Create 3D model from equation

Hello my friends,

I am a beginner with Rhino and Grasshopper. Some years ago I went to an exhibition (can’t remember what was about, nor who the author was), this exhibition had on display, various models that were generated from mathematic equations. I only took this two photographs, one i was lucky to include the equation.

So, how do I even start to model this one with Rhino? I guess I must use Grasshopper.
I also believe that the model is composed of two intersecting parts, hence the break in the equation. I wasn’t that great with mathematics, but I can see it’s composed with parabolic equations, giving the U shapes, I just don’t understand how the cubic exponent influences the model.

Thank you.

I want to learn how to make other model from equations so I am grateful if you provide links or tutorials on how to start with the basics.

(y^2)(z^3) - 4xz^4 + 6.75y^4 - (36xy^2)z +
+ (32x^2)(z^2) - 64x^3 = 0

(can’t type superscript numbers here so I hope I wrote it correctly. down below is a correct screenshot)

These are the graphs that are generated using Being z a variable, the graph changes accordingly.

PS: I found the exhibition links, here they are: (sorry it’s in portuguese)

Hi Lois - this may help-


for version 4 and 5 there is a plugin which lets you produce equational shapes i believe even based on nurbs. i have no idea if that works with rhino 6, also i never got it to run on the mac version of rhino. but i believe it should be exactly what you are after.

other than that i am pretty sure grasshopper should do the trick.
maybe this could be a starting point.

maybe @Mahdiyar could shed some light either.

also you can actually use LATEX to embed equations into the forum.

Thank you @pascal and @encephalon. I will read your suggestions carefully and see what I can do for now. I will post feedback later.

Meanwhile, I found the authors of those shapes, which are very interesting to take a look at. All shapes and model come from algebraic geometry like Kummer Surface, Togliatti Surface.

But I was, specifically, more interested in the Swallowtail surface, which can be seen in 3D here:


Hi @lois,

In addition to @pascal’s link to Mathematica, Wolfram have an online solver, Alpha. This will solve your equation for Z:

Taking the result on trust :innocent:, in Grasshopper you can use the two resulting equations to plot points for domains in X and Y and then fit curves through them.


p.s. Your typed equation does include an error: there are two consecutive + signs

Math_3DE.rhp (v 1.2.3) for Rhino 5 mentioned above by encephalon works with Rhino 6 also.