Appendix 4

Program of the course "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 to 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. 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 Binet's formulas for continues domain. A new approach to the Fibonacci numbers theory.
  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 (measurement, calculation and harmony). 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) the Golden p-Sections as a new class of irrationals and new number theory; (3) Hyperbolic Fibonacci and Lucas functions as generalization of Binet's formulas. Structure of Harmony mathematics. Applications of Harmony mathematics.

References:

  1. Hoggat, V., Jr., Fibonacci and Lucas numbers. Houghrton-Mifflin, Palo Alto, California, 1969.
  2. Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited, 1989.
  3. Vorob'ev, N.N. Fibonacci numbers. Moscow, Publisher "Nauka", 1978.
  4. Bondarenko, B.A. Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs, and Applications. Fibonacci Association, 1993.
  5. Stakhov, A.P., Massingue, V., Sluchenkova, A.A. Introduction into Fibonacci coding and cryptography. Kharkov, Publisher "Osnova", 1999.
  6. Stakhov, O.P. Computer Arithmetic based on Fibonacci numbers and Golden Section: New Information and Arithmetic Computer Foundations. Toronto, SKILLSET Training, 1997.
  7. Stakhov A.P. A New Mathematics for the Alive Nature: Hyperbolic Fibonacci and Lucas Functions. Moscow, Publisher "Kirillica", 2003.
  8. Stakhov A.P. Sacred Geometry and Harmony Mathematics. Vinnitsa, Publisher "ITI", 2003.
  9. Stakhov A.P., Sluchenkova A.A. Museum of Harmony and Golden Section: Mathematical Connections in Nature, Science and Art. Vinnitsa, Publisher "ITI", 2003.

Program of the course "Fibonacci Arithmetic, Coding and Computers"

  1. Number systems. Basic concepts and definitions. Positional and non-positional notations. A little of a history. Radix of notation. Requirements to notations. Canonical positional notations. Binary notations and binary arithmetic. Special binary notations. Conversion algorithms of numbers from one notation to other one. Optimal radix for notations. Symmetrical notations. Ternary symmetrical notation and ternary arithmetic. Nega-positional notations and arithmetic. Notations with the complex radix. Factorial notation. Notation in residual classes.
  2. Golden Section and Fibonacci Numbers. Golden Section problem. Algebraic and geometric properties of the Golden Section. Fibonacci and Lucas numbers. Generalized Fibonacci numbers. Hyperbolic Fibonacci and Lucas functions. Generalized Golden Sections. Fibonacci Q-matrix. Generalized Fibonacci matrices.
  3. Algorithmic measurement theory. Fibonacci's "weighing problem". Asymmetry principle of measurement. Classical measurement algorithms. "Binary" measurement algorithm and its generalization. Measurement algorithms based on Pascal Triangle. Fibonacci measurement algorithms.
  4. Fibonacci codes and arithmetic. Fibonacci representation. Zeckendorf representation. Redundancy of Fibonacci codes. Fibonacci addition and subtraction. Fibonacci multiplication and division. Basic micro operations. Noise-tolerant Fibonacci arithmetic. Concept of Fibonacci computer. Fibonacci self-synchronization codes.
  5. Bergman's notation. Mathematical properties of the "Tau-system". Multiplicity of number representation. Z-property of natural numbers. F- and L-codes. "Golden" arithmetic. Notations based on generalized Golden Proportions. New definition of a number. Concept of Harmony mathematics.
  6. Ternary mirror-symmetrical notation. Number conversion from the "Tau-system to the ternary "golden" representation. Ternary F- and L-codes. Property of "mirror symmetry". Representation of negative numbers. Range of number representation. Redundancy of the ternary mirror-symmetrical representation. Ternary mirror-symmetrical addition and subtraction. "Swing"-phenomenon. Ternary mirror-symmetrical multiplication and division. Mirror-symmetrical representation with the floating point. Ternary flip-flap-flop. Ternary mirror-symmetrical adder.
  7. "Golden" analog-to-digit and digit-to-analog converters. Resistive divisor for the "Tau-system". Resistive divisor for the ternary mirror-symmetrical number system. Digit-to-analog converters for number systems with irrational radices. Self-correcting analog-to-digit converters.
  8. Coding theory based on Fibonacci matrices. Coding based on the Q-matrix. Basic check correlation for Fibonacci coding. Error detection and correction. Fibonacci coding based on the generalized Fibonacci matrices. Application to cryptography.

References:

  1. Hoggat, V., Jr., Fibonacci and Lucas numbers. Houghrton-Mifflin, Palo Alto, California, 1969.
  2. Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited, 1989.
  3. Vorob'ev, N.N. Fibonacci numbers. Moscow, Publisher "Nauka", 1978.
  4. Bondarenko, B.A. Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs, and Applications. Fibonacci Association, 1993.
  5. Stakhov, O.P. Computer Arithmetic based on Fibonacci numbers and Golden Section: New Information and Arithmetic Computer Foundations. Toronto, SKILLSET Training, 1997.
  6. Stakhov, A.P., Massingue, V., Sluchenkova, A.A. Introduction into Fibonacci coding and cryptography. Kharkiv, Publishers "Osnova", 1999.
  7. Stakhov A.P. A New Mathematics for the Alive Nature: Hyperbolic Fibonacci and Lucas Functions. Moscow, Publisher "Kirillica", 2003.
  8. Stakhov A.P. Sacred Geometry and Harmony Mathematics. Vinnitsa, Publisher "ITI", 2003.
  9. Stakhov A.P., Sluchenkova A.A. Museum of Harmony and Golden Section: Mathematical Connections in Nature, Science and Art. Vinnitsa, Publisher "ITI", 2003.
Doctor of Sciences in Computer Science, Professor
Scientific coordinator of the International Conference "Problems of Harmony,
Symmetry and the Golden Section in Nature, Science and Art"
Academician of the Ukrainian Academy of Engineering Sciences
President of the International Club of the Golden Section
Alexey Stakhov