Curvy 4D objects with non-orientable surface elements?

Higher-dimensional geometry (previously "Polyshapes").

Curvy 4D objects with non-orientable surface elements?

Postby quickfur » Fri Apr 27, 2018 11:57 pm

I'm not sure what the correct term is, but I'm thinking of objects that may contain polytope-like flat cells, but in addition may also contain curved elements like 3-manifolds. I was going to say toratopes, but then I wasn't sure if the category of shapes I have in mind is restricted to toratopes, since they could include more things like the convex subset of Bowers' polytwisters. Basically, I'm thinking of shapes that are approximable by polytopes, e.g., the duocylinder is approximated by m,n-duoprisms as m and n approach infinity; the bicircular tegum is approximated by m,n-tegums as m, n approach infinity, etc..

Anyway, terminology isn't my primary concern here. What I wanted to ask was, is it possible for such an object to be both convex and contain a 2-manifold surface element that's non-orientable, like a Mobius strip? I.e., in the approximating polytopes, you'd have a circuit of polygons that approximate a Mobius strip, but everything else would still fit together "traditionally" and form a convex polytope (as an approximation to the limiting shape). Can such an object exist?
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Re: Curvy 4D objects with non-orientable surface elements?

Postby wendy » Sat Apr 28, 2018 3:30 am

The term you are looking for is 'solid'. Polytopes are solids with plane faces. Solids includes the likes of spheres and torotopes and cones and anything else that has a definite boundary. Non-solids are things like exp{-r^2}, where the volume is differentiable, but not discrete.

Solids can have surtopes, and the usual incidence rules don't apply. The cone in 3d, has two faces, separated by a line, and an isolated point. These are preserved in the product. A cone is a point-circle pyramid, for example.

I'm pretty sure that the various out-vectors prevent non-orientable features in any dimension.

Let's look at the Thah. It has three crossing squares, and four triangles. It is possible to implement a volume, where the surface is the gradiant of volume, by using skew marginoids. These are divisions of the surface (aka margins), where the outward direction reverses. In this way, the squares support no volume, and the volumes are due entirely to the triangles, (where the volume is the moment of surface).

In four dimensions it is possible to convert this figure into a surface that does not cross. The three squares, suffice to say, pass at different heights and never see each other. But doing this would invalidate both the skew marginoids formed where the edges did cross, and the ability to contain chorage (3d volume). What happens is that the hedrix (2fabric) exists, but never in the same chorix (3-space).

The problem in 3d, would be like trying to span a knot with a single non-winding surface. We suppose this is true for the pentagram etc, but this is because we don't suppose the pentagram leaves the hedrix (2-space), and instead the edges cross without vertix (rather like interferring waves). But the moment you start over-and-under stuff, it no longer holds a surface.
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Re: Curvy 4D objects with non-orientable surface elements?

Postby quickfur » Tue May 01, 2018 3:52 am

Thanks, so it's safe to assume that any solid would have orientable surtopes / sur-hedrixes / sur-chorixes, etc?
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