Polydrafters

Higher-dimensional geometry (previously "Polyshapes").

Polydrafters

Postby Keiji » Wed Apr 08, 2009 5:03 pm

I came across these while reading through Wikipedia's Polyforms category - wiki article.

Wiki says there are 14 tridrafters, but while trying to find them myself, I found only 10. Later I realized this 14 number comes from the fact that for no good reason you're allowed to attach edges of different lengths together. o_o

Well anyway, I found all the polydrafters up to order 4 that I could which don't allow this stupid behaviour, and here they are. Does anyone feel like checking these? I did them all manually, so I may have missed or duplicated a few. Note that I'm counting reflections as identical.

Also, the angles are not quite exact, since I wanted to make the triangles 3x5 just to make editing a little easier on myself. It's only to illustrate the forms.

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Re: Polydrafters

Postby zero » Sun Apr 12, 2009 2:21 am

I'd prefer to see the component shapes as well as the completed outlines.

Given only the silhouettes to view, the third and fourth tridrafters look like reflected rotations of the same structure. However, there are two different internal configurations to obtain that shape. This is something we don't encounter with polyominos (because they are composed entirely of squares, which are regular).

If you are going to use silhouettes to designate these polydrafters, obscuring the internal structure of how they are connected, then it makes sense, I suppose, to attach additional triangles wherever they fit. If you connect the longest side of a triangle to a polydrafter edge comprised of two shortest component triangle sides, what does it matter if the final shape is all that's of interest? On the other hand, if the component connections are of interest, they ought to be shown.

Let's call these two different approaches by the names of "silhouette" versus "structure" polydrafters. I wonder which approach leads to a greater number of n-drafters for sufficiently large n? In the long run, which matters more? The ability to add component triangles wherever they fit on the edge of a figure? Or the number of different ways the same shape may be built with component triangles matching edge for edge? Note that my assumption for the silhouette-polydrafter approach is that two with the same outline shape are to be considered identical, no matter how many different ways one might build them piece by piece, whereas the structure-polydrafters are distinguished by their internal structure even when they end up looking the same externally.
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Re: Polydrafters

Postby Keiji » Sun Apr 12, 2009 12:27 pm

Aha, I see your point. I am indeed counting polydrafters with the same border as identical, hence why I didn't show the internal structure. That does make those two tridrafters the same. So I guess I should be counting those "mismatched" ones, as well.
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