Dissections And Configurations Of Polygons

Laczkovich proved that if a square can be cut into a finite number of similar triangles, then either those triangles are right triangles or their interior angles are rational multiples of In this dissertation we discuss the problems of dissections of regular n-gons, parallelograms and orthogonal polygons into similar triangles.Stein suggested many famous open problems about the dissections of polygons into triangles with equal areas. We study, in particular, the related dissection problems about trapezoid, quadrilateral, polyomino, hyperpolyomino, orthogonal polygon and special polygon and the Stein conjectures.Stein and Szabo posed the following open problem:When does an m1 x m2 x ... xmn brick tile an a1 x a2 x ...?x an n-dimensional box?We give some related necessary and sufficient conditions.About the tilings of rectangles we prove that the rectangle R(t) (t Q+) can be decomposed into finitely many rectangles similar to R(u) if and only if u is algebraic and the real part of each of its conjugates is positive, and that if u is algebraic and each of its conjugates lies in the open half-plane Re(z) > 0, then the rectangle R(m) (m N) can be decomposed into finitely many rectangles similar to R(um). As for the tilings of rectangles with similar right triangles we obtain the similar results.One of the important problems in the study of polygon configuration is theproblem on the relative distance. In this paper we prove the Doliwka - Lassak conjecture about relatively short sides of convex hexagons, and obtain some results about relatively short sides of convex polygons, and results about relatively long sides of convex polygons, and we generalize a result of Zsolt La"ngi about relatively short sides of convex heptagons to arbitrary convex n-gons (n > 7).