This interesting book illustrates the use of space-filling curves in scientific computing. The basic reason for this use is given by the need of sequentialization of multidimensional data, in many applications. After presenting the definition of a space-filling curve and two classical examples (the Hilbert curve and the Peano curve), the author illustrates a grammar-based description and an arithmetic representation of space-filling curves, the concept of approximating polygons, the Sierpinski curve and other space-filling curves. The extension of the definition of these curves to the 3D case is then discussed.
In order to approximate space-filling curves, the concepts of quadtree and octree (and their generalization to higher dimensions) are introduced. Parallelization and partitioning based on space-filling curves, the problem of Hölder continuity of space-filling curves and the Sierpinski curves on triangular and tetrahedral meshes are also considered. Finally, two case studies concerning space-filling curves are illustrated (cache efficient algorithms for matrix operations and numerical simulation on spacetree grids).
An extended bibliography is given (including, in particular, [H. Sagan, Space-filling curves. New York: Springer (1994; Zbl 0806.01019)]).
The exposition is rather informal and the mathematical content is limited to an elementary treatment.
Several exercises are proposed and for many of them a solution is presented at the end of the book.