ubersketch wrote:Title. I have come up with a good reason which has to do with ambiguity but I have since forgotten it.
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.In the latter case then you still have 3 different interpretations:
- 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.
For that latter case you then would have to distinguish between the remaining 2 options of
- 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.
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!
- 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).
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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