An Application of Number Theory to the Organization of Raster-Graphics Memory

A high-resolution raster-graphics display is usually combined with processing power and a memory organization that facilitates basic graphics operations. For many applications, including interactive text processing, the ability to quickly move or copy small rectangles of pixels is essential. This paper proposes a novel organization of raster-graphics memory that permits all small rectangles to be moved efficiently. The memory organization is based on a doubly periodic assignment of pixels to M memory chips according to a “Fibonacci” lattice. The memory organization guarantees that, if a rectilinearly oriented rectangle contains fewer than M/ @@@@5 pixels, then all pixels will reside in different memory chips and thus can be accessed simultaneously. Moreover, any M consecutive pixels, arranged either horizontally or vertically, can be accessed simultaneously. We also define a continuous analog of the problem, which can be posed as: “What is the maximum density of a set of points in the plane such that no two points are contained in the interior of a rectilinearly oriented rectangle of unit area?” We show the existence of such a set with density 1/ @@@@5, and prove this is optimal by giving a matching upper bound.