Ways the universe could be closed

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Ways the universe could be closed

Postby anderscolingustafson » Fri Oct 16, 2015 5:41 am

I was thinking about the shape of the universe and how normally when people think of a closed universe they think of the universe as being a four dimensional hypersphere. In 4d it's possible for the universe to close back on it self but not without distorting space time and so it would in principle be possible to figure out if the universe is a hypersphere as a triangle drawn on a hypersphere would have angles that add up to more than 180 degrees. It is possible though for the universe to be closed without being a hypersphere.

Instead of being the 4d equivalent of a sphere another way the universe could be closed is if it is the 6d equivalent of a duocylinder. While hypersphers have distorted surfaces "hyperduocylinders" do not have distorted surfaces and the only difference between their surface and a regular flat surface is that they close on themselves and so they can be cut into flat surfaces without any stretching or squeezing. If you draw a triangle on the 6d equivalent of a duocylinder it's angles will add up to 180 degrees meaning that if the universe is the six dimensional equivalent of the duocylinder it will pass the triangle test for flatness. A closed universe that is the six dimensional equivalent of a duocylinder could be much smaller than a universe that is the 4d equivalent of a sphere without us figuring out that the universe is a 6d equivalent of a duocylinder as the only way we could figure out that the universe is the 6d equivalent of a duocylinder is if we can either detect extra dimensions, or by having the 6d equivalent of a duocylinder have a circumference smaller than the diameter of the visible universe

It's also possible that the universe could close on itself but in such a way that if you go around the universe once everything is inverted to it's mirror image relative to you.
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Re: Ways the universe could be closed

Postby ICN5D » Fri Oct 16, 2015 3:25 pm

What you're thinking of is the 3D edge if a 6D tri-circular prism, the triocylinder. It would in fact be a closed surface, with zero intrinsic curvature. Another way to describe it is a 3-torus T^3 embedded in R^6. This surface is a flat torus, which can't be visualized completely in 3D, since it curves into 4,5, and 6D. There is one particular way it can intersect 3D space, which comes out as 8 points in the vertices of a cube. All other slices are empty intersections. Each point is joined to its adjacent 3 neighbors along the x, y, or z axes. If you label the six axes as x, y, z, w, v, u, the circular connections for any two points are on the xw , yv, and zu planes.
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Re: Ways the universe could be closed

Postby wendy » Sat Oct 17, 2015 6:41 am

The current thinking of a closed universe is that it is a single face of a twelftychoron, the opposite faces linked by a half-edge turn, in the clockwise direction on each face. This is called a poincare dodecahedron.

That is, you take a dodecahedron, and draw a clockwise arrow on its faces. Pairs of opposite faces join, so you get for a 18-degree twist on each face, a 36deg displacement of the two sides, and this the pentagons join, and this is repeated for each of the six pairs of faces,

The truncated cube of an octagonny works like this, as well as the octahedra of a 24ch, or the cubes of a tesseract, all do this.
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Re: Ways the universe could be closed

Postby Klitzing » Sun Oct 18, 2015 9:53 pm

The elliptical version of spherical polytopes, i.e. their identification of antipodal points, is well-known. For most of the platonics this point inversion is an intrinsic symmetry. But not so for the simplices, esp. not for the tetrahedron. Accordingly you can apply a mod-wrap e.g. to x3o5o3o and x5o3o5o (both being honeycombs of hyperbolic space) by identification of antipodal points of both the cells and of the vertex figures, which then result in the famous 11-cell of Grünbaum resp. the 57-cell of Coxeter. - There are other such figures too, but there comes in some degeneration, whenever there are used monostratic cells or vertex figures. - But surely simplices - for either purpose - are to be excluded already right for the above mentioned reason.

Now you mention x5o3o3o as pre-image for some mod-wrap. Sure, here the dodecahedral cells would not have antipodal points being identified, rather there is some further twist to be applied as well. Using this swirl symmetry, it is also known, that this hecatonicosachoron decomposes into 12 cycles of 10 dodecahedra each. It is further known, that the 12 * 10 connecting pentagons of all cycles are isolated in such a way, that at any edge of that 120-cell there will be at most one such pentagon, while at any vertex there will be incident exactly one. - Therefore you would need exactly 5 colors for the vertices of those pentagons of either cycle. And as there are 12 such cycles, you might need 12 times as many, provided there is no further repetition required. By means of a later color identification it follows moreover that at least all dodecahedra of any of those cycles get mod-wrapped into the same mod-cell. - But then, assuming 12 * 5 = 60 vertex colors and 12 mod-cells, one results in a contradiction of elliptical hemitetrahedra for vertex figures. Therefore you might postulate just 30 vertex colors and 12 mod-cells, or anything the like, which re-allows full spherical tetrahedra.

Best you could further outline your intended construction(s).

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