My definition of polytope.

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

My definition of polytope.

Postby ubersketch » Thu Dec 14, 2017 12:59 am

So, it's quite hard to define a polytope but I'll try.

A polytope must have flat elements (no curves), cannot have a completely concealed hole (concealed means that you cannot see it in n-1 vision where n is the number of dimensions the shape is in and isn't connected to any other element), must have polytope elements, and finally uncapped faces are only allowed when they are on an imaginary vertex which is where theoretically, is where the vertices should be on a stellation to infinity or an unbounded shape meant to have a terminating vertex but cannot as the sides are all parallel or a similar case.
What about compounds and stuff like that?
Coincidics, compounds, and combofaceteds are part of a subgroup of polytopes called combopolytopes, non combopolytopes are called monopolytopes.
Now, if you remove the flat elements part, you get what I simply call topes which can have curved elements> If they are coincidic, compounds, or combofaceted, they are called combotopes. If they aren't, then they are monotopes.
Last edited by ubersketch on Fri Dec 15, 2017 11:22 pm, edited 4 times in total.
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Re: My definition of polytope.

Postby Mercurial, the Spectre » Thu Dec 14, 2017 11:30 am

Interesting!
Although the "topes" you meant could be simply called shapes.
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Re: My definition of polytope.

Postby wendy » Thu Dec 14, 2017 1:04 pm

I use the word 'solid' to refer to figures with a definite boundary, such as polytopes, spheres, cones, torotopes &c. You can have non-solid figures, such as a gaussian ball (ie a density of d_0 e^{-r^2}.

But a lot of brilliant minds here set to define 'polytope' in the context of multi-density figures (such as the kepler-poisont figures), and pretty much ran into a dead end. It is true that there are polytopes, and things that are not polytopes, but the boundary is fuzzy. One can find things that make various definitions hard to work. It's kind of like when red merges into orange. There are red things, and there are orange things, but where do you cut off red and start using orange?
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Re: My definition of polytope.

Postby ubersketch » Thu Dec 14, 2017 10:01 pm

Mercurial, the Spectre wrote:Interesting!
Although the "topes" you meant could be simply called shapes.

Topes actually can't have concealed holes inside them like a circle with a circle removed in the center or have non-tope elements. So no carved out cylinders. Uncapped areas also have to be where imaginary vertices are.
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