Ch. 3: Isomorphism

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

…can be symmetric.

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

…can be symmetric. More immediate Goal: Decide what this means.What should it mean to say that two objects/patternsare symmetric in the same way?

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

…can be symmetric. More immediate Goal: Decide what this means.What should it mean to say that two objects/patternsare symmetric in the same way?

Symmetric in the same way?

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

…can be symmetric. More immediate Goal: Decide what this means.What should it mean to say that two objects/patternsare symmetric in the same way?

Symmetric in the same way?

• Same # rotations. • Same # flips.• Similar Cayley tables.

• Different colors. • Different centers.• Different tilts.

What really matters?

(1) These two squares have exactly the same symmetry groups:

(1) These two squares have exactly the same symmetry groups:

(Each symmetry of one is a symmetry of the other)

(2) These two squares do NOT have exactly the same symmetry groups:

(2) These two squares do NOT have exactly the same symmetry groups:

(Symmetries of the red square are NOT also symmetries of the green square because they move the green square around.)

(2) These two squares do NOT have exactly the same symmetry groups:

BUT these two squares have essentially the same symmetry groups.

What should this mean? How are their Cayley tables similar?

Rene’s red square:

{I, R90, R180, R270, H, V, D, D’}. * I R90 R180 R270 H V D D’

I I R90 R180 R270 H V D D’

R90 R90 R180 R270 I D’ D H V

R180 R180 R270 I R90 V H D’ D

R270 R270 I R90 R180 D D’ V H

H H D V D’ I R180 R90 R270

V V D’ H D R180 I R270 R90

D D V D’ H R270 R90 I R180

D’ D’ H D V R90 R270 R180 I

(Same Cayley table we built in Ch. 2)

Rene’s red square:

{I, R90, R180, R270, H, V, D, D’}. * I R90 R180 R270 H V D D’

I I R90 R180 R270 H V D D’

R90 R90 R180 R270 I D’ D H V

R180 R180 R270 I R90 V H D’ D

R270 R270 I R90 R180 D D’ V H

H H D V D’ I R180 R90 R270

V V D’ H D R180 I R270 R90

D D V D’ H R270 R90 I R180

D’ D’ H D V R90 R270 R180 I

How can Gretchen copy Rene’s work to build her Cayley table?

Gretchen’s green square:

{I, R90, R180, R270, F1, F2, F3, F4}.

*

Rene’s red square:

{I, R90, R180, R270, H, V, D, D’}. * I R90 R180 R270 H V D D’

I I R90 R180 R270 H V D D’

R90 R90 R180 R270 I D’ D H V

R180 R180 R270 I R90 V H D’ D

R270 R270 I R90 R180 D D’ V H

H H D V D’ I R180 R90 R270

V V D’ H D R180 I R270 R90

D D V D’ H R270 R90 I R180

D’ D’ H D V R90 R270 R180 I

Gretchen’s green square:

{I, R90, R180, R270, F1, F2, F3, F4}.

*

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

She should use this dictionary to convert as she copies!

Rene’s red square:

{I, R90, R180, R270, H, V, D, D’}. * I R90 R180 R270 H V D D’

I I R90 R180 R270 H V D D’

R90 R90 R180 R270 I D’ D H V

R180 R180 R270 I R90 V H D’ D

R270 R270 I R90 R180 D D’ V H

H H D V D’ I R180 R90 R270

V V D’ H D R180 I R270 R90

D D V D’ H R270 R90 I R180

D’ D’ H D V R90 R270 R180 I

Gretchen’s green square:

{I, R90, R180, R270, F1, F2, F3, F4}.

* I R90 R180 R270 F1 F3 F4 F2

I I R90 R180 R270 F1 F3 F4 F2

R90 R90 R180 R270 I F2 F4 F1 F3

R180 R180 R270 I R90 F3 F1 F2 F4

R270 R270 I R90 R180 F4 F2 F3 F1

F1 F1 F4 F3 F2 I R180 R90 R270

F3 F3 F2 F1 F4 R180 I R270 R90

F4 F4 F3 F2 F1 R270 R90 I R180

F2 F2 F1 F4 F3 R90 R270 R180 I

This creates a valid Cayley table for Gretchen (each cell is filled in correctly)!

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

* I R90 R180 R270 H V D D’

I I R90 R180 R270 H V D D’

R90 R90 R180 R270 I D’ D H V

R180 R180 R270 I R90 V H D’ D

R270 R270 I R90 R180 D D’ V H

H H D V D’ I R180 R90 R270

V V D’ H D R180 I R270 R90

D D V D’ H R270 R90 I R180

D’ D’ H D V R90 R270 R180 I

* I R90 R180 R270 F1 F3 F4 F2

I I R90 R180 R270 F1 F3 F4 F2

R90 R90 R180 R270 I F2 F4 F1 F3

R180 R180 R270 I R90 F3 F1 F2 F4

R270 R270 I R90 R180 F4 F2 F3 F1

F1 F1 F4 F3 F2 I R180 R90 R270

F3 F3 F2 F1 F4 R180 I R270 R90

F4 F4 F3 F2 F1 R270 R90 I R180

F2 F2 F1 F4 F3 R90 R270 R180 I

This dictionary converts every true red equation into a true green equation!

