symmetry in the plane chapter 8. imprecise language what is a figure? definition: any collection of...
TRANSCRIPT
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 Example was wallpaper
• Better term is “unbounded” No boundary to stop the pattern
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
Symmetries
• Regular polygons are symmetric figures Rotations and reflections
• How many symmetries of each type are there for a regular n-gon?
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
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))
Compositions of Symmetries
• Complete the table for Activity 5
• Identity? Inverses?
R0 R120 R240 V L R
R0
R120
R240
V
L
R
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
4
2
3
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
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
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
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
Symmetry in Design
• Architecture
• Nature
SnowChrystals.com
http://oldgeezer.info/bloom/poplar/poplar.htm
http://www.nationmaster.com/encyclopedia/Beauty
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)
Friezes and Symmetry
• Consider Activity 6. . . ZZZZZZZZZZZZZZZZZZZZZ . . .. . . XXXXXXXXXXXXXXXXXXX . . .. . . WWWWWWWWWWWWW . . .
• Definition : friezeA pattern unbounded along one line
Line known as the midline of the pattern
Friezes and Symmetry
• Examples of a frieze in woodcarving
Friezes and Symmetry
• Examples of a frieze in quilting
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
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
Friezes and Symmetry
• Consider all possible combinations
• Consider all possible combinations
• Note seven possibilities
Friezes and Symmetry
Wallpaper Symmetry
• Consider allowing translations as symmetries
• Results in wallpaper symmetry Reflections in both horizontal, vertical
directions
. . .
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
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
Tilings
• Escher’s tilings in a circle Using Poincaré disk model All figures are “congruent”
Tilings
• Elementary tiling All regions are congruent to one basic shape
• Theorem 8.7Any quadrilateral can be used to create an elementary tiling
Tilings
• Given arbitrary quadrilateral Note sequence of steps to tile the plane
Rotate initial figure 180 about midpoint of side
Repeat for successive
results
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
Tilings
• Which regular polygons can be used to tile the plane? Tiling based on a regular polygon called a
regular tiling
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?
2 180n
n
2 180360
nk
n
Tilings
• Semiregular tilings When every vertex in a tiling is identical
• Demiregular tilings Any number of edge to edge tilings by regular
polygons
Tilings
• Penrose tiles Constructed from a rhombus Divide into two quadrilaterals – a kite and a dart
Tilings
• Here the = golden ratio
• Possible to tile plane in nonperiodic way No transllational symmetry
11 5
2
Tilings
• Combinations used for Penrose tiling
Tilings
• Penrose tilings
Symmetry in the Plane
Chapter 8