Math + Making

A student blog for Math 189AH: Making Mathematics at Harvey Mudd College

Frieze Groups on a Circle

Kaeshav Danesh

For this project, I explored visualizations of all seven frieze groups on a circle. I created a bracelet using lettered beads and laser engraved water bottles for the Math department.

What are the Frieze Groups?

Frieze Groups describe the possible symmetries of patterns on a 1D strip of paper. There are only seven of them. They are generated by several transformations:

  1. Translation
  2. Glide Reflection
  3. Vertical Reflection
  4. Horizontal Relfection
  5. 180 Degree Rotations

Then the seven frieze groups are generated by

  1. Translations
  2. Translations and Glide Reflections
  3. Translations and Vertical Line Reflections
  4. Translations and 180 Degree Rotations
  5. Glide Reflections and Vertical Reflection / 180 Degree Rotation
  6. Translation and Horizontal Reflection
  7. Translation, Horizontal, and Vertical Reflections

The Process

First, I made an vector file of the design using inkscape. I saved it as an svg and as a pdf because sometimes the laser cutter has problems with very detailed svgs. The seven frieze groups listed earlier are shown from right to left.

I used the rotary tool on the laser cutter to make this. It can be checked out from the cage. When using the laser cutter, it is important to only engrave powder coated metal water bottles because some plastic ones could create nasty fuemes. For this water bottle, I used these settings:

80% speed
45% power
100& current

But, these settings could be different if you are using a raster file and if you have a different colored water bottle.

Setting up the water bottle on the rotary tool is tricky because you want to minimize the wiggle room such that it lines up correctly at the end. You also want to make sure the water bottle itself is flat.

Once it gets going, it is satisfying to watch!

Here is what the final product looks like!



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