Potential extension of polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Potential extension of polytopes

Postby ubersketch » Tue Dec 25, 2018 9:38 pm

Imagine a bicone, it’s two cones stacked on top of each other from their bases. Now we divide this shape into several elements. 2 points, 1 ring, and 2 faces. Now what are these elements made of? Well a ring can be referred to as a pointless edge, and the 2 faces being an edgeless face. These are elements with 0 subelements, only defined by their dimension. This naturally lends itself to ordinal dimension and elements, and possibly an extension of dimension, that I talk about in “Cartesian geometry beyond Dimensions.” Honestly though I think that the only useful and interesting part is the part about 0-subelements.
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Re: Potential extension of polytopes

Postby mr_e_man » Thu Dec 27, 2018 12:18 am

You might be interested in the sphericon series. Take a regular polygon, rotate it around an axis of symmetry to make a solid of revolution (sweeping each edge along a conical frustum), cut the solid in half along the original polygon, rotate one half relative to the other, and reconnect the two polygons.

For the standard sphericon, start with a square, rotate it around a diagonal to make a bicone, cut the bicone in half along the square, rotate one half 90deg and reconnect. This makes a solid with only one surface, two semicircular edges, and four vertices. It can roll continuously, wobbling back and forth, but moving in a straight line on average.

I wouldn't call these "polytopes". It is a new concept, so it should have a new name.

Surfaces with curved edges could be called "polytopes" by stretching definitions (not requiring straight lines), as long as the combinatorial structure is an abstract polytope. The bicone and sphericons don't qualify. I suppose they could be called "piecewise-smooth manifolds" or something similar.
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Re: Potential extension of polytopes

Postby ubersketch » Sat Dec 29, 2018 10:53 pm

Yeah, I would call them discrete manifolds or something silly like that.
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Re: Potential extension of polytopes

Postby ubersketch » Sat Dec 29, 2018 10:54 pm

Also I am aware of the sphericon series. They are actually my inspiration, as well as Bowers' coiloids.
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Re: Potential extension of polytopes

Postby wendy » Sun Dec 30, 2018 7:52 am

A figure with a definite surface is a solid. Polytopes in the simplest form are solids with flat faces. The names I use for these is to take the fabric (hedron), and make it a solid (hedrid = 2d, chorid = 3d, terid = 4d, petid = 5d, ectid = 6d, zettid = 7d, yottid = 8d)

Something like the cone is a solid, with 1 vertex, 1 edge, and 2 faces. It's the pyramid-product of a point (1,1) and a circle (1,1,0,1).

The point has a point and nulloid, but no surface. The circle has an edge. We add to these surfaces, a body and nulloid, and multiply the two.

So point (1,1) * circle (1,1,0,1) = cone (1,2,1,1,1). Removing the terminating 1's, gives 2 faces, an edge and a vertex.

A bicurcular tegum is the product of surfaces of a circles, ie 1,0,1 * 1,0,1 = 1,0,2,0,1. Remove the trailing 1, gives a chorid surface (3d), plus two non-intersecting edges.

Multitopes

These are an assembly of polytopes without a closure. For example, the net or unfolded cube would be a multitope. These still have the same 'complete joining' of polytopes, (ie an edge is shared entirely with everything that it is part of). These are used to demonstrate the idea of 'mounting', or correctly joining polytope bits to make a full-size model.
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the dream we dream together is reality.

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