Question

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??

Answer 1

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.