The Mathematics of Symmetry Beth Kirby and Carl Lee University of - - PowerPoint PPT Presentation

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The Mathematics of Symmetry Beth Kirby and Carl Lee University of - - PowerPoint PPT Presentation

Apr 03, 2023 270 likes •1.9k views

Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying The Mathematics of Symmetry Beth Kirby and Carl Lee University of Kentucky MA 111 Fall 2009 Symmetry UK Info Symmetry Finite Shapes

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Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying

The Mathematics of Symmetry

Beth Kirby and Carl Lee

University of Kentucky MA 111

Fall 2009

Symmetry UK

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Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying

Symmetry UK

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Course Information

Text: Peter Tannenbaum, Excursions in Modern Mathematics, second custom edition for the University of Kentucky, Pearson. Course Website: http://www.ms.uky.edu/∼lee/ma111fa09/ma111fa09.html

Symmetry UK

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11.0 Introduction to Symmetry

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Are These Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is This Symmetrical?

Assume this extends forever to the left and right

Symmetry UK

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Is This Symmetrical?

Assume this extends forever to the left and right

Symmetry UK

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Is This Symmetrical?

Assume this extends forever in all directions

Symmetry UK

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Is This Symmetrical?

Assume this extends forever in all directions

Symmetry UK

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Is This Symmetrical?

Assume this extends forever in all directions

Symmetry UK

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Is This Symmetrical?

Assume this extends forever in all directions

Symmetry UK

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Is This Symmetrical?

Assume this extends forever in all directions

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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Is the Image of the Sun Symmetrical?

Symmetry UK

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Is This Symmetrical?

Symmetry UK

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11.6 Symmetry of Finite Shapes

Symmetry UK

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Symmetries of Finite Shapes

Let’s look at the symmetries of some finite shapes — shapes that do not extend forever in any direction, but are confined to a bounded region of the plane.

Symmetry UK

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Symmetries of Finite Shapes

Symmetry UK

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Symmetries of Finite Shapes

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Symmetries of Finite Shapes

This shape has 1 line or axis of reflectional symmetry. It has symmetry of type D1.

Symmetry UK

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Symmetries of Finite Shapes

Symmetry UK

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Symmetries of Finite Shapes

This shape has 3 axes of reflectional symmetry.

Symmetry UK

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Symmetries of Finite Shapes

It has a 120 degree angle of rotational symmetry. (Rotate counterclockwise for positive angles.)

Symmetry UK

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Symmetries of Finite Shapes

By performing this rotation again we have a 240 degree angle

  • f rotational symmetry.

Symmetry UK

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Symmetries of Finite Shapes

If we perform the basic 120 degree rotation 3 times, we bring the shape back to its starting position. We say that this shape has 3-fold rotational symmetry. With 3 reflections and 3-fold rotational symmetry, this shape has symmetry type D3.

Symmetry UK

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Symmetries of Finite Shapes

Symmetry UK

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Symmetries of Finite Shapes

This shape has 2 axes of reflectional symmetry.

Symmetry UK

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Symmetries of Finite Shapes

It has a 180 degree angle of rotational symmetry.

Symmetry UK

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Symmetries of Finite Shapes

If we perform the basic 180 degree rotation 2 times, we bring the shape back to its starting position. We say that this shape has 2-fold rotational symmetry. With 2 reflections and 2-fold rotational symmetry, this shape has symmetry type D2.

Symmetry UK

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Symmetries of Finite Shapes

Can you think of some common objects that have symmetry type:

◮ D4? ◮ D5? ◮ D6? ◮ D8?

Symmetry UK

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Symmetries of Finite Shapes

This shape has no axes of reflectional symmetry.

Symmetry UK

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Symmetries of Finite Shapes

It has a 72 degree angle of rotational symmetry. If we perform the basic 72 degree rotation 5 times, we bring the shape back to its starting position. We say that this shape has 5-fold rotational symmetry. With 0 reflections and 5-fold rotational symmetry, this shape has symmetry type Z5.

