An irregular octagon R has vertices 6, 2), (2, 6), (−1, 5), (−5, 1), (−6, −2),
(−2, −6), (1, −5) and (5, −1).
Using standard notation, write down the elements of the symmetry group S(R) of R, giving a brief description of the
geometric effect of each symmetry on points in the plane.
To find the elements of the symmetry group S(R) of the irregular octagon R, we need to identify all the transformations (symmetries) of R that preserve its shape and position in the plane. These transformations include rotations, reflections, and combinations of these.
1. Identity Transformation (E):
This is the "do nothing" transformation where every point stays in its original position.
2. Rotations:
- Rotation of 45 degrees (R1): This rotation takes each point of R and rotates it 45 degrees counterclockwise around the origin.
- Rotation of 90 degrees (R2): This rotation takes each point of R and rotates it 90 degrees counterclockwise around the origin.
- Rotation of 135 degrees (R3): This rotation takes each point of R and rotates it 135 degrees counterclockwise around the origin.
- Rotation of 180 degrees (R4): This rotation takes each point of R and rotates it 180 degrees counterclockwise around the origin.
- Rotation of 225 degrees (R5): This rotation takes each point of R and rotates it 225 degrees counterclockwise around the origin.
- Rotation of 270 degrees (R6): This rotation takes each point of R and rotates it 270 degrees counterclockwise around the origin.
- Rotation of 315 degrees (R7): This rotation takes each point of R and rotates it 315 degrees counterclockwise around the origin.
3. Reflections:
- Horizontal Reflection (H): This reflection reflects R across the x-axis.
- Vertical Reflection (V): This reflection reflects R across the y-axis.
- Diagonal Reflection (D): This reflection reflects R across the line that passes through the origin and has a slope of -1.
- Antidiagonal Reflection (A): This reflection reflects R across the line that passes through the origin and has a slope of 1.
Note: As we have an irregular octagon, it does not have any rotational symmetry less than 45 degrees or any reflectional symmetry other than those mentioned above.
So, the elements of the symmetry group S(R) are:
S(R) = {E, R1, R2, R3, R4, R5, R6, R7, H, V, D, A}
Each symmetry in S(R) has a specific effect on the points in the plane. For rotations, each point in R is rotated around the origin by the specified angle. For reflections, each point is reflected across the specified axis or line. Both rotations and reflections preserve the shape and position of R.