MMW

• Leonardo of Pisa or Pisano


  Itallian Mathematician who is famous for Fibonacci Sequence and published Liber Abaci and Book of Calculations around 1200 AD
• The terms in Fibonacci sequence is called Fibonacci numbers.
• Golden Rectangle


  is known as one of the most visually satisfying of all geometric forms - hence, the appearance of the Golden Ratio in art. It is also related to the Golden Spiral, which is created by making adjacent squares of Fibonacci dimensions.
• Fibonacci Spiral


  It approximates the golden spiral, using Fibonacci sequence square sizes up to 34. Take a golden rectangle, break it down into smaller squares based from Fibonacci Sequence and divide each with an arc.
• The Golden Ratio is the limit of the ratios of successive terms of Fibonacci Sequence, as originally shown by Johannes Kepler (1571 - 1630).
• The golden ratio is equals to 1.618 or (1 + √5)/2
• Fibonacci can be seen in

  Flower petals

  Sunflower seeds

  Pineapple spirals

  Pinecone spirals
• Luca Pacioli 


  is the one who found the relationship between Fibonacci numbers and the golden ratio.
• The golden ratio was first called as the Divine Proportion in the early 1500s in Leonardo Da Vinci’s work was explored by Luca Pacioli (italian mathematician) entitled “De Divina Proportione” in 1509
• Da Vinci’s drawings of the 5 platonic solids and it was probably da Vinci who first called it “section aurea” latin for golden section.
• Golden Ratio can be seen in

  Nature

  Body Parts

  Arts
• the greek symbol for golden ration is Phi, from Phidias who widely used the golden ratio in his works of sculpture like the exterior dimension of Parthenon in Greece
• In the book Timateus by Plato, he describes 5 regular solids that relates to golden ratio


  Tetrahedron also known as triangular pyramid with 4 faces

  Cube with 6 faces

  Octahedron, a polyhedron with 8 faces

  Dodecahedron, a polyhedron with 12 faces

  Icosahedron, a polyhedron with 20 faces
• Euclid


  He was the first to give definition of the golden ratio as “a dividing line in the extreme and mean ratio” in his book the “Elements.”

  He proved the link of the number to the construction of the pentagram, known as golden ratio
• Leonardo de ser Piero da Vinci's works are

  Monalisa

  St. Jerome

  the Last Supper

  Vitruvian Ma
• Michelangelo or Michelangelo di Lodovico Simon's work is

  The Creation of Adam
• Raphael or Raffaello Sanzio da Urbinoworks are


  School of Athens

  Stairs

  Two Men

  Upper Arch

  Holy Family

  Crucifixion
• Rembrandt or Rembrandt Harmenszoon Van Rijn is


  Dutch painter who applied golden triangle in his painting "self portrait"
• Sandro Botticelli or Alessandro di Mariano di Yanni Filipepi


  the Birth of Venus is one of the world’s famous and appreciated work of art and it was painted between 1482 1485.
• Georges Seurat works are


  Barthers of Asinieres

  Bridge of Courbevoie

  A Sunday on La Grande Jatte
• Salvador Dali who framed his paintings using the golden ratio in his masterpiece, “The Sacrament of the Last Supper.”
• The Great Pyramid of Giza, a Golden ratio of Architecture is built 4700 BC in Ahmes Papyrus of Egypt is with proportion according to a golden ratio.
• Notre Dame, a Golden ratio of Architecture is Gothic Cathedral in Paris, built in between 1163 and 1250.
• Cathedral of Our Lady of Chartes, an architecture in Paris, France also exhibits the golden ratio.
• Taj Mahal in India used the golden ratio in its construction and was completed in 1648.
• United Nations Building, the window configuration reveal golden proportion.
• Golden Ratio in Architecture

  The Great Pyramid of Giza

  Notre Dame

  Cathedral of Our Lady of Chartes

  Taj Mahal

  United Nations Building

  Eiffel Tower

  CN Tower
• Golden Ratio in Designs

  Pepsi

  Toyota

  Apple

  National Geographic
• Types of Patterns

  Symmetry

  Fractals

  Spirals
• Symmetry is a sense of harmonious and beautiful proportion of balance or an object is invariant to any of various transformations (reflection, rotation, or scaling).


  2 types of Symmetry are

  • Bilateral - in which the left and right sides of organism can be divided into approximately mirror image of each other along the midline.
  • Radial - around a fixed point known as the center
• FRACTALS


  It is a class of highly irregular shapes and is a curve or geometric figure, each part of which has the same statistical character as the whole.
• SPIRALS


  It was first described by Rene Descartes and was later investigated by Jacob Bernoulli. A logarithmic spiral (or growth spiral) is a self-similar spiral curve which often appears in nature.
• Applications of Mathematics


  Application of Calculus

  Mathematical Modeling

  Engineering

  Information Technology

  Social Sciences

  Political Science

  Music and Arts

  Cryptography

  Archaeology
• Mathematics in the Modern World by Winston S. Sirug, Ph.D. ... what is physical is subject to the laws of mathematics, and what is spiritual to the laws of God, and the laws of mathematics are but the expressions of the thoughts of God.
• MATHEMATICAL LANGUAGE


  is the system used to communicate mathematical ideas.
• Mathematical notation


  used for formulas has its own grammar and is shared by mathematicians anywhere in the globe.
• An expression (or mathematical expression) is a finite combination of symbols that is well-defined according to rules that depend on the context.
• Symbols can designate numbers, variables, operations, functions, brackets, punctuations, and groupings to help determine order of operations, and the other aspects of mathematical syntax.
• Expression - correct arrangement of mathematical symbols to represent the object of interest, does not contain a complete thought, and cannot be determined if it's true or false
• Sentence (or mathematical sentence) - a statement about two expressions, either using numbers, variables, or a combination of both.