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape? Type Z2.

Symmetry UK

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Symmetries of Finite Shapes

Can you think of some common objects that have symmetry type:

◮ Z2? ◮ Z3? ◮ Z4?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape? Type D9.

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape? Type D4.

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape? Type D1.

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this image of the sun?

Symmetry UK

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Symmetries of Finite Shapes

Because it has infinitely many axes of reflectional symmetry and infinitely many angles of rotational symmetry, this symmetry type is D∞.

Symmetry UK

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Symmetries of Finite Shapes

What are the symmetry types of these various names?

Symmetry UK

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Symmetries of Finite Shapes

What is the symmetry type of this shape?

Symmetry UK

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Symmetries of Finite Shapes

Because this has no symmetry other than the one trivial one (don’t move it at all, or rotate it by an angle of 0 degrees), it has symmetry type Z1.

Symmetry UK

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11.7 Symmetries of Patterns

Symmetry UK

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Symmetries of Border Patterns

Now let’s look at symmetries of border patterns — these are patterns in which a basic motif repeats itself indefinitely (forever) in a single direction (say, horizontally), as in an architectural frieze, a ribbon, or the border design of a ceramic pot.

Symmetry UK

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Symmetries of Border Patterns

What symmetries does this pattern have?

Symmetry UK

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Symmetries of Border Patterns

You can slide, or translate, this pattern by the basic translation shown above.

Symmetry UK

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Symmetries of Border Patterns

You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left.

Symmetry UK

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Symmetries of Border Patterns

You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left. So this border pattern only has translational symmetry.

Symmetry UK

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Symmetries of Border Patterns

What symmetries does this pattern have?

Symmetry UK

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Symmetries of Border Patterns

You can translate this pattern by the basic translation shown above.

Symmetry UK

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Symmetries of Border Patterns

You can translate this pattern by the basic translation shown

  • above. This translation is the smallest translation possible; all
  • thers are multiples of this one, forward and backward.

Symmetry UK

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Symmetries of Border Patterns

You can also match the pattern up with itself by a combination of a reflection followed by a translation parallel to the reflection. This is called a glide reflection.

Symmetry UK

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Symmetries of Border Patterns

You can also match the pattern up with itself by a combination of a reflection followed by a translation parallel to the reflection. This is called a glide reflection. So this border pattern has both translational symmetry and glide reflectional symmetry.

Symmetry UK

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Symmetries of Border Patterns

What symmetries does this pattern have?

Symmetry UK

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Symmetries of Border Patterns

You can slide, or translate, this pattern by the basic translation shown above.

Symmetry UK

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Symmetries of Border Patterns

You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left.

Symmetry UK

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Symmetries of Border Patterns

There are infinitely many centers of 180 degree rotational

  • symmetry. Here is one type of location of a rotocenter.

Symmetry UK

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Symmetries of Border Patterns

And here is another type of location of a rotocenter.

Symmetry UK

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Symmetries of Border Patterns

And here is another type of location of a rotocenter. But this pattern has no reflectional symmetry or glide reflectional symmetry.

Symmetry UK

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Symmetries of Border Patterns

And here is another type of location of a rotocenter. But this pattern has no reflectional symmetry or glide reflectional

  • symmetry. So this border pattern only has translational

symmetry and 2-fold (or half-turn) rotational symmetry.

Symmetry UK

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Symmetries of Border Patterns

What symmetries does this pattern have?

Symmetry UK

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Symmetries of Border Patterns

You can translate this pattern by the basic translation shown above.

Symmetry UK

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Symmetries of Border Patterns

You can translate this pattern by the basic translation shown

  • above. This translation is the smallest translation possible; all
  • thers are multiples of this one, to the right and to the left.

