Octopus Saddle

by: Khalilah Hamer

This week, the Math 401: Mathematics Through 3D Printing class created 3D figures inspired by multivariable calculus concepts. Inspired by the hyperbolic paraboloid or saddle and the dubbed “monkey saddle” I decided to recreate an “octopus saddle” containing 8 maxima and 8 minima. 

I used the application Mathematica in creating this saddle and the equation f_8 (x,y) = (x^8 - 28 x^6 y^2 + 70 x^4 y^4 - 28 x^2 y^6 + y^8)/7.

I derived this equation from  z(r, θ) = r^n e^(i n θ) = r ^n [cos(n θ) + i sin(n θ) =

 With the knowledge of i^2 being -1, I kept k at even numbers to derive the equation.  Having the final equation = f_8 (x,y) = (x^8 - 28 x^6 y^2 + 70 x^4 y^4 - 28 x^2 y^6 + y^8). The final equation is divided by 7 which will be discussed when I examine the problems I’ve had when creating the print. 

Going about coding the project. I examined the example codes and saw that it was much easier and made more sense to define everything at the beginning of the code so when I go back through the code to troubleshoot if any problems occur, I can read the words like "range" and "tubess" then I will know what to change or edit as opposed to a bunch of numbers and equations trying to figure out what will help shrink the image. Then I worked on the the tubes first. The code is below.

 Below is a comparison of the original equation vs when I divided the equation by 7, as you can see the print turned out to be really tall and if I did try to change the range a bit, the minima would be cut off.

Overall the coding was simple and the print looked good. I decided not to add the “feet” to the bottom of the graph since it seemed with eight legs to be quite stable already. The only problem I encountered was that the sides of the print was very tall. I didn't want to necessarily scale the graph and from the way the code was, editing the plotRange did not actually affect the range when the code was opened on the ultimaker. The way I solved this problem was dividing the equation to flatten the graph. The end result was a divide by 7 which did flatten the graph quite a bit but it wasn’t nearly as tall and the print didn't take over a day. Below is a figure of my final print.