# 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

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