Governor’s School for the Sciences Mathematics Day 11.
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Transcript of Governor’s School for the Sciences Mathematics Day 11.
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