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Topic 2 Plane Symmetries and Tranformations

2.1 Synopsis

This topic is an extention of the Topic 1, Plane Tessellation. In producing tessellation, we

should know how the tessellations were formed. Starting by a given motif, the entire pattern can

be made up by using transformations which retain the shape and size of the original motif but

move to the new positions in the plane. In this topic, only two sub-topics will be discussed, i.e.

isometry of the plane (rotation, reflection, translation and glide reflection) and finite symmetries

groups and seven frieze patterns. Plane symmetries will be dicussed during lecture session.

2.2 Learning Outcomes

1. Enhance your knowledge in plane symmetries and transformations.

2. Understand the formation of seven frieze patterns.

2.3 Conceptual Framework

Plane Symmetries

and

Transformations

Isometry of thePlane Plane Symmetry

Finite Symmetries

Groups and Seven

Frieze Patterns


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2.4 Isometry of the Plane

Isometry is a transformation which retain the shape and size of the original motif. Isometries

came from the Greek words isos(equal) and metron(measure). There are only four isometries

of the plane, i.e. translations, reflections, rotations and glide reflections.

2.4.1 Translation

Translation is a transformation which slide everything to a fixed distance in a fixed

direction. So, we need to spesify its directionand distancewhich each points move. A

simple translation shown in figure 2.4 (1):

(i) image

Object

(ii)

Figure 2.4 (2) : A translation whichcan be found in nature: Thisphotograph shows a microscopicview of the scales of a butterfly.

Figure 2.4 (1) : Translation


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(iii)

2.4.2 Reflection

Reflection is a transformation which is basically a 'flip' of a shape over the line of

reflection. The object and the image have the same size and shape, but in the image

face in opposite direction. Reflection has infinitely many invariant point (points which

remain unchanged under the application of the isometry). Besides, the distancefrom a

point to the line of reflection is the sameas the distance from the point's image to the

line of reflection. In other word, the line of reflection is place in the middle of the object

and image.

(i)

Figure 2.4 (3) : When you are slidingdown a water slide, you areexperiencing a translation. Yourbody is moving a given distance (thelength of the slide) in a givendirection. You do not change yoursize and shape.

Line of reflectionObject Image

Figure 2.4 (4)


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(ii)

Why the word AMBULANCE is put in such a way as shown in figure 2.4 (5) ?

2.4.3 Rotation

A rotation is a transformation that turns a figure about a fixed point called the center of

rotation. An object and its rotation are the same shape and size, but the figures may be

turned in different directions.

Figure 2.4 (6): When you are riding on aFerris wheel, you are experiencing arotation.

Figure 2.4 (5)


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2.4.4 Glide Reflection

Glide reflection is a combination of a translation in a given direction and reflection in a

line parelel to that direction. Whether the reflection happens first or not, it does not

matter. This transformation will produce the same image. The same terms that apply to

reflections and translations apply to glide reflections: an axis is needed to perform the

reflection, a magnitude and direction are needed to perform the translation. Thus to tell

spesifically for glide reflection, the translation need to spesify as well as the line of

reflection.

(i)

(ii)

Figure 2.4 (7): A simple glide reflection.

Figure 2.4 (8): Footsteps in the sand.


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2.5 Plane Symmetries

This sub-topic will be discussed in the lecture.

2.6 The seven frieze (strip) patterns

In this section we will take a brief look at some infinite symmetry groups which can be used to

classify the symmetries of frieze or strip patterns. What are the frieze patterns? Frieze

patterns are classified under infinite discrete symmetry groups. A frieze is a horizontal band of

sculptures or decoration appearing on buildings and walls, often near the ceiling, laces and

borders.

Determine the transformation that used to form each tessellation below:

a. b.


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2.6.1 Types of transformations involved

Because the patterns are along a strip only, the translations which can be included are

those which shift points alongthe strip. The only rotationwhich allowed is half turns

i.e. rotation through 1800 .

Patterns along the strip only allowed 2 types of reflection:

(i) the reflection line (mirror line) perpendicular to the strip (if the strip is assumed in

horizontal position)

(ii) The reflection line (mirror line) horizontalalong the midline of the strip.

Translation and glide reflection can be used by themselves to generate infinite patterns

since the square of both a half turn (rotation through 1800) and a reflection is the

identity.

To summarize, the transformations involved to form infinite patterns in a strip are

(i) Translation

(ii) Glide reflection (combination of 2 types of transformation: reflection followed by

translation in the same direction)

(iii) Rotation through 1800only and

(iv) Reflection (mirror line perpendicular to the strip or horizontal along the midline of

the strip)

Reflection line

Reflection line


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There are only seven frieze patterns. It is shown in the following Table 2.6(1):

Pattern Type of

transformation

Pattern Examples

1. C Translation

2. C Glide reflection

3. D 2 half turns (1800)

4. D 2 reflections (mirrorperpendicular and

horizontal)

5. D 1 reflection and 1 half

turn(1800)

6. C x D1 1 translation dan 1

reflection (mirror

horizontal along the

midline of the strip)7. Dx D1 3 reflections

Table 2.6(1)

Note : C = Cyclic

D = Dihedral(Combination of rotation and reflection)


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2.6.2 Flow chart to distinguish frieze patterns

Diagram adapted from Hayley Rintel, Melissa Shearer, and the 1999

Exploring Symmetry Class

Is there a vertical reflection?

Yes NoIs there a horizontal reflection?

Yes

7

Is there a vertical reflection orglide reflection?

Is there a half turn?

No

Yes No

5 3

Is there a horizontalreflection?

Is there a half turn?

Yes

6 2

No

No

Yes

Yes No

4 1


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Additional Notes for Students:

1. All frieze patterns have translation symmetry

2. When a frieze pattern has vertical reflection symmetry that means we can draw at least

one vertical line so that one side of the pattern is a mirror image of the other side. There

is often more than one possible vertical line.

3. When a frieze pattern has horizontal symmetry, the only possible horizontal line is the

line through the center of the pattern.

4. The best way to determine if a frieze pattern has rotation symmetry is to look at it upside

down and see if it looks the same.

5. The best way to recognize glide reflection symmetry is to picture a set of footprints in the

sand.

Collect information (Scrap book)

Collect materials that are related to frieze pattern. Examples of materials that can be

collected are laces, embroidered cloth, and pictures of bracelets, necklaces or

bangles, pattern on prayer mat, ironworks, frame, cornices, borders and others that

shows pattern along the strip.

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