Symmetry UK

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Symmetries of Border Patterns

This pattern has one horizontal axis of reflectional symmetry

Symmetry UK

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Symmetries of Border Patterns

This pattern has one horizontal axis of reflectional symmetry but infinitely many vertical axes of reflectional symmetry, that have two types of locations.

Symmetry UK

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Symmetries of Border Patterns

Because there are translations and horizontal reflections, we can combine them to get glide reflections. Here is one type of glide reflection. Others use the same reflection axis but multiples of this translation, to the right and to the left.

Symmetry UK

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Symmetries of Border Patterns

There are infinitely many centers of 180 degree rotational

  • symmetry. Here the two types of locations of rotocenters.

Symmetry UK

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Symmetries of Border Patterns

So this border pattern has translational, horizontal reflectional, vertical reflectional, and 2-fold rotational symmetry. Even though glide reflections also work, our text states that we don’t say this pattern has “glide reflectional symmetry” because the glide reflections are in this case just a consequence

  • f translational and the horizontal reflectional symmetry.

Symmetry UK

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11.2 Reflections

Symmetry UK

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What is a Reflection?

A reflection is a motion that moves an object to a mirror image of itself. The “mirror” is called the axis of reflection, and is given by a line m in the plane.

Symmetry UK

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What is a Reflection?

To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection

  • m. The image P′ will be the point on this line whose distance

from m is the same as that between P and m.

Symmetry UK

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What is a Reflection?

To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection

  • m. The image P′ will be the point on this line whose distance

from m is the same as that between P and m.

bP

m

Symmetry UK

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What is a Reflection?

To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection

  • m. The image P′ will be the point on this line whose distance

from m is the same as that between P and m.

bP

m

bP

m

bP′ b

2 2

Symmetry UK

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Examples

1.

b P

m

Symmetry UK

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Examples

1.

b P

m

b P

m

b P′

0.89 0.89

Symmetry UK

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Examples

1.

b P

m

b P

m

b P′

0.89 0.89 2.

bQ

m

Symmetry UK

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Examples

1.

b P

m

b P

m

b P′

0.89 0.89 2.

bQ

m

bQ

m

b

Q′

Symmetry UK

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Examples

bA bB b

C m

Symmetry UK

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Examples

bA bB b

C m

bA bB b

C m

bA′ bB′ bC ′

Symmetry UK

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Properties of Reflections

  • 1. A reflection is completely determined by its axis of

reflection.

Symmetry UK

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Properties of Reflections

  • 1. A reflection is completely determined by its axis of

reflection. ...or...

  • 2. A reflection is completely determined by a single

point-image pair P and P′ (if P = P′).

Symmetry UK

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Properties of Reflections

Given a point P and its image P′, the axis of reflection is the perpendicular bisector of the line segment PP′.

b

P′

b

P

Symmetry UK

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Properties of Reflections

Given a point P and its image P′, the axis of reflection is the perpendicular bisector of the line segment PP′.

b

P′

b

P

b

P′

b

P m

Symmetry UK

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Properties of Reflections

A fixed point of a motion is a point that is moved onto itself. For a reflection, any point on the axis of reflection is a fixed point.

  • 3. Therefore, a reflection has infinitely many fixed points (all

points on the line m).

Symmetry UK

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Properties of Reflections

bA bB bC b

D m

bA′ b

B′

b

C ′

bD′

The orientation of the original object is clockwise: read ABCDA going in the clockwise direction. The orientation of the image under the reflection is counterclockwise: A′B′C ′D′A′ is read in the counterclockwise direction.

Symmetry UK

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Properties of Reflections

  • 4. A reflection is an improper motion because it reverses the
  • rientation of objects.
  • 5. Applying the same reflection twice is equivalent to not

moving the object at all. So applying a reflection twice results in the identity motion.

Symmetry UK

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11.3 Rotations

Symmetry UK

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What is a Rotation?

