Fun introduction to symmetry and a common geometry lies here. Parallelograms also include rhombuses when sides are equal, rectangle when angles are equal, and square when both sides and angles are equal. Once you see the rotational symmetry of a parallelogram, you can see the rotational symmetry of all 4 four-forms much simpler.
How To find Rotational Symmetry of Any shape
A shape has rotational symmetry if you can rotate it by some amount and it will be identical to its original self. More precisely, the order of rotational symmetry enumerates the total ways an object can be rotated to match itself. You can see rotational symmetry in mandalas here.
For parallelogram you might think the rotational symmetry would be the same as a square, but the fact that the angles are not equal changes things.
Rotational Symmetry of a Parallelogram
The parallelogram, when rotated 360 degrees, only matches its original form once, at the 180 degrees mark. This is the least amount of symmetry something can have.
The order of symmetry of a parallelogram is 2. There are two degrees of rotation that will result in isomorphism (matching the original form).
Applying to Rhombus, Rectangle, and Square
The rhombus is a parallelogram with equal sides. Well, the sides will match after 90 degrees but not the angles as consecutive angles are not equal. The rhombus has rotational symmetry of 180 degrees, order 2.
For a rectangle we experience the same but the angles match and the lengths do not. Rectangles as well have order 2 symmetry, under 180 degrees rotation.
For square, we have a regular polygon. Regular polygons you can be sure always have rotational symmetry. In fact the rotational symmetry of regular polygons has an order equal to the number of sides / angles. So a regular pentagon has rotational symmetry of order 5, for example. Read about why pentagons are cool.
So for a square the angle of rotational symmetry is 90 degrees, and also 180 degrees, and 270 degrees. The order of symmetry of a square is 4 then.
Why is rotational symmetry important
In math, if there is rotational symmetry determined to exist in the problem, then usually matters simplify greatly. In physics, space is generally figured to be isotropic, the same in all directions, so many physical laws require symmetry under rotations. So whether or not a system has rotational symmetry can help support the application to various models.
“In addition to being a well-studied concept mathematically, rotational symmetry is also a far-reaching notion due to the prevalence of such symmetry among many naturally-occurring objects including snowflakes, crystals, and flowers.”
WolframAlpha’s take on the importance of rotational symmetry
WolframAlpha’s take on the importance of rotational symmetry
Later we will do rotational symmetry in 3D forms like tetrahedron. We will also talk about symmetry isometric laws in physics and rotational symmetry in fractals so follow me and check back. And here you can read more about FOUR-ness.
