Types of Regular Polytopes

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

Types of Regular Polytopes

Postby ubersketch » Wed Dec 13, 2017 10:06 pm

So, after reading up on the differences between the Kepler solids and the Poinsot solids I came up with types of regular polytopes.

1
Each face meets n at a corner where n is the number of vertices each facet has. Some examples are the tet and squat.

2
Each face meets n at a vertex where n is the number of dimensions. This means that the vertex figures are simplectic. This includes the tet, cube, and doe.

3
Each face meets n at a vertex, all faces are simplices. This includes the tet, oct, and ike.

4
Anything that's not the above. All of these are tilings.
Last edited by ubersketch on Thu Dec 21, 2017 1:23 am, edited 3 times in total.
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Re: Types of Regular Polytopes

Postby Mercurial, the Spectre » Thu Dec 14, 2017 11:26 am

So...
for (1) you have the 3D {n,n} because there is no well-defined ridge (n-2 facet) in 2D. The faces are n-gons with n vertices (its ridge) and the whole figure itself is self-dual.
In 4D, if you mean n facets meeting a vertex, then among the regular 4-polytopes you have none. The same applies to higher dimensions.

for (2) you have the polytopes with simplectic vertex figures. The Schläfli symbol for such an n-d polytope would be [n,3,3,...,3,3} where the number of 3s depends on dimension.

for (3) you have the polytopes with simplectic facets. These are duals of (2) and are of the form {3,3,...,3,3,n}.

Also in n-d the final element in the Schläfli symbol determines the number of facets meeting in a peak (n-3 facet), so it is generally not a vertex. The statement you made applies mostly to polyhedra and planar tilings.

I think you should clarify more :D.
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Re: Types of Regular Polytopes

Postby ubersketch » Thu Dec 14, 2017 9:57 pm

Mercurial, the Spectre wrote:So...
for (1) you have the 3D {n,n} because there is no well-defined ridge (n-2 facet) in 2D. The faces are n-gons with n vertices (its ridge) and the whole figure itself is self-dual.
In 4D, if you mean n facets meeting a vertex, then among the regular 4-polytopes you have none. The same applies to higher dimensions.

for (2) you have the polytopes with simplectic vertex figures. The Schläfli symbol for such an n-d polytope would be [n,3,3,...,3,3} where the number of 3s depends on dimension.

for (3) you have the polytopes with simplectic facets. These are duals of (2) and are of the form {3,3,...,3,3,n}.

Also in n-d the final element in the Schläfli symbol determines the number of facets meeting in a peak (n-3 facet), so it is generally not a vertex. The statement you made applies mostly to polyhedra and planar tilings.

I think you should clarify more :D.

I revised the ridges part in 1 to vertices because that was too exclusive and also that wasn't what I was aiming for.
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Re: Types of Regular Polytopes

Postby Mercurial, the Spectre » Fri Dec 15, 2017 4:24 pm

There.
However, regular polytopes have common features, but the only thing separating them is their Schläfli symbol and duality relationships.

It would be best to define two kinds of regular polytopes:
- Paired regular polytopes (it has a different dual, which forms a pair)
- Self-dual regular polytopes (it is its own dual)

And not only that, there are generalizations such as infinite polytopes and complex polytopes. In fact, it can also broadly include honeycombs and tessellations.
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