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Symmetry in the Plane Chapter 8
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Imprecise Language What is a “figure”? Definition: Any collection of points in a plane Three figures – instances of the constellation Orion
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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
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Symmetries Activity 8.1 Isometries of rotation Square congruent to itself at rotations of 0, 90, 180, 270 Definition: Symmetry An isometry f for which f(S) = S
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Symmetries Regular polygons are symmetric figures
Rotations and reflections How many symmetries of each type are there for a regular n-gon?
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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
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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))
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Compositions of Symmetries
Complete the table for Activity 5 Identity? Inverses? R R R V L R R0 R120 R240 V L R
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Compositions of Symmetries
Try it out for a square … What are the results of this composition? (1 4) (2 3) ( ) What is the end result symmetry? 1 2 4 3
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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
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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
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Classifying Figures by Symmetries
Theorem 8.1 Leonardo’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
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Classifying Figures by Symmetries
Proof of (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
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Symmetry in Design Architecture Nature
SnowChrystals.com
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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)
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Friezes and Symmetry Consider Activity ZZZZZZZZZZZZZZZZZZZZZ XXXXXXXXXXXXXXXXXXX WWWWWWWWWWWWW . . . Definition : frieze A pattern unbounded along one line Line known as the midline of the pattern
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Friezes and Symmetry Examples of a frieze in woodcarving
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Friezes and Symmetry Examples of a frieze in quilting
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Friezes and Symmetry Theorem 8.3 Only 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
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Friezes and Symmetry Theorem 8.4 There 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
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Friezes and Symmetry Consider all possible combinations
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Friezes and Symmetry Consider all possible combinations
Note seven possibilities
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Wallpaper Symmetry Consider allowing translations as symmetries
Results in wallpaper symmetry Reflections in both horizontal, vertical directions . . .
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Wallpaper Symmetry Theorem 8.5 Crystallographic Restriction The 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 8.6 There are exactly 17 wallpaper groups
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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
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Tilings Escher’s tilings in a circle Using Poincaré disk model
All figures are “congruent”
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Tilings Elementary tiling
All regions are congruent to one basic shape Theorem 8.7 Any quadrilateral can be used to create an elementary tiling
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Tilings Given arbitrary quadrilateral
Note sequence of steps to tile the plane Rotate initial figure 180 about midpoint of side Repeat for successive results
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Tilings Corollary 8.8 Any triangle can be used to tile the plane Proof
Rotate original triangle about midpoint of a side Result is quadrilateral – use Theorem 8.7
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Tilings Which regular polygons can be used to tile the plane?
Tiling based on a regular polygon called a regular tiling
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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?
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Tilings Semiregular tilings Demiregular tilings
When every vertex in a tiling is identical Demiregular tilings Any number of edge to edge tilings by regular polygons
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Tilings Penrose tiles Constructed from a rhombus
Divide into two quadrilaterals – a kite and a dart
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Tilings Here the = golden ratio
Possible to tile plane in nonperiodic way No transllational symmetry
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Tilings Combinations used for Penrose tiling
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Tilings Penrose tilings
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Symmetry in the Plane Chapter 8
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