Course:Harris, Fall 07: Diary Week 6

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  • laid out general plan for next some weeks:
    • first part of course (up to now) concentrated on geometry and algebra of tessellations, as that underlies Escher's work
    • practical part is being able to recognize patterns and build patterns
    • theoretical part is the algebra of group multiplication
    • next part of course will concentrate on practical way of turning tessellations into Escher-like art (but still some emphasis on the theoretical side of things)
  • encouraged everyone to turn in Alhambra extra credit and corrections to exercises or Cathedral Project by Wednesday, so that I can return them by Friday in advance of Exam 1 on Monday
  • class worked on Why There Are No More Than 3 Regular Tessellations Exploration
    • need to look beneath the surface, just a bit, to answer the "why" question (parenthetical) in #8
    • need to consider the algebraic nature of things to answer #9
  • added the first six questions of that exploration to topics for Exam 1
  • found that Geometer's Sketchpad is available on class computers; that will be for next class


  • collected extra credit and corrected exercises and projects
  • showed class the basic steps in employing Geometer's Sketchpad
  • groups worked the rest of the time on using rotations (and translations, if desired) to create a tessellation of the plane by an irregular non-convex quadrilateral
    • noted that they can expect a question on Exam 2, "Show how this quadrilateral can be used to tessellate the plane."


  • answered question on how to show an n-sided polygon has an angle-sum of (n-2)x180 degrees:
    • assume a triangle has an angle-sum of 180
    • show how you can divide a polygon into triangles, vertex-to-vertex
    • count (in examples), showing that it's n-2 triangles if the polygon has n sides
    • notice that the sum of all the triangle angle-sums amounts to the angle-sum of the polygon (see how it works in some examples)
    • thus, polygon angle-sum is # of triangles times angle-sum of each triangle
  • looked carefully at how a tessellation is built up from an irregular quadrilateral:
    • rotate 180 degrees, around midpoints on the sides
    • note how this puts all four angles of the quadrilateral together around one vertex
    • since the angle-sum on a quardilateral is 360, that works!
  • reviewed definition of regular tessellation
    • vertex-to-vertex
    • formed out of only one polygonal shape
    • that has to be a regular polygon
      • equilateral
      • equiangular
    • noted that Monday's exploration showed there are only three such, using
      • triangle
      • square
      • hexagon
  • semi-regular (Archimedean) tessellations:
    • vertex-to-vertex
    • formed out of multiple regular polygons
    • every vertex has to look like every other vertex
    • there are only eight such, shown in the text