--- In sfconsim-l@yahoogroups.com, Erik Max Francis <max@...> wrote:
>
> Jonathan wrote:
> > In spherical and hyperbolic geometry we see spaces that are Euclidean on
> > infinitesimal scales but become very non-euclidean on larger scales.
> > However, the only non-euclidean geometry I know of that, like Euclidean
> > geometry, is consistent on all scales is Lorentzian geometry, which is
> > derived by flipping signs around in the Euclidean metric tensor.
> > (Actually, come to think of it, I can think of Taxicab and Chebyshev
> > geometry too). But other than that, are there any other non-Euclidean
> > geometries that are the same on all scales? For instance, where the
> > angles of a triangle add up to a constant other than 180, or where the
> > ratio of the circumference of a circle to its diameter is something
> > other than pi?
>
> I think you'd have to be more specific than this to get at exactly what
> it is you're asking. Non-Euclidean spaces include all kinds of other
> weird cases that you probably don't want, such as fractal or other
> non-differential spaces that aren't of much use here.

Well, maybe. Depends on if they have interesting properties that I can easily
wrap my mind around. This is more a mental voyage of exploration than anything
else.

> So first I'm guessing you're talking about a Riemannian manifold, which
> requires differentiability (and thus "smoothness"). It's hard to
> imagine a curved space(time) where the shapes of circles or triangles
> are not distorted over distances, since that's what curvature _is_.
> However, since you invoked Lorentzian geometry, I'm guessing you're
> including four-dimensional spacetimes and thus also Minkowski space.

Well, really I'm talking about metrics of any dimension with mixed signs in
their signatures, whether that be (-+) or (-++) or (--++) or (---++++). Such
metrics form the only ones I know of with a geometry that is neither Euclidean
nor varies with scale. I'm looking for anything else that combines those two
properties.