H*D = R90 F1*F4 = R90

convert

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

DEFINITION: An isomorphism between two groups means a one-to-one matching (dictionary)between their members that converts each true equation in one group into a true equationin the other.

This is the same as saying that it converts an entire Cayley table for one group into a validCayley table for the other.

We say two groups are isomorphic there exists an isomorphism between them.

Thus, the symmetry group of the red square is isomorphic to the symmetry group of the green square.

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

QUESTION: Do the star and the moth have isomorphic symmetry groups?

QUESTION: Do the star and the moth have isomorphic symmetry groups?

ANSWER: Yes! Here are their Cayley tables:

And here is the isomorphism:

star I R180

I I R180

R180 R180 I

moth I V

I I VV V I

I R180

↕ ↕I V

QUESTION: Do the star and the moth have isomorphic symmetry groups?

ANSWER: Yes! Here are their Cayley tables:

star I R180

I I R180

R180 R180 I

moth I V

I I VV V I

And here is the isomorphism:

(It doesn’t matter that a flip is geometrically different from a rotation.)

I R180

↕ ↕I V

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

YES! Here is an isomorphism:

… T–4 T–3 T–2 T–1 T0 T1 T2 T3 T4 …↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

… T–8 T–6 T–4 T–2 T0 T2 T4 T6 T8 …

Symmetries of G-pattern

Symmetries of P-pattern

“T” means translate the subscripted number of centimeters (right if positive, left if negative).

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

YES! Here is an isomorphism:

Symmetries of G-pattern

Symmetries of P-pattern

T5 * T8 = T13

↓ ↓ ↓T10 * T16 = T26

Watch this dictionary turn a true greenequation into a true purple equation:

… T–4 T–3 T–2 T–1 T0 T1 T2 T3 T4 …↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

… T–8 T–6 T–4 T–2 T0 T2 T4 T6 T8 …

G G G G G G G G G G G G G G G G G G G G GQ: Are these two groups isomorphic:

G = the symmetry group of the G-border pattern above

Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

G G G G G G G G G G G G G G G G G G G G GQ: Are these two groups isomorphic:

G = the symmetry group of the G-border pattern above

Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

YES! Here is an isomorphism:

… T–4 T–3 T–2 T–1 T0 T1 T2 T3 T4 …↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

… -4 -3 -2 -1 0 1 2 3 4 …

Symmetries of G-pattern

The integers

G G G G G G G G G G G G G G G G G G G G GQ: Are these two groups isomorphic:

G = the symmetry group of the G-border pattern above

Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

YES! Here is an isomorphism:

… T–4 T–3 T–2 T–1 T0 T1 T2 T3 T4 …↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

… -4 -3 -2 -1 0 1 2 3 4 …

Symmetries of G-pattern

The integers

Watch this dictionary turn a true greenequation into a true purple equation:

T5 * T8 = T13

↓ ↓ ↓ 5 + 8 = 13

This “slide-and-tilt” is a rigid motion of the plane.

We used it to create our isomorphism between their symmetry groups.I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

This “slide-and-tilt” is a rigid motion of the plane.

We used it to create our isomorphism between their symmetry groups.

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

This “slide-and-tilt” is a rigid motion of the plane.

We used it to create our isomorphism between their symmetry groups.

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

This theorem is the real reason that the red and green squares have isomorphic symmetry groups.

I R90 R180 R270 H D’ V D↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕I R90 R180 R270 F1 F2 F3 F4

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two stars rigidlyequivalent?

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two stars rigidlyequivalent?

YES!(so their symmetry groupsmust be isomorphic)

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are the star and the moth rigidlyequivalent?

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are the star and the moth rigidlyequivalent?

NO!(but their symmetry groups are isomorphic anyways)

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two border patterns rigidly equivalent?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

DEFINITION: Two objects in the plane are called rigidly equivalent if there exists a rigid motion of the plane which, when applied to the first object, repositions it so that afterwards the two objects have exactly the same symmetries.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two border patterns rigidly equivalent?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

NO! But each is rigidly equivalent to a rescaling (enlarging/shrinking) of the other.

Q: Are these two groups isomorphic:

D4 = The symmetry group of a square.

C5 = The symmetry group of an oriented 5-gon (or star).

Q: Are these two groups isomorphic:

D4 = The symmetry group of a square.

C5 = The symmetry group of an oriented 5-gon (or star).

NO! They have different sizes.

Q: Are these two groups isomorphic:

D4 = The symmetry group of a square.

C8 = The symmetry group of an oriented 8-gon (or star).

Q: Are these two groups isomorphic:

D4 = The symmetry group of a square.

C8 = The symmetry group of an oriented 8-gon (or star).

NO! Because only C8 is commutative.

Q: Are these two groups isomorphic:

D2 = The symmetry group of a rectangle.

C4 = The symmetry group of an oriented 4-gon (or star).

Q: Are these two groups isomorphic:

D2 = The symmetry group of a rectangle.

C4 = The symmetry group of an oriented 4-gon (or star).

NO! Think about why no matching could work.

REVIEW: Our eventual goal is toclassify all of the ways in which… (1) bounded objects (2) border patterns (3) wallpaper patterns

…can be symmetric. Now we can make this question precise...What should it mean to say that two objects/patternsare symmetric in the same way?

REVIEW: Our eventual goal is toclassify all of the ways in which… (1) bounded objects (2) border patterns (3) wallpaper patterns

…can be symmetric. Now we can make this question precise...What should it mean to say that two objects/patternsare symmetric in the same way?

(A) That they are rigidly equivalent.(B) That they have isomorphic symmetry groups.

Either choice is good, and we’ll discuss other possibilities too.

REVIEW: Our eventual goal is toclassify all of the ways in which… (1) bounded objects (2) border patterns (3) wallpaper patterns

…can be symmetric.

So a typical classification theorem might look like this:

• “Any bounded object is rigidly equivalent to one of these model objects…”

• “The symmetry group of any border pattern is isomorphic to the symmetry group of one of these model patterns...”

Or like this:

Two notation systems for C5 = this star’s five rotation symmetries

1 2C5 = {I, R72, R144, R216, R288} C5 = {0, 1, 2, 3, 4}

How many “turns” How many degrees

Two notation systems for C5 = this star’s five rotation symmetries

1 2C5 = {I, R72, R144, R216, R288} C5 = {0, 1, 2, 3, 4}

How many degrees How many “turns”

* I R72 R144 R216 R288 I I R72 R144 R216 R288

R72 R72 R144 R216 R288 I R144 R144 R216 R288 I R72 R216 R216 R288 I R72 R144 R288 R288 I R72 R144 R216

+ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3

Which system do you prefer?

Two notation systems for C5 = this star’s five rotation symmetries

1 2C5 = {I, R72, R144, R216, R288} C5 = {0, 1, 2, 3, 4}

How many degrees How many “turns”

* I R72 R144 R216 R288 I I R72 R144 R216 R288

R72 R72 R144 R216 R288 I R144 R144 R216 R288 I R72 R216 R216 R288 I R72 R144 R288 R288 I R72 R144 R216

+ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3

R216* R288 = R144 3 + 4 = 2 (in C5)

Two notation systems for C5 = this star’s five rotation symmetries

1 2C5 = {I, R72, R144, R216, R288} C5 = {0, 1, 2, 3, 4}

How many degrees How many “turns”

* I R72 R144 R216 R288 I I R72 R144 R216 R288

R72 R72 R144 R216 R288 I R144 R144 R216 R288 I R72 R216 R216 R288 I R72 R144 R288 R288 I R72 R144 R216

+ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3

R216* R288 = R144 3 + 4 = 2 (in C5)

Convention: We’ll use this simpler notation system from now on.

The members of Cn will be denoted:

Cn = {0, 1, 2, 3, …, n-1}.

We’ll write “+” for the algebraic operation in Cn.

We’ll think of this as “addition with wrap-around”.

* R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7)

R0 R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7)

R1(360/7) R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0

R2(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7)

R3(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7)

R4(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7)

R5(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7)

R6(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7)

This new system is much better forC7 = the symmetry group of an oriented seven-pointed star.

* 0 1 2 3 4 5 60 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

R2(360/7) * R3(360/7) = R5(360/7) 2 + 3 = 5 (in C7)

OLDNEW

* R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7)

R0 R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7)

R1(360/7) R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0

R2(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7)

R3(360/7) R3(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7)

R4(360/7) R4(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7)

R5(360/7) R5(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7)

R6(360/7) R6(360/7) R0 R1(360/7) R2(360/7) R3(360/7) R4(360/7) R5(360/7)

This new system is much better forC7 = the symmetry group of an oriented seven-pointed star.

* 0 1 2 3 4 5 60 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

R2(360/7) * R3(360/7) = R5(360/7) 2 + 3 = 5 (in C7)

OLDNEW

When a pair of groups is isomorphic, it is often best to think of them as a single group described in two different notation systems.

Vocabulary Review

IsomorphismIsomorphic

Rigidly equivalenta + b = c (in Cn)

Theorem Review“If two objects are rigidly equivalent

then they have isomorphic symmetry groups.”