Exploring Polygons, Tilings, and Symmetry

1. Overview 0 dimensions: points 1 dimension: lines and curves between 1 and 2 dimensions: fractals 2 dimensions: polygon tilings/tesselations symmetry

3. Polygons A polygon is a “many-angled” figure (often called an n-gon for specific values of n). A polygon is called regular if all sides and vertex angles are congruent.

4. Angles in Polygons Theorem: The angles in any triangle add to 180 degrees. Use this fact to find the vertex angles for regular triangles, squares, and regular pentagons. Fill in the list of vertex angles on page 86.

5. Tilings A tiling (or tessellation) is a filling-up of the plane with a shape or shapes that meet edge to edge and vertex to vertex. A tiling is regular if the only shape used is a regular polygon and the basic pattern is repeated. Basic example: tilings with squares. Notation: we call this tiling 4.4.4.4 (indicating that 4 squares meet at a vertex)

6. Regular Tilings To find all regular tilings, note that one shape is used to surround a point. The vertex angle must evenly divide 360 degrees. Find and notate all other regular tilings.

7. Semiregular tilings A tiling is semiregular if it can be formed in a regular way (i.e., each vertex has the same configuration) with two or more regular polygons. Example: 4.8.8 Why does this work?

8. Our first goal We need to find the different configurations of regular polygons that can fit around a point. Build the justifications for Rules 1, 2, 3, 4, and 5 on pages 86-88. Build the list of possible configurations on page 88.