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Governor’s School for the Sciences

Governor’s School for the Sciences. Mathematics. Day 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 by Chaim Goodman-Strauss Univ. of Arkansas. Isometries.

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Governor’s School for the Sciences

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  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 byChaim Goodman-StraussUniv. 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

  9. Visual Notation Worksheet

  10. 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?

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

  12. Possible tiles

  13. 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!

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

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