[math-fun] Para-uniform polytopes
There is something about the classic definition of a uniform polytope that has always bugged me. There are three clauses to the definition: 1. The polytope must be vertex-transitive; informally, there is only one "kind" of vertex. 2. The edges must all be of the same length. 3. All the facets (that is, (n-1)-dimensional cells) of the polytope must themselves be uniform. Clause 3 establishes a recursive requirement that _all_ the cells be uniform, but these cells aren't required to be _identical_ until you get down to dimension 1 and clause 2 takes over. It feels to me like clause 2 is a weird addendum. I define a para-uniform polytope as follows: 1. The polytope must be vertex-transitive. 2. All its facets must themselves be para-uniform. This makes the edges behave like all the other cells: there are allowed to be different kinds. The para-uniform polygons, for instance, are all the equi-angular polygons that alternate two edge lengths. For odd numbers of sides, the two edge lengths are forced to be equal by global structure. Here's my question: do we get any novel structures? Or are all para-uniform polytopes just uniform polytopes with variously-stretched edges?
Can I just say, without having anything of real value to contribute to the discussion, that I find Allan's definition very attractive? I guess the answer to the preceding question is yes, because I just did. :-) Jim Propp On Thursday, August 10, 2017, Allan Wechsler <acwacw@gmail.com> wrote:
There is something about the classic definition of a uniform polytope that has always bugged me. There are three clauses to the definition:
1. The polytope must be vertex-transitive; informally, there is only one "kind" of vertex. 2. The edges must all be of the same length. 3. All the facets (that is, (n-1)-dimensional cells) of the polytope must themselves be uniform.
Clause 3 establishes a recursive requirement that _all_ the cells be uniform, but these cells aren't required to be _identical_ until you get down to dimension 1 and clause 2 takes over.
It feels to me like clause 2 is a weird addendum. I define a para-uniform polytope as follows:
1. The polytope must be vertex-transitive. 2. All its facets must themselves be para-uniform.
This makes the edges behave like all the other cells: there are allowed to be different kinds. The para-uniform polygons, for instance, are all the equi-angular polygons that alternate two edge lengths. For odd numbers of sides, the two edge lengths are forced to be equal by global structure.
Here's my question: do we get any novel structures? Or are all para-uniform polytopes just uniform polytopes with variously-stretched edges? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Aren't all bricks para-uniform? And generally [0,a]×[0,b]×[0,c]...? --Michael On Aug 10, 2017 12:50 PM, "James Propp" <jamespropp@gmail.com> wrote:
Can I just say, without having anything of real value to contribute to the discussion, that I find Allan's definition very attractive?
I guess the answer to the preceding question is yes, because I just did. :-)
Jim Propp
On Thursday, August 10, 2017, Allan Wechsler <acwacw@gmail.com> wrote:
There is something about the classic definition of a uniform polytope that has always bugged me. There are three clauses to the definition:
1. The polytope must be vertex-transitive; informally, there is only one "kind" of vertex. 2. The edges must all be of the same length. 3. All the facets (that is, (n-1)-dimensional cells) of the polytope must themselves be uniform.
Clause 3 establishes a recursive requirement that _all_ the cells be uniform, but these cells aren't required to be _identical_ until you get down to dimension 1 and clause 2 takes over.
It feels to me like clause 2 is a weird addendum. I define a para-uniform polytope as follows:
1. The polytope must be vertex-transitive. 2. All its facets must themselves be para-uniform.
This makes the edges behave like all the other cells: there are allowed to be different kinds. The para-uniform polygons, for instance, are all the equi-angular polygons that alternate two edge lengths. For odd numbers of sides, the two edge lengths are forced to be equal by global structure.
Here's my question: do we get any novel structures? Or are all para-uniform polytopes just uniform polytopes with variously-stretched edges? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Yes, these are examples, but they don't clear the bar of being "novel structures". They are just distorted cubes. I suspect that generalizing to para-uniform only allows things that are derived from uniform polyhedra by changing some edge lengths. But I can't quite prove it. On Thu, Aug 10, 2017 at 4:01 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
Aren't all bricks para-uniform? And generally [0,a]×[0,b]×[0,c]...?
--Michael
On Aug 10, 2017 12:50 PM, "James Propp" <jamespropp@gmail.com> wrote:
Can I just say, without having anything of real value to contribute to the discussion, that I find Allan's definition very attractive?
I guess the answer to the preceding question is yes, because I just did. :-)
Jim Propp
On Thursday, August 10, 2017, Allan Wechsler <acwacw@gmail.com> wrote:
There is something about the classic definition of a uniform polytope that has always bugged me. There are three clauses to the definition:
1. The polytope must be vertex-transitive; informally, there is only one "kind" of vertex. 2. The edges must all be of the same length. 3. All the facets (that is, (n-1)-dimensional cells) of the polytope must themselves be uniform.
Clause 3 establishes a recursive requirement that _all_ the cells be uniform, but these cells aren't required to be _identical_ until you get down to dimension 1 and clause 2 takes over.
It feels to me like clause 2 is a weird addendum. I define a para-uniform polytope as follows:
1. The polytope must be vertex-transitive. 2. All its facets must themselves be para-uniform.
This makes the edges behave like all the other cells: there are allowed to be different kinds. The para-uniform polygons, for instance, are all the equi-angular polygons that alternate two edge lengths. For odd numbers of sides, the two edge lengths are forced to be equal by global structure.
Here's my question: do we get any novel structures? Or are all para-uniform polytopes just uniform polytopes with variously-stretched edges? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (3)
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Allan Wechsler -
James Propp -
Michael Kleber