





  • $\begingroup$ To apply Sperner's lemma to Brouwer's fixed-point theorem, you have to let the size of the polytopes tend to zero, which breaks down in a real-world context. I would be surprised if there were any kind of approximate fixed-point result that would apply to a real cup of tea. $\endgroup$ – TonyK May 29 '11 at 17:56
  • $\begingroup$ Yes, there are problems. Sperner's lemma will let you find a point such that nearby points are being stirred in every possible direction, or something like that, but with a fixed triangulation that doesn't necessarily tell you anything about the point itself. I agree that the basic issue is that it's unclear what reasonable notion of continuity might be preserved in the discrete setting. $\endgroup$ – Qiaochu Yuan May 29 '11 at 18:36
  • $\begingroup$ Sperner's lemma allows you to find almost-fixed points, points that are mapped with epsilon distance to itself. What you can not guarantee is that the point is close to a real fixed point. $\endgroup$ – Michael Greinecker Jan 4 '12 at 15:07


  • $\begingroup$ I don't see how this answers the question. Why does this preclude the possibility that some points that were initially less than an atomic radius apart are still less than an atomic radius apart? $\endgroup$ – Qiaochu Yuan May 29 '11 at 16:28
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  • $\begingroup$ @Qiaochu: The possibility is not precluded by my argument. But Brouwer's theorem relies on the displacement function being continuous, which is not the case at the molecular level. $\endgroup$ – TonyK May 29 '11 at 17:48

Brouwers fixed point theorem doesn't apply practically or even theoretically at all for the simple reason that stirring is not a continuous transformation.

For instance if you stirred vigourously and a drop of liquid jumped out and then fell back in, it's no longer continuous. If some liquid sticks to the spoon as you withdraw it then nope.

The cup of water is topologically equivalent to a sphere, and any transformation to a non-equivalent volume is discontinuous. If the spoon touches the bottom, then now the fluid is equivalent to a torus temporarily and nope. If you suck down a single air bubble, or as you swirl a vortex forms which collapses on itself then nope.

If you were to carefully perform some action in a way to guarantee continuity then I doubt anyone would identify that action with 'stirring'.


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