Course:Harris, Fall 07: Diary Week 6

Mon:

• 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

Wed:

• 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."

Fri:

• 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