Rigid Motions &
Symmetry
Math 203J
11 November 2011
(11-11-11 is a cool date!)
Rigid Motions & Symmetry
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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?
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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?
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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?
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

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?
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Art can be very geometric
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Example(s):
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M.C. Escher – tesselations
How does this relate to art?
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Art can be very geometric

Example(s):
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M.C. Escher – tesselations
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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
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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!
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Quilt blocks
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Anything else that repeats – wallpaper
Quilt Block Examples!
90 degree
clockwise
rotation
Back to Start!
Reflection
Across a
Horizontal Line
What's Symmetry?
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Ways in which a rigid motion doesn't change
what the image looks like
This time there are only two types!
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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)!
Reflection
About
Vertical
Axis
Back to
Start!
Reflection
About
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
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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!
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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
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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
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3
By 120 degrees or
240 degrees or by
360 degrees about
the point in the middle
1
2
3
2
1
1
2
3
Reflections
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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?
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What relationships can we find between our
rigid motions of the triangle?
3
2
3
1
2
1
2
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1
Here, we did a reflection, and then rotated 1
back to its starting point.
3
Relationships?
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What relationships can we find between our
rigid motions of the triangle?
2
3
1
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2
1
Here, we just did a reflection, but got to the
same position as before.
3
Relationships?
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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):
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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
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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?