Beauties of Broken Symmetry

One simple kind of beauty is symmetry. But is perfect symmetry the most beautiful form?

Walking through fog or watching the steam from your breath on a cold morning reveals water vapor with much symmetry, meaning that it looks the same from any direction. Turn your heads and it still looks like a white cloud-like form. Some call this symmetry “perfect.” But when water vapor crystalizes into a snowflake, it loses some of the symmetry. The six-sided snowflake looks the same from any of six directions, but a gas looks the same from all sides. Yet because a flat (almost 2-dimensional) snowflake has less symmetry than a more 3-dimensional cloud, we strike a balance between a distinctive form and symmetry.

You can turn a snowflake any of six times and it looks the same. You can also look at a snowflake in a mirror and it looks the same; It has reflectional symmetry. Typically, one learns about mirror symmetry in elementary school, but nature often breaks this simple mirror symmetry, to create new kinds of symmetry. If we break the mirror symmetry of a snowflake, then we can see other patterns emerge. Although these patterns are not “perfectly symmetrical,” they still retain varying amounts of symmetry. 

So, while symmetry is associated with beauty, reducing symmetry reveals new kinds of beauty. A circle has more symmetry than a hexagon, because you can turn a circle any and all amounts and it still looks the same, but there’s only six turns of the hexagon that leave it looking the same. But despite having less symmetry than the circle, the snowflake, and other hexagonal structures, have new beauties and new functionalities than the circle. For example, the hexagonal honeycomb is a more efficient space filler than a lattice of circles. Hexagons link up in 2D to completely fill the space.

Likewise, a pentagon has less symmetry than a hexagon, but a pentagon opens new functionalities. If we reduce symmetries of a hexagon from a six-fold to a five-fold, that is transforming it from a hexagon to a pentagon, this adds new functional possibilities. Whereas a hexagon is very efficient at filling space in two dimensions (2D), like a flat honeycomb sheet, a cluster of pentagons leads to a shape, the dodecahedron that can fill space using three-dimensional (3D) symmetry. That is, the dodecahedrons can stack up to fill space.

In all these cases, the symmetrical nature of the structures means they can be built with very simple rules. You can connect all the edges of pentagons, to make a dodecahedron, in which all the edges and faces are the same and are joined with the same rules. Even if we reduce the symmetry of the dodecahedron by elongating one of each of the five sides (as in the enclosed gold pyritohedron), the structure results from relatively simple rules.

Similarly, the icosahedron, joins up triangles all with the same rules of connection, just repeated over and over. The simplicity of the rules means that we can build the structures with minimal instruction. For this reason, nature can and does build virus shells in the icosahedral and dodecahedral shapes. To build these virus shells requires minimal rules, which means minimal genetic code (DNA). Although an icosahedral virus shell has lots of symmetry, it lacks mirror symmetry because the proteins that make up each triangle unit of the shell turn in one direction.

Each of the triangles in this virus icosahedron is made of three proteins. The trio of proteins creates a shape that turns in one direction, so there is no mirror symmetry. That twist in the proteins breaks the mirror symmetry. But like any regular icosahedron, each corner joins five triangles, so there is a five-fold rotational symmetry. You can rotate the virus around a corner one fifth of a turn and end up with the same pattern (spin the virus around the little yellow 5 in the diagram)

You can play around with the enclosed model of an icosahedral virus to see the other symmetries. You can rotate the virus around the center of a triangle a third of a turn or two thirds of a turn, and you will see the same pattern as before (rotate around the yellow 3 as indicated in the diagram). Also, you can rotate the shape at the point in the middle of the side of any triangle (shown as a small yellow number 2 in the picture) and end up with the same pattern. So, there is a five-fold, three-fold, and a two-fold symmetry in this icosahedron, called a 5 3 2 symmetry.

An interesting feature of the triangles in the icosahedron is that you could also lay them flat to make a two-dimensional tiling, like a simple jigsaw puzzle on a table. Then you would see the same three-fold and two-fold symmetry, but now the five-fold symmetry becomes six-fold because you can fit in an extra triangle. We can call this a 6 3 2 symmetry.

 

This 6 3 2 symmetry is similar to a decorative pattern found in the Alhambra, a gorgeously patterned palace in Granada, Spain.

You can see this 6 3 2 symmetry too by using the transparent sheet with red outline and the colored Alhambra pattern shown above. To rotate, place the colored sheet on top of your cardboard box (or any material you can puncture). Place the transparent outline on top and align the patterns. Push a toothpick or small rod through the pre-made holes in the sheets and into the foamboard. This secures the sheets and serves as a rotation point.

Notice that this Alhambra pattern does not have mirror symmetry. The curves bend in a clockwise direction. If you look through the transparency on the other side, then the curves bend in a counterclockwise direction. This broken mirror symmetry brings out new beauty.

Breaking mirror symmetry, while retaining some rotational symmetry, provides many possibilities for creativity.  Here is a beautiful three-fold rotational symmetry made by simply drawing three squiggly lines rotated around a central point. Imagine you make a squiggly line (perhaps made from a pipe cleaner) and pin it at one end. Next, rotate it one third of a turn, draw the line, then rotate it another third turn, and so on. Here we see this done with three different squiggly lines at three points, creating a 3 3 3 symmetry (3 triads of shapes, in this case squiggly lines).

You might recognize this as the famous reptile tiling of M.C Escher. Repeating the pattern creates outlines of reptiles.

The pattern of points forms hexagons.

 

 

 

 

We made these hexagons so that you can match up the sides when you lay down one next to another to create a honeycomb pattern that also brings together the correct sides.

 

Because regular hexagons don’t link up in 3D, we cannot use this hexagon to create a 3D shape. That is, unless we transform the hexagon shape into a pentagon. We tried this and it was quite difficult to create the lizard to fit nicely onto a pentagon, but it did work. The easy part is turning the hexagon into a pentagon, by folding in one of the sides of a hexagon. Specifically, we folded in one of the sides of our rainbow hexagon to make a 3D pentagon that, in turn, fits together to make a dodecahedron. From here, we also can make an icosahedron.