Maybe the problem is constructing a metric that makes this diagram true. Something like d(x,y) = | |x| - |y| | might work but I’m too lazy to check triangle inequality.
Triangle inequality for your metric follows directly from the triangle inequality for the Euclidean metric. However, you don’t need a metric for the Pythagorean Theorem, you need an inner product and, by definition, an inner product doesn’t allow non-real values.
Maybe the problem is constructing a metric that makes this diagram true. Something like d(x,y) = | |x| - |y| | might work but I’m too lazy to check triangle inequality.
Triangle inequality for your metric follows directly from the triangle inequality for the Euclidean metric. However, you don’t need a metric for the Pythagorean Theorem, you need an inner product and, by definition, an inner product doesn’t allow non-real values.