A rotation is a motion that swings an object around a fixed point. The fixed center point of the rotation is called the rotocenter. The amount of swing is given by the angle of rotation.

bA b

B

bC bO b bA b

B

b

C

bO b

A′

b B′ bC ′ b

90◦

Symmetry UK

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What is a Rotation?

Notice that the distance of each point from the rotocenter O does not change under the rotation:

bA b

B

b

C

bO b

A′

b B′ bC ′ b

Symmetry UK

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The Angle of Rotation

As a convention, any angle in the counterclockwise direction has a positive angle measure. Any angle in the clockwise direction has a negative angle measure.

bA b

B

bC

45◦

bD bE bF

  • 45◦

Symmetry UK

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Examples

The rotation with rotocenter O and angle of rotation 135◦:

bA b

B

b

C

bD bO

135◦

Symmetry UK

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Examples

The rotation with rotocenter O and angle of rotation 135◦:

bA b

B

b

C

bD bO

135◦

bA b

B

b

C

bD bO b

A′

b

B′

b

C ′

b

D′ 135◦

Symmetry UK

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Examples

The rotation with rotocenter O and angle of rotation −45◦:

bA b

B

b

O

bD bE b

Symmetry UK

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Examples

The rotation with rotocenter O and angle of rotation −45◦:

bA b

B

b

O

bD bE b bA b

B

b

O

bD bE bA′ bB′ b bD′ b

E ′

  • 45◦

Symmetry UK

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Properties of Rotations

A rotation is completely determined by .

Symmetry UK

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Properties of Rotations

A rotation is completely determined by . If we know one point-image pair...

b

P

b

P′

Symmetry UK

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Properties of Rotations

A rotation is completely determined by . If we know one point-image pair...

b

P

b

P′ there are infinitely many possible rotocenters:

b

P

b

P′

b b b

Symmetry UK

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Properties of Rotations

  • 1. A rotation is completely determined by two point-image

pairs. The rotocenter is at the intersection of the two perpendicular bisectors:

b

P

b

P′

1

bQ b b bQ′

What is the angle of rotation of this rotation?

Symmetry UK

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Properties of Rotations

  • 1. A rotation is completely determined by two point-image

pairs. The rotocenter is at the intersection of the two perpendicular bisectors:

b

P

b

P′

1

bQ b b bQ′

What is the angle of rotation of this rotation? 180◦.

Symmetry UK

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Properties of Rotations

Once we know the rotocenter, the angle of rotation is the measure of angle ∠POP′. This will be the same as the measure of angle ∠QOQ′.

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Properties of Rotations

What are the fixed points of a rotation?

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Properties of Rotations

What are the fixed points of a rotation?

  • 2. A rotation has one fixed point, the rotocenter.

Is a rotation a proper or improper motion?

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Properties of Rotations

What are the fixed points of a rotation?

  • 2. A rotation has one fixed point, the rotocenter.

Is a rotation a proper or improper motion?

  • 3. A rotation is a proper motion. The orientation of the
  • bject is maintained.
  • 4. A 360◦ rotation around any rotocenter is equivalent to

the identity motion.

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Properties of Rotations

Notice that any rotation is equivalent to one with angle of rotation between 0◦ and 360◦:

bA′ b

O

bA

90◦

b

450◦ 270◦

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11.4 Translations

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What is a Translation?

A translation is a motion which drags an object in a specified direction for a specified length. The direction and length of the translation are given by the vector of translation, usually denoted by v.

bA bA′

v

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What is a Translation?

The placement of the vector of translation on the plane does not matter. We only need to see how long it is and in what direction it’s pointing to know the image of an object.

b

A

bB bC

v

bA′ bB′ bC ′

The vector v indicates that each point moves down one unit and to the right two units.

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Examples

b

A

b

B

bC bD

v

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Examples

b

A

b

B

bC bD

v

b

A

b

B

bC bD

v

bA′ bB′ bC ′ bD′

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Examples

b

P

b

Q

bR b

S v

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Examples

b

P

b

Q

bR b

S v

b

P

b

Q

bR b

S

bP′ bQ′ bR′ bS′

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Examples

Given a vector v, the vector −v has the same length but the

  • pposite direction.

