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Given an arbitrary polygon, transform it by a finite sequence of “vertex-centroid” moves to a regular polygon. A vertex-centroid move is a translation of a vertex v along the line vm, where m is the centroid of the vertices of the polygon, i.e., 1/n-th of the sum of the vertex coordinates. Vertices may move only one at a time, but in any order and any number of times.
Steve Gray, 2003.
Let v(t) and m(t) be the positions of the moving vertex and centroid as a function of time t, where t runs from 0 to 1 during the vertex translation. Let L be the line containing v(0) m(0). As v(t) moves on L, m(t) remains on L.
For n=3, a triangle can be made equilateral in two moves. Already for n=4 the situation is less clear.
One could set many other transformational goals besides achieving regularity: scale the polygon by s > 0, rotate the polygon, etc. The notion generalizes to arbitrary dimensions.
A more difficult variant would be to use the area centroid rather than the vertex centroid, in which case m(t) does not remain on L, so that a vertex move would have the flavor of pursuit of a moving target.
J. O’Rourke, 1 Aug. 2005; S. Gray, 15 Aug. 2005.