# 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.