Borsuk-Ulam Circle Antipodal Values

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.

Experiment

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

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