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Application of the mathematics of harmony -- Golden non-Euclidean geometry in modern math

World Scientific


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Credit: World Scientific

Have you heard of Pythagoras's idea regarding the main components of mathematics? According to Pythagoras, mathematics consists of four main components called MATEMs, namely arithmetic, geometry, harmonics and spherics. Unfortunately, the MATEM of harmonics was lost in the process of historical development of mathematics.

The question arises: how was the MATEM of harmonics reflected in Euclid's Elements? This question is answered by the Proclus hypothesis, which was formulated in the 5th Century (A.D.) by outstanding Greek philosopher, Proclus, who was also one of the best commentators of Euclid's Elements. By analysing the Euclidean Elements, Proclus drew attention to the fact that Euclidean Elements began with the "golden section" (Proposition II.11) and ended with Platonic solids, which were associated with the "Harmony of Universe" in Plato's cosmology. This observation led him to the conclusion that Euclid wrote his Elements with the goal to create the geometric theory of Platonic solids. This means that the basis idea of Euclid's Elements is the mathematical simulation of the Harmony of the Universe, based on Platonic solids. Thus Euclidean Elements can be regarded, historically, the first mathematical theory of the Harmony of the Universe.

The "Golden" Non-Euclidean Geometry: Hilbert's Fourth Problem, "Golden" Dynamical Systems, and the Fine-Structure Constant along with its prequel The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science seek to restore the MATEM of harmonics in modern mathematics. They go back to ancient mathematics and affect our idea of the development of the history of mathematics, starting from Pythagoras, Plato and Euclid, in unexpected ways.

Famous German scientist Professor Volkmar Weiss commented that the prequel was 'a breakthrough' and that it 'may well change not only the way we view the history of mathematics, but the future development of mathematics in its application to the natural sciences and computer design'.

For those interested in the development of modern science and mathematics, this sequel to the highly praised The Mathematics of Harmony is not to be missed.

The "Golden" Non-Euclidean Geometry outputs the initial concept on the level of scientific Millennium Problems. In essence, it opens a new stage in the development of non-Euclidean geometry. In contrast to classical non-Euclidean geometry, the principle of self-similarity, which is used by Nature in the development of natural structures, is embodied in the "Golden" non-Euclidean geometry.

This book retails at US$118 / £98 (hardback). More information can be found at


About the Authors

Alexey Stakhov is a Ukrainian mathematician who has lived in Canada since 2004. Stakhov obtained his Doctorate in Computer Science in 1972 and his Professorship in 1974. He is the author of over 500 publications and 14 books. He is also the creator of many original theories in mathematics and computer science including algorithmic measurement theory, the mathematics of harmony, and codes of the golden p-proportions; and holds 65 international patents. In 2009, Stakhov was awarded the Certificate of Honour and the medal of "Knight of Science and Art" by the Russian Academy of Natural Sciences. Together with Aranson, they were the first awardees of the Commemorative Medal of Leonardo Fibonacci.

Samuil Aranson is a prominent Russian mathematician who lives in the USA. He is a Doctor of Physical-Mathematical Sciences (in differential equations, geometry and topology), Professor, Honored Worker of Science of Russia, and Academician of the Russian Academy of Natural Sciences. He has authored more than 200 scientific works, including monographs that were published in Russia, USA, Germany and other countries. In 2016, the Euro Chamber awarded Aranson the Golden Medal for outstanding achievements in science. Together with Stakhov, they were the first awardees of the Commemorative Medal of Leonardo Fibonacci.

About World Scientific Publishing Co.

World Scientific Publishing is a leading independent publisher of books and journals for the scholarly, research, professional and educational communities. The company publishes about 600 books annually and about 130 journals in various fields. World Scientific collaborates with prestigious organizations like the Nobel Foundation, US National Academies Press, as well as its subsidiary, the Imperial College Press, amongst others, to bring high quality academic and professional content to researchers and academics worldwide. To find out more about World Scientific, please visit For more information, contact Amanda Yun at

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