Harmonic Geometry: Harmonic Involutions, Harmonic Multilaterals and Harmonic Lines

The present edition is also available in print, in 3 PAPERBACK volumes with ISBN 978-618-83803-1-8 (first volume), ISBN 978-618-83803-2-5 (second volume), ISBN 978-618-83803-3-2 (third volume). History Harmonic Geometry is a New Geometry, pioneering, innovative and novel, which emerged after many years of research. Thus, in addition to our well-known Geometries: Descriptive, Analytic, Projective, Differential, etc., we now also have Harmonic Geometry. Definition Harmonic Geometry is that field of Euclidean Geometry that examines only Geometrical Elements with Harmonic Properties. Main Subject Until now, we were acquainted with the Harmonic Quadruple and the Harmonic Quadrilateral, whose various properties and constructions we knew. With the creation of Harmonic Geometry, we sought and succeeded in extending these properties and constructions of the Harmonic Quadruple and the Harmonic Quadrilateral to the Harmonic n-tuples and to the Harmonic Polygons. That is, with Harmonic Geometry, among other things, we succeed in constructing Harmonic n-tuples and Harmonic Polygons, which were unknown until recently. The basis of this achievement was the Regular Polygon and the use of our amazing New Method, called Harmonic Transformation, which has proved to be a powerful tool for the solution of related Geometrical Problems and the proof of related Geometrical Theorems. Also, an additional topic developed in this book is the invention and proof of the main properties of the Harmonic n-tuples and the Harmonic Polygons, which we have achieved with the help of the Harmonic Involutions, found at the beginning of this book. This book is intended for those who love Euclidean Geometry, who are keen and are looking for something new beyond the binds of the trivial. We believe it is possible for Harmonic Geometry to be introduced into other fields of Mathematics, such as Analytic Geometry, Descriptive Geometry, Differential Geometry, etc. but by other researchers, as I, because of my advanced age, will not pursue this issue. That is my vision. Comments We believe that this book deals with a great scientific breakthrough in Mathematics and that is why we consider it a book of great merit. The book has been published and released in Greece, in Greek. It has also been published in English and has been released abroad (apart from Amazon), while the same book will be sent to universities and international award-granting institutions for assessment with the ultimate goal of receiving an award. It can also be found on Amazon, in hard copy, in three volumes and in digital Kindle form, condensed into one volume. We highly recommend the digital Kindle edition because of its very low price. Volumes 1 and 2 of the printed edition are the main book, while Volume 3 is auxiliary and includes all the figures of the book. Along with the digital Kindle edition of the book, we also recommend the purchase of the 3rd volume of the book with all its auxiliary figures as an additional help to the reader when studying it. In the preface, in the epilogue and in the main text, there is also abundant relevant historical and informative data. Let us note that in the last pages (768-779) of the main book there are many comments of praise about the author and his work by approximately fifty mathematicians. I am convinced that anyone who will carefully study this book will really enjoy the Magic, the Harmony and the Greatness of Geometry, and will find in it Geometric Treasures that have remained hidden until today. There are many great challenges in this book and we have given all the relevant tools to those wishing to expand on it. Once again, it has been proved that Geometry is alive and well, in defiance of its enemies. Read more