Reflexive Ghost

thingiverse

George Mason University Math 401: Mathematics Through 3D PrintingThis print was designed to replicate the fascinating optical illusion known as a Kokichi Sugihara Reflexivity-Fused Object. As Sugihara himself described it, "The objects themselves have meaningless shapes, but if they are placed on a horizontally oriented mirror, meaningful shapes appear." Using the power of Mathematica, I began creating my version of this intriguing object.To begin, I employed Mathematica to plot two separate equations, namely g1 and f1. The f1 equation is positioned on top, while g1 represents the reflected image in the mirror. In this particular case, the equations took the form of:f1 = 2.7 Sin[t] - 1.2g1 = 0.5 Cos[8 t] + Sin[t]These equations were then skillfully woven into the following command: "ParametricPlot[{{Cos[t], f1}, {Cos[t], g1}}, {t, 0.418, 2.733}]" within Mathematica.With these mathematical foundations in place, I proceeded by setting alpha to 1/Sqrt[2] and the height to 2. I then executed the following commands:a = ParametricPlot3D[{t, sum1, h + diff1}, {t, 0.418, 2.733}, AspectRatio -> Equal, ViewPoint -> {0, Infinity, Infinity}]Subsequently, I entered:a3 = ParametricPlot3D[scale {{t, sum1, u (h + diff1)}}, {u, 0, 1}, {t, 0.418, 2.733}, PlotStyle -> Thickness[2], Mesh -> False, PlotPoints -> 100, AspectRatio -> Equal, ViewPoint -> {0, Infinity, Infinity}]Once this intricate process was successfully completed, the next step was to export the resultant object into an STL file.When it came time to bring this creation to life, I used a Creality Ender-3 Pro 3D printer. My settings remained mostly default, with the exception of the temperature, which I set at 215 degrees Celsius. I opted not to use any support structures, and I added a skirt to ensure a smooth start. The entire printing process took a total of 1 hour and 20 minutes.In the end, my efforts had culminated in a tangible piece of art, showcasing the fascinating interplay of mathematics and 3D printing technology, just as Kokichi Sugihara's work had intended.Sugihara, K. (n.d.). “reflexibly-fused objects” whose shapes are completed only when they are fused with their mirror images. Invisible Objects (Japanese). http://www.isc.meiji.ac.jp/~kokichis/reflexivelyfusedobjects/reflexivelyfusedobjectse.html