my terminolgy!

Higher-dimensional geometry (previously "Polyshapes").

my terminolgy!

Postby wintersolstice » Sun Aug 16, 2009 2:34 pm

I have some of my own terminolgy for polytopes.

Mountainform: Polytope in 3D and beyond that has regular faces.(not the same as "scaliform" which has the extra property of being "vertex-transitive"

Sub-regular: all the facets are regular (I know that would probably be semi-regular but I prefer to use that term for non-indentical regular facets)
Hyper3-regular: all facets are sub-regular

Super-scaliform: all the cells are uniform(or regular thus creating two types!)
Hyper3-scaliform: membranes are uniform (or regular or sub-regular, three types)
etc

Membrane: 4-face

a lot of these are just until I can find something better! :D
What do you think

I'm hoping to have a full system for catogrising "mountainforms"
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Re: my terminolgy!

Postby wendy » Mon Aug 17, 2009 8:39 am

The trouble to aluding to mountians, ridges etc, in polytopes, is that tilings are also polytopes, and these features become flat!

Semi-regular already permits non-regular "facets". These have been enumerated in the 19C by Thorald Gosset.

The current splits are

1. Uniform = iso-vertices + equal edges + decent
iso-vertices = all vertices are alike as far to reflection symmetry
equal edges = all edges are equal in measure
decent = all surtopes are also uniform

This includes of course Platonic (regular by reflections), + Prisms (prism-product of uniforms) + archimedean (all others)

Its rather unfortunate to tie words to particular meanings, like your "membrane". When you get to higher dimensions, it's going to bite you. A choron in four dimensions bounds the polytope (ie a face), but in six dimensions, it can appear in section as a point! The words in use here are

2d hedron, 3d choron, 4d teron, 5d peton, 6d ecton, 7d zetton, 8d yotton are patches. For the general fabric, replace -on with -ix eg hedrix is a 2d fabric in any dimension. It only divides space in 3d. The series descends as 0d teelix, 1d latrix, although "vertex" and "edge" are permitted.

N-1d face (something that faces you), N-2d margin.

I have an on-line dictionary of this.
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Re: my terminolgy!

Postby wintersolstice » Tue Aug 18, 2009 11:06 am

One thing I will say is that this terminolgy is "incomplete" I need to seperate the groups better (refine the groups) and come up with "snazzy" names! (I got that off "hedrondude! :D )

Edit: It's also based soley on "uniformity" of the elements (not on transitivity of the elements)
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Re: my terminolgy!

Postby wendy » Wed Aug 19, 2009 10:25 am

The importance of uniformity being transitive is that it discounts the many different cases where the vertex-figure is the same at each vertex, but is not transitive. For example, you can take something like a rhombo-cuboctahedron, and rotate one of the octagons by 45 degrees. The vertex figures are still the same, but are not transitive.

The rule of descent requires that all of the faces etc follow the same rule. You can have in five dimensions, a polytope xo3oo4oo3ox*#t, "24-choron antiprism" comprising of dual 24-chora, in 5d, with a variety of pentachora and octahedral-pyramids stretching from top to bottom. The thing is vertex-transitive but does not have uniform descent: the octahedral-prisms are not themselves transitive.
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