Astrid Ludwig

What you gave here is a specific example of a group. It illustrates the key properties of groups, but I was also looking for a bit more detail about what those properties are. Recall that we said a group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing that your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Peter Zona

What you gave here is a specific example of a group. It illustrates the key properties of groups, but I was also looking for a bit more detail about what those properties are. Recall that we said a group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing that your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Kylee Sullivan

The main thing that your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Lucia Bostrom

Most of the essay is very good. The only thing that your essay did not describe very clearly is the idea of what a group is. Recall that we said a group is a collection of numbers or transformations, or ... with an operation, and the operation has to satisfy several properties (see class notes handout).

Also, as you said, it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Catherine Caracciolo

The first thing that your essay did not describe very clearly is the idea of what a group is. Recall that we said a group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

Also, your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Matt Kasuba

What you gave here is some specific examples of how a particular collection of rotations with each other. They illustrate the key properties of groups, but I was also looking for a bit more detail about what those properties are. Recall that we said a group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing that your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type just have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Andrew Marzo

You certainly have almost all of the right elements I was looking for here. The reasons I took off the two points are that there seems to be a pair of slight misunderstandings here, if I’m “reading you” correctly.

First, it is not really correct to say that the mathematical concept of a group is the algebraic generalization of “a symmetry transformation.” The group properties can be thought of a a generalization of the properties of the collection of all transformations that leave an object or pattern invariant. It’s the properties of a collection of transformations that are important here, not just the properties of a single transformation.

The second thing that your essay did not describe very clearly is the idea that it is possible to classify the periodic plane patterns by saying two patterns have the same type if their groups of symmetries are “the same” (made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different, though.

Kyle Kadlick

The last property in the definition of a group is the requirement that every element has an inverse – property 4 below:

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing was that I was not quite sure what you meant by “degrees of the transformation” toward the end of your essay. Two patterns of the same type have “the same” symmetries in the sense that their groups of symmetries have the same structure and are made up of the same types of transformations. That means in particular that every p4 pattern, for instance, has rotational symmetries only for 0, 90, 180, 270-degree rotations. The motifs or shapes that make up the pattern, the colors, and so forth can be different.

Xuan Ha Nghiem

The property in the definition of a group you did not mention is the requirement that a group must have an identity element – property 3 below:

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing was that I was not quite sure what you meant when you said the “pattern might hinder a transformation in one but not in the other.” If two patterns are of the same type, then the transformations in their groups of symmetries are “the same.” If one pattern has a 90-degree rotation symmetry about a point, then there has to be a corresponding point in the other pattern around which the other also has 90-degree rotational symmetry.

Kaitlin O’Hagan

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

You had a lot of this, but I was looking for a bit more detail. Also, notice that commutativity (the statement that x ∗ y = y ∗ x for all x, y) is not included. In fact, the groups of symmetries of 16 out of the 17 types of plane period patterns are not commutative. Translations (shifts) do not commute with rotations and reflections.

Max Pettinelli

This is mostly very good. However, I was looking for some more detail in the discussion of the idea of a group.

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing is that your essay did not come out clearly and state the basic idea of classification the periodic plane patterns (although you were certainly saying parts of this, so that’s why I only took off one point here). Remember we said that the two patterns are put into the same type if their groups of symmetries are “the same” (having the same structure as groups and made up of the same types of transformations). This is where the classification into 17 types comes from. Two patterns of the same type have “the same” symmetries. The motifs or shapes that make up the pattern, the colors, and so forth can be different, though.

Kevin Sutman

What you have in the essay is good, but there are two particular things that you did not discuss in enough detail. First, part of the question asked about particular types of symmetry transformations that we looked at. I was looking for the types of transformations we applied to the patterns from the Alhambra or the Escher drawings to see the invariance – translations (shifts), rotations, reflections, glide reflections. Since you brought up the musical analogies, you might have mentioned transposition, inversion, etc. there.

The second thing that your essay did not describe very clearly is the idea of what a group is. Recall that we said a group is a collection G of numbers or transformations, or ... with an operation ∗. To give a group, the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

John Macomber

This is mostly very good (although your handwriting is very hard to read!). However, I was looking for some more detail in the discussion of the idea of a group.

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

Thomas Santa Maria

This is mostly good. The deductions were based on the following other things I was looking for.

First, I was looking for more detail about the mathematical definition of what a group is. Recall we said:

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing is your essay did not come out clearly and state the basic idea of classification the periodic plane patterns (although you were certainly saying parts of this, so that’s why I only took off one point here). Remember we said that the two patterns are put into the same type if their groups of symmetries are “the same” (having the same structure as groups and made up of the same types of transformations). So there is no “probably” about it – this is actually where the classification into 17 types comes from. Two patterns of the same type have “the same” group of symmetries (same algebraic structure, same types of symmetry transformations). The motifs or shapes that make up the pattern, the colors, and so forth can be different, though.

Muhammad Javaid

This is not bad. The deductions were based on the following other things I was looking for.

First, I was looking for more detail about the mathematical definition of exactly what a group is. Recall we said:

A group is a collection G of numbers or transformations, or ... with an operation ∗, and the operation has to satisfy several properties:

1) For all x, y in G, x ∗ y is in G.

2) For all x, y, z in G, (x ∗ y) ∗ z = x ∗ (y ∗ z) (associativity)

3) There is an identity element e in G that satisfies e ∗ x = x ∗ e = x for all x in G.

4) For each x in G there is an inverse element y in G that satisfies x ∗ y = y ∗ x = e.

(See class notes handout.)

The other thing was that I was looking for more of a discussion of what types of transformations we looked at as symmetries of the Escher drawings or Alhambra tilings. The types of transformations (hence the types of symmetries we found) were based on translations (shifts), rotations, reflections, and also glide reflections.