# Snowflakes and symmetries

Thursday 9th December 2021

We’ve been looking how we can use TED Talks to inspire and engage our students within the curriculum.

Symmetry, reality’s riddle is a talk by Marcus du Sautoy who is a mathematician, science communicator, and author of The Music of the Primes. In his talk, Marcus du Sautoy looks at lines of symmetry, rotational symmetry, and the symmetries that can be found at the Alhambra Palace in Granada. The question is then posed: what happens if one symmetry is applied after another?

So, here’s an investigation into this that you can try with your class, to go along with clips from the talk.

In the California Institute of Technology in Pasadena, scientists have been creating triangular snowflakes to study how they are being formed in nature. Below is a diagram of one of these snowflakes. What symmetries does this snowflake have?

Well, the snowflake has three lines of symmetry, and has order of rotation 3.

Let’s give these symmetries letters:

- U reflection on the line of symmetry through point a
- V reflection on the line of symmetry through point b
- X reflection on the line of symmetry through point c
- Y Rotation by \(\frac{1}{3}\) of a turn
- Z Rotation by \(\frac{2}{3}\) of a turn
- I Rotation by 0

What happens if we perform Y followed by Z? Can the result be described by just one symmetry? How about Z then Y? Does the order matter? Cut out the snowflake and investigate what happens, then complete the following table.

* | U | V | X | Y | Z | I |
---|---|---|---|---|---|---|

U |
||||||

V |
||||||

X |
||||||

Y |
||||||

Z |
||||||

I |

Look at each row and column. Is there anything that you notice?

What about this rather unusual snowflake? What symmetries does it have? Assign letters to these symmetries and fill out the table above.

What happens with other shapes?