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