Rigid Motions & Symmetry

Math 203J11 November 2011

(11-11-11 is a cool date!)

Rigid Motions & Symmetry

What's a rigid motion?

Examples of rigid motions.

What kinds of symmetry are there?

Examples of symmetry.

What are Rigid Motions?

Think: My shape is a solid object (like a piece of wood) how can I move it in space?

Even better: My shape is a thin solid object so that there is a clear way to lie it down in a plane.

Only three kinds!

What are Rigid Motions?

Rotation – turn a given angle about a point

Reflection – flip over a given line – like a mirror

Translation – move a given amount in a given direction

What are Rigid Motions?

Rotation – turn a given angle about a point

Reflection – flip over a given line – like a mirror

Translation – move a given amount in a given direction

What are Rigid Motions?

Rotation – turn a given angle about a point

Reflection – flip over a given line – like a mirror

Translation – move a given amount in a given direction

How does this relate to art?

Art can be very geometric Example(s):

M.C. Escher – tesselations

How does this relate to art?

Art can be very geometric Example(s):

M.C. Escher – tesselations Goal is to fill the plane with one (or more) identical figures Saw a few examples last time Easy to do with equilateral triangles, rectangles, squares,

or regular hexagons – ask me to draw small examples of any of these!

How does this relate to art?

Art can be very geometric Example(s):

M.C. Escher – tesselations Goal is to fill the plane with one (or more) identical figures Saw a few examples last time Easy to do with equilateral triangles, rectangles, squares,

or regular hexagons – ask me to draw small examples of any of these!

Quilt blocks Anything else that repeats – wallpaper

Quilt Block Examples!

90 degree clockwise

rotation

Back to Start!

ReflectionAcross a

Horizontal Line

What's Symmetry?

Ways in which a rigid motion doesn't change what the image looks like

This time there are only two types! Rotational Symmetry – rotating the image gets you

back where you started Reflectional Symmetry – reflecting the image gets

you back where you started What examples can you come up with???

New (quilt block)!

ReflectionAbout

Vertical Axis

Back to Start!

ReflectionAbout

Horizontal Axis

Back to start!

Doesn't have 90º clockwise (or counter clockwise) rotational symmetry!

Is there any rotational symmetry???

Goal: Complete the picture

Knowing we have a given type of symmetry, can we complete an image?

Example

We'll complete the picture knowing that there's 90 degree rotational symmetry. Direction doesn't actually matter – why not?

Note to Kat: Draw these examples on the whiteboard since OpenOffice Impress isn't very impressive software!

Note to students: Take notes on how I did this if you want examples to take home with you!

Another Example!

This time we'll complete the picture knowing that there's both horizontal and vertical reflectional symmetry.

Find the Rigid Motions Used

Find More Rigid Motions

What's the Basic Shape?

Zoomed In

Real Example!

Rigid Motions of an (Equilateral) Triangle

How can I use rigid motions and put the triangle back down where it is?

Which rigid motions work, and what's the relationship between them?

Rotations

By 120 degrees or 240 degrees or by 360 degrees about the point in the middle

1 2

3

3 1

2

2 3

1

Reflections

About the lines of symmetry – there are 3 of them

Translations

Can I translate my triangle and have it land exactly on top of itself (as if it hadn't moved)???

Translations

Can I translate my triangle and have it land exactly on top of itself (as if it hadn't moved)???

NOPE!

Relationships?

What relationships can we find between our rigid motions of the triangle?

1 2

3

1 3

2

Here, we did a reflection, and then rotated 1 back to its starting point.

2 1

3

Relationships?

What relationships can we find between our rigid motions of the triangle?

1 2

3

1 3

2

Here, we just did a reflection, but got to the same position as before.

Relationships?

What relationships can we find between our rigid motions of the triangle?

There are other relationships that can be found. Most importantly (if you ask me): doing the same rotation 3 times gets you back

where you started, and doing the same reflection twice gets you back

where you started.

More on Groups

The rigid motions we found for the triangle form something called a group. The group is called D

3.

The three indicates that we're working with a triangle.

So what's the name of the group of rigid motions of a square?

What about a pentagon? hexagon?