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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. *H =H H D’ V D I R90 R180 R270 I =D’ R90 H R180 =V R270 ONE FLIP IS ENOUGH: “Composing with H” matches the 4 rotations with the 4 flips! =D 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**Leonardo Da Vinci’s self-portrait The model bounded objects Any bounded object is “symmetric in the same way” as one of these model objects. More precisely…**(1) Bounded Objects**Leonardo Da Vinci’s self-portrait The model bounded objects Any bounded object is “symmetric in the same way” as one of these model objects. More precisely… 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)**(1) Bounded Objects**Leonardo Da Vinci’s self-portrait The model bounded objects Any bounded object is “symmetric in the same way” as one of these model objects. More precisely… 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…**(1) Bounded Objects**Leonardo Da Vinci’s self-portrait The model bounded objects Any bounded object is “symmetric in the same way” as one of these model objects. More precisely… 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?**(1) Bounded Objects**Leonardo Da Vinci’s self-portrait The model bounded objects Any bounded object is “symmetric in the same way” as one of these model objects. More precisely… 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.**(1) Bounded Objects**Leonardo Da Vinci’s self-portrait 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. 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.**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 R10 were the smallest. • This means R20, R30, R40,…,R350 are also symmetries. • Something else, like R37 could not also be a symmetry because • that would make (R30-1)*R37 = R7 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). Border Pattern Identification Card 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? 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). Border Pattern Identification Card 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? 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). Border Pattern Identification Card 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? Y Y Y Y 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). Border Pattern Identification Card 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? 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). Border Pattern Identification Card 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? N Y N N 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). Border Pattern Identification Card 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? 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). Border Pattern Identification Card 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? N Y Y Y Classify this border pattern as type 1-7.**(3) Wallpaper Patterns**Qubbah Ba'adiyim in Marrakesh photo by amerune, Flickr.com WoodCut QBert Block Texture by Patrick Hoesly, Flickr.com Many of M. C. Escher’s art pieces are wallpaper patterns (click here)**(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.**Wallpaper Pattern Identification Card**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? The 17 model wallpaper patterns: diagram by Brian Sanderson,http://www.warwick.ac.uk/~maaac/**Vocabulary Review**“indecomposable glide-reflection” “order of a rotation” Classification Theorem Review**Vocabulary Review**“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