Governor’s School for the Sciences Mathematics Day 11

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

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

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

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.

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

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

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

Visual Notation Worksheet

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?

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?

Possible tiles

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!

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