Apply the translation with vector v and then the translation with vector −v.

b

A

b

B v

  • v

b b

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Examples

Given a vector v, the vector −v has the same length but the

  • pposite direction.

Apply the translation with vector v and then the translation with vector −v.

b

A

b

B v

  • v

b b b

A

b

B v

b

A′

bB′

  • v

b b

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Examples

Given a vector v, the vector −v has the same length but the

  • pposite direction.

Apply the translation with vector v and then the translation with vector −v.

b

A

b

B v

  • v

b b b

A

b

B v

b

A′

bB′

  • v

b b b

A

b

B v

  • v

b b

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Properties of Translations

How many point-image pairs do we need to determine the translation?

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Properties of Translations

How many point-image pairs do we need to determine the translation? Given one point-image pair, the vector of translation is the arrow that connects the point to its image.

b

A

b

A′ v

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Properties of Translations

How many point-image pairs do we need to determine the translation? Given one point-image pair, the vector of translation is the arrow that connects the point to its image.

b

A

b

A′ v

  • 1. A translation is completely determined by one

point-image pair.

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Properties of Translations

  • 2. A translation has no fixed points. Why?
  • 3. A translation is a proper motion because the orientation
  • f the object is preserved.
  • 4. By applying the translation with vector −v after the

translation with vector v, we obtain a motion that is equivalent to the identity motion.

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11.5 Glide Reflections

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What is a Glide Reflection?

A glide reflection is a combination of a translation and a reflection. The vector of translation v and the axis of reflection m must be parallel to each other.

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Examples

b

P

b

Q

b

R v m

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Examples

b

P

b

Q

b

R v m

b

P

b

Q

b

R v m

b P* bQ* bR*

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Examples

b

P

b

Q

b

R v m

b

P

b

Q

b

R v m

b P* bQ* bR* b

P

b

Q

b

R v m

b P* bQ* bR* b

P′

bQ′ bR′

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Examples

Since the vector of translation and the axis of reflection are parallel, it does not matter which motion is done first in the glide reflection.

b

P

b

Q

b

R v m

b

P′

bQ′ bR′ b

P*

b

Q*

bR*

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Properties of Glide Reflections

  • 1. A glide reflection is completely determined by two

point-image pairs.

◮ The axis of reflection is the line passing through the two

midpoints of the segments PP′ and QQ′.

◮ Use the axis of reflection to find an intermediate point.

For example, the image of P under the reflection is the intermediate point P∗.

◮ Finally, the vector of translation is the vector connecting

P to P∗.

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Two Point-Image Pairs Determine a Glide Reflection

b

P

bQ bP′ bQ′

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Two Point-Image Pairs Determine a Glide Reflection

b

P

bQ bP′ bQ′ b

P

bQ bP′ bQ′ b b

m

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Two Point-Image Pairs Determine a Glide Reflection

b

P

bQ bP′ bQ′ b

P

bQ bP′ bQ′ b b

m

b

P

bQ bP′ bQ′ b b

m

bQ*

v

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Properties of Glide Reflections

Does a glide reflection have any fixed points?

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Properties of Glide Reflections

Does a glide reflection have any fixed points? No.

  • 2. Since a translation has no fixed points, a glide reflection

has no fixed points.

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Properties of Glide Reflections

Does a glide reflection have any fixed points? No.

  • 2. Since a translation has no fixed points, a glide reflection

has no fixed points. Is a glide reflection a proper or improper motion?

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Properties of Glide Reflections

Does a glide reflection have any fixed points? No.

  • 2. Since a translation has no fixed points, a glide reflection

has no fixed points. Is a glide reflection a proper or improper motion? Improper. Why?

  • 3. A glide reflection is an improper motion.

