Find matrix representation for rotating/reflecting hexagon

In summary, the problem involves finding a matrix representation of the group of transformations (rotations plus reflections) that leaves a snowflake invariant. This group has 12 elements, with 6 rotations and 6 reflections. The group is non-Abelian, as the rotations do not commute with the reflections. To include all symmetries, the rotations must be multiplied by any reflection to get all 12 elements. The other reflections are products of a rotation and a reflection.
  • #1
c3po
2
0

Homework Statement


Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves the snowflake invariant. Is the group Abelian or non-Abelian?

Homework Equations


R = [cosθ sinθ]
[-sinθ cosθ]

Reflection so that x → -x:
[-1 0][x] = [-x]
[0 1][y] [y]

Reflection so that y → -y:
[1 0][x] = [x]
[0 -1][y] [-y]

Abelian: A.B = B.A

The Attempt at a Solution


I know that I will be rotating the regular hexagon by increments of 60° so that each time I rotate, the hexagon is invariant. I know that the regular hexagon has 2n = 12 different symmetries, so I will have 12 2x2 matrices in total. I began by constructing my rotated matrices (0°, 60°, 120°, 180°, 240°, and 300°). I know that I will have 6 axes of symmetry to reflect by . . . 3 that stretch from vertex to vertex and 3 that stretch from the midpoint on one side of the hexagon to the opposite.

I think I am finding that this group will be non-Abelian, but I am not sure. I have calculated the 6 matrices for the rotations but am unsure of what to do next. If I reflect each rotation so that the +x values become -x I get 6 matrices, but those are only for the symmetries along the axes from vertex to vertex. This cannot be correct since I am leaving out the symmetries that exist from midpoint to midpoint. How do I correct my approach so that it includes these symmetries?

I have a midterm exam in the class this material appears in tomorrow morning and I know for certain that a problem very similar to this one will be on my exam. I went to my professor for help and he explained the general method for solving it, but I need some extra guidance! I am stuck. Please help!
 
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  • #2
What would you need to rotate by to reflect parallel to the edges? Maybe 1/2 of the angle to get from vertex to vertex?
Reflecting each rotation is not the same as reflecting over a line of symmetry...you need to rotate back when you are done. In that way, the matrices corresponding to opposing vertices will be redundant, so you will only have 3 (as expected) for the vertex-vertex reflections.
 
  • #3
c3po said:

Homework Statement


Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves the snowflake invariant. Is the group Abelian or non-Abelian?

Homework Equations


R = [cosθ sinθ]
[-sinθ cosθ]

Reflection so that x → -x:
[-1 0][x] = [-x]
[0 1][y] [y]

Reflection so that y → -y:
[1 0][x] = [x]
[0 -1][y] [-y]

Abelian: A.B = B.A

The Attempt at a Solution


I know that I will be rotating the regular hexagon by increments of 60° so that each time I rotate, the hexagon is invariant. I know that the regular hexagon has 2n = 12 different symmetries, so I will have 12 2x2 matrices in total. I began by constructing my rotated matrices (0°, 60°, 120°, 180°, 240°, and 300°). I know that I will have 6 axes of symmetry to reflect by . . . 3 that stretch from vertex to vertex and 3 that stretch from the midpoint on one side of the hexagon to the opposite.

I think I am finding that this group will be non-Abelian, but I am not sure. I have calculated the 6 matrices for the rotations but am unsure of what to do next. If I reflect each rotation so that the +x values become -x I get 6 matrices, but those are only for the symmetries along the axes from vertex to vertex. This cannot be correct since I am leaving out the symmetries that exist from midpoint to midpoint. How do I correct my approach so that it includes these symmetries?

I have a midterm exam in the class this material appears in tomorrow morning and I know for certain that a problem very similar to this one will be on my exam. I went to my professor for help and he explained the general method for solving it, but I need some extra guidance! I am stuck. Please help!

I'm not exactly sure what you are asking. But you say you know there are 12 elements in the group. The 6 rotations form an abelian subgroup. If you multiply those by ANY reflection ##r## then you will get all 12 group elements. The other reflections are products of a rotation and ##r##. To determine if the group is abelian just check if the rotations commute with ##r##.
 

1. What is a matrix representation for rotating/reflecting a hexagon?

A matrix representation for rotating/reflecting a hexagon is a mathematical way of representing the transformation of a hexagon. It involves creating a matrix that contains the coordinates of the vertices of the original hexagon and applying a rotation or reflection transformation to those coordinates.

2. How is a matrix representation for rotating/reflecting a hexagon calculated?

A matrix representation for rotating/reflecting a hexagon is calculated by multiplying the coordinates of each vertex of the original hexagon by a transformation matrix. The transformation matrix is determined by the type of rotation or reflection being applied and the angle of rotation or axis of reflection.

3. What are the steps for finding a matrix representation for rotating/reflecting a hexagon?

The steps for finding a matrix representation for rotating/reflecting a hexagon are:
1. Determine the type of transformation (rotation or reflection) and the angle or axis of rotation/reflection.
2. Write out the coordinates of the original hexagon in a matrix form.
3. Create a transformation matrix based on the type of transformation and angle/axis.
4. Multiply the original hexagon matrix by the transformation matrix.
5. The resulting matrix will be the matrix representation for the rotated/reflected hexagon.

4. How do you use a matrix representation for rotating/reflecting a hexagon?

To use a matrix representation for rotating/reflecting a hexagon, you can multiply the coordinates of the original hexagon by the transformation matrix to get the coordinates of the rotated/reflected hexagon. These new coordinates can then be plotted to visualize the transformation.

5. Can a matrix representation for rotating/reflecting a hexagon be used for any size or shape of hexagon?

Yes, a matrix representation for rotating/reflecting a hexagon can be used for any size or shape of hexagon as long as the coordinates of the vertices are known and the transformation matrix is calculated correctly.

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