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Discrete Structures Project

Discrete Structures Project. ZAINAB EJAZ BSE-2014-027. Symmetry. Zainab Ejaz. Contents: What is symmetry? Important Terms about symmetry Symmetry groups Types of symmetry Examples of symmetry. 5/30/2015 4:12 PM. Zainab Ejaz. 4. What is symmetry ?.

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Discrete Structures Project

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  1. ZainabEjaz

  2. Discrete Structures Project ZAINAB EJAZ BSE-2014-027

  3. Symmetry • ZainabEjaz

  4. Contents: • What is symmetry? • Important Terms about symmetry • Symmetry groups • Types of symmetry • Examples of symmetry 5/30/2015 4:12 PM ZainabEjaz 4

  5. What is symmetry ? The quality of being made up of exactly similar parts facing each other or around an axis is called symmetry. • ZainabEjaz

  6. Important Terms: • Line of symmetry • Center of symmetry • Rotation • Reflection • Translation • ZainabEjaz

  7. Line of symmetry: • A line that divides a figure into two halves that are mirror images of each other (axis, plane of symmetry). • ZainabEjaz

  8. This smiley face has 1 line of symmetry Line of symmetry • ZainabEjaz 5/30/2015 4:12 PM 8

  9. Center of symmetry: • The center of symmetry is a point in space such that if a line is drawn from any part of the object to that point and extended an equal distance beyond it, an analogous part will be encountered (inversion center). • ZainabEjaz

  10. Center of symmetry Center of symmetry • ZainabEjaz 5/30/2015 4:12 PM 10

  11. Rotation (Turn): The action of turning a figure around a point or a vertex. • ZainabEjaz

  12. Turning a figure around a point or a vertex Rotation • ZainabEjaz 5/30/2015 4:12 PM 12

  13. Reflection (flip): • The result of a figure flipped over a line. • ZainabEjaz

  14. Reflection The result of a figure flipped over a line. • ZainabEjaz

  15. Translation: • The action of sliding a figure in any direction. • ZainabEjaz

  16. Translation The act of sliding a figure in any direction. • ZainabEjaz

  17. Symmetry group: A symmetry operation, for a subset X of the plane, is an isometry f of the plane that maps X to itself, meaning that f(X) = X. The set of all symmetry operations of X is denoted by Sym(X) and is called the symmetry group of X. • ZainabEjaz

  18. Symmetry groups: • The Symmetry Group of a Square • The Symmetry Group of a Hexagon • The Frieze Patterns • The 17 Wallpaper Patterns • ZainabEjaz

  19. The Symmetry Group of a Square: The square has eight symmetries - four rotations, two mirror images, and two diagonal flips. These eight form a group under composition. • ZainabEjaz

  20. The Symmetry Group of a hexagon: The hexagon has twelve symmetries - six rotations and six reflections.

  21. The Frieze Patterns: A frieze pattern is any linear pattern that repeats in one direction. …FFFFFFFFFFFFFFFFFF… This has no symmetry, other than translational symmetry in one direction. …NNNNNNNNNNNNNN… This pattern has rotational symmetry about the centre of each letter N.

  22. The Wallpaper Patterns: A wallpaper pattern is any patterns that cover the whole plane and is repetitive in two independent directions. There are 17 different types of wallpaper patterns.

  23. ZainabEjaz

  24. Types of symmetry: • Reflection / Bilateral symmetry • Rotational / Radial symmetry • ZainabEjaz

  25. Reflection symmetry: A shape has reflection symmetry when one half of it is the mirror image of the other half. This is also called plane, mirror, bilateral or line symmetry. • ZainabEjaz

  26. 1 horizontal line of symmetry. • ZainabEjaz 5/30/2015 4:25 PM 26

  27. one vertical line of symmetry • ZainabEjaz

  28. One vertical line of symmetry. • ZainabEjaz

  29. a horizontal line of symmetry. • ZainabEjaz

  30. 1 horizontal line of symmetry. • ZainabEjaz

  31. one vertical and one horizontal line of symmetry. • ZainabEjaz

  32. 8 lines of symmetry. • ZainabEjaz

  33. 4 lines of symmetry. • ZainabEjaz

  34. 2 lines of symmetry. • ZainabEjaz

  35. one line of symmetry. • ZainabEjaz

  36. 4 lines of symmetry • ZainabEjaz

  37. one line of symmetry. • ZainabEjaz

  38. 5 lines of symmetry. • ZainabEjaz

  39. one vertical line of symmetry • ZainabEjaz

  40. 8 lines of symmetry. • ZainabEjaz

  41. one line of symmetry. • ZainabEjaz

  42. one line of symmetry. • ZainabEjaz

  43. How many lines of symmetry to these regular polygons have? 5 4 3 8 6 • ZainabEjaz

  44. Circle has infinite lines of symmetry • ZainabEjaz

  45. Rotational symmetry: An object with rotational symmetry is an object that looks the same after a certain amount of rotation. Object may have more than one rotational symmetry. This is also called radial symmetry. • ZainabEjaz

  46. Order 7 Order 3 Order 4 Order 4 • ZainabEjaz

  47. rotational symmetry at order 2 • ZainabEjaz

  48. rotational symmetry at order 2 • ZainabEjaz

  49. rotational symmetry at order 2 • ZainabEjaz

  50. rotational symmetry at order 2 • ZainabEjaz

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