Multiple spherical space?

Higher-dimensional geometry (previously "Polyshapes").

Multiple spherical space?

Postby Marek14 » Fri Sep 23, 2016 8:24 pm

This is one idea that got me thinking.

In a space with positive curvature, we can have two basic geometries: spherical, where two straight lines intersect in two points, and elliptic, where they only intersect in one point.

But could this be extended higher? Would it be possible to have a self-consistent geometry of, say, double-winded sphere, where two straight lines intersect in four points and the total area of the world is double the area of a normal sphere?
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Re: Multiple spherical space?

Postby Klitzing » Sat Sep 24, 2016 4:26 pm

Ain't you have to switsh from one sheet onto the next?
This then asks for some punchhole(s).
And thus your geometry no longer would be the same throughout.

For polytopes this clearly is different.
There you have vertices, edges, etc.
No need for homogenity throughout.

For complete space geometries that one is required, I thought.

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Re: Multiple spherical space?

Postby wendy » Sun Sep 25, 2016 8:22 am

Marek's question is interesting, i should imagine. I've dealt with similar kinds of space in the past, but there have been implicit winding points.

There was a conversation between John Conway and myself as to whether E3 could be globally double-wound without winding points. Rather like E1. The hairy ball rule rules out even spaces like E2 and E4, but it is less clear on E3 or E5. My own suspicion here is that only E7 might work, but that's chancingly speculation.

Still, the idea is to cover an even sphere with elements so there is no winding. A start might be to do something like a bi-pentagrammic prism, which leads to a four-fold cover of the surface.

The other attack is to try and grasp what something like 5(5/2)5 might look like. According to my calculations, this has a density of 11.
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