Cardinality estimates for piecewise congruences of convex polygons

Christian Richter

Abstract: Two convex polygons $P,P^\prime \subseteq {\mathbb R}^2$ are congruent by dissection with respect to a given group $G$ of transformations of ${\mathbb R}^2$ if both can be dissected into the same finite number $k$ of polygonal pieces $Q_1,\ldots,Q_k$ and $Q_1^\prime,\ldots,Q_k^\prime$ such that corresponding pieces $Q_i,Q_i^\prime$ are congruent with respect to $G$, $1 \le i \le k$. In that case $\DEG_G(P,P^\prime)$ denotes the smallest $k$ with the above property. For the group ${\rm Isom}^+$ of proper Euclidean isometries we prove two general upper estimates for $\DEG_{{\rm Isom}^+}(P,P^\prime)$, the first one in terms of the numbers of vertices and the diameters of $P,P^\prime$, the second one depending moreover on the radii of inscribed circles. A particular result concerns regular polygons $P,P^\prime$. For the groups ${\rm Sim}^+$ and ${\rm Sim}$ of proper and general similarities we give upper bounds for $\DEG_{{\rm Sim}^+}(P,P^\prime)$ and $\DEG_{\rm Sim}(P,P^\prime)$ in terms of the numbers of vertices.

Keywords: congruence by dissection, scissors congruence, piecewise congruence, equidissectable, convex polygon, isometry, similarity, translation, number of pieces

Classification (MSC2000): 52B45; 52B05

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