This second edition of the author’s excellent book on ‘Regular Complex Polytopes’ is always still updated. The term “complex polytope” was first used by D. M. Y. Sommerville [Geometry of n dimensions’ (1929)]. Sommerville presumably uses this word in its colloquial sense. But that idea becomes natural when the coordinates are complex numbers and a Hermitian form is used to define a unitary metric. Thus the generalization of regular polytopes to complex ones can be derived from complex numbers.
The complete definition of a complex polytope and the content of the first edition of this book (1974) is given in a review in Zbl 0296.50009). The new edition contains a further chapter (14) on “Almost regular polytopes”: The symmetry group of a regular polytope is transitive on the vertices while its facets are congruent and regular so that adjacent facets are related by unitary reflections. Relaxing this definition by allowing the transformation to be anti-unitary (this is a semi-linear transformation which involves replacing each coordinate by its complex conjugate) the resulting polytopes are called almost regular. They are still combinatorially regular.
Now new exercises and discussions have been added throughout the book including an introduction to Hopf fibration and real representations for two complex polytopes. While a preponderance of the figures have fivefold symmetry (which became again interest since the discovery of quasicrystals) the new chapter 14 introduces figures having sevenfold symmetry.