Transforming polygons into regular polygons by vertex-centroid moves
Needs verification: A solution is claimed (Guan, 2026); held until the record of the claim is listed here.
Let $P$ be a polygon with $n$ vertices, and let $m$ be the centroid of its vertices, the average of their coordinates. A vertex-centroid move translates one vertex $v$ along the line through $v$ and $m$. Vertices move one at a time, in any order and as often as desired.
During such a move the centroid $m$ also stays on the line through the original positions of $v$ and $m$. A triangle can be made equilateral with two moves; for quadrilaterals the situation is already less clear.
Problem. Can every polygon be transformed into a regular polygon by a finite sequence of vertex-centroid moves?
Source: Gray (2003); Problem 60 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Keywords
- polygon transformation, regular polygon, The Open Problems Project, vertex centroid
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01098, "Transforming polygons into regular polygons by vertex-centroid moves", clopen.solutions.