Numeral Systems with Irrational Bases for Mission-Critical Applications

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• Publisher: WSPC
• eBook ISBN: 9789813228634

Chapter 1

Preliminary Historical and Mathematical Information

1.1. The idea of harmony in its historical development

1.1.1. Harmonic ideas of Pythagoras, Plato and Euclid

What is the major idea of the ancient Greek science? A majority of researchers give the following answer: the idea of Harmony associated with the golden ratio and Platonic Solids. Pythagoras, Plato and Euclid were the most outstanding representatives of this trend in the ancient Greek science, philosophy and mathematics. The greatest interest to the concept of harmony, that is, to the ideas of Pythagoras, Plato and Euclid, always arose in the periods of greatest prosperity of the “human spirit.” From this point of view, in the studying of the Mathematics of Harmony [4] we can highlight the following critical periods.

1.1.2. The ancient Greek period

Conventionally, it can be assumed that this period starts with the research of Pythagoras and Plato. Euclid’ Elements became a final event of this important period. According to Proclus’ hypothesis [34], Euclid created his Elements in order to create the complete geometric theory of the five Platonic solids, which have been associated in the ancient Greek science with the Universe Harmony. The geometric fundamentals of the theory of Platonic solids (Fig. 1.1) were described by Euclid in the concluding Book (Book XIII) of the Elements.

In addition, Euclid simultaneously introduced in Elements some advanced achievements of ancient Greek mathematics, in particular, the golden section (Book II), which was used by Euclid for the creation of the geometric theory of the Platonic solids (Book XIII).

Figure 1.1. Platonic solids: tetrahedron, octahedron, cube, dodecahedron, icosahedron.

1.1.3. The Middle Ages

In the Middle Ages, very important mathematical discovery was made. The famous Italian mathematician Leonardo of Pisa (Fibonacci) wrote the book “Liber Abaci” (1202). In this book, he described the task of rabbits reproduction. By solving this problem, he found the remarkable numerical sequence — the Fibonacci numbers F_n:

which are given by the recurrent relation:

1.1.4. The Renaissance

This period is connected with the names of the prominent figures of the Renaissance: Piero della Francesca (1412–1492), Leon Battista Alberti (1404–1472), Leonardo da Vinci (1452–1519), Luca Pacioli (1445–1517), Johannes Kepler (1571–1630). In that period three books, which were the best reflection of the idea of the “Universe Harmony,” were published. The first of them is the book Divina Proportione (“The Divine Proportion”) (1509). This book had been written by the outstanding Italian mathematician and scholar monk Luca Pacioli under the direct influence of Leonardo da Vinci, who illustrated Pacioli’s book.

Also the brilliant astronomer of 17th century Johannes Kepler made an enormous contribution to the development of the “harmonic ideas” of Pythagoras, Plato and Euclid.

Figure 1.2. Kepler’s Cosmic Cup.

In his first book Mysterium Cosmographicum (1596) he built the so-called Cosmic Cup (Fig. 1.2), the original model of the Solar system, based on the Platonic solids. The book Harmonice Mundi (Harmony of the World) (1619) is the main Kepler’s contribution into the Doctrine of the Universe Harmony. In the Harmony, he attempted to explain the proportions of the Universe — particularly the astronomical and astrological aspects — by using musical terms. The Musica Universalis or Music of the Spheres, which had been studied by Pythagoras, Ptolemy, was the central idea of Kepler’s Harmony.

1.1.5. The 19th century

This period is connected with the works of the French mathematicians Jacques Philippe Marie Binet (1786–1856), Francois Edouard Anatole Lucas (1842–1891), German poet and philosopher Adolf Zeising (1810–1876) and the German mathematician Felix Klein (1849–1925).

Jacques Philippe Marie Binet derived a mathematical formula (Binet’s formula) to represent the Fibonacci numbers through the “golden ratio”

(see below).

Francois Edouard Anatole Lucas introduced the Lucas numbers L_n similar to the Fibonacci numbers F_n (1.1); the Lucas numbers L_n are calculated by the recurrent relation similar to (1.2), but with other seeds:

The recurrent relation (1.3) generates the Lucas numbers:

The merit of Binet and Lucas consists of the fact that their researches became a launching pad for the researches of American Fibonacci Association, established in 1963.

German poet Adolf Zeising in 1854 published the book Neue Lehre von den Proportionen des menschlichen Körpers aus einem bisher unerkannt gebliebenen, die ganze Natur und Kunst durchdringenden morphologischen Grundgesetze entwickelt. The basic idea of Zeising is to formulate the Law of proportionality. He formulated this Law as follows:

“A division of the whole into unequal parts is proportional, when the ratio between the parts is the same as the ratio of the bigger part to the whole; this ratio is equal to the golden mean.”

The famous German mathematician Felix Klein in 1984 published the book “Lectures on the icosahedron and the solution of equations of the fifth degree” [50] dedicated to the geometric theory of the icosahedron and its role in the general theory of mathematics. Klein treats the icosahedron as a mathematical object, which is a source for the five mathematical theories: geometry, Galois theory, group theory, invariant theory and differential equations.

What is the significance of Klein’s ideas from the point of view of the Mathematics of Harmon...

Table of contents

• Cover page
• Title page
• Copyright
• Contents
• Preface
• Introduction
• Acknowledgements
• Chapter 1. Preliminary Historical and Mathematical Information
• Chapter 2. A New View on Numeral Systems: Unusual Hypotheses, Surprising Properties and Applications
• Chapter 3. Bergman’s System, “Golden” Number Theory and Mirror-Symmetrical Arithmetic
• Chapter 4. Fibonacci p-Codes and Concept of Fibonacci Computers
• Chapter 5. Codes of the Golden p-Proportions and Their Applications in Computer Science and “Golden” Metrology
• Bibliography
• Index