Comment on Fuck geometry
OrganicMustard@lemmy.world 3 weeks ago
You can make something like this properly by defining a different metric. For example with metric dl^2^ = dx^2^ - dy^2^ the vector (1, 1) has length 0, so you can make a “triangle” with sides of lengths 1, -1 and 0.
kogasa@programming.dev 2 weeks ago
That’s not a metric. In any metric, distances are positive between distinct points and 0 between equal points
OrganicMustard@lemmy.world 2 weeks ago
It depends which metric definition are you using. The one I wrote is a pseudo-Riemannian metric that is not positive defined. Normally physicists use that generalized metric definition because spacetime in most cases has a metric signature of (-1, 1, 1, 1). Points with zero distance are not necessarily the same point, they just are in the same null geodesic.
kogasa@programming.dev 2 weeks ago
You’re talking about a metric tensor on a pseudo-Riemannian manifold, I’m talking about a metric space. A metric in the sense of a metric space takes nonnegative real values. If you relax the condition that distinct points have nonzero distance, it’s a pseudometric.
technocrit@lemmy.dbzer0.com 2 weeks ago
This is true for real-valued metrics but not complex-valued metrics.
en.wikipedia.org/wiki/Complex_measure
kogasa@programming.dev 2 weeks ago
Metric, not measure. Metrics are real by definition.