Rectangle tilings, connectivity and associated covariants

In this work, we focus on Tiling by rectangles, its 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. Then the notion of rectangle tiling is extended to tiling rectangles inside some other shapes. This generalization is achieved through another object of a mathematical nature, namely the allocation matrix. This problem also involves the study of another mathematical concept the extra spaces which are introduced in the geometric distribution process. When tiling by rectangles, an effort has been made to obtain a tiling which is best from the point of view of connectivity. To refine the number of solutions originating from the algorithm, we have studied and developed several covariants related to tilings and their graph. The presented research work opens a new field for applied mathematicians, in that it combines various aspects of geometry, topology, graph theory and optimization theory, towards the solution of a problem which is well known to architects, but which has rarely been attacked by mathematical methods.