The Pentagonal Tiling Type 13
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The Pentagonal Tiling Type 13 Heegyung Kim September 10, 2021 George Mason University MATH 401: Mathematics Through 3D Printing A pentagonal tiling is a process of covering a plane or space with geometric shapes of the pentagon. It cannot be overlap nor have gaps between the shapes. Including the recent discovery by Mann/McLoud/Von Derau in 2015, there are 15 types of pentagonal tiling. Eight of them are edge-to-edge (types 1, 2, 4, 5, 6, 7, 8, and 9) and seven of them are non-edge-to-edge (types 3, 10, 11, 12, 13, 14, and 15). This 3D-printed object represents a pentagonal tiling of type 13. It is discovered by Marjorie Rice in 1977. This is a 2-isohedral and contains an 8-tile primitive unit. For type 13, the side d=2a=2e and the angle B=E=90° and angle 2A+D=360°. For this object, the dimensions are 40 x 40 x 4.5 in mm. I used OpenSCad to build this object and exported it as an STL file. Then, for printing, two 3D printers were used: Ultimaker Cura and MakerBot Replication. Each printer was used to print 4 tiles to get a variation of colors of the PLA filaments and to shorten the time. The printing time for both printers was about 3 hours. It used the skirt for the build plate adhesion on Ultimaker Cura and the raft on MakerBot Replication which created a height difference between the green and yellow tiles. Sources: 1. https://en.wikipedia.org/wiki/Pentagonal_tiling 2. https://sinews.siam.org/About-the-Author/pent-up-using-pentagons-to-tile-a-plane
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