Problem 60: Transforming Polygons via Vertex-Centroid Moves

**Statement**- 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. **Origin**- Steve Gray, 2003.
**Status/Conjectures**- Open.
**Partial and Related Results**- 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. **Appearances****Categories**- polygons
**Entry Revision History**- J. O'Rourke, 1 Aug. 2005; S. Gray, 15 Aug. 2005.