This book comprises, in addition to auxiliary material, the research on which I have worked for over 50 years. Some of the results appear here for the first time. The impetus for writing the book came from the late Victor Klee, after my talk in Minneapolis in 1991. The main subject is simplex geometry, a topic which has fascinated me since my student days, caused, in fact, by the richness of triangle and tetrahedron geometry on one side and matrix theory on the other side. A large part of the content is concerned with qualitative properties of a simplex. This can be understood as studying relations not only of equalities but also of inequalities. It seems that this direction is starting to have important consequences in practical (and important) applications, such as finite element methods.

Another feature of the book is using terminology and sometimes even more specific topics from graph theory. In fact, the interplay between Euclidean geometry, matrices, graphs, and even applications in some parts of electrical networks theory, can be considered as the basic feature of the book.

In the first chapter, the matricial methods are introduced and used for building the geometry of a simplex; the generalization of the triangle and tetrahedron to higher dimensions is also discussed. The geometric interpretations and a detailed description of basic relationships and of distinguished points in an n-simplex are given in the second chapter.