3D printing the Lorenz Attractor using MATLAB

Question

I'm using MATLAB to plot the Lorenz attractor and was wondering how I could export the XYZ coordinates to a 3D printable file! I'm having trouble going from the XYZ coordinates to a surface (should I even do that?). In addition, even thinking of the Lorenz Attractor as a "surface" seems not right to me, since my understanding is that it's more of a "path."

My code so far:

clear all; clc; clf;

dt = .01;
n = 100; %Number of Drawings
t = 0:dt:n; %Time Scale

%INITIAL CONDITIONS
x = 1;
y = 1;
z = 1;
xyz0 = [x y z];

%SOLVE ODE
[t,xyz] = ode45(@rhs, t, xyz0);

x = xyz(:,1);
y = xyz(:,2);
z = xyz(:,3);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%     PLOTTING    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%plot3(x,y,z);

rotate3d on;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%         RHS FOR ODE45     %%%%%%%%%%%%%%%%%%%%%%%%
function xyzdot = rhs(t,xyz)

    sigma = 10;
    b     = 8/3;
    r     = 100;

    x = xyz(1);
    y = xyz(2);
    z = xyz(3);

    dxdt = sigma*(y - x);
    dydt = r*x - y - x*z;
    dzdt = x*y - b*z; 

    xyzdot = [dxdt; dydt; dzdt];

end

I tried this earlier:

[X,Y] = meshgrid(x,y);
surf(X, Y, z