Week 5 Recap: Symmetry

This week we began to look at symmetry, starting by talking about what we mean by symmetry, and trying to group images with similar types of symmetry. For math folks, a symmetry is a rigid motion (no breaking, bending, or shrinking) that leaves a figure looking the same — you can rotate, reflect in a mirror, slide (translate), or do a so-called “glide-reflection.” This week we looked at rotations and reflections. Figures that have finitely many symmetries are called “rosettes” and then can have rotations and reflections. These are easy to classify! For shapes with only rotational symmetry, we say these have a “cyclic” symmetry group (the cyclic is because they spin around in a cycle) and we classify them as C1, C2, C3, etc, where the number is the order of the rotation, how many times you can move it around before you end up back where you started. You can also think of degrees, a C3 can be rotated around by 360/3=120 degrees. So for instance, the image from the Hong Kong flag below is a C5, which can be rotated through 360/5=72 degrees, and rotating 5 times brings you back to the start.

"Flag of Hong Kong" by Tao Ho - http://www.protocol.gov.hk/flags/chi/r_flag/index.html.
C5: “Flag of Hong Kong” by Tao Ho – http://www.protocol.gov.hk/flags/chi/r_flag/index.html.

When there is mirror symmetry, one mirror indicates a figure with “bilateral symmetry” — the symmetry we see in many animals, including humans. When there are two or more mirrors, there is also a rotational symmetry, and the order of the rotation is the same as the number of mirrors. The figures with mirror symmetry are said to have a “dihedral” symmetry group and are classified according to the number of mirrors, D1, D2, D3, and so on. For example, this is a D6 snowflake, which is the symmetry type of most snowflakes.

Snowflakes by Wilson Bentley with D6 symmetry group.
Snowflakes by Wilson Bentley with D6 symmetry group.

In class, we also tried our hand at cutting our own “snowflakes” and “swirlflakes” after watching this Vi Hart video: Snowflakes, Starflakes, and Swirlflakes. We also tried making rotational and reflectional symmetry by hand (sketching) and using pattern blocks.

Other sources

What’s next? Well, your next project will be a portfolio of symmetries and is due April 20th. And you should write up a response to this week’s class and email it to me or hand it to me in next week’s class. Since we are talking about symmetry it is particularly nice to start noticing symmetry in the world or in your work, so feel free to use that as part of your response! Next week we will start talking about wallpaper patterns and frieze patterns.


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