Hybrid geometries?

Higher-dimensional geometry (previously "Polyshapes").

Hybrid geometries?

Postby quickfur » Thu Mar 31, 2016 1:19 am

All sources on non-Euclidean geometry that I've found so far only consider geometries in which curvature is constant and homogenous in all directions. But what about non-homogenous, hybrid geometries? By that I mean, things like, for example, a 3-dimensional space in which the first dimension is Euclidean, the second dimension is elliptical, and the 3rd dimension is hyperbolic. Is such a thing possible? One could think of such a space as a 3-manifold in the shape of a cylindrical hyperbolic tube, i.e., cross-sections perpendicular to the hyperbolic dimension (of the manifold) are cylinders, while cross-sections parallel to the hyperbolic dimension are hyperboloids of various kinds.

An interesting feature of such geometries, if they are in fact self-consistent, is that orientation matters, and figures in certain orientations may not be able to fit into the space in certain other orientations. For example, a "straight line" parallel to the elliptical dimension is topologically a circle, and you can't rotate it into the Euclidean or hyperbolic dimensions without fundamentally altering its nature. Smaller objects may be rotatable between the inequivalent dimensions, but will experience deformation. Such spaces will also exhibit "preferred" directions, and possibly an absolute reference frame (or at least, an absolute orientation).

An even more interesting question is what happens in higher dimensional hybrid geometries. If we restrict ourselves to only hyperbolic, elliptical, and Euclidean as the only options, then we have the peculiar case that all higher-dimensional hybrid geometries can be partitioned into 3 homogenous subspaces (i.e., a homogenously hyperbolic subspace, a homogenously elliptical subspace, and a homogenously Euclidean subspace, with the sum of their dimensions equal to the dimensionality of the entire space). This is strange because no matter which dimension is chosen to be hyperbolic/elliptical/flat this decomposition is always possible. Objects rotating within each subspace will experience no deformation.

However, if the relative curvatures of, say, the hyperbolic dimensions differ across dimensions, then multiple {hyperbolic,elliptical} dimensions no longer are freely interchangeable, and this ternary subspace decomposition is no longer possible. Rotations between two elliptical dimensions of unequal curvature will experience deformation (and probably resistance due to said deformation -- a kind of deformative resistance?).

Even more interesting things arise if we consider what kind of shapes may exist in such hybrid geometries. We might get a polytope (perhaps even a regular one??!) that's closed along one dimension but unbounded along another dimension, like toroidal, cylindrical, or other such strange things.

Of course, once we step out of the confines of "traditional" non-Euclidean geometry (i.e., homogenous geometries), there's no telling what else we can hybridize geometries with. What about a 5D space produced by the product of an Euclidean line, a circle, a hyperbola, and a projective plane? It could contain polytopes that are part-Euclidean, part bounded, part hyperbolic, and part projective? What manner of strange shapes would be possible in such a hybrid geometry?
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Re: Hybrid geometries?

Postby ICN5D » Thu Mar 31, 2016 2:29 am

Well, that's a wild idea. ( I wonder if Mantis is a creature of this unknown realm??? ) A space with different curvatures per axis/plane ... hmm ....
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Re: Hybrid geometries?

Postby Klitzing » Thu Mar 31, 2016 8:29 am

Haha, the only regular polytope within such a space would be a single point! Cause the only symmetry operation which transforms a shape into itself then can be the identity. Thus, in order to have a transient operation of incidency flags, then there can be just a single one.

Btw. the same holds for uniformity and even for scaliformity. By the same argument not even CRFs would be possible there. At most you could consider polytopes with a single edge size throughout. But note, edges would never fall into classes of common symmetry, every one will be unique.

But then, already the number of polygons (within usual spaces), which undergo only that constraint of equal sized sides, is way too wild. Only a triangle then is fixed. All other n-gons are fully deformable, in fact with increasing degrees of freedom.

--- rk

Edit: correction here.
Last edited by Klitzing on Thu Mar 31, 2016 4:26 pm, edited 1 time in total.
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Re: Hybrid geometries?

Postby wendy » Thu Mar 31, 2016 10:51 am

You do realise that some polytopes exist only at exact positions. For example, the 11-cell {3,5,3} and 57-cell {5,3,5} appear to be tied to particular points, because the vertex-symmetry is different to the symmetries everywhere else.

I've been playing with Einstein's space-curvature using the schwarzchild metric, and there are points of spherical symmetry do not generally pass over all space.
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Re: Hybrid geometries?

Postby quickfur » Thu Mar 31, 2016 3:37 pm

I'm pretty sure polytopes larger than a single point can exist in a hybrid geometry. Take for example a 2D space consisting of an elliptical dimension and an Euclidean dimension. You can embed this in 3-space as a cylinder of infinite height. Consider then a tesselation of this cylinder with regular hexagons, i.e., {6,3}. Wouldn't this qualify as a regular polytope in that space?

Or at least, some kind of uniform polytope, since you have at least rotational symmetry in the plane perpendicular to the cylinder, and translational symmetry along the axis of the cylinder.
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Re: Hybrid geometries?

Postby Klitzing » Thu Mar 31, 2016 4:09 pm

Hmmm, your indeed right, quickfur.
I was upset by considering rotations only. Those are clearly impossible within non-homogenous spaces.

But we have to consider translations as well, I forgot. Cyclical ones within the spherical geometry and lattice like ones in flat dimensions. And probably such similar stuff for hyperbolic space too. And you might even combine such movements, e.g. producing some helical movement on your infinite cylinder. And point inversion could be considered as well. And therefore your one sided mod-wrap of {6,3} indeed is at least uniform there. (Probably the slanted and the parallel edges have to be distinguished nonetheless.) In fact, there even ought be some reflection in euclidean subspaces. In your cylinder example this would combine left helical and right helical edges into one single class.

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Re: Hybrid geometries?

Postby Marek14 » Fri Apr 01, 2016 3:03 pm

My "Hyperbolic world" thread is arguably a case of this since the time is Euclidean while the three spatial dimensions are hyperbolic. This then leads to weird effects when moving, as a body will always feel a force when moving, even in a straight line and at constant speed, and so rest and motion are empirically distinguishable (which is not possible in our spacetime).
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