Savannah Crawford - MATH 401 F19
Over the summer I went to a summer school program in Vancouver, BC that was hosted by the Pacific Institute for the Mathematical Sciences. One of the courses in the program was on symmetries, specifically the 17 wallpaper groups for tiling the Euclidean plane. So when Sander announced the topic for the assignment, this seemed like a natural fit for a project on symmetries.
In 1891, Evgraf Fedorov proved that there are 17 wallpaper groups which tile the Euclidean plane. The 17 groups are classified by their symmetries: mirror lines, points of rotation, and transitional symmetry. Using this approach, you can reduce seemingly different patterns to their symmetries and see that they are the same geometrically. Additionally, there's a cost function associated with each group, and the cost for all 17 groups is 2.
[This pattern belongs to *442 since there are two pints where 4 mirror lines intersect and one where 2 mirror lines intersect.]
My pattern is in *442 (Conway notation). It's one of the simplest groups. Graph paper is another example of *442, but my base shape is not a square so the pattern looks pretty interesting despite its group. Essentially, the base shape of my print is a spiky ring.
[Here is the spiky ring that generated the wallpaper.]
I tiled the base shape such that the edges overlap creating a grid with spikes. I left the edges round to showcase the geometry the original base shape.
[Final object to print]
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