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It is known that the amount of irrational (incommensurable) numbers is infinite. However, some of them occupy a special place in the history of mathematics, science and education. Their significance lies in the fact that they are expressing some fundamental relationships, which are universal by their nature and appear in the most unexpected places.

The first of them is the irrational number equal to the ratio of the diagonal to the side of a square. This number is associated with the discovery of “incommensurable segments” and the history of the most dramatic period in ancient mathematics that led to the development of the theory of irrationalities and irrational numbers and, ultimately, to the creation of modern “continuous” mathematics.

The next two irrational (transcendental) numbers are as follows: the number of π, which is equal to the ratio of the length of circumference to its diameter (this number lies at the basis of the trigonometric functions) and the Naperian number of e (this number underlies the hyperbolic functions and is the basis of natural logarithms). Between π and e, that is, between the two irrational numbers that dominate over the analysis, there is the following elegant relation derived by Euler:

where is an imaginary unit, another amazing creation of the mathematical mind.

Another famous irrational number is the “golden proportion” Φ = (1 + )/2, which arises as a result of solving the geometric task of “dividing a segment in the extreme and mean ratio” [32]. This task is described in Book II of Euclid’s Elements (Proposition II.11).

In the preface, we already mentioned about the brilliant German astronomer, Johannes Kepler, who named the golden ratio as one of the “treasures of geometry” and compared it to the Pythagoras theorem. A prominent Soviet philosopher Alexey Losev, a researcher of the aesthetics of antiquity and the Renaissance, in his citation (see the preface) argues that “from Plato’s point of view, and in general in terms of the entire ancient cosmology, the Universe is determined as a certain proportional whole, which obeys to the law of harmonic division, the golden section.”

It is well known that the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . . ., introduced in the 13th century by the famous Italian mathematician Leonardo of Pisa (Fibonacci) for solving the “task of rabbits’ reproduction” are closely related to the golden ratio. The deep mathematical connection between the Fibonacci numbers and the golden ratio is that the ratio of the two neighboring Fibonacci numbers in the limit tends to be the golden ratio, which implies that this numerical sequence is also expressing Harmony.

The so-called Pascal’s triangle, the special table for the location of binomial coefficients, is one of the highly harmonious objects of mathematics. This table was proposed in the 17th century by the outstanding French mathematician and physicist Blaise Pascal (1623–1662). In the second half of the 20th century, the famous American mathematician George Polya (1887–1995) in his book [111] had described the connection of Fibonacci numbers to the so-called diagonal sums of Pascal’s triangle. The development of these ideas led to a generalization of the task of rabbit reproduction and the introduction of the so-called Fibonacci p-numbers [6].

Volume I of this three-volume book is devoted to the presentation of the foundations of the theory of these extremely beautiful mathematical objects (Fibonacci p-numbers), the interest in which will not fade for centuries or even millennia.

Volume I consists of four chapters. Chapter 1 “The Golden Section: History and Applications” begins with the analysis of a sensational hypothesis, the Proclus hypothesis, which overturns our ideas about Euclid’s Elements and the entire history of origin of the mathematics. According to this hypothesis, Euclid’s Elements were created under the powerful influence of the “Harmony idea”, which was the basic concept of ancient Greek science. The main goal of Elements was to create a complete geometric theory of Platonic solids. This theory was described by Euclid in the final (Book XIII) book. To give the completed theory of dodecahedron, Euclid, already in Book II, introduces and solves the task of “dividing the segment in the extreme and mean ratio” (Proposition II.11) [32], which in modern science is called the golden section.

Chapter 1 deals with the following: the geometric method of constructing the golden section, the algebraic equation of the golden section, the most famous algebraic identities for the golden ratio and also the geometric figures associated with the golden section (“golden” triangles, pentagon and pentagram, “golden” ellipse, decagon, etc.). Chapter 1 ends with examples of using the golden section in works of fine arts and culture (Cheops pyramid, the arts of ancient Greece and the Renaissance).

Chapter 2 is a popular introduction to the “theory of Fibonacci and Lucas numbers”, which actively began to develop in the second half of the 20th century in the works of Soviet and Western mathematicians and philosophers [7–11]. In Chapter 2, we expound the little-known results and applications of the Fibonacci and Lucas numbers, such as Steinhaus’s Iron Table, the connection of Fibonacci numbers with Pythagorean triangles, numerological properties of Fibonacci and Lucas numbers and we also consider the examples of applications of Fibonacci numbers in Nature (pentagonal symmetry, Fibonacci spirals, phyllotaxis phenomenon, and so on).

In Chapter 2, we also describe the original theory of Fibonacci p-numbers, introduced by Alexey Stakhov in the middle of 1960s. The foundations of this theory were expounded by Stakhov in Refs. [6, 16, 17]. Also, the relationship of the Fibonacci p-numbers to the Pascal triangle and binomial coefficients is shown.

Chapter 3 is devoted to the discussion of diagonal sums of Pascal’s triangle to Fibonacci p-numbers. In Chapter 3, we consider a generalization of the problem of the golden section [6, 16, 17, 60] and introduce an important concept of the “golden p-proportion”, which is a positive root of the algebraic equation of the golden p-proportion and generalization of the classical golden proportion. We consider the algebraic equations for the golden p-proportion, based on Vieta’s formulas, and also Binet’s formulas for the Fibonacci p-numbers and for the Lucas p-numbers.

Chapter 4 is devoted to the Platonic solids and Plato’s cosmology and to the discussion of the historical role of the Platonic solids in the two outstanding discoveries of modern theoretical natural sciences, fullerenes and quasicrystals, which were awarded the Nobel Prize. We consider the Archimedean truncated icosahedron as the most important geometric model of the fullerenes, as the mystery of the Egyptian calendar and its connection to the dodecahedron and also Klein’s conception of icosahedron as the main geometric object of mathematics [113]. In concluding part of Chapter 4, we consider the new ideas in the theory of elementary particles, based on the Platonic solids.