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File: BlowSoapBubbles-Step-5.jpg (62KB, 800x532px) Image search: [Google]

62KB, 800x532px

>>4906944
>44
>dubs

OP, here's some pretty weird shit. Consider a graph on R^(n-1) in R^n. So when n = 2, you have a normal y = f(x) function, when n = 3, you have a surface z = f(x,y), etc. Now imagine that the function locally minimizes its area (we're in R^n, so we have a notion of distance, thus area). For example, a curvy y = f(x) doesn't minimize area because the shortest distance between any two points is a straight line. So for n = 2, f(x) has to be a straight line.

Likewise, in n = 3, the function z = f(x, y) has to be a flat plane: if there were a bump or a saddle or something, you could smush it just a little flatter. You can see this if you take a soap bubble and stretch it over a circle as in the picture related: it will try to form as flat a surface as the circle allows, because surface tension of water leads to area minimization.

So you might think "Of course! The shape that minimizes area is as close to flat as possible!" This holds for n=2, 3, 4, 5, 6, 7, although the higher cases are harder to visualize. But for n = 8, it fails: a weird shape called Simons' Cone exists which isn't flat, yet minimizes area.

>>4906978
Very interesting!