Rigid Motions & Symmetry Math 203J 11 November 2011 (11-11-11 is a cool date!)
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Transcript of Rigid Motions & Symmetry Math 203J 11 November 2011 (11-11-11 is a cool date!)
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Rigid Motions & Symmetry
Math 203J11 November 2011
(11-11-11 is a cool date!)
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Rigid Motions & Symmetry
What's a rigid motion?
Examples of rigid motions.
What kinds of symmetry are there?
Examples of symmetry.
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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!
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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
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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
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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
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How does this relate to art?
Art can be very geometric Example(s):
M.C. Escher – tesselations
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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!
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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
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Quilt Block Examples!
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90 degree clockwise
rotation
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Back to Start!
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ReflectionAcross a
Horizontal Line
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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???
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New (quilt block)!
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ReflectionAbout
Vertical Axis
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Back to Start!
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ReflectionAbout
Horizontal Axis
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Back to start!
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Doesn't have 90º clockwise (or counter clockwise) rotational symmetry!
Is there any rotational symmetry???
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Goal: Complete the picture
Knowing we have a given type of symmetry, can we complete an image?
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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!
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Another Example!
This time we'll complete the picture knowing that there's both horizontal and vertical reflectional symmetry.
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Find the Rigid Motions Used
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Find More Rigid Motions
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What's the Basic Shape?
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Zoomed In
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Real Example!
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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?
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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
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Reflections
About the lines of symmetry – there are 3 of them
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Translations
Can I translate my triangle and have it land exactly on top of itself (as if it hadn't moved)???
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Translations
Can I translate my triangle and have it land exactly on top of itself (as if it hadn't moved)???
NOPE!
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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
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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.
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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.
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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?