Weird Geometric Construction

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Loganfs
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Weird Geometric Construction

Postby Loganfs » Thu May 24, 2007 1:44 am UTC

I was messing around in Geometer's Sketchpad and found a construction, but I can't figure out what it is. Any ideas?

Image

That is connecting the midpoints of the outer polygon, then connecting the midpoints of the midpoints, etc. It seems to "regularize," but I can't see exactly what's happening or where it will stop.

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adlaiff6
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Postby adlaiff6 » Thu May 24, 2007 3:18 am UTC

Every two iterations, you're essentially moving each vertex to the average point between the midpoints on either side of it (think vectors). If you instead move each vertex to the average of the point on either side (or all adjacent points, in a 3d polyhedron), you get an operation I've seen in Maya called, drumroll please, Average Vertices. It has the tendency to smooth a figure and decrease the volume, although for some convex shapes, the volume can increase. If you keep iterating that, they will converge to a point.
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The LuigiManiac
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Postby The LuigiManiac » Thu May 24, 2007 3:23 am UTC

Trippy. What version of Geometer's Sketchpad did you use? Or is there only the one really old version?
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