becomes $f(\theta)$.
3. We interpolate your points into a continuous function and find where $g(\theta) = f(\theta) - f(\theta + \pi) = 0$.

The theorem guarantees that zero always exists. Try to draw one where it doesn't!

Draw your function

Draw points below. The x-coordinate becomes the angle, y becomes the value.

Why does this always work?

Define $g(\theta) = f(\theta) - f(\theta + \pi)$. Then:

$g(0) = f(0) - f(\pi)$ $g(\pi) = f(\pi) - f(2\pi) = f(\pi) - f(0) = -g(0)$

So $g(0)$ and $g(\pi)$ have opposite signs (unless one is already zero). By the Intermediate Value Theorem, $g$ must cross zero somewhere in between.

That zero is your antipodal pair: $f(\theta^*) = f(\theta^* + \pi)$. ∎

This is the 1D case of the Borsuk-Ulam theorem. The full theorem says: for any continuous map $f: S^n \to \mathbb{R}^n$, there exists a point $x$ where $f(x) = f(-x)$. The 2D version implies that right now, there are two diametrically opposite points on Earth with the same temperature and pressure.