Louis Vuitton Meets P4m

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Louis Vuitton Meets P4m Sarah Boyt George Mason University Math 401 Mathematics and 3D Printing Most people have not heard of wallpaper patterns, but there is no doubt that they have seen them. Islamic art and architecture, textiles, bathroom tiles and wallpaper are real life examples of wallpaper patterns. A wallpaper group is a mathematical classification for a repetitive 2D pattern based upon symmetries in the pattern. The 17 wallpaper groups are classified by reflections, glide reflections and the highest order of rotation. I was assigned the p4m wallpaper group. P4m patterns have a square lattice and a maximum of 4 orders of rotation. Additionally, p4m patterns have 4 axes of reflection (horizontal, vertical, and diagonal) with mirroring lines at 45 degrees. They also have two glide reflections that pass through each order 2 rotation that form 45 degree angles to the reflection axes. P4m is one of the few wallpaper patterns that requires two tiles to tile the plane uniquely (meaning it only tiles for p4m group) . To design my piece, I started with this website [https://math.hws.edu/eck/js/symmetry/wallpaper.html]. This site allows the user to select a group and begin drawing. My strategy was to take shapes the I can easily create in openscad and combine them into an irregular shape. I included a picture of my design above. As you can see, for my first piece I took a square and added equilateral triangles to each side. This piece ended up looking like a p4m version of the Louis Vuitton logo. My second piece was a rhombus that fit in between the Louis Vuitton logos. Our professor told us to use offset = -.5 to ensure that our pieces fit together. I took this as an absolute and used offset = -.5 for both pieces. It took 4.5 hours to print 4 Louis Vuitton logos and 8 rhombuses. I learned that offset has a bigger impact on smaller pieces ( like my rhombus) than larger pieces (like my Louis Vuitton logos). As you can see in the attached photo, the original rhombuses came out too small. Thanks to our wonderful LAs, I was able to reprint the rhombuses with offset = -.2. References : [https://mathstat.slu.edu/escher/index.php/Wallpaper_Patterns] [https://www.math.fsu.edu/~quine/MB_10/schattschneider.pdf]