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Properties of Glide Reflections

Given a glide reflection with translation vector v and axis of reflection m, how can we “undo” the motion?

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Properties of Glide Reflections

Given a glide reflection with translation vector v and axis of reflection m, how can we “undo” the motion? To undo the translation, we must apply the translation with vector −v. To undo the reflection, we must apply the reflection with the same axis of reflection m.

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Properties of Glide Reflections

Given a glide reflection with translation vector v and axis of reflection m, how can we “undo” the motion? To undo the translation, we must apply the translation with vector −v. To undo the reflection, we must apply the reflection with the same axis of reflection m.

  • 4. When a glide reflection with vector v and axis of reflection

m is followed by a glide reflection with vector −v and axis

  • f reflection m, we obtain the identity motion.

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Summary of Motions

Point-Image Rigid Motion Specified by Proper/Improper Fixed Points Pairs Needed Reflection Axis of reflection ℓ Improper All points on ℓ One Rotation* Rotocenter O and angle α Proper O only Two Translation Vector of translation v Proper None One Glide Reflection Vector of translation v and Improper None Two axis of reflection ℓ *Identity Proper All points (0◦ rotation) Symmetry UK

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11.7 Classifying Symmetries of Border Patterns

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Classifying Symmetries of Border Patterns

By definition, border patterns always have translational symmetry—there is always a basic, smallest, translation that can be repeated to the right and to the left as many times as

  • desired. There is a basic design or motif that repeats

indefinitely in one direction (e.g., to the right and the left).

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Classifying Symmetries of Border Patterns

We have seen some border patterns that have no other symmetry, some that have glide reflectional symmetry, some that have rotational (half-turn) symmetry, and even some that have horizontal reflectional symmetry, vertical reflectional symmetry, and half-turn (2-fold rotational) symmetry.

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Classifying Symmetries of Border Patterns

We can classify border patterns according to the combinations

  • f symmetries that they possess. It turns out that there are
  • nly seven different possibilities:

Symmetry Horizontal Vertical Glide Type Translation Reflection Reflection Half-Turn Reflection 11 Yes No No No No 1m Yes Yes No No No* m1 Yes No Yes No No mm Yes Yes Yes Yes No* 12 Yes No No Yes No 1g Yes No No No Yes mg Yes No Yes Yes Yes

*These patterns do have glide reflections, but only as a result

  • f having translational and horizontal reflectional symmetry.

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Classify These Seven Patterns

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Classifying Symmetries of Border Patterns

Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is?

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Classifying Symmetries of Border Patterns

Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is? A half turn.

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Classifying Symmetries of Border Patterns

Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is? A half turn. So if a border pattern admits a horizontal reflection and a vertical reflection, it must also have a half turn. You cannot have a border pattern with a horizontal reflection, a vertical reflection, and no half turn.

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Classifying Symmetries of Border Patterns

The net result of a horizontal reflection followed by a half turn is?

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Classifying Symmetries of Border Patterns

The net result of a horizontal reflection followed by a half turn is? A vertical reflection.

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Classifying Symmetries of Border Patterns

The net result of a horizontal reflection followed by a half turn is? A vertical reflection. So if a border pattern admits a horizontal reflection and a half turn, it must also have a vertical reflection. You cannot have a border pattern with a horizontal reflection, a half turn, and no vertical reflection.

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Classifying Symmetries of Border Patterns

Other combinations of possible symmetries can be considered to rule out other cases, leaving only seven remaining.

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Classifying Wallpaper Patterns

Border patterns have a repeated basic design or motif that repeats indefinitely in a single direction (e.g., to the right and left). Wallpaper patterns have a basic design or motif that repeats indefinitely in at least two different directions. It turns out that there are 17 different types of symmetry for wallpaper

  • patterns. A flowchart for classifying wallpaper patterns

appears on page 627 of the text. Try classifying the wallpaper patterns in the file “More Patterns” on the course website.

Symmetry UK