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.
*H
=H
I
H
R90
D’
R180
V
R270
D
I
R90
=D’
H
R180
=V
R270
=D
ONE FLIP IS ENOUGH:
“Composing with H” matches
the 4 rotations with the 4 flips!
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
The model bounded objects
Any bounded object is “symmetric in the same way”
as one of these model objects. More precisely…
Leonardo Da Vinci’s self-portrait
(1) Bounded Objects
The model bounded objects
Any bounded object is “symmetric in the same way”
as one of these model objects. More precisely…
Leonardo Da Vinci’s self-portrait
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
The model bounded objects
Any bounded object is “symmetric in the same way”
as one of these model objects. More precisely…
Leonardo Da Vinci’s self-portrait
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
The model bounded objects
Any bounded object is “symmetric in the same way”
as one of these model objects. More precisely…
Leonardo Da Vinci’s self-portrait
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
The model bounded objects
Any bounded object is “symmetric in the same way”
as one of these model objects. More precisely…
Leonardo Da Vinci’s self-portrait
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
The only pair that has isomorphic symmetry groups
even though they are not rigidly equivalent.
Leonardo Da Vinci’s self-portrait
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
(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?
Classify this border pattern as type 1-7.
Y
Y
Y
Y
(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?
Classify this border pattern as type 1-7.
N
Y
N
N
(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?
Classify this border pattern as type 1-7.
N
Y
Y
Y
(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
“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 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”
Theorem Review
Da Vinci’s Theorem (group version)
Da Vinci’s Theorem (rigid version)
The Classification of Border Patterns
The Classification of Wallpaper Patterns