L1 & L2 Reviewer

• National Council of Teachers of Mathematics (1991): 'Mathematics is a study of patterns and relationship, a way of thinking, an art, a language, and a tool. It Is about patterns and relationship. Numbers are just a way to express those patterns and relationships.'
• Symmetry


  • Bilateral or reflection symmetry
  • Radial symmetry
  • Rosette patterns
  • Frieze or border patterns
  • Wallpaper pattern
• Bilateral or reflection symmetry


  Simplest kind of symmetry, most common in the natural world, mirror symmetry where objects have a left side and a right side that are mirror images of each other, bilateral-symmetric objects have at least one line or axis of symmetry
• Radial symmetry


  Rotation symmetry around a fixed point known as the center, images with more than one lines of symmetry meeting at a common point
• Rosette patterns


  Consist of taking a motif or an element and rotating and/or reflecting that element, can be cyclic (only admits rotational symmetries) or dihedral (admits both rotational and bilateral/reflectional symmetries)
• Frieze or border patterns


  Basic motif repeats itself over and over in one direction, can be mapped onto itself by horizontal translation, 7 types: hop, step, sidle, spinning hop, spinning siddle, jump, spinning jump
• Wallpaper pattern


  Pattern with translation symmetry in two directions, arrangement of friezes stacked upon on another to fill the entire plane, must have at least the basic unit, one copy by translation, and a copy of these two by translation in the second direction, there must be at least two rows, each one of at least two units long
• Tessellation


  A pattern of one or more shapes that do not overlap or have space between them, a repeating pattern of figures that covers a plane with no gaps or overlaps, can be created with translations, rotations, and reflections
• Tessellation

  • Honeycomb - perfect example of a natural tessellation
• Waves


  Patterns of ripples in water or sand, created through wind that create dunes
• Fractals


  Never-ending patterns, repetition of simple equations, any small part resembles the whole
• Spiral


  Curved pattern that focuses on a center point, series of circular shapes that revolve around it
• Meanders, flow, chaos


  Meanders - bends in a sinuous form that appear in rivers or other channels, Flows - the water that flows, Chaos - study of how simple patterns can be generated from complicated underlying behavior
• Spots, Stripes


  Evolutionary explanation, functions which increase the chances that the offspring of the patterned animal will survive to reproduce, for camouflage
• Cracks


  Linear openings that form in material to relieve stress, pattern engineers want to avoid
• Foam and Bubbles


  Foam - mass of bubbles, substance made by trapping air, Bubble - spherically contained volume of air or other gas
• Fibonacci Sequence


  Invented by Leonardo Pisano Bigollo (1180 - 1250), Fn = Fn-1 + Fn-2, recursive definition starting with initial values
• Golden Ratio


  First called Divine Proportion in the early 1500s, formally defined as the limit as n approaches infinity of (Fn+1/Fn), where Fn is the nth Fibonacci number
• Golden Rectangle


  Most visually satisfying of all geometric forms, related to golden spiral, created by making adjacent squares of Fibonacci dimensions
• Fibonacci sequence has captivated mathematicians, scientists, artists, and designers for centuries, it is a sequence with many interesting properties
• Fibonacci Spiral


  Take a golden rectangle, break it down into smaller squares
• Roger Bacon (1214 - 1294): 'Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world'
• Places where Fibonacci numbers appear

  • Pinecones, seed heads, vegetables, fruits, flowers, branches, honeybees, the human body, geography, weather, galaxies
• Golden ratio and/or the golden spiral in

  • Architecture: Great Pyramid of Giza, Parthenon
  • Arts: Mona Lisa, An Old Man, Vetruvian Man, Holy Family, Crucifixion, The Sacrament of the Last Supper
• Applications of Mathematics in our world

  Organize patterns and regularities, predict behavior of nature and phenomena, control nature and occurrences for our own good, indispensable in many human endeavors
• Sequence


  A set of numbers in a specific order, the numbers in the sequence are called terms
• Arithmetic sequence


  Difference between successive terms is constant, an = a1 + (n-1)d to find nth term
• Geometric sequence


  Ratios of consecutive terms are the same, common ratio r, an = a1(r)^(n-1) to find nth term
• Difference table


  Shows the difference between successive terms of a sequence, may have first, second, third differences
• Proposition


  A declarative sentence that can be objectively identified as either true or false, but not both
• Negation of a proposition


  The proposition which is false when the original proposition is true, and vice versa
• Compound proposition


  Propositions formed by combining simple propositions using logical connectives like conjunction, disjunction, conditional, biconditional
• Tautology


  A compound proposition whose truth value remains true regardless of the truth values of its component propositions
• Contradiction


  A compound proposition whose truth value remains false regardless of the truth values of its component propositions