The PG Tile

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The PG Tile by Krista Cimbalista for Math Through 3D Printing/MATH401 at George Mason University on 9/20/2020. There exist 17 distinct plane symmetry groups. Each group defines a lattice, cells, units, axes of reflection, axes of glide reflections, the highest order of rotation, centers of rotation, reflections, and generating groups. These groups all perfectly tile the plane. You can tell what the symmetry group of a tile is with this list characteristics, because it is unique to the group.[1] You can find a great flowchart midway down this page https://mathstat.slu.edu/escher/index.php/Wallpaper_Patterns that will help identify the group that you are looking at. My tile for this assignment is part of the group called pg. The notation pg is short for p1g1, which stands for primitive cell (meaning that the highest order of rotation is at the vertices), the first 1 denotes the highest order of rotation, g indicates that there is no reflection but that there is a glide reflection axis, and the second 1 represents that there is no symmetry axis for this group. The lattice that is used is the rectangular lattice, there are three axes of glide-reflections, one non-trivial glide reflection, and no centers of rotation. You can generate the tile with only a half unit.[1] Please note that these characteristics do not necessarily apply to the decorations on the top of the tile, they are added strictly for visual interest. To design my shape, I started with the program that you can find at this link: http://math.hws.edu/eck/js/symmetry/wallpaper.html. I then chose the “pg” selection and started designing. After some frustration, I saw that what I needed to do was make a shape that did not allow for any holes in the tiling. So, when creating my tile, it was very important to understand the symmetries in order to put something together that would do this. With some trial and error, I found that you can put two offset, mirrored rhombi together and successfully tile the plane. Citations: [1] https://www.math.fsu.edu/~quine/MB_10/schattschneider.pdf OpenSCAD code: module shape() { linear_extrude(5) offset (delta = -.4) { difference() { union() { translate([3,-3]) square([4,6]); polygon([[0,0],[14,0],[20,20],[6,20]]); translate([13,20]) polygon([[0,0],[-14,0],[-20,20],[-6,20]]); } //union bracket translate([-1,37]) square([4,6]); } //difference bracket } //delta bracket translate([4,2,5]) cube([2,2,3]); translate([4,36,5]) cube([2,2,3]); translate([15,10,5]) rotate([0,0,61]) cube([2,10,3]); translate([-1,27,5]) rotate([0,0,-61]) cube([2,10,3]); translate([7,14.5,8]) rotate([0,0,33]) cube([2,15,2]); } //module bracket shape();