Summary

George Mason University Math 401: Mathematics Through 3D Printing
Created on October 11th, 2023 by S.T.

This cylinder is augmented to show a circular opening from a certain angle and a four-pointed star when next to a mirror. The cylinder’s stretched-out points create 2 star points, where the curve of the cylinder create the next 2 points.

Print Settings

Printer Brand:

Creality

                                            <p class="detail-setting printer"><strong>Printer: </strong>                     <div><p>Ender 3</p></div></p>                                                                                <p class="detail-setting rafts"><strong>Rafts: </strong>                     <div><p>No</p></div></p>                                                                                <p class="detail-setting supports"><strong>Supports: </strong>                     <div><p>No</p></div></p>                                                                                <p class="detail-setting infill"><strong>Infill: </strong>                     <div><p>20%</p></div></p>                                                    <strong>Filament:</strong> Overture 1.75 mm PLA                White

How I Designed This

This 3D print creates an optical illusion when facing a mirror. The hollowed cylinder has a flat base with a uniquely curved top and two distinct ridges on the sides of its curves, creating an oval-like shape when viewing it from most angles. However, from one specific angle, the cylinder’s top looks like a perfect circle, and when that viewpoint is facing a mirror, the cylinder’s reflection will show a 4-pointed star instead of the circle top.

To create this shape in Mathematica, I first created the 4-pointed star using its polar coordinates to create a symmetric shape that can fit on the cylinder. However, this method had it’s limitations as it was difficult to create the right proportions for each point of the star so that it would print correctly, thus I switched to using some simpler algebraic expressions using absolute value to create the points of the star, then create their mirror inverses by negating each of them. Before, I had 8 different functions that would create the 4-pointed star, but the algebraic expressions were able to create a similar star with just 6 different functions.

I then used the sums and differences between the circle’s expressions to the star’s, then created the 3D parametric plot of the main curve of the cylinder and its two opposing points on each side of it. The ranges for the expressions were determined by the intersections between the star’s 6 expressions and its intersections to the circle. The final step was to model the shape with its desired length, thickness, and mesh to create a smooth 3D model that was exported as a STL to Cura for printing. Due to the filament and printer type, no supports or bed adhesion types were necessary.

Ambiguous Cylinder: A Star Inside A Circle