BROUWER stir the coffee — some point ends where it began; a fixed point must exist
Take any continuous map of a disk (or ball) to itself. Brouwer's theorem: it must have a fixed point — some point the map leaves exactly where it was. No formula, no construction; the fixed point is forced to exist by continuity and shape alone. Stir a cup of coffee and, however you swirl it, some speck returns to its start. Lay a map of the country on the ground anywhere in the country and one point of the paper lies directly above the spot it names. It is the ancestor of Nash equilibria and general-equilibrium economics — existence proved without ever exhibiting the point.
THE TECHNIQUE continuous self-map of a ball ⇒ a fixed point exists (non-constructive)
The demo shows a self-map of [0,1] crossing the diagonal y=x — where it crosses is a fixed point that must exist: live demo
HISTORY & CREDIT L. E. J. Brouwer, 1911
“To know a fixed point exists you must find it.” — Brouwer proves existence from continuity and shape alone; the point is forced, even when no formula can name it. cited
1911 · L. E. J. Brouwer — the fixed-point theorem for continuous self-maps of a ball (having, ironically, doubted non-constructive proof). 1-D case · the intermediate value theorem: g(x)−x changes sign, so it is zero somewhere. now · Nash equilibria (Kakutani's extension), general equilibrium, topology.
Continuity and a shape are enough — some point comes home, whether or not you can say which. Existence without construction. Brouwer 1911
RECOMMEND FOR I-13 the forced crossing, on the compiler
On the canonical compiler, g(x)=1−x²/2 sits above the diagonal at 0 and below it at 1, so it must cross — iterating lands on the fixed point √3−1 ≈ 0.732:
$ i13 run f_brouwer.i13 # g(x) = 1 - x^2/2 ; crosses y=x
RUN OK · 738 step(s) · peak stack 4 · call depth 41
fp = 0.7320500348247456 -- the fixed point
crossing_exists = 1 -- above at 0, below at 1: it MUST cross (Brouwer / IVT)
Recommend: Brouwer is the fixed point as an existence theorem — continuity forces one, even with no formula — and i13 both locates the crossing (0.732) and confirms the sign change that guarantees it. Not a keeper (an existence theorem is the purest witnessed fact — it holds of every continuous self-map, and i13 confirms, it does not cause). The dart that shows a fixed point can be certain before it is findable.