Transforming Polygons via Vertex-Centroid Moves

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For vertices $v_1,\ldots,v_n\in\mathbb R^2$, the vertex centroid is $m=n^{-1}\sum_i v_i$. In one vertex-centroid move, a single vertex $v_i$ is translated along the line through $v_i$ and the current $m$, with all other vertices fixed. Recompute the centroid after each move. Vertices can be moved in any order and can be moved repeatedly. The same vertex-centroid definition applies in $\mathbb R^d$. A scaling target uses a prescribed factor $s>0$; a rotation target applies a prescribed Euclidean rotation. When a polygon bounds a region $P$ of positive area, its area centroid is $m_{\rm area}=\operatorname{area}(P)^{-1}\int_P x\,dx$. In the area-centroid variant, the moving vertex moves along the instantaneous line through it and the current area centroid, which can itself leave the initial line during the move.

Statement recorded in the supplied source.

Can every polygon be transformed exactly into a regular polygon by a finite sequence of vertex-centroid moves? Investigate also the associated transformation goals of scaling by a prescribed $s>0$ or rotating the polygon, and vertex-centroid transformations in arbitrary dimensions. Does the regularization question hold with the area centroid in place of the vertex centroid? [S2]

Short English statement

Can every polygon be transformed exactly into a regular polygon by a finite sequence of vertex-centroid moves? Investigate also the associated transformation goals of scaling by a prescribed $s>0$ or rotating the polygon, and vertex-centroid transformations in arbitrary dimensions. Does the regularization question hold with the area centroid in place of the vertex centroid?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5500060 (AMR-054-0060). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke (editors), The Open Problems Project, maintained original source, Problem 60, Statement and complete Partial/Related Results. https://github.com/edemaine/topp/blob/main/Problems/P.000060.