Pythagoras' theorem

Pythagoras' theorem is a natural consequence of having the typical metric on a plane. However, in his proof, he never mentions what metric he uses - somehow, he must have snuck the metric in. Where does his proof assume the usual metric?

http://en.wikipedia.org/wiki/Pythagorus's_theorem#Proof_using_similar_triangles

"The shortest path between any two points is a straight line"

I'm pretty sure that's one of the axioms of classical geometry. Too lazy to look.

>>2

That's common to all metrics [ d(a,c) <= d(a,b) + d(a,c) ]

>>3
A line, in the context of classical geometry, has a meaning aside from "the shortest path."

>>4
I guess I should clarify this. The "straight line" defined in Euclid's geometry is what it is because the geometry of space is locally indistinguishable from Euclidean space. In short, the proof of Pythagoras' theorem works because Euclid defined the shortest path between two points to mean a geodesic in the real world, and geodesics in the real world are (for all practical purposes, on the scale that Euclid was working at) Euclidean geodesics (straight lines).

>>1
Euclidean geometry (assumption) = usual metric.

>>2
I'm not sure if it's an axiom, but it is only true for Euclidean geometry.

>>3
phail for metrics that describe pseudoriemannian manifolds (ie GR)

>>4
Probably.

that's more specific, not more in general

>>6
If it doesn't satisfy the triangle inequality (which was mistyped by >>3, but I'm guessing you get the idea), it isn't a metric.

At any rate, the Euclidean metric is brought into geometry through: rotational invariance (of distance), translational invariance (of distance), and the parallel postulate.

>>8
Notice how I mentioned "pseudoriemannian manifolds" instead of "riemannian manifolds".  Inner product spaces on the former are not positive definite.

http://en.wikipedia.org/wiki/Triangle_inequality (near the bottom) shows that the triangle inequality reverses (with some constraints) in the Minkowski metric (or "metric" if you prefer).  It also includes Lorentz invariance.

>>9
http://en.wikipedia.org/wiki/Inner_product
http://en.wikipedia.org/wiki/Metric_space

Minkowski spaces are not inner product spaces, and the "weak inner product" does not give a metric. Calling them such doesn't change the fact that they do not satisfy the appropriate axioms.

>>10
Thus, it will be called "metric" (with quotes) instead of metric.

The word pirahna, is all I can think of that rhymes with marijuana

Marijuana MUST be legalized.