"Mathematics of the Golden Section"

  1. Introduction into Pascal triangle. Pascal triangle and its properties. A little of history. Pascal operation. Binomial coefficients. A number of parts of the given set. Connection with factorials. Generalized Pascal triangles and generalized binomial coefficients. Pascal pyramids and trinomial coefficients. Multinomial coefficients and Pascal hyperpiramids.
  2. Introduction to Fibonacci numbers. Fibonacci numbers in the Pascal triangle. Lucas and Catalan numbers. A little of history. Relationships for Fibonacci and Lucas numbers. Application of Fibonacci numbers. Generalized Fibonacci numbers and their algebraic properties.
  3. Golden Section. Connection of Fibonacci numbers with the Golden Section. Algebraic properties of the Golden Section. Geometry of the Golden Section. Pentagon. Decagon. Golden triangles. Golden rhombuses. Platonic solids. Golden brick. Golden masonry. Generalized Golden Sections.
  4. Theory of Fibonacci matrices. The Fibonacci Two by Two Matrix, the Q-matrix, and its generalizations, the Fibonacci (p + 1) ´ (p + 1) matrices, based on the p-Fibonacci numbers. The theorems for powers of the Fibonacci matrices and their determinants.
  5. Hyperbolic Fibonacci and Lucas functions. Role of the fundamental mathematical constants in the mathematics history. The p-number and trigonometric functions. The Euler's number e and the exponential function. The hyperbola theory. Hyperbolic rotation. Geometric theory of natural logarithms. Hyperbolic functions. Euler's formulas. Binet's formulas. Hyperbolic Fibonacci and Lucas functions as a generalization of the Binet's formulas for continuos domain.
  6. Golden Section and Fibonacci numbers in modern science. Resonance theory of the Solar system. Quasi-crystals. New geometric theory of phyllotaxis. Law of structural harmony of systems.
  7. Application of the Golden Section to computers and modern information technologies. Algorithmic measurement theory. Coding theory based on Fibonacci numbers and Golden Section. Fibonacci arithmetic and Fibonacci computer concept. Fibonacci cryptography.
  8. Concept of Harmony Mathematics. Mathematics: the search for knowledge. Three fundamental problems of science in the mathematics history. Look to the mathematics history since Golden Section's point of view. Foundations of the classical mathematics. New approach to the mathematics foundations since Golden Section point's of view: (1) algorithmic measurement theory; (2) new geometric number definition based on the Golden p-Sections; (3) the Golden Section generating hyperbolic Fibonacci and Lucas functions. Structure of Harmony mathematics.


  1. Stakhov, A.P., Massingue, V., Sluchenkova, A.A. Introduction into Fibonacci coding and cryptography. Kharkiv, Publishers "Osnova", 1999.
  2. Stakhov, O.P. Computer Arithmetic based on Fibonacci numbers and Golden Section: New Information and Arithmetic Computer Foundations. Toronto, SKILLSET Training, 1997.
  3. Hoggat, V., Jr., Fibonacci and Lucas numbers. Houghrton-Mifflin, Palo Alto, California, 1969.
  4. Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited, 1989.
  5. Vorobjov, N.N. Fibonacci numbers. Moscow, Nauka, 1978.
  6. Bondarenko, B.A. Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs, and Applications. Fibonacci Association, 1993.