Type 4 Pentagonal Tiile

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Type 4 Pentagonal Tile Lucas Newman George Mason University - Math 401 - Mathematics Through 3D Printing September 10, 2021 This tile is an example of an irregular pentagon that is capable of tessellating with itself and tiling the whole plane. When a shape can "tile the whole plane", this means that many copies of the shape can be placed edge to edge alongside each other, and that in doing this a pattern is created which can be extended in all two dimensional directions indefinitely. Regular pentagons do NOT have this property - if you try to place them into a tessellation they crowd each other and leave small gaps in the pattern. In order for a pentagon to tile in this way it must be instead constructed so that the sides and angles are NOT all identical, but DO adhere to constraints that endow it with this property. Tiling pentagons can be placed into categories according to the specific set of geometric constraints by which they abide. These constraints consist of equations - in each set, expectations are expressed that certain subsets of the angles and certain subsets of the sides will relate to each other in a prescribed manner. As a result, the creation of a tiling pentagon has these steps: 1) choose a category of constraints, 2) identify which parameters of the pentagon you are free to customize given those constraints, 3) make sure that the constrained parameters (which will often reference the unconstrained ones) have been calculated according to proper instruction. This particular pentagon is created with the "Type 4" constrains, in which two pairs of sides are set equal to each other, and two of the angles are fixed at 90 degrees. Note that this textual description does not specify the pentagon in and of itself, as it is important which of the angles and which of the sides receive this instruction. The type is fully specified in one of the included pictures. This object was crafted using OpenSCAD, and is a straightforwards linear extrude on the underlying pentagonal shape, which has been configured by adjusting the variables of some preexisting code. The eight pieces above were printed on an Ultimaker Cura, and took roughly 40 minutes. No raft or supports were needed for something this simple, and only a skirt was used. Infill was %10, and all eight pieces were printed together. Citations: https://en.wikipedia.org/wiki/Pentagonal_tiling#Types_3,_4,_and_5 https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/ https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/ https://sinews.siam.org/About-the-Author/pent-up-using-pentagons-to-tile-a-plane