euclidean

Is euclidean geometry any more real than non-euclidean geometry simply because we find the former in objective physical reality?

So you can fabricate an Euclidean space of colors where the Euclidean distance is close to the perception distance, but it may never be perfect.

There is also an Euclidean mode, where you can see for yourself the things which do not work.

They seem to solve the Euclidean distance problem by embedding in a field with more dimensions.

In an urban setting, euclidean tends to underestimate walking distance, so manhattan would compare even better.

I mean, as long as the game forms a manifold it is locally euclidean and if we think games like snake where the very euclidean looking board loops around it still has (non-euclidean) geometry of torus.

Learning non-Euclidean geometry is harder if early math teachers don't mention that our usual school lessons in geometry assume Euclidean geometry with the parallel postulate is the only possible true description of the world.

If you meant in the more mathematical sense we perform our clustering in a normalized euclidean space.

A d-dimensional manifold is a topological space with the property that all small enough bits of it look like ordinary d-dimensional Euclidean space.

As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature).

Another example is energy vs work, or Euclidean space vs vectors.

> Manhattan distance approximation is seriously wrong The Manhattan (1-norm) metric is strongly equivalent to euclidean (2-norm) metric.

The game looks good, but it makes me wonder are there any clear gameplay advantages/specialties that come from non-euclidean game "board".

Cut the Knot has another good page, "The Pythagorean Theorem is Equivalent to the Parallel Postulate," pointing out that the conventional form of the Pythagorean Theorem is a theorem only of Euclidean geometry, and the theorem's statement has to be modified in non-Euclidean geometry.

For instance, you might hope to be able to put a "differentiable structure" on it, which is equivalent (not trivially equivalent, given the usual definitions, but never mind) to saying that you can instantiate it as a subset of a nice Euclidean space, defined by saying that it's where some nice well-behaved functions on that space are zero.

He figured out that spacetime may be non-Euclidean before he was aware that the math had been extensively studied.

The Dover book 'Beyond Geometry' collects a bunch of the major papers in non-Euclidean geometry leading up to relativity, and is a fantastic read.

Sometimes the theorems that are developed are not applicable to their original subject of study, but can be applied to another by re-interpreting their meaning, like non-Euclidean geometry.