Course:Harris, Fall 08: Diary Week 1

Mon:

• written statements, "Why are you in the class?" (not very long answers)
• free time, look at Escher works
• What's interesting about Escher?
  • repetitions
  • changing perspectives
  • interesting shapes, interlocking (some based on polygons)
• Tessellation implies geometry. (e.g., Switzerland and Belgium > Development II in Galleries)
• What is a tessellation?
  • fills plane with shapes
  • no overlaps among shapes
  • no gaps between shapes
• What makes for an interesting tessellation?
  • recognizable shapes
  • transformations of figures
    • rotations
    • flips
    • translations

Wed:

• Started in on Quadrilaterals Exploration, but numerous excursions along the way:
  • circle
    • definition
      • pick center point P, radius R
      • circle of center P and radius R is all points in the plane distance R from P
    • related
      • oval
        • distorted from circle by being narrower at one point than all others, wider at another point than at all others
      • ellipse
        • distorted from circle by being stretched symmetrically
  • polygon
    • definition
      • plane figure made up edges (line segments), each of which has two vertices (endpoints)
      • edges meet only at vertices
      • each vertex must have exactly two edges meeting there
    • Does definition imply a polygon divides the plane into interior and exterior?
      • Yes, but this is a subtle point, not easily seen in complex cases.
    • Does # edges = # vertices?
      • Yes: Suppose there are n edges.
        • Each edge specifies 2 vertices.
        • Thus, there are up to 2n edge-specified vertices (before taking account of how many edges specify a given vertex).
        • In point of fact, each edge-specified vertex has 2 edges specifiying it, so there are only n edge-specified vertices.
        • That accounts for all vertices, since every vertex is specified by edges. So there are n vertices total.
• Quadrilaterals Exploration to be finished Friday
  • groups (of 2, communicating in pairs) about finished question 1, got started on some of the others

Fri:

• 20 minutes finishing up Quads Exploration
• 15 minutes looking at Exercises quesiion #3: Are polygons wtih congruent opposite angles the same thing as parallellograms?
  • examined implication, congruent opposite angles ==> parallel sides
    • We thought we might use "opposite interior angles" theorem:
      • If lines L1 and L2 are cut by line L3, with congruent opposite "interior angles", then L1 and L2 are parallel.
      • Seems plausible, by looking at what happens if L1 and L2 intersect: We then get a triangle with > 180 degrees for angle-sum.
    • To use the theorem, looked at sides L1 and L2 (opposite) in quadrilateral, L3 side cutting both of those, forming angles A and B.
    • Opposite interior angles are A and supplement of B; so need A and B supplementary to use theorem (i.e., to have congruent oppoiste interior angles).
    • To get A + B = 180, again used triangle angle-sum is 180, applying that to quadrilateral made up of 2 triangles.
      • Quadrilateral angle-sum is A + B + A + B (by congruent opposite angles), so we have A + B + A + B = 360 (two triangle angle-sums).
      • Dividing by 2 yields A + B = 180, and we're done.
• Groups started in on Tessellations Exploration (15 minutes); didn't manage to finish even the blocks questions.
  • two kinds of tessellations:
    • vertices have to meet other verticies
    • vertices allowed to touch interiors of edges--allows for lots more possibilities
  • Will finish this up next class, 10-15 minutes.