Editing Brouwer fixed-point theorem (section)

==Intuitive approach==

===Explanations attributed to Brouwer===
The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee.<ref>The interest of this anecdote rests in its intuitive and didactic character, but its accuracy is dubious. As the history section shows, the origin of the theorem is not Brouwer's work. More than 20 years earlier [[Henri Poincaré]] had proved an equivalent result, and 5 years before Brouwer P.&nbsp;Bohl had proved the three-dimensional case.</ref>
If one stirs to dissolve a lump of sugar, it appears there is always a point without motion.
He drew the conclusion that at any moment, there is a point on the surface that is not moving.<ref name=Arte>This citation comes originally from a television broadcast: ''[https://archive.today/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html Archimède]'', [[Arte]], 21 septembre 1999</ref>
The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit.
The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."<ref name=Arte />
Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as [[Stefan Banach]]'s, that guarantee uniqueness.

===One-dimensional case===
[[File:Théorème-de-Brouwer-dim-1.svg|200px|right]]
In one dimension, the result is intuitive and easy to prove.  The continuous function ''f'' is defined on a closed interval [''a'',&nbsp;''b''] and takes values in the same interval.  Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [''a'',&nbsp;''b''] which maps ''x'' to ''x'' (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal.  To prove this, consider the function ''g'' which maps ''x'' to ''f''(''x'')&nbsp;−&nbsp;''x''. It is ≥&nbsp;0 on ''a'' and ≤&nbsp;0 on&nbsp;''b''.  By the [[intermediate value theorem]], ''g'' has a [[Root of a function|zero]] in [''a'',&nbsp;''b'']; this zero is a fixed point.

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."<ref name=Arte />