This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway.
The book is intended both for the use of undergraduate students (especially future teachers of mathematics) and for the informed public interested to learn more about geometry viewed as part of our "cultural heritage." To attain this goal, the author divided the text in two distinct parts, very different and at the same time very well connected to each other.
Part 1 is called "A Cultural Heritage" and contains material usually not included in a mathematical book; it is not a history of geometry, but it refers to some stories and historical connections with the goal of explaining the beginnings, "the roots of the themes to be treated in Part 2." Although this first part of the book is intended for the general public, it has some rigorous mathematical treatments (many of them not quite complete). Certainly the "walk through geometry" offered by this first part of the book is very interesting and fun to read and provides a very appealing and concise view of the development of geometry, without using many deep mathematical arguments (which might discourage a reader not interested too much in the rigorous mathematical treatment of geometry.)
This first part includes six chapters: "Early Beginnings", "The Great River Civilizations", "Greek and Hellenic Geometry", "Geometry in the Hellenistic Era", "The Geometry of Yesterday and Today", "Geometry and the Real World." It takes the reader through the important moments in the development of geometry, from prehistory to such modern day creations as catastrophe theory and fractals. It also allows the reader "to meet" renowned geometers and their important contributions to geometry: Thales of Miletus, Pythagoras, Plato, Archytas, Euclid, Archimedes, Eratosthenes, Nicomedes, Apollonius, Heron of Alexandria, Menelaus of Alexandria, Claudius Ptolemy, Pappus of Alexandria, Hypatia, Desargues, Blaise and Étienne Pascal, Descartes, Cantor, Grothendieck, René Thom, V. I. Arnold, C. Zeeman, Helge von Koch (and these are just some of the names mentioned.)
I found section 3.4 on "The Discovery of Irrational Numbers" very beautiful and (I think) enlightening for all readers. Also of special interest is the treatment of the classical problems — "Constructions by Compass and Straightedge": "Squaring the Circle", "Doubling the Cube", "Trisecting any Angle." While these problems (with a little history) are introduced in Part 1 of the text, a more rigorous treatment is given in chapter 16, included in Part 2 of the book.
Part 2, "Introduction to Geometry", is a true mathematics textbook that develops geometry beginning with Euclid's postulates and ending with fractal geometry and catastrophe theory. It has 12 chapters: "Axiomatic Geometry", "Axiomatic Projective Geometry", "Models for Non-Euclidean Geometry", "Making Things Precise", "Projective Space", "Geometry in the Affine and the Projective Plane", "Algebraic Curves of Higher Degrees in the Affine Plane ", "Higher Geometry in the Projective Plane", "Sharpening the Sword of Algebra", "Construction with Straightedge and Compass", "Fractal Geometry", "Catastrophe Theory."
The treatment of these topics is quite rigorous, although not all (complete) proofs are included and most topics are not treated exhaustively. Most of the proofs are very clear and detailed.
One of the most interesting chapters is 16 "Constructions with Straightedge and Compass", both because of its interesting and beautifully explained content and because it makes a very good connection with the first part of the book. The last two chapters, 17 "Fractal Geometry" and 18 "Catastrophe Theory" provide short but poignant introductions to fractal dimension and control theory respectively, thus "opening the door" to two very important and relatively new areas of geometry.
The book also contains an extensive list of references, including important and beautiful books that could lead the reader to a deeper understanding and knowledge of the topics treated or just mentioned.
Summarizing: I think that this book is a beautiful treatment of geometry. It might very well serve as textbook for geometry courses (the author suggests two such courses: Geometry 1: "Historical Topics in Geometry"; Geometry 2: "Introduction to Modern Geometry".) The only remark I would make is about the absence of any problems or applications that might help the student readers understand the topics better and practice. But an instructor's insight or many other books may be used to supplement Audun Holme's book.
I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course, or for lecturers considering a presentation on geometry for the general public. The book will be very useful also for high-school and undergraduate interested in studying geometry. A receptive and informed general public interested in mathematics will find in the first (and, maybe, second) part of the book a very readable and fun approach to geometry.
Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.