Tiling Rectangles, Connectivity and Associated Covariants

Tiling by rectangles is the geometric problem of arranging a given family of rectangles in a larger specific frame. A primitive version is rectangle tiling, i.e., tiling rectangles inside a rectangular frame. Then the notion of rectangle tiling is extended to tiling rectangles inside some other shapes. This generalization is achieved through mathematical objects, namely the allocation matrix and the extra spaces. When tiling by rectangles, an effort has been made to obtain a best connected tiling. Connectivity is defined in terms of adjacency for both the given rectangles and the extra spaces. As far as the applications are concerned, the concept of tiling by rectangles has been applied to different fields, with special emphasis on architectural designing, where the aim is to obtain floor plans of some assigned shapes. The presented research work opens a new field for applied mathematicians, in that it combines various aspects of geometry, topology and optimization theory, towards the solution of a problem which is well known to architects, but which has rarely been attacked by mathematical methods.