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Spain 2007

Spain 2007. Ralph Abraham Talk #4. Islamic Patterns. Ref: Syed Jan Abas & Amer Shaker Salman Symmetries of Islamic Geometrical Patterns Singapore: World Scientific, 1995 pp. 57-66. The Euclidean Plane. Descartes, Geometry + Algebra, ca 1630 AD

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Spain 2007

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  1. Spain 2007 • Ralph Abraham Talk #4

  2. Islamic Patterns • Ref: Syed Jan Abas & Amer Shaker Salman • Symmetries of Islamic Geometrical Patterns • Singapore: World Scientific, 1995 • pp. 57-66

  3. The Euclidean Plane • Descartes, Geometry + Algebra, ca 1630 AD • A point in E2 is defined by coordinates (x, y) • Distance from (xa, ya) to (xb, yb) = • Square root of sum of squares

  4. Isometries of E2 • An isometry is a function from E2 to itself • preserving distances • Theorem: there are only four types: • translation, rotation, reflection, and • glide (translation plus reflection)

  5. Symmetries • A symmetry of a pattern (subset) P of E2: • an isometry of E2 • that maps P exactly onto itself

  6. Symmetry Groups • The symmetry group of a pattern is the set of all symmetries of the pattern • It as a group: • closed under composition • composition is associative • Each symmetry has an inverse • There is an identity

  7. The Dihedral Group • I = Identity, R1 = rotate 90 degrees CCW • M1 = flip 42, M2 = flip DB, etc • D8 = {I, R1, R2, R3, M1, M2, M3, M4} • combination T2.T1 means apply T2 after T1

  8. Repeating Patterns • Crystallographic Theorem: • the only rotational symmetries are • 2, 3, 4, or 6-fold

  9. Crystallographic Groups • Theorem: • There are only 17. • p6m, p4m, cmm, pmm, and p6 (later ...) • are the most common symmetries • of Islamic patterns

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