How do you create mathematical models?

I am also a newcomer to this forum. I've been thinking about 3D printing some of the things I've been researching. To make a long story short, I stumbled across (on the internet) an unknown application of graph theory that describes higher dimensional tori. It is unpublished original research, so you won't find any of this on wikipedia. Unless you are interested, I won't bore you with the mathematical details. To put it simply, there is a method that derives an algebraic equation (implicit or parametric) for many different kinds of hypertoroidal objects. You can build them as simple or as complex and many dimensional as you want.

More to the point, the 3D cross sections are as bizarrely beautiful as the are strangely alien:








These are oblique 3D slices of some 6D tori. After taking many pictures, I soon discovered how to make animated gifs:


this animation is a 3D slice of a 7D torus, doing a double rotation. Made by the equation (vary 'a' from 0 to pi/2) :

(sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 5)^2 + (y*sin(a))^2) - 2.5)^2) - 1.25)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(a))^2) - 2.5)^2) - 1.25)^2 = 0.5^2




a triple rotation of a 6D torus, animation made by :

(sqrt((sqrt((sqrt((x*cos(a))^2 + (y*sin(a))^2) - 8)^2 + (z*sin(a))^2) - 4)^2 + (sqrt((y*cos(a))^2 + (x*sin(a))^2) - 4)^2) - 2)^2 + (z*cos(a))^2 = 1




This one is a more complex position and rotation of an 8D hypertorus. Animation made by:

a=5 , animate 0 < b < 2pi , c=pi/4 , d=0 , t=0

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

Come to find out, the program I use to plot these surfaces can export an STL file! My ideas are to 3D print the cross section arrays of passing a hypertorus through a 3-plane. And, also some of the more exotic slices seen in the first images. The most recent idea I had was to sort of document the slices in real-time, and make a youtube channel. Here is a 3-torus, the most well-known 4D hypertorus:

Blender version of my previous 3dsmax tutorial. Hope you enjoy.

Interlocked Star Nest (Blender Version 2.77)

The last time I taught mathematical modeling we used Introduction to Mathematical Modeling, by Giordano, Fox, Horton, and Weir. The first part of Chapter 2 contains a nice introduction to the modeling process.

I make quaternion fractals: https://www.shapeways.com/designer/shawn_halayka/creations

I use my own software to make the models.