1. Governor’s School for the Sciences Mathematics Day 11

2. MOTD: A-L Cauchy 1789-1857 (French) Worked in analysis Formed the definition of limit that forms the foundation of the Calculus Published 789 papers

3. Tilings Based on notes by Chaim Goodman-Strauss Univ. of Arkansas

4. Isometries The rigid transformations: translation, relfection, rotation, are called isometries as they preserve the size and shape of figures Products of rigid transformations are also rigid transformations (and isometries) 2 figures are congruent if there is an isometry that takes one to the other

5. Theorem 0: The only isometries are combinations of translations, rotations and reflections Proof: Given two congruent figures you can transform on to the other by first reflecting (if necc.) then rotating (if necc) then translating.

6. Theorem 1: The product of two reflections is either a rotation or a translation Theorem 2: A translation is the product of two reflections Theorem 3: A rotation is the product of two reflections Theorem 4: Any isometry is the product of 3 reflections Draw Examples

7. Regular Patterns A regular pattern is a pattern that extends through out the entire plane in some regular fashion

8. Rules for Patterns Start with a figure and a set of isometries 0 A figure and its images are tiles; they must fit together exactly and fill the entire plane 1 Isometries must act on all the tiles, centers of rotation, reflection lines and translation vectors 2 If two copies of the figure land on top of each other, they must completely overlap

10. Visual Notation Worksheet

11. Results If we can apply these isometries and cover the plane: we have a tiling If we get a conflict, then the tile and the generating isometries are “illegal” What types of tiles and generators are legal?

12. Theorems 5 A center of rotation must have an angle of 2  /n for some n 6 Two reflections must be parallel lines or meet at an angle of 2  /n 7 If a pattern has 2 or more rotations they must both must be 2  /n for n = 2, 3, 4 or 6 What shapes are available for tiles?

13. Possible tiles

14. Possible transformations? For a given tile, what transformations are possible? Which combinations of tiles and transformations are equivalent? How many different tilings are possible? Homework and Tomorrow!

15. Fun Stuff Group origami: Modular Dimpled Dodecahedron Ball  bracelet/necklace/etc. Exploratory lab (optional) Catch-up lab time