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

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 http://www.worldscientific.com/worldscibooks/10.1142/9603.

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