# Course:Harris, Fall 08: Diary Week 1

From EscherMath

Jump to navigationJump to searchMon:

- 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

- oval

- definition
- 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.

- Yes: Suppose there are n edges.

- definition

- circle
- 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.

- We thought we might use "opposite interior angles" theorem:

- examined implication, congruent opposite angles ==> parallel sides
- 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.

- two kinds of tessellations: