CM Tile Pattern with Lemniscate

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Vincent Vu September 20, 2021 George Mason University MATH 401: Mathematics Through 3D Printing This print is of a cm wallpaper group pattern that tiles the xy-plane. For centuries, mathematicians have been searching for tessellations, which is when a repeated pattern filles the plane without any gaps. There are seventeen distinct wallpaper groups that tile the plane, and each of them belongs to their own symmetry group. Symmetry groups are defined by their transformations, which include translations, reflections, and glide reflections. I was assigned Symmetry group 5, or cm wallpaper group, and used OpenSCAD to design my tile. The cm wallpaper group only has a glide reflection, which means that the figure is translated before being reflected. A good example of a glide reflection is a trail of footprints (assuming a person’s feet are mostly similar). The tile itself was created using OpenSCAD’s polygon() function, and every points besides the tip and the bottom of the polygon was a reflection across the x-axis of another point. Since there wasn’t a whole lot that went into designing the tile itself, most the work went into creating the lemniscate design on top, which was inspired by my older brother’s idea to put some sort of infinity design on top. The lemniscate was created by intersecting two tear drops, then using difference() in OpenSCAD on two more interior tear drops to create the holes. Due to the perfect roundness of a circle not really working with something elongated like a lemniscate, the figure had to be scaled in a way to make it lengthier in order to make it more like a lemniscate; since the tear drops were reflected across the x-axis, the scale was increased in the x-direction. Once the lemniscate was moved atop the tile, I wanted to make it shrink as it went into the tile to add some more to the design. This was achieved by using a loop of one-hundred iterations that gradually decreased each lemniscate’s scaling. Then, the difference() function was used between these lemniscates and the tile. Print settings listed below. References: - https://www2.clarku.edu/faculty/djoyce/wallpaper/seventeen.html (Information regarding wallpaper groups) - https://www.publicdomainpictures.net/en/view-image.php?image=54551 (Footprint image)