From Earth to the Stars

thingiverse

Earth to Stars Victor Guzman Duran September 10th, 2021 George Mason University Math 401: Mathematics Through 3D Printing Looking at the mathematical question of polygons spanning a plane or as it’s commonly said in everyday colloquial language “tiling a space”, we can quickly see that there are only a certain number of convex pentagons that are able to tile a given space. By restricting our study to just convex pentagons which are simply pentagons with interior angles being less than 180 degrees (Mann, 2021). While also using the usual properties of linear transformations which are rotations, reflections, and translations which correspond with scalar multiplication along with the composition of vectors. If we combine these two concepts on the usual two-dimensional space (RxR) we can see that there are a set number (15) of convex pentagons that can span this space (Honner, 2017). A fun fact about this specific pentagon is that it has the characterization of being a chiral shape because of its inability to mirror itself and be the same shape. In other words, you cannot create the mirror image of the shape by just using the normal transformations such as translations and rotations (Carmichael,2013). By using the given tile code in OpenSCAD I was able to create the convex pentagon from which by adjusting certain variables, I was able to create a unique, yet within the same class, polygon from which we could span the plane. After taking these points and mapping them onto the compiler I could then move into the design phase since I want this to be a personalized tile to my own taste. The concept of using a star came to me from the first prototype of my design because I wanted to place a concave object within a convex one (star inside the convex pentagon). The star that I made was created using a set of transcendental functions (sine and cosine) in order to plot the points of the star and then merge them as they were made closer and closer to each other. After this, it was a matter of combining the two shapes and taking the difference so as to create a cut from them. I printed my final object on the Monoprice Mini in the MIX Lab with thanks to the people that work there after going to the training session. I used a raft as recommended by one of the instructors there and no supports as it was just a solid flat tile. I had one small error in which Ultimaker Cura was bugging out and not letting me slice even though I have already successfully completed two separate print jobs with it and even the instructor had a hard time figuring out the issue. We ended up just going back through the code and seeing what the problem was but there really was no issue with the code and instead, it was just a glitch since we simply restarted Cura and it worked fine.