MMW

• Fibonacci


  • Born in Pisa, Italy in 1175 AD
  • Full name was Leonardo Pisano
  • Grew up with a North African education under the Moors
  • Traveled extensively around the Mediterranean coast
  • Met with many merchants and learned their systems of arithmetic
  • Realized the advantages of the Hindu-Arabic system
• Fibonacci's Mathematical Contributions


  • Persuaded mathematicians to use the Hindu-Arabic number system
  • Introduced the Hindu-Arabic number system into Europe
  • Based on ten digits and a decimal point
  • Europe previously used the Roman number system
  • Consisted of Roman numerals
  • Wrote five mathematical works
• Fibonacci's Mathematical Works


  • Liber Abbaci (The Book of Calculating) written in 1202
  • Practica geometriae (Practical Geometry) written in 1220
  • Flos written in 1225
  • Liber quadratorum (The Book of Squares) written in 1225
  • A letter to Master Theodorus written around 1225
• Fibonacci Numbers


  • Series begins with 0 and 1
  • Next number is found by adding the last two numbers together
  • Pattern is repeated over and over
• Fibonacci Numbers


  • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
• Fibonacci Numbers in Nature


  • Fibonacci spiral found in both snail and sea shells
  • Lilies and irises = 3 petals
  • Black-eyed Susan's = 21 petals
  • Corn marigolds = 13 petals
  • Buttercups and wild roses = 5 petals
  • Bananas have 3 or 5 flat sides
  • Pineapple scales have Fibonacci spirals in sets of 8, 13, 21
• Golden Section


  • Represented by the Greek letter Phi
  • Phi equals ± 1.6180339887 ... and ± 0.6180339887 ...
  • Ratio of Phi is 1 : 1.618 or 0.618 : 1
  • Mathematical definition is Phi^2 = Phi + 1
  • Euclid showed how to find the golden section of a line
• Fibonacci Numbers and Golden Section


  • The Fibonacci numbers arise from the golden section
  • The graph shows a line whose gradient is Phi
  • The coordinates are successive Fibonacci numbers
  • The golden section arises from the Fibonacci numbers
  • Obtained by taking the ratio of successive terms in the Fibonacci series
  • Limit is the positive root of a quadratic equation and is called the golden section
• Golden Section in Geometry


  • Is the ratio of the side of a regular pentagon to its diagonal
  • The diagonals cut each other with the golden ratio
  • Pentagram describes a star which forms parts of many flags
• Golden Rectangle


  • Sides are in golden proportion to each other
  • Considered to be the most visually pleasing of all rectangles
  • Used extensively in design, art, architecture, advertising, packaging, and engineering
• The ratio of the length of the longer side to the length of the shorter side of a Golden Rectangle is approximately 1.6
• Golden Section in Architecture


  • Appears in many of the proportions of the Parthenon in Greece
  • Front elevation is built on the golden section (0.618 times as wide as it is tall)
  • Can be found in the Great pyramid in Egypt
  • Perimeter of the pyramid, divided by twice its vertical height is the value of Phi
  • Can be found in the design of Notre Dame in Paris
  • Continues to be used today in modern architecture
• Golden Section in Music


  • Stradivari used the golden section to place the f-holes in his famous violins
  • Baginsky used the golden section to construct the contour and arch of violins
  • Mozart used the golden section when composing music
  • Divided sonatas according to the golden section
  • Beethoven used the golden section in his famous Fifth Symphony
• The Golden Mean Gauge was developed by Dr. Eddy Levin DDS for use in dentistry and is now used as the standard for the dental profession
• Examples of the Golden Ratio

  • The Bagdad City Gate
  • Dome of St. Paul: London, England
  • The Great Wall of China
  • The Parthenon: Greece
  • Windson Castle
• Examples of the Golden Ratio in Nature


  • Animals
  • Plants
• Graph Theory


  A branch of Mathematics concerned with networks of points connected by lines
• Graph Theory had its beginnings in recreational math problems, but it has grown into a significant area of mathematical research, with applications in Chemistry, Operations Research, Social Sciences, and Computer Sciences
• Seven Bridges of Konigsberg


  An eighteenth century problem, where the city of Konigsberg (Kaliningrad, Russia) has four districts divided by the Pregel River, and people were puzzled if there is a travel route that would only cross each of the seven bridges exactly one
• Leonhard Euler


  A Swiss mathematician who proved in 1736 that it is IMPOSSIBLE to take a stroll that would lead them across each bridge and return to the starting point without traversing the same bridge twice
• Graph


  A collection of points called vertices or nodes and line segments or curves called edges that connect the vertices
• The position of the vertices, the lengths of the edges, and the shape of the edges do not matter in a graph. The number of vertices and which of them are joined by edges that matter most
• Constructing a Graph


  Draw a graph that represents the information where each vertex represents a city and an edge connects two vertices if the two cities have a direct flight
• Loop


  An edge connecting a vertex to itself
• Multiple Edges


  Two vertices that are connected by more than one edge
• Simple Graph


  A graph with no loops and no multiple edges
• Connected


  A graph is connected if there is a path connecting all the vertices
• Adjacent


  Two vertices are adjacent if there is an edge joining them
• Degree


  The degree of a vertex is the number of edges attached to it
• Examples of graphs


  • Null or Disconnected Graphs
  • Graph with a Loop
  • Graph with Multiple Edges
• Complete Graph


  A graph is complete if every pair of vertices of a graph are adjacent. A connected graph in which every possible edge is drawn between vertices
• A complete graph with n vertices is denoted by Kn
• Complete Graphs


  • K1: One Vertex
  • K2: Two Vertices
  • K3: Three Vertices
  • K4: Four Vertices
  • K5: Five Vertices
  • K10: Ten Vertices
• Equivalent graphs


  A graph whose edges form the same connections of vertices
• Path


  An alternating sequence of vertices and edges. It can be seen as a trip from one vertex to another using the edges of the graph
• Circuit / Cycle


  If a path begins and ends with the same vertex, it is a closed path
• Euler Circuits


  A closed path that uses every edge, but never uses the same edge twice. The path may cross through vertices more than once. A graph that contains an Euler Circuit is called Eulerian
• Determining if a graph is Eulerian

  1. Determine whether the graph is Eulerian. If it is, Find an Eulerian circuit. If it is NOT, explain why.

  2. Determine whether the graph is Eulerian. If it is, Find an Eulerian circuit. If it is NOT, explain why.

  3. Determine whether the graph is Eulerian. If it is, Find an Eulerian circuit. If it is NOT, explain why.
• Eulerian Graph Theorem


  A connected graph is Eulerian if and only if every vertex of the graph is of even degree
• Determining if a graph has an Euler Circuit


  Determine whether the graph shown has an Euler Circuit. If it is, find an Euler circuit. If it is not, explain how you know. The number beside each vertex indicates the degree of the vertex.