📍 Brouwer Fixed Point Theorem
Every continuous map from a disk to itself has at least one fixed point — a point that the map sends back to itself. Drag the target point or pick a preset to explore the vector field.
Brouwer Fixed Point Theorem (1911)
Proved by L.E.J. Brouwer in 1911: every continuous function from a closed disk
D² to itself has at least one fixed point — a point x where f(x) = x.
The visualization shows the displacement vector field v(x) = f(x) − x.
A fixed point is exactly where v = 0 (no arrow). The theorem guarantees such
a zero always exists inside the disk. The glowing star ✦ marks discovered fixed points.
Contraction: the entire disk contracts toward the draggable target — fixed point is exactly the target. Rotation: every point rotates by an angle that shrinks to zero at the centre — only the centre is fixed. Squeeze: x-axis contracted, y-axis stretched, yet the origin remains fixed. Twist: a vortex-like swirling that also pins the centre. Inversion: maps x → lerp(x, −x, t) — as t→1 the only fixed point is the origin.
Drag anywhere inside the disk while in Contraction mode to move the fixed point.