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

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

(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, R 90, R 180, R 270, H, V, D, D’}. *IR 90 R 180 R 270 HVDD’ IIR 90 R 180 R 270 HVDD’ R 90 R 180 R 270 ID’DHV R 180 R 270 IR 90 VHD’D R 270 IR 90 R 180 DD’VH HHDV IR 180 R 90 R 270 VVD’HDR 180 IR 270 R 90 DDVD’HR 270 R 90 IR 180 D’ HDVR 90 R 270 R 180 I (Same Cayley table we built in Ch. 2)

Rene’s red square: {I, R 90, R 180, R 270, H, V, D, D’}. *IR 90 R 180 R 270 HVDD’ IIR 90 R 180 R 270 HVDD’ R 90 R 180 R 270 ID’DHV R 180 R 270 IR 90 VHD’D R 270 IR 90 R 180 DD’VH HHDV IR 180 R 90 R 270 VVD’HDR 180 IR 270 R 90 DDVD’HR 270 R 90 IR 180 D’ HDVR 90 R 270 R 180 I How can Gretchen copy Rene’s work to build her Cayley table? Gretchen’s green square: {I, R 90, R 180, R 270, F 1, F 2, F 3, F 4 }. *

Rene’s red square: {I, R 90, R 180, R 270, H, V, D, D’}. *IR 90 R 180 R 270 HVDD’ IIR 90 R 180 R 270 HVDD’ R 90 R 180 R 270 ID’DHV R 180 R 270 IR 90 VHD’D R 270 IR 90 R 180 DD’VH HHDV IR 180 R 90 R 270 VVD’HDR 180 IR 270 R 90 DDVD’HR 270 R 90 IR 180 D’ HDVR 90 R 270 R 180 I Gretchen’s green square: {I, R 90, R 180, R 270, F 1, F 2, F 3, F 4 }. * IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4 She should use this dictionary to convert as she copies!

Rene’s red square: {I, R 90, R 180, R 270, H, V, D, D’}. *IR 90 R 180 R 270 HVDD’ IIR 90 R 180 R 270 HVDD’ R 90 R 180 R 270 ID’DHV R 180 R 270 IR 90 VHD’D R 270 IR 90 R 180 DD’VH HHDV IR 180 R 90 R 270 VVD’HDR 180 IR 270 R 90 DDVD’HR 270 R 90 IR 180 D’ HDVR 90 R 270 R 180 I Gretchen’s green square: {I, R 90, R 180, R 270, F 1, F 2, F 3, F 4 }. *IR 90 R 180 R 270 F1F1 F3F3 F4F4 F2F2 IIR 90 R 180 R 270 F1F1 F3F3 F4F4 F2F2 R 90 R 180 R 270 IF2F2 F4F4 F1F1 F3F3 R 180 R 270 IR 90 F3F3 F1F1 F2F2 F4F4 R 270 IR 90 R 180 F4F4 F2F2 F3F3 F1F1 F1F1 F1F1 F4F4 F3F3 F2F2 I R 90 R 270 F3F3 F3F3 F2F2 F1F1 F4F4 R 180 IR 270 R 90 F4F4 F4F4 F3F3 F2F2 F1F1 R 270 R 90 IR 180 F2F2 F2F2 F1F1 F4F4 F3F3 R 90 R 270 R 180 I This creates a valid Cayley table for Gretchen (each cell is filled in correctly)! IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

*IR 90 R 180 R 270 HVDD’ IIR 90 R 180 R 270 HVDD’ R 90 R 180 R 270 ID’DHV R 180 R 270 IR 90 VHD’D R 270 IR 90 R 180 DD’VH HHDV IR 180 R 90 R 270 VVD’HDR 180 IR 270 R 90 DDVD’HR 270 R 90 IR 180 D’ HDVR 90 R 270 R 180 I *IR 90 R 180 R 270 F1F1 F3F3 F4F4 F2F2 IIR 90 R 180 R 270 F1F1 F3F3 F4F4 F2F2 R 90 R 180 R 270 IF2F2 F4F4 F1F1 F3F3 R 180 R 270 IR 90 F3F3 F1F1 F2F2 F4F4 R 270 IR 90 R 180 F4F4 F2F2 F3F3 F1F1 F1F1 F1F1 F4F4 F3F3 F2F2 I R 90 R 270 F3F3 F3F3 F2F2 F1F1 F4F4 R 180 IR 270 R 90 F4F4 F4F4 F3F3 F2F2 F1F1 R 270 R 90 IR 180 F2F2 F2F2 F1F1 F4F4 F3F3 R 90 R 270 R 180 I This dictionary converts every true red equation into a true green equation! H*D = R 90 F 1 *F 4 = R 90 convert IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

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 equation in the other. This is the same as saying that it converts an entire Cayley table for one group into a valid Cayley table for the other. We say two groups are isomorphic there exists an isomorphism between them. 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 equation in the other. This is the same as saying that it converts an entire Cayley table for one group into a valid Cayley 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. IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

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

ANSWER: Yes! Here are their Cayley tables: And here is the isomorphism: starIR 180 II I mothIV IIV VVI IR 180 ↕↕ IV

QUESTION: Do the star and the moth have isomorphic symmetry groups? ANSWER: Yes! Here are their Cayley tables: starIR 180 II I mothIV IIV VVI And here is the isomorphism: (It doesn’t matter that a flip is geometrically different from a rotation.) IR 180 ↕↕ IV

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 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 1 cm between Gs 2 cm between Ps YES! Here is an isomorphism: …T –4 T –3 T –2 T –1 T0T0 T1T1 T2T2 T3T3 T4T4 … ↕ ↕↕↕↕↕↕↕↕ …T –8 T –6 T –4 T –2 T0T0 T2T2 T4T4 T6T6 T8T8 … 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 1 cm between Gs 2 cm between Ps YES! Here is an isomorphism: Symmetries of G-pattern Symmetries of P-pattern T 5 * T 8 = T 13 ↓ ↓ ↓ T 10 * T 16 = T 26 Watch this dictionary turn a true green equation into a true purple equation: …T –4 T –3 T –2 T –1 T0T0 T1T1 T2T2 T3T3 T4T4 … ↕ ↕↕↕↕↕↕↕↕ …T –8 T –6 T –4 T –2 T0T0 T2T2 T4T4 T6T6 T8T8 …

G G G G G G G G G G G G G G G G G G G G G Q: 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 G Q: 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 T0T0 T1T1 T2T2 T3T3 T4T4 … ↕ ↕↕↕↕↕↕↕↕ … … 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 G Q: 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 T0T0 T1T1 T2T2 T3T3 T4T4 … ↕ ↕↕↕↕↕↕↕↕ … … Symmetries of G-pattern The integers Watch this dictionary turn a true green equation into a true purple equation: T 5 * T 8 = T 13 ↓ ↓ ↓ = 13

This “slide-and-tilt” is a rigid motion of the plane. We used it to create our isomorphism between their symmetry groups. IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

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. IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

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. IR 90 R 180 R 270 HD’VD ↕↕↕↕↕↕↕↕ IR 90 R 180 R 270 F1F1 F2F2 F3F3 F4F4

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 rigidly equivalent? Q: Are these two stars rigidly equivalent?

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 rigidly equivalent? Q: Are these two stars rigidly equivalent? YES! (so their symmetry groups must 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 rigidly equivalent? Q: Are the star and the moth rigidly equivalent?

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 rigidly equivalent? Q: Are the star and the moth rigidly equivalent? 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

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

Q: Are these two groups isomorphic: D 4 = The symmetry group of a square. C 5 = The symmetry group of an oriented 5-gon (or star).

Q: Are these two groups isomorphic: D 4 = The symmetry group of a square. C 5 = The symmetry group of an oriented 5-gon (or star). NO! They have different sizes.

Q: Are these two groups isomorphic: D 4 = The symmetry group of a square. C 8 = The symmetry group of an oriented 8-gon (or star).

Q: Are these two groups isomorphic: D 4 = The symmetry group of a square. C 8 = The symmetry group of an oriented 8-gon (or star). NO! Because only C 8 is commutative.

Q: Are these two groups isomorphic: D 2 = The symmetry group of a rectangle. C 4 = The symmetry group of an oriented 4-gon (or star).

Q: Are these two groups isomorphic: D 2 = The symmetry group of a rectangle. C 4 = The symmetry group of an oriented 4-gon (or star). NO! Think about why no matching could work.

REVIEW: Our eventual goal is to classify 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/patterns are symmetric in the same way? Now we can make this question precise... What should it mean to say that two objects/patterns are symmetric in the same way?

REVIEW: Our eventual goal is to classify 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/patterns are symmetric in the same way? Now we can make this question precise... What should it mean to say that two objects/patterns are 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 to classify 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 C 5 = this star’s five rotation symmetries C 5 = {I, R 72, R 144, R 216, R 288 } C 5 = {0, 1, 2, 3, 4} How many “turns”How many degrees

Two notation systems for C 5 = this star’s five rotation symmetries C 5 = {I, R 72, R 144, R 216, R 288 } C 5 = {0, 1, 2, 3, 4} How many degreesHow many “turns” Which system do you prefer?

Two notation systems for C 5 = this star’s five rotation symmetries C 5 = {I, R 72, R 144, R 216, R 288 } C 5 = {0, 1, 2, 3, 4} How many degreesHow many “turns” R 216 * R 288 = R = 2 (in C 5 )

Two notation systems for C 5 = this star’s five rotation symmetries C 5 = {I, R 72, R 144, R 216, R 288 } C 5 = {0, 1, 2, 3, 4} How many degreesHow many “turns” R 216 * R 288 = R = 2 (in C 5 ) Convention: We’ll use this simpler notation system from now on. The members of C n will be denoted: C n = {0, 1, 2, 3, …, n-1}. We’ll write “+” for the algebraic operation in C n. We’ll think of this as “addition with wrap-around”. Convention: We’ll use this simpler notation system from now on. The members of C n will be denoted: C n = {0, 1, 2, 3, …, n-1}. We’ll write “+” for the algebraic operation in C n. We’ll think of this as “addition with wrap-around”.

*R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) This new system is much better for C 7 = the symmetry group of an oriented seven-pointed star. * R 2(360/7) * R 3(360/7) = R 5(360/7) = 5 (in C 7 ) OLD NEW

*R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 3(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 4(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 5(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 6(360/7) R0R0 R 1(360/7) R 2(360/7) R 3(360/7) R 4(360/7) R 5(360/7) This new system is much better for C 7 = the symmetry group of an oriented seven-pointed star. * R 2(360/7) * R 3(360/7) = R 5(360/7) = 5 (in C 7 ) OLD NEW 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. 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 Isomorphism Isomorphic Rigidly equivalent a + b = c (in C n ) Isomorphism Isomorphic Rigidly equivalent a + b = c (in C n ) Theorem Review “If two objects are rigidly equivalent then they have isomorphic symmetry groups.” “If two objects are rigidly equivalent then they have isomorphic symmetry groups.”