Different Types of Lines of Symmetry to be Aware of

Since learning about rotational symmetry, you may wonder if there’s even more different types of lines of symmetry. Besides the elementary school butterfly symmetry (which is is a reflectional symmetry), there is a more general way to describe “symmetry” – anything conserved under change.

Abstractly defined, symmetry is the property of something to stay constant in some way after transformation. Say you take the original image and apply a transformation, or an operation to the image to create a new image. The things that are identical about the old and new images, is the way that that image is symmetric under that operation. This is how symmetry is applied abstractly in math and theoretical physics.

Aside from math, symmetry is a prevalent part of critiquing and discussing artwork. Great artists knowingly and unknowingly invoke blends of different symmetries creating emotional effects. Then we also have the fact that symmetry is INESCAPABLE in nature, not only in plants but every single part of Earth and abroad. So let’s look at all the different types of lines symmetry (including ones without lines!).

1. Translational Symmetry: Gridlines

“Translation” (in geometry) is the transformation that changes the location. If you have a flat square and you slide it over, that’s translating it (however many units in whatever direction). If you take an image and translate your field of view over it, we can look for translational symmetry every time the screen shows an identical image.

Translational symmetry is spoken of in the context of repeating patterns. For a tiled floor using all the same square tiles, there is translational symmetry in up/down and left/right directions. A line of translational symmetry would usually be grid-like, cutting the picture into identical “puzzle” pieces. But a line of translational could be any shape in the plane that can be repeated with no gaps. For example, hexagonal tilings. A pattern can also have multiple grids offset to show the translational symmetry pattern.

A related concept is tessellation. Tessellation is used in artwork a lot to create unique patterning. M.C. Escher had some famous tessellation art and surrealists loved them for the sense of wavy calmness. Here is a collection of tessellation pictures.

2. Rotational Symmetry: Radial lines

Rotational symmetry is the congruence of an image to the original after rotation. For a five-pointed star, every 72 degrees of turn will give an identical image to the original. Since there are five angles we can do this for (72, 144, 216, 288, 360), the star has five-fold symmetry. Also, we can say “the star is symmetric under n*72 degree rotations”. I can use n=9 for example, for an equivalent of a 288 degree rotation, and the image would be identical.

A line of rotational symmetry (cutting the picture into identical parts), would be from the center and out radially, repeating the order of symmetry number of times. Here is my whole article about how to see rotational symmetry.

3. Reflectional Symmetry: Slice Lines

Reflectional symmetry is when the image stays constant under reflection. Many letters in English, like T, H, V, M, have lines of reflectional symmetry. Notice the difference between those, and a letter like S. Does S have reflectional symmetry?

(No, but S has rotational symmetry for 180 degrees.)

4. Dilation Symmetry: Telescoping Patterns

Dilation symmetry will be the most important type of symmetry for studying fractals. Another name for dilation symmetry is “self-similarity” and is the same property as fractality. When called “dilation symmetry” it just emphasizes more the fact of congruence under the transformation of dilation. Dilation is to make smaller or bigger with the same proportions. So dilation is like zooming in or zooming out. Can we zoom in or zoom our and find a congruent picture to the original? Prime example is broccoli.

Fractal zooms are strikingly hypnotizing because periodically the pattern repeats after sporadic periods of change and dissimilarity.

5. Helical Symmetry: Twirly Patterns

Helical symmetry: one of the rarest types. It’s only valid in 3D geometry and higher. For helical symmetry you can imagine a screw or spring that repeats identically when it is rotated about an axis. That’s why helical symmetry is only in 3D – because you have to have an axis in one dimension and then the rotation happens in a circle in the plane formed by the two perpendicular directions to the axis.

Some regular items have helical symmetry for their functions: springs, screws, slinkies. Then we also have helical molecule, including biomolecules like dna and collagen (triple helix). What’s neat about the line of symmetry for these shapes is, like a circle, they are always identical (as long as they are homogenous), so there are infinitely many lines and planes of symmetry.

Those are Not All the Different types of Lines of Symmetry

For 3D shapes you can also have planes of symmetry, and that extends to 4D and higher mathematical objects as well. Other types of symmetry are found from studying certain analysis to the descriptions of things mathematical objects There is a whole system for defining more subtle types of symmetry in any number of dimensions, and this is the work of Galois theory, with related fields of Abstract Algebra. This gives fun templates for mental structures.

These we can slowly introduce as we go over the platonic solids. Now that you see how symmetry can be applied in 2 dimensions in a number of ways, we can look at the platonic solids and probe for different analogous symmetries in 3D.