3D Folium of Descartes

thingiverse

3D Folium of Descartes George Mason University 401: Mathematics Through 3D Printing Something common we learn in the average Calculus 3 course is a concept called the saddle point. A saddle point intuitively is simply a point on a 3D surface that appears like a saddle. A more specific mathematical definition is a point where it is a local maximum from one axis and is a local maximum from the orthogonal direction. These points serve as an interesting way to learn about the many different types of surfaces that exist in 3D. For this, I created my surface using what is called the folium of Descartes. The folium of Descartes is a 2D line that is defined by the equation x^3+y^3-3xy=0. Although this line is interesting to study on its own, I found that if we instead set the left side of the equation equal to z instead of 0 we get a 3D surface. This surface results in a saddle point and gives the surface shown above. The code in the images describes the manner in which I created the surface along side some contour lines to showcase the way the surface changes in the x or y directions. Needless to say, when printing this object, it is important to recognize that code provided alongside with the files do not include any proper supports for the object, so it is important to either include supports with the Mathematica code or to print the object in the given orientation using tree supports. For cooling, a brim was used. Finally, in my code, a sphere was placed at the bottom tip of the object. The reasoning behind this is that when creating the lines that sit on the original surface, the 3D printer used may confuse the intersections and leave holes where the object should be filled in. So as a result, I found such an occurence would take place at the bottom tip, thus a sphere was placed.