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

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!

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

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 groupseven 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 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!

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 ofrotations with the one selected flip!

(2) Border Patterns

(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).

(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.

(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.

(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

(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.

(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

(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.

(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

(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)

(3) Wallpaper Patterns

Here are the 17 model wallpaper patterns!

(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).

(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.

(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.

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”

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

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