The Truth About Seeing Angle Of Rotational Symmetry

There are actually many different types of symmetry. The typical, butterfly-style symmetry is “bilateral” reflectional symmetry. Rotational symmetry emanates from the center. (In 2D shapes, there are four kinds of symmetry you can have: reflectional, rotational, translational, and by transflection.)

What is the Angle of Rotational Symmetry?

The angle of rotational symmetry (also called radial symmetry) is the angle you’d need to turn the figure by to achieve an identical figure. Imagine rotating the figure, and when the “after” matches the “before,” the angle you’ve turned through is an angle of rotational symmetry for that figure.

Reflectional Symmetry vs. Rotational Symmetry

Symmetry is a self-repeating quality. A butterfly has two-fold symmetry, because if you take a 360 degree circle around it, and cut it into two pieces, that would be at the 180 degree mark. But it isn’t the same as rotational symmetry. In rotating the butterfly 180, you will not get the same image, but a vertically flipped image. The butterfly has reflectional symmetry. If you hold a mirror to it on its axis of symmetry, you will see the rest of the butterfly reflected in the mirror.

In two-fold rotational symmetry, if you start at 0 degrees, and then rotate your perspective by 180 degrees, you’ll see the image repeat itself. This would be a true 180 degree angle of rotation. Rectangles (discussed below) have 180 degree angle of rotational symmetry.

Examples of Rotational Symmetry for Different shapes

To see the angle of rotational symmetry, let’s look at common shapes. There are, of course, ways to find the angle of rotational symmetry using the mathematical functions that draw the figure. But for many common shapes, you can tell right away by looking at it. If there are repeating units about the center of the figure, then dividing 360 degrees by the number of “lobes”, will tell you the angles of rotational symmetry.

Example: Circles

Circles are very symmetrical. If you can draw a circle around it and divide it into identical wedges, the angle of each wedge is an angle of rotational symmetry. I also think of these wedges as “lobes.” This is one way a circular design can be imbued with other numerical archetypes.

In the nature altar above, the wedges are not identical. Even so, the image has a nice symmetrical feel to it, showing how far this principle goes psychologically and aesthetically. The altar is closest to having six lobes, as there are 6 ferns and ivy leaves. But the pinecones and inner stones are giving 5, it’s simply not as symmetrical as it could be.

Rotational Symmetry in Squares, Rectangles & Parallelograms

Squares are the most symmetrical, like a circle with four lobes. Some other quadrilaterals have 2 fold symmetry, but that’s it.

The square has four-fold rotational symmetry, the rectangle and parallelogram only have 2-fold the lowest number possible. Notice rectangle has folding bilateral symmetry but the parallelogram does not.

Trapezoids will generally not have any rotational symmetry. The rhombus is no better than the paralellogram. The angle of rotational symmetry of the rhombus will be 180 degrees also, because the matching angles (acute to acute, for example) are 180 degrees apart.

Example: Triangles

Equilateral triangles have the highest rotational symmetry out of all the triangles, at 3 angles of rotational symmetry. The equilateral triangle’s angle of rotational symmetry is 360 degrees divided by 3, 120 degrees. Isosceles triangles have an angle of rotational symmetry of 180 degrees like the rectangles. Other triangles with all sides unequal have no symmetry.

Angles of Rotational Symmetry in Polygons

For five-fold symmetry, you’d take a circle around, for example, a pentagram, or a flower. 360 degrees divided into 5 equal parts is 72 degrees, so every 72 degrees you’d see the picture repeating itself.In a regular polygon you can just divide by the number of sides.

So in mandelas, or anything with identical slices from the center, the angle of rotational symmetry is 360 degrees divided by the number of identical wedges. Flower petals usually, like the pentagon, have an angle of rotation smmetry of 72 degrees. Actually, every edible fruit comes from a five-petaled flower. Read my article about five-ness.

Fractional Orders of Symmetry

The nature altar above has some 5-ness and some 6-ness (ignoring rest). It feels like a pentagon (angle of rotation 360/5=72) and a hexagon (angle of rotational symmetry 360/6=60). You can combine these angles to get somewhere in between. Some fractals (and spirals) have rotational symmetry that is in between numbers, like 5.5-fold, for example. I can let you know when that article drops.This is often the true character of rotational symmetry in nature.

Some images and pictures are more nuanced, but now you can see angles of rotational symmetry in your daily life. Here is a gallery of images found in nature and around cities of radial symmetry. Much of the art has integer-fold rotational symmetry, while in nature the more spiral patterns have fractional-fold symmetry. And now with this knowledge we can start talking about symmetry in 3D solids.