Governor’s School for the Sciences

MathematicsMathematicsDay 11

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

Tilings

Based on notes byChaim 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 Draw ExamplesExamples

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 isometries0 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

WorksheetWorksheet

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