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Voronoi + desired pleat outcomes

Not sure how to describe it, but you can't have two 45 degree angles meet in a 45.135.45.135 intersection and have the two tips meet up without vertical distance separating the two of them. You can see this with an interactive Voronoi tessellation generator, which shows you that such a pleat situation cannot arise regardless of how much you fiddle with it. (or so I'm thinking, please prove me wrong, I would much appreciate it!)

 

However, you *CAN* create something like a 45.90.135.90 intersection, or other variants of the sort- where the upper polygon has a much wider angle. this comes together quite nicely, and the dual of such a pleat intersection is our old friend, the deltoid (kite shape).

 

If we follow through on the pleat arrangements- by folding the Voronoi tessellation lines as a valley fold, and the 1/2 points between that line and the original coordinates as mountain folds (as seen in the picture) we get a fully twistable pleat intersection which realizes the dual as the final twist polygon. In this case, that means the twist will be a kite shape, and it matches up to the geometry of the Delaunay triangulation (the drawn lines in the photo, or in this case the dual of the Voronoi tessellation.)

 

(and yes, the Delaunay triangulation isn't exactly what I have drawn here, but I'm not a math person so it's close enough for my purposes.)

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Taken on June 17, 2006