Hypersphericons

Discussion of shapes with curves and holes in various dimensions.

Hypersphericons

Postby ubersketch » Sat Dec 29, 2018 11:07 pm

I've encountered quite a few interesting solids during my time. Some of them, are the sphericons. They are taken by rotating a regular polygon around an axis, and twisting the resulting figure to reach a snaking appearance. I'd like to explore higher dimensional analogs, since I feel that you could use a polyhedron instead of a polygon to create a 4d version. On another note, I've noticed their similarities to slices of coiloids though this may only be superficial.

http://www.mathias.org/steve/sphericons/series.htm
Here are some pictures of sphericons.
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Re: Hypersphericons

Postby wendy » Wed Jan 02, 2019 10:39 am

I imagine the bi-circular tegum would fit the bill here. It is the convex hull of two circles, in perpendicular 2-spaces (hedrixes).
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Re: Hypersphericons

Postby ubersketch » Sun Jan 06, 2019 2:52 am

wendy wrote:I imagine the bi-circular tegum would fit the bill here. It is the convex hull of two circles, in perpendicular 2-spaces (hedrixes).

Does it have any sort of twist?
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Re: Hypersphericons

Postby wendy » Sun Jan 06, 2019 8:55 am

The sorts of twist that make interesting things in 3d don't really exist in 4d.

If you look at the web-page in the OP, you will see that the cut is made to expose a polygon, like a square or octagon. These can be glued back together in different ways.

In four dimensions, the cut would be a polyhedron, or chorid (3d solid) in general. These are not really good at joining back togethe in a different way.
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Re: Hypersphericons

Postby Klitzing » Sun Jan 06, 2019 11:24 am

wendy wrote:In four dimensions, the cut would be a polyhedron, or chorid (3d solid) in general. These are not really good at joining back togethe in a different way.

Oh well, there surely are several bistratic polychora of type aba-P-cdc-P-ebe-&#xt which allow for gyration into abe-P-cdc-P-eba-&#xt (whenever a<>e).

--- rk
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Re: Hypersphericons

Postby ubersketch » Tue Feb 26, 2019 1:55 pm

I have noticed that the coiloid slices and sphericon slices look super similar as previously mention. And several of the sphericons are composed of a single spiraling edge as well, similar to the contruction of the coiloid. Another thing, is those bicircular tegums also have similar looking slices.
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Re: Hypersphericons

Postby ICN5D » Sat Mar 02, 2019 2:54 am

The bi-circular tegum doesn't have any twists. It's more of a double pyramid/cone-like thing made by stretching a tent over two orthogonal intersecting circles in 4-space. I think it is hypersphericon-worthy.
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