# Ch. 4: The Classification Theorems. THE ALL-OR-HALF THEOREM: If an object has a finite symmetry group, then either all or half of its symmetries are proper.

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Ch. 4: The Classification Theorems

THE ALL-OR-HALF THEOREM: If an object has a finite symmetry group, then either all or half of its symmetries are proper. I R 90 =H R 180 R 270 =D’ =V =D H I R 90 R 180 R 270 H D’ V D ONE FLIP IS ENOUGH: “Composing with H” matches the 4 rotations with the 4 flips! *H Recall from Chapter 2: All flips are obtained by composing a single flip with all of the rotations! That’s why the All-Or-Half Theorem was true! Recall from Chapter 2: All flips are obtained by composing a single flip with all of the rotations! That’s why the All-Or-Half Theorem was true!

Goal: Classify all of the ways in which… (1) bounded objects (2) border patterns (3) wallpaper patterns …can be symmetric.

(1) Bounded Objects 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

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)

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…

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?

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.

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 groups even though they are not rigidly equivalent.

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.

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)

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?

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 R 10 were the smallest. This means R 20, R 30, R 40,…,R 350 are also symmetries. Something else, like R 37 could not also be a symmetry because that would make (R 30 -1 )*R 37 = R 7 be a smaller one! Example of why: Suppose R 10 were the smallest. This means R 20, R 30, R 40,…,R 350 are also symmetries. Something else, like R 37 could not also be a symmetry because that would make (R 30 -1 )*R 37 = R 7 be a smaller one!

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.)

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.

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’.

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?

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 of rotations with the one selected flip!

(2) Border Patterns

THE 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).

(2) Border Patterns THE 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? 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 rescaling of the model pattern with the same 4 answers.

(2) Border Patterns THE 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? 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.

(2) Border Patterns THE 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? 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 YYYY YYYY YYYY

(2) Border Patterns THE 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? 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.

(2) Border Patterns THE 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? 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 NYNN NYNN NYNN

(2) Border Patterns THE 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? 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.

(2) Border Patterns THE 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? 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 NYYY NYYY NYYY

(3) Wallpaper Patterns Here are the 17 model wallpaper patterns!

(3) Wallpaper Patterns THE 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).

(3) Wallpaper Patterns THE 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 be altered to become rigidly equivalent to the model pattern that it matches.

(3) Wallpaper Patterns THE 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 be altered to become rigidly equivalent to the model pattern that it matches.

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? 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

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?

Vocabulary Review “indecomposable glide-reflection” “order of a rotation” “indecomposable glide-reflection” “order of a rotation” Classification Theorem Review “symmetric in the same way” means… Number of model objects The fine print Bounded ObjectsRigid equivalenceInfinitely many Must have a finite symmetry group Border Patterns Rigid equivalence after rescaling 7 Must have a smallest translation Wallpaper Patterns Isomorphic symmetry groups 17 Must have a smallest translation

Vocabulary Review “indecomposable glide-reflection” “order of a rotation” “indecomposable glide-reflection” “order of a rotation” Theorem Review Da Vinci’s Theorem (group version) Da Vinci’s Theorem (rigid version) The Classification of Border Patterns The Classification of Wallpaper Patterns Da Vinci’s Theorem (group version) Da Vinci’s Theorem (rigid version) The Classification of Border Patterns The Classification of Wallpaper Patterns

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