This is visualization of Borsuk-Ulam Theorem. Colors from blue to red denote continuous function on a sphere. Sliders Time and Scale allow to modify it. Note, this function is not symmetrical.

White pixels denote points on the opposite sides of the sphere that have the same value of this function. Note, not all white points have the same value, but if a point is white, this means that there is a point on the opposite side of the sphere that has the same value.

Surprising property of these points is that they are symmetrical. All shapes created by them are repeated (mirrored) on the other side of the sphere. It is true even though the function is not symmetrical.

Use mouse to rotate/move the sphere.

This page was created with great help of Copilot and chat GPT. Random continuous function is modelled by Perlin Noise GLSL code.