From flowers to crystals, viruses, and atomic particles, nature abounds with broken symmetries. With your box or course on broken symmetries, you can play with floral shapes, making 3D models with broken symmetry, the Trillium flower, and the pentagonal Mountain Laurel. The Mountain Laurel often forms clusters that sometimes suggests little dodecahedrons.

Consider the pyritohedral crystal of iron pyrite or a model of the same pyritohedron (paper model enclosed in box, and as a download in the course). The pyritohedron is a dodecahedron with less symmetry than the regular dodecahedron. Fascinatingly, the pyritohedron still possesses much symmetry, each of the twelve faces is the same, but one of the edges of each face is elongated. Another dodecahedron with reduced symmetry (compared to the regular dodecahedron) is seen in the enclosed garnet crystal, the twelve faces are the same as each other, but the faces are rhombic (four equal sides).

You can experiment with our method for producing beautiful tiles that fit together perfectly, yet do not possess mirror symmetry. We illustrate this specifically with our Trillium flower exercise. More generally, with the method of cutting hexagons, you can create countless other tessellations (tiles linked together like a jigsaw puzzle).

We also will see a similarity between the symmetries of certain viruses with the patterns adorning the walls of the Alhambra, a gorgeous palace in Granada.

See how reducing symmetry allows us to tile lizards in 3D, in dodecahedral and icosahedral patterns lacking mirror symmetry. You will see how reducing symmetry opens new practical possibilities and new beauty.

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. Here is a model of the pyritohedron.  

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). Here is an online model of that viral icosahedron.  Here is a model of the underlying geometry of the icosahedron. 

You can play around with the model of an icosahedral virus (included in your box, if you subscribed) 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.

Here is our online model of the same Alhambra tiling pattern, with which you can likewise rotate the outline and see the underlying symmetries.

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.

Here’s a playful online model of the lizard tesselation.

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.

Here is our animated model of the tiles fitting together.

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 (here is a model of the dodecahedron). From here, we also can make an icosahedron. Here is our online model of how to make square tilings. Here is a model that gives samples of the kinds of square-based tiling you can make and a model serving as an instructional demo. 

Below is a video introduction to the course on tiling, 3D symmetries and broken symmetries. This visually reviews material presented above. (Remember that with subscriptions of 6 months or more you have free access to the entire set of video lectures and downloads). 

Below is a video focusing on 3D tiling from the 2D shapes. 

Discussion Questions

To answer the following questions, try experimenting with actual cut shapes.  If you are having difficulty, don’t worry! Simply review the video lecture and material above. Also, note that we include a glossary at the end of this page.

Which of the following can you tile together on a flat table, so that no spaces are left in between the shapes? 

    triangle, square, pentagon, hexagon, circle  (hint: Some of those flat shapes, polygons, do not tile together in 2D)

Give an example of translational symmetry?  

Give an example of rotational symmetry?  

Which of the following can you connect together to make a 3D shape?

  triangle, square, pentagon, hexagon, circle  

What kind of symmetry does a pentagon have?

What kind of symmetry does a dodecahedron have?

Is a dodecahedron a polygon or polyhedron?

Does the pyritohedron have more symmetry or less symmetry than the dodecahedron?

What kinds of symmetries or broken symmetries are most beautiful to you?

Glossary

Polygon – a polygon is flat shape with straight edges. In more technical terms, a polygon is a plane figure described by straight line segments connected to form a closed chain (or polygonal circuit). The segments of a polygonal circuit are called its edges or sides. The points where two edges meet are the polygon’s vertices (singular: vertex) or corners.

Polyhedron – a polyhedron (the plural is polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The regular polyhedra are made up of connected flat polygons.

Dodecahedron – In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. The plural is dodecahedra (or, sometimes dodecahedrons). Some dodecahedra have the same basic structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry.

Pyritohedron – Like the regular dodecahedron, the pyritohedron has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular. One side of the pentagon is longer than the rest.

Rhombic Dodecahedron – The rhombic dodecahedron is similar to the pyritohedron and dodechedron in structure but the faces are diamond shaped (rhombus shaped). It has 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types.

Symmetry – in everyday language, symmetry refers to a sense of harmonious and beautiful proportion and balance. In mathematics and geometry, “symmetry” refers to an object that is unchanging in shape when you turn it or flip it–invariant under some transformations such as translation, reflection, rotation or scaling. In other words, symmetry is when a shape looks the same after some change such as rotation, or mirror image flipping, or even shrunk to a smaller size.

Rotational Symmetry – Rotational symmetry, also known as radial symmetry, is when a shape looks the same after some rotation by a partial turn. An object’s degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.

Translational Symmetry – To translate a geometric figure is to move it from one place to another (without rotating it). Many tilings and wallpaper patterns have translational symmetry such that when you shift it over, you see the same pattern repeating. That is easier to see after watching our movie.

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