Question

Consider a regular hexagon orientated so that two of its vertices lie on the \(x\)-axis. Find matrix representations of a rotation \(R\) through \(\pi / 6\) and a reflection \(m_{y}\) in the \(y\)-axis by determining their effects on vectors lying in the \(x y\)-plane. Show that a reflection \(m_{x}\) in the \(x\)-axis can be written as \(m_{x}=m_{y} R^{3}\) and that the \((12)\) elements of the symmetry group of the hexagon are given by \(R^{n}\) or \(R^{n} m_{y} .\) Using the representations of \(R\) and \(m_{y}\) as generators, find a two- dimensional representation of the symmetry group, \(C_{6}\), of the regular hexagon. Is it a faithful representation?

Short answer

Matrix representations are \( R = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \) and \( m_{y} = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \). The symmetry group \( C_{6} \) is faithfully represented. The reflection in the x-axis can be written as \( m_{y} R^3 \).

Step-by-step solution

Matrix Representation of Rotation R

A rotation through \( \frac{\pi}{6} \) can be represented by the matrix \[ R = \begin{pmatrix} \cos(\frac{\pi}{6}) & -\sin(\frac{\pi}{6}) \ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) \end{pmatrix} \]Calculate \( \cos( \frac{\pi}{6} ) = \frac{\sqrt{3}}{2} \) and \( \sin( \frac{\pi}{6} ) = \frac{1}{2} \).Thus, \[ R = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \]

Matrix Representation of Reflection my

A reflection in the \( y \)-axis can be represented by the matrix \[ m_{y} = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \]

Calculation of R^3

Compute the third power of the rotation matrix: \[ R^3 = \left( \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \right)^3 \]Using matrix multiplication, we find: \[ R^3 = \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} \]

Verification of mx

Using the relation \( m_x = m_y R^3 \), compute \[ m_{y} R^3 = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} \]This results in \[ m_{x} = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \]

Generators of Symmetry Group

The 12 elements of the symmetry group can be written using rotations and reflections as \( R^{n} \) or \( R^{n} m_{y} \) for \( n = 0, 1, 2, 3, 4, 5 \).

Two-Dimensional Representation

To represent the symmetry group \( C_{6} \), use matrices \( R \) and \( m_{y} \) as generators. For this representation:\[ R = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}, \quad m_{y} = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \]This representation is faithful because it maps each group element to a distinct matrix.

Key concepts

Hexagon Symmetry

A regular hexagon exhibits fascinating symmetrical properties. It has six sides of equal length and is symmetrical about multiple axes. When dealing with symmetry, we consider both rotations that move it around its center and reflections that flip it across a line.
For a regular hexagon:

• There are 6 rotational symmetries
• There are 6 reflectional symmetries

Rotation symmetries include rotating the hexagon by multiples of \(\frac{\frac{\tau}{6}}\) while reflections are done across lines through its vertices or midpoints of sides.
Symmetry elements form a group: the \(C_6\) symmetry group.

Matrix Representations

Matrix representations help us understand transformations in a structured way. For a 2D plane, rotations and reflections can be represented by 2x2 matrices. These matrices change a vector's position while preserving the shape of the object it represents.
In the case of a hexagon, we think about how each corner vector is transformed by a rotation or reflection matrix.
Some key points to remember:

• A rotation matrix rotates vectors by a specified angle.
• A reflection matrix flips vectors across a specified axis.

This structured approach makes it easier to apply and combine transformations.

Rotation Matrix

A rotation matrix is used to rotate points in the plane around the origin.
For a rotation by an angle \(\theta\), the matrix is: \[ R = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \] For example, for a rotation by \(\frac{\frac{\tau}}{6}\), \(\theta = \frac{\frac{\tau}}{6}\), we get: \[ R = \begin{pmatrix} \(\frac{\frac{\sqrt{3}}}{2}\) & -\(\frac{\frac{1}}{2}\) \ \(\frac{\frac{1}}{2}\) & \(\frac{\frac{\sqrt{3}}}{2}\) \end{pmatrix} \] This matrix, when applied to a vector, rotates it by \(\frac{\frac{\tau}}{6}\).
Rotations help in understanding cyclic patterns in symmetrical objects like the hexagon.

Reflection Matrix

Reflection matrices flip vectors over a chosen axis. For a reflection in the y-axis, the matrix is: \[ m_{y} = \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \] This turns any point from \( (x, y) \right( (-x, y) \).
Similarly, a reflection in the x-axis is: \[ m_{x} = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \] which transforms \( (x, y) \right( (x, -y) \). Reflections are mirror operations valuable in symmetry analysis.
Finding reflections using matrices help in combining multiple reflections to understand complex changes.