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Symmetry in the Plane

Symmetry in the Plane. Chapter 8. Imprecise Language. What is a “figure”? Definition: Any collection of points in a plane Three figures – instances of the constellation Orion. Imprecise Language. What about “infinite along a line”? Suggests a pattern indefinitely in one direction

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Symmetry in the Plane

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  1. Symmetry in the Plane Chapter 8

  2. Imprecise Language • What is a “figure”?Definition:Any collection of points in a plane • Three figures – instances of the constellation Orion

  3. Imprecise Language • What about “infinite along a line”? • Suggests a pattern indefinitely in one direction • Example was wallpaper • Better term is “unbounded” • No boundary to stop the pattern

  4. Symmetries • Activity 8.1 • Isometries of rotation • Square congruentto itself at rotationsof 0, 90, 180, 270 • Definition: Symmetry • An isometry f for which f(S) = S

  5. Symmetries • Regular polygons are symmetric figures • Rotations and reflections • How many symmetries of each type are there for a regular n-gon?

  6. Groups of Symmetries • Abstract algebra : group • A set G with binary operator  with properties • Closure • Associativity • An identity • An inverse for every element in G • (Note, commutativity not necessary) • The operation  is composition of symmetries

  7. Compositions of Symmetries • Cycle notation • Label vertices of triangle • R120 = (1 2 3)Rotation of 120 • V = (1)(2 3)Reflection in altitude through 1 • Thus V  R120 = (1)(2 3)  (1 2 3)(apply transformation right to left) • V  R120 (P) = V(R120 (P))

  8. Compositions of Symmetries • Complete the table for Activity 5 • Identity? Inverses?

  9. Compositions of Symmetries • Try it out for a square … • What are the results of this composition? • (1 4) (2 3)  (1 2 3 4) • What is the end result symmetry? 1 2 4 3

  10. Classifying Figures by Symmetries • What were the symmetry groups for the letters of the alphabet?A B C D E F G H I J K L M N O P Q R S T U V W X Y Z • Identity only • Identity + one rotation • Identity + one reflection • Identity + multiple rotations + multiple reflections

  11. Classifying Figures by Symmetries • Types of symmetric groups • Cyclic group – only rotations • Dihedral group – half rotations, half reflections • We classify these types of groups by how many rotations, how many reflections • Cyclic group – C3 • Dihedral group – D4

  12. Classifying Figures by Symmetries • Theorem 8.1Leonardo’s Theorem • Finite symmetry group for figure in the plane must be either • Cyclic group Cn • Dihedral group Dn • Lemma 8.2 • Finite symmetry group has a point that is fixed for each of its symmetries • Note proof in text

  13. Classifying Figures by Symmetries • Proof of 8.1 (Finite symmetry for a group is either Cn or Dn ) • Case 1 – single rotation • Case 2 – one rotation, one reflection • Case 3 – single rotation, multiple reflections • Case 4 – Multiple rotations, no reflections • Case 5 – Multiple rotations, at least 1 reflection

  14. Symmetry in Design • Architecture • Nature http://www.nationmaster.com/encyclopedia/Beauty http://oldgeezer.info/bloom/poplar/poplar.htm SnowChrystals.com

  15. Friezes and Symmetry • Previous symmetry groups considered bounded • Do not continue indefinitely • Also they use only rotations, reflections • Translations not used • Figure would be unbounded in direction of translation (infinte)

  16. Friezes and Symmetry • Consider Activity 6. . . ZZZZZZZZZZZZZZZZZZZZZ . . .. . . XXXXXXXXXXXXXXXXXXX . . .. . . WWWWWWWWWWWWW . . . • Definition : friezeA pattern unbounded along one line • Line known as the midline of the pattern

  17. Friezes and Symmetry • Examples of a frieze in woodcarving

  18. Friezes and Symmetry • Examples of a frieze in quilting

  19. Friezes and Symmetry • Theorem 8.3Only possible symmetries for frieze pattern are • Horizontal translations along midline • Rotations of 180 around points on midline • Reflections in vertical lines  to midline • Reflection in horizontal midline • Glide reflections using midline

  20. Friezes and Symmetry • Theorem 8.4There exist exactly seven symmetry groups for friezes • We use abbreviations for types of symmetries • H = reflection, horizontal midline • V = reflection in vertical line • R = rotation 180 about center on midline • G = glide reflection using midline

  21. Friezes and Symmetry • Consider all possible combinations

  22. Friezes and Symmetry • Consider all possible combinations • Note seven possibilities

  23. Wallpaper Symmetry • Consider allowing translations as symmetries • Results in wallpaper symmetry • Reflections in both horizontal, vertical directions . . .

  24. Wallpaper Symmetry • Theorem 8.5 Crystallographic RestrictionThe minimal angle of rotation for wallpaper symmetry is 60, 90, 120, 180, 360. All others must be multiples of the minimal angle for that pattern • Theorem 8.6There are exactly 17 wallpaper groups

  25. Tilings • Definition:Collection of non-overlapping polygons • Laid edge to edge • Covering the whole plane • Edge of one polygon must be an edge of an adjacent polygon • Contrast to tessellation

  26. Tilings • Escher’s tilings in a circle • Using Poincaré disk model • All figures are “congruent”

  27. Tilings • Elementary tiling • All regions are congruent to one basic shape • Theorem 8.7Any quadrilateral can be used to create an elementary tiling

  28. Tilings • Given arbitrary quadrilateral • Note sequence of steps to tile the plane Rotate initial figure 180 about midpoint of side Repeat for successive results

  29. Tilings • Corollary 8.8Any triangle can be used to tile the plane • Proof • Rotate original triangle about midpoint of a side • Result isquadrilateral – useTheorem 8.7

  30. Tilings • Which regular polygons can be used to tile the plane? • Tiling based on a regular polygon called a regular tiling

  31. Tilings • A useful piece of information • Given number of sides of regular polygon • What is measure of vertex angles? • So, how many regular n-gons around the vertex of a tiling?

  32. Tilings • Semiregular tilings • When every vertex in a tiling is identical • Demiregular tilings • Any number of edge to edge tilings by regular polygons

  33. Tilings • Penrose tiles • Constructed from a rhombus • Divide into two quadrilaterals – a kite and a dart

  34. Tilings • Here the  = golden ratio • Possible to tile plane in nonperiodic way • No transllational symmetry

  35. Tilings • Combinations used for Penrose tiling

  36. Tilings • Penrose tilings

  37. Symmetry in the Plane Chapter 8

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