krtisfranks wrote: > krtisfranks wrote: > > Why restrict this to polygons (two-dimensional space)? I would see this
> > most as star polytopes, which is to say that the domain of reference is
> at > > least the set of all shapes given by Schlafli symbols that are > well-formed > > and which contain at least one positive non-integer rational number > (quite > > possibly in coprime ("lowest") terms); furthermore, it is possible that
> > various trivial reductions could be allowed (for example, lines and > convex > > polytopes could be considered to be stars without indentations) - I would > > > not disallow these possibilities. > > If we keep it to two-dimensions and keep them regular, than all we need
> is > > the number of outer vertices and how they are connected (is it every > other, > > every three, etc.?); my comments about trivial cases still applies. But I > > > also think that that it could be useful to have the terbri for outer and > > inner vertices (as you basically already do, although I am not sure that > > they actually are vertices) because they can now easily be referenced via > > > conversion. > > Oh, here is a difference: This word is for the shape formed by tracing out > the outline rather than from outer vertex to outer vertex (in my previous
> definition, there actually are only the outer vertices; the inner ones are > illusions).
Okay. I have tried to formalize the concept as follows:
Define a "polytopal hull" to be the minimal hypervolume that is connected, has a polygonal/polytopal boundary, and which contains (either within its interior or its boundary) all of the vertices and edges of the Euclidean-spatial embedding of a graph.
Then this word is the polytopal hull of a graph given by a Schlafli symbol as described before.
I think that that works.
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