how can you determine whether a design has rotational symmetry? if a design has a rotational symmetry, how many can you find the angle of roation??
rotation matrix for rotation angle T
|cos T |-sin T|
|sin T | cos T|
That matrix, call it M
then M (column x|y) gives new x|y
for example if T = 90 degrees
| 0 |-1 | [x]
| 1 | 0 | [y]
gives
0 x - y = -y for new x value
and
1 x + 0 y = x for new y value
SO you can reverse that
new x = old x cos T - old y sin T
new y = old x sin T + old y cos T
If you have a bunch of old x and y values and their corresponding new values, you can solve for sin T and Cos T and therefore for T, the rotation angle.
To determine whether a design has rotational symmetry, you need to examine whether there are one or more lines or points in the design that act as a center of rotation, meaning that when you rotate the design around that line or point, it appears unchanged or repeats itself.
Here's how you can approach it:
1. Start by visualizing the design and identifying any lines or points around which you think the design may have rotational symmetry.
2. Choose a line or point as a potential center of rotation and mentally rotate the design around it. Observe if the design repeats itself in any way or if it appears unchanged after the rotation.
3. Repeat this process for other lines or points in the design until you have examined all possible options.
If you find that a design appears unchanged or repeats itself after a rotation of a certain angle, it means the design has rotational symmetry.
To determine the angle of rotation, you can measure the angle of the rotation needed to make the design appear unchanged or repeat. You can use a protractor or any measuring tool that can measure angles accurately. Note that the angle of rotation can be a fraction of a full rotation (e.g., 90 degrees, 180 degrees, or even smaller fractions).
It is possible for a design to have multiple angles of rotation if there are multiple lines or points of rotational symmetry. Each center of rotation will have its own unique angle of rotation.
Remember, the best way to confirm rotational symmetry and determine the angle(s) of rotation is to physically manipulate or visualize the design and observe its repeated patterns.