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.


What is the etymology?


Aside: I was actually just thinking about how this should be a gismu the
other day. I even told my friend about it. Stars have been on my mind
lately because of my research.

Comment #2: Re: Generalizing and Etymology

Curtis W Franks (Mon Jul 13 15:54:54 2015)

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. > > > What is the etymology? Ignore this > > > Aside: I was actually just thinking about how this should be a gismu the > other day. I even told my friend about it. Stars have been on my mind > lately because of my research.

Comment #3: Re: Generalizing and Etymology

Curtis W Franks (Mon Jul 13 16:04:35 2015)

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).

Comment #4: Re: Generalizing and Etymology Curtis W Franks (Mon Jul 13 16:34:29 2015) 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. Comment #5: Re: Generalizing and Etymology Jonathan (Mon Jul 13 22:50:06 2015) It all comes back to that canlu problem doesn't it? We need to first find a way to deal with dimensionality, then we can deal with distinguishing shapes by dimensionality. Perhaps lujvo can be built out of this for star polytopes.