1 Ch 17:Classification Modified from Massengale, biology junction.
Ch. 4: The Classification Theorems
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Transcript of Ch. 4: The Classification Theorems
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Ch. 4: The Classification Theorems
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THE ALL-OR-HALF THEOREM: If an object has a finite symmetry group, then either all or half of its symmetries are proper.
I
R90
=H
R180
R270
=D’
=V
=D
H
I
R90
R180
R270
H
D’
V
D
ONE FLIP IS ENOUGH:“Composing with H” matchesthe 4 rotations with the 4 flips!
*H
Recall from Chapter 2: All flips are obtained by composing a single flip withall of the rotations! That’s why the All-Or-Half Theorem was true!
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Goal: Classify all of the ways in which… (1) bounded objects (2) border patterns (3) wallpaper patterns
…can be symmetric.
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(1) Bounded Objects
Leonardo Da Vinci’s self-portrait
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
Any bounded object is “symmetric in the same way”as one of these model objects. More precisely…
The model bounded objects
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
Any bounded object is “symmetric in the same way”as one of these model objects. More precisely…
The model bounded objects
What you already knew: Any bounded object (with a finite symmetry group)has the same number of rotations & flips as one of these model objects.(by the All-or-Half Theorem)
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
Any bounded object is “symmetric in the same way”as one of these model objects. More precisely…
The model bounded objects
What you already knew: Any bounded object (with a finite symmetry group)has the same number of rotations & flips as one of these model objects.(by the All-or-Half Theorem)
But does it have the same rotation angles?Does it have the same arrangement of reflection lines?
Da Vinci answered these questions…
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
Any bounded object is “symmetric in the same way”as one of these model objects. More precisely…
The model bounded objects
What you already knew: Any bounded object (with a finite symmetry group)has the same number of rotations & flips as one of these model objects.
RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
What does this imply about its symmetry group?
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
Any bounded object is “symmetric in the same way”as one of these model objects. More precisely…
The model bounded objects
What you already knew: Any bounded object (with a finite symmetry group)has the same number of rotations & flips as one of these model objects.
RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
GROUP VERSION OF DA VINCI’S THEOREM: The symmetry group of any bounded object in the plane is either infinite or is isomorphic to a dihedral or cyclic group.
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Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
GROUP VERSION OF DA VINCI’S THEOREM: The symmetry group of any bounded object in the plane is either infinite or is isomorphic to a dihedral or cyclic group.
The only pair that has isomorphic symmetry groupseven though they are not rigidly equivalent.
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group.
Like maybe one of these shapes,or anything else your Google image search turned up.
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point.
(by the Center Point Theorem)
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one.
WHY?
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one.
Example of why:Suppose R10 were the smallest.This means R20, R30, R40,…,R350 are also symmetries.Something else, like R37 could not also be a symmetry
becausethat would make (R30
-1)*R37 = R7 be a smaller one!
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one. Let n denote the number of rotations your object has.
(Notice it has the same n rotation angles as a regular n-gon.)
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one. Let n denote the number of rotations your object has.
(Notice it has the same n rotation angles as a regular n-gon.) If your object has NO FLIPS, then it is rigidly equivalent to an oriented n-gon
via any rigid motion that matches up their center points.
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one. Let n denote the number of rotations your object has.
(Notice it has the same n rotation angles as a regular n-gon.) If your object has NO FLIPS, then it is rigidly equivalent to an oriented n-gon
via any rigid motion that matches up their center points. If your object has some flips, then choose one and call it F.
Also choose a flip of the regular n-gon and call it F’. Your object is rigidly equivalent to the regular n-gon via any rigid motion, M,
that matches up their center points and the reflection lines of F with F’.
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one. Let n denote the number of rotations your object has.
(Notice it has the same n rotation angles as a regular n-gon.) If your object has NO FLIPS, then it is rigidly equivalent to an oriented n-gon
via any rigid motion that matches up their center points. If your object has some flips, then choose one and call it F.
Also choose a flip of the regular n-gon and call it F’. Your object is rigidly equivalent to the regular n-gon via any rigid motion, M,
that matches up their center points and the reflection lines of F with F’.
Why will the remaining flips also match?
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RIGID VERSION OF DA VINCI’S THEOREM: Any bounded object in the plane with a finite symmetry group is rigidly equivalent to one of these model objects.
PROOF: Imagine you have a bounded object with a finite symmetry group. All of your object’s rotations have the same center point. All of your object’s rotation angles are multiples of the smallest one. Let n denote the number of rotations your object has.
