Fundamentals of Geometry


"Ptolemy once asked Euclid if there was not

a shorter road to geometry than through the Elements,

and Euclid replied that there was no royal road to geometry.''


            While there are no royal roads to geometry, I invite you on a hitchhiking trip into this wonderful land. Recently it has come into fashion to take readers only to its modern part where it borders on the amazing realm of algebra. In my opinion, however, to restrict teaching geometry to its modern advances is like limiting the study of architecture to the buildings made of glass, steel, and concrete. While the latter are more likely to inspire awe in the young technologically oriented minds than the austere creations of the ancients, something that is newer is not always more effective or more beautiful…

            Like architecture, geometry has long and exciting history. In Part 1, entitled Classical Geometry, I take you to a history museum. But the goal of this visit is not to show you dusty specimens that would come apart at a touch of a straw. Rather, it is to show you the intellectual power, esthetic beauty, and the ongoing practical utility of methods whose ancestry can be traced to the Greeks. Just as we freely use modern terminology to express our appreciation of ancient visual arts, we can better appreciate the ancient art of geometry using modern axiomatic method. Following these considerations, Part 1 is built on the solid foundation of Hilbert's axioms…        

            Another noticeable feature of this book is the extensive use of formal notation - letters and quantifiers instead of words, etc. Granted, this requires some initial effort on the part of the reader ''to acclimatize.'' But, once the notation is mastered,  significant  economy of type/pen ink/paper, as well as eye movements, can be achieved.  Since so many proofs in geometry are long and tedious, my concern was to present them in a most compact manner. Proofs that occupy several pages tend to frustrate the reader (this is especially true about those with modest mathematical background, such as high schoolers or college juniors), by their sheer extent.

            In proofs, it is very hard to keep track  of every minute detail. The notorious words like ''it is easy to show that'', ''obviously'', ''evidently'', ''trivial'', can sometimes be useful. They inform the reader that it would not take an extraordinary intellectual feat to supply missing arguments. On the other hand, the author has tried to exercise his utmost discretion in calling obvious and trivial only those things that are obvious and trivial. He realizes (from both his own and his students' bitter experience) that when used improperly such labels can foster the intellectual inferiority complex. Learned helplessness growing into exasperation forces the learner to put aside the offending book. Then, the best intent of the writer defeats itself…

            This book, and even this foreword, is not, and is unlikely to ever become, complete. Nor is it likely to ever start glittering with much typographic gloss. So anyone reading it may envision himself a hardy beta-tester! However, I believe this should not discourage interested people from reading it. Besides, geometry itself, like any other area of scientific knowledge, will never become a finished book, which does not prevent some people from studying it with ardent enthusiasm!   

            In contrast to many pre-press versions appearing on the sites of publishers and/or individuals writing them, this work is not intended for publication. In writing the book, the author had in mind a noble purpose of sharing the knowledge with everyone seeking it. He realizes that publishing it on paper might interfere with its free distribution because of various laws concerning ''intellectual property'' - the term that the author living in a country where some people have intellect and the others property - finds offensive. My book carries no copyright, copywrong, or whatever, attached. You are completely free to reproduce it or any part of it in any form and by any means without begging me or anyone else to grant you a permission.

            On the other hand, the present book aims to dissolve a myth that has been around as long as the online writing itself. Namely, the perception of a free internet book as a poor man's version of a printed book, both in its coverage and the quality of content.    

            It seems necessary to put in a couple of words about possible use of this text in teaching. Although much of the book is devoted to what has traditionally come to be termed ''elementary geometry'', the present monograph is emphatically not suitable to serve as someone's first geometry book. This does not mean, however, that this book, addressed to lovers of geometry of all ages, cannot be used in teaching at secondary level. Quite to the contrary. High school students with serious interest in geometry can use my book both to lay a solid basis for the material learned from their textbooks and as a starting point in their further exploration of this wonderful subject.  

            Originally this text was intended to supply geometric background for another project by the author (still in its germinal state), a book on symmetry. That is where the mysterious words "Introduction to Symmetry" in the title have come from. Soon as I started writing the present text, however, I immediately got carried away by the beauty of the subject matter...

            Well, I guess this foreword is already getting long and boring. So…

            Here is the current (February 2007) version of my book.

            Please note that this is a work in progress and accept my apologies for any inconveniences resulting from this. Adobe Acrobat (Reader) 5.0+ is required to view the book properly. As the file is over 2MB in size, you might wish to download it for offline viewing.



            Oleg A. Belyaev,

            Moscow State University,

            Dept. of  Physics.


You are welcome to contact the author by the following e-mail: