Why aren't exotics considered true polytopes?

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

Why aren't exotics considered true polytopes?

Postby ubersketch » Fri Mar 02, 2018 12:31 am

Title. I have come up with a good reason which has to do with ambiguity but I have since forgotten it.
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Re: Why aren't exotics considered true polytopes?

Postby Klitzing » Fri Mar 02, 2018 1:09 pm

ubersketch wrote:Title. I have come up with a good reason which has to do with ambiguity but I have since forgotten it.

Yes, ambiguity is the main reason.

Thus the 10-edged complete graph of the pentagon e.g.
  • either could be said to have (case A) 5 vertices and 10 edges, 4 being incident per vertex,
  • or could be said to have 10 vertices instead, which just happen to be placed as 5 coincident pairs.
In the latter case then you still have 3 different interpretations:
  • either the graph then would fall apart (case B) into a tau scaled pentagram plus a unit scaled convex pentagon as its hull,
  • or you could say you'd alternate between the larger and smaller edges each.
For that latter case you then would have to distinguish between the remaining 2 options of
  • either running around always in the prograde sense (case C),
  • or you would take the longer edges as prograde and the smaller ones as retrograde (case D).
All these options are fully valide abstract polytopes, at least in the Grünbaumian sense. E.g. the first case does not follow the dyadicity restriction usually being understood. The second case usually rather is being considered a compound. And the other 2 cases just are different realisations of the same abstract polytope (in fact, a decagon with alternatingly to be distinguished sides), which just happen to show up the same graph!

So it is more a matter of conveniance to disallow such figures when not wanting to have to argue about all these exceptional cases all the time. None the less you are free to consider such figures, if you like! But then you would have to be very precise in telling, which Interpretation you are currently dealing with.

--- rk
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Re: Why aren't exotics considered true polytopes?

Postby ubersketch » Fri Mar 02, 2018 10:11 pm

Klitzing wrote:
ubersketch wrote:Title. I have come up with a good reason which has to do with ambiguity but I have since forgotten it.

Yes, ambiguity is the main reason.

Thus the 10-edged complete graph of the pentagon e.g.
  • either could be said to have (case A) 5 vertices and 10 edges, 4 being incident per vertex,
  • or could be said to have 10 vertices instead, which just happen to be placed as 5 coincident pairs.
In the latter case then you still have 3 different interpretations:
  • either the graph then would fall apart (case B) into a tau scaled pentagram plus a unit scaled convex pentagon as its hull,
  • or you could say you'd alternate between the larger and smaller edges each.
For that latter case you then would have to distinguish between the remaining 2 options of
  • either running around always in the prograde sense (case C),
  • or you would take the longer edges as prograde and the smaller ones as retrograde (case D).
All these options are fully valide abstract polytopes, at least in the Grünbaumian sense. E.g. the first case does not follow the dyadicity restriction usually being understood. The second case usually rather is being considered a compound. And the other 2 cases just are different realisations of the same abstract polytope (in fact, a decagon with alternatingly to be distinguished sides), which just happen to show up the same graph!
So it is more a matter of conveniance to disallow such figures when not wanting to have to argue about all these exceptional cases all the time. None the less you are free to consider such figures, if you like! But then you would have to be very precise in telling, which Interpretation you are currently dealing with.

--- rk

I was talking specifically about case A.
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Re: Why aren't exotics considered true polytopes?

Postby ubersketch » Mon Mar 05, 2018 10:40 pm

I thought this would be fitting for this thread.
So, I was thinking about this alot, and I decided that true polytopes have exactly 2 facets per ridge, all of their elements are connected, they aren't coincidic, and they have true polytope elements.
Also I'm not sure what coincide means but I'm guessing it means an element shares every lower dimensional element with another element.
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