Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 2, pp. 389403 (2009) 

Cardinality estimates for piecewise congruences of convex polygonsChristian RichterMathematisches Institut, FriedrichSchillerUniversität, D07737 Jena, Germany, email: richterc@minet.unijena.deAbstract: 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 Full text of the article:
Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.
© 2009 Heldermann Verlag