(Notice it has the same n rotation angles as a regular n-gon.) If your object has NO FLIPS, then it is rigidly equivalent to an oriented n-gon
via any rigid motion that matches up their center points. If your object has some flips, then choose one and call it F.
Also choose a flip of the regular n-gon and call it F’. Your object is rigidly equivalent to the regular n-gon via any rigid motion, M,
that matches up their center points and the reflection lines of F with F’.
Why will the remaining flips also match?
Because they are compositions ofrotations with the one selected flip!
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(2) Border Patterns
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Any border pattern is rigidly equivalent to a rescalingof the model pattern with the same 4 answers.
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
YYYY
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
![Page 26: Ch. 4: The Classification Theorems](https://reader036.fdocuments.in/reader036/viewer/2022062501/56816320550346895dd399bd/html5/thumbnails/26.jpg)
(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
NYNN
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(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
![Page 28: Ch. 4: The Classification Theorems](https://reader036.fdocuments.in/reader036/viewer/2022062501/56816320550346895dd399bd/html5/thumbnails/28.jpg)
(2) Border PatternsTHE CLASSIFICATION OF BORDER PATTERNS: Any border pattern is rigidly equivalent to a rescaling of one of the seven model border patterns illustrated below (provided it has a smallest non-identity translation).
Q1 – Does it have any horizontal reflection symmetry?Q2 – Does it have any vertical reflection symmetry?Q3 – Does it have any 180 degree rotation symmetry?Q4 – Does it have any glide reflection symmetry?
Border Pattern Identification Card
Classify this border pattern as type 1-7.
NYYY
![Page 29: Ch. 4: The Classification Theorems](https://reader036.fdocuments.in/reader036/viewer/2022062501/56816320550346895dd399bd/html5/thumbnails/29.jpg)
(3) Wallpaper Patterns
Qubbah Ba'adiyim in Marrakeshphoto by amerune, Flickr.com
WoodCut QBert Block Textureby Patrick Hoesly, Flickr.com
Many of M. C. Escher’s art pieces are wallpaper patterns (click here)
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(3) Wallpaper Patterns
Here are the 17 model wallpaper patterns!
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(3) Wallpaper PatternsTHE CLASSIFICATION OF WALLPAPER PATTERNS: The symmetry group of any wallpaper pattern is isomorphic to the symmetry group of one of the 17 model patterns (provided it has a smallest non-identity translation).
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(3) Wallpaper PatternsTHE CLASSIFICATION OF WALLPAPER PATTERNS: The symmetry group of any wallpaper pattern is isomorphic to the symmetry group of one of the 17 model patterns (provided it has a smallest non-identity translation).
In fact, any wallpaper pattern can be altered by a “linear transformation” to become rigidly equivalent to one of the 17 model patterns.
EXAMPLE: This pattern must bealtered to become rigidly equivalentto the model pattern that itmatches.
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(3) Wallpaper PatternsTHE CLASSIFICATION OF WALLPAPER PATTERNS: The symmetry group of any wallpaper pattern is isomorphic to the symmetry group of one of the 17 model patterns (provided it has a smallest non-identity translation).
In fact, any wallpaper pattern can be altered by a “linear transformation” to become rigidly equivalent to one of the 17 model patterns.
EXAMPLE: This pattern must bealtered to become rigidly equivalentto the model pattern that itmatches.
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O – What is the maximum Order of a rotation symmetry?R – Does it have any Reflection symmetries?G – Does it have an indecomposable Glide-reflection symmetries?ON – Does it have any rotations centered ON reflection lines? OFF – Does it have any rotations centered OFF reflection lines?
Wallpaper Pattern Identification Card
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The 17 model wallpaper patterns: diagram by Brian Sanderson,http://www.warwick.ac.uk/~maaac/
O – What is the maximum Order of a rotation symmetry?R – Does it have any Reflection symmetries?G – Does it have an indecomposable Glide-reflection symmetries?ON – Does it have any rotations centered ON reflection lines? OFF – Does it have any rotations centered OFF reflection lines?
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Vocabulary Review
“indecomposable glide-reflection”“order of a rotation”
Classification Theorem Review“symmetric in the
same way” means…Number of model
objects The fine print
Bounded Objects Rigid equivalence Infinitely many Must have a finite symmetry group
Border Patterns Rigid equivalenceafter rescaling 7 Must have a smallest
translation
Wallpaper Patterns Isomorphic symmetry groups 17 Must have a smallest
translation
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Vocabulary Review
“indecomposable glide-reflection”“order of a rotation”
Theorem ReviewDa Vinci’s Theorem (group version)Da Vinci’s Theorem (rigid version)
The Classification of Border PatternsThe Classification of Wallpaper Patterns