The first question one may have about The Concrete Tetrahedron is what the authors, Manuel Kauers and Peter Paule, have in mind by a concrete tetrahedron.
The word concrete in the title is an allusion to Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik. It is also a blend of con- from continuous and -crete from discrete. That is, the book deals with a blend of continuous and discrete mathematics, applying continuous methods to discrete problems.
The tetrahedron in the title alludes to the four recurring themes of the book, each connected to the other as are the vertices of a tetrahedron. These themes are
- symbolic sums,
- recurrence equations,
- generating functions, and
- asymptotic estimates.
After two introductory chapters, the rest of the chapters in the book are organized according to the tetrahedron motif: the tetrahedron for polynomials, the tetrahedron for hypergeometric functions, etc.
How does The Concrete Tetrahedron related to Concrete Mathematics? Here is the authors’ explanation.
In most stages of our discourse, the book by Graham, Knuth, and Patashnik can be used as a reference to many additional techniques, applications, stories, etc. However, the present book is not meant to be merely a summary of “Concrete Mathematics.” We have a new twist to add to the matter, and this is computer algebra. In the last decade of the 20th century [i.e. after the publication of Concrete Mathematics], many algorithms have been discovered…
As the authors say, The Concrete Tetrahedron is not a summary of Concrete Mathematics, despite extensive overlap and despite their relative sizes, the former being about a third the size of the latter. Tetrahedron has a unique perspective and unique material. It is shorter, but also more focused and more compactly written. Finally, although the subtitle of Concrete Mathematics is “A Foundation for Computer Science”, there is more explicit computer science in The Concrete Tetrahedron.
Those who are interested in the subject matter of Concrete Mathematics will enjoy The Concrete Tetrahedron for an independent perspective, some more recent results, and for new applications.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.