Borsuk-Ulam Circle Antipodal Values

Two opposite points can share the same height even when the function never crosses zero.

Borsuk-Ulam Circle Antipodal Values

Concept Overview

The Borsuk-Ulam theorem guarantees that any continuous function from a sphere into Euclidean space of the same dimension maps some pair of antipodal (opposite) points to the same value. For the 1-dimensional circle S¹ mapping to the real line R, we examine the family of functions f(θ) = 5 + A cos(θ) + B sin(θ) + C cos(2θ). By considering the gap function g(θ) = f(θ) - f(θ + π) = 2(A cos(θ) + B sin(θ)), we observe it is continuous and odd: g(θ + π) = -g(θ). The Intermediate Value Theorem thus ensures g(θ) crosses zero, meaning f(θ) = f(θ + π) for some witness angle θ, even if f(θ) is always strictly positive.

Worked Example

Consider the function f(θ) = 5 + 1 · cos(θ) + 0 · sin(θ) + 1 · cos(2θ) with coefficients A = 1, B = 0, C = 1. We evaluate this at a probe angle of 45°.

Calculate f(45°): 5 + cos(45°) + cos(90°) = 5 + √2/2 + 0 ≈ 5.7071.
Calculate the opposite point f(225°): 5 + cos(225°) + cos(450°) = 5 - √2/2 + 0 ≈ 4.2929.
The antipodal gap is f(45°) - f(225°) ≈ 1.4142. Since this gap is non-zero, the probe is not the guaranteed witness point.
We can find the witness point by setting g(θ) = 2(1 · cos(θ) + 0 · sin(θ)) = 2 cos(θ) = 0. This gives θ = 90°.
Check the witness point: f(90°) = 5 + cos(90°) + cos(180°) = 5 + 0 - 1 = 4. The opposite point is f(270°) = 5 + cos(270°) + cos(540°) = 5 + 0 - 1 = 4. Both share the common value 4.

Experiment

You can interactively explore the function and its antipodal values in the Borsuk-Ulam Circle Antipodal Values visualization.

Select Cosine coefficient to 1, Sine coefficient to 0, and Even harmonic to 1.
Set the Probe degrees to 45. The visualization shows the probe point and its opposite with a non-zero gap.
Change the Probe degrees to 90. The probe point aligns with the witness marker, and the gap becomes exactly zero, demonstrating the theorem.

Limitations

This module demonstrates the 1-dimensional case of the Borsuk-Ulam theorem (S¹ → R). While it visually confirms that antipodal points with equal values must exist on a circle for any continuous function, this does not serve as a proof for arbitrary higher-dimensional cases, such as the geographic claim that there must be antipodal points on Earth with exactly the same temperature and pressure (S² → R²). Additionally, this visualization uses a restricted family of functions based on low-order harmonics, which cannot represent every possible continuous function on a circle.

References

The Borsuk-Ulam theorem and related topological concepts are discussed in detail in the UCLA Math Circle handout:UCLA Math Circle Handout.

Borsuk-Ulam Circle Antipodal Values