Chapter 11. Symmetry

Review of Sections 1-3, 6

A rigid motion is a motion that doesn’t change distances or angles. A symmetry of a figure is any rigid motion which moves the figure exactly on top of itself. So far we have studied reflections and rotations.

Reflections have a line of fixed points, called the axis of the reflection. Every other point moves to the point directly opposite it across this line: A A*

C B B* C*

The reflection about the vertical line takes triangle ABC to A*B*C*. Notice that clockwise motion around ABC is changed into counterclockwise motion around A*B*C*. Motions with this property (“reversing orientation”) are called improper. Proper motions are those which preserve the orientation.

A rotation has exactly one fixed point, the rotocenter. The rotation moves each line through the center to another line through the center, and each line is moved by the same angle. Below we show a 90º clockwise rotation about A; it sends ABC to AB*C*. Notice that rotations are proper motions. B C A B*

C*

1 Symmetry types of finite shapes. Dn (n = 1, 2, 3, ...): This is the symmetry type of an object with n reflection symmetries and n rotation symmetries. Zn (n = 1, 2, 3, ...): This is the symmetry type of an object with n rotation symmetries but no reflection symmetries. D∞: This is the symmetry type of a circle.

These are the only possible symmetry types of finite figures!

Sections 4, 5, 7

The other types of rigid motions are translations and glide reflections. These have no fixed points, which is why finite figures can’t have symmetries involving these.

A vector is an arrow: it has length and direction. A vector can be moved anywhere in the plane; as long as you don’t change its direction or length, it remains the same vector.

Each vector defines a translation: every point A in an object is moved to a new point A*, with the arrow pointing from A to A* being our given vector. The vector sends the triangle ABC to the triangle A*B*C*:

C C* A* A B B*

Notice that translations are proper motions.

2 A glide reflection consists of a translation by a vector v followed by a reflection about an axis which is parallel to the vector. Below we describe a glide reflection which sends triangle ABC to A’B’C’; notice that the glide reflection is an improper motion.

C v C* A* A B B* axis: B’ A’

C’ Infinite shapes can have symmetries involving translations and glide reflections. If all the translation symmetries are in the same direction, the shapes are called border patterns. If the translation symmetries are in two different directions, the shapes are called wallpaper patterns. See the two translation directions in the following wallpaper pattern:

On your homework and exam we will have border patterns but no wallpaper patterns.

3 Here are 3 ancient Greek border patterns:

The possible symmetries for border patterns are: • the identity • translations (always in the direction of the pattern) • horizontal reflections (the axis runs through the pattern) • vertical reflections (the axis is perpendicular to the pattern) • half-turns (180º rotations) • glide reflections (the axis runs through the pattern)

In the left example above, there is a glide reflection with vector w half the length of the translation symmetry with vector v. Note there is no horizontal symmetry. On the right there is a horizontal reflection symmetry, and applying this after the translation symmetry produces a glide reflection symmetry. In the table below we only count the glide reflections which don't come from a translation symmetry followed by a horizontal reflection symmetry.

4 There are only seven types of border pattern symmetries: symmetry types m1 mm mg 11 12 1m 1g translations yes yes yes yes yes yes yes vertical reflections yes yes yes no no no no horizontal reflections no yes no no no yes no glide reflections no no* yes no no no* yes half-turns no yes yes no yes no no * = because we don't count glide reflection symmetries if there is also a horizontal reflection symmetry.

The symbols used to describe the types were created by crystallographers (scientists who study crystals). The first symbol is an m (for mirror) if there is a vertical reflection, otherwise it is a 1. The second symbol is m: there is a horizontal reflection, g: there is a glide reflection, but no horizontal reflection 2: there is a half-turn (but no horizontal or glide reflection), 1: none of these.

What symmetry types do the following border patterns have? 1. ... JJJJJJJJ ... Answer: 11 2. ... TTTTTT ... Answer: m1 3. ... CCCCCC ... Answer: 1m 4. ... NNNNNN ... Answer: 12 5. ... qbqbqbqb ... Answer: 12 6. ... qdqdqdqd ... Answer: 1g 7. ... dbdbdbdb ... Answer: m1 8. ... qpdbqpdb ... Answer: mg You don't have to memorize these symbols; but you should be able to recognize which symmetries a border pattern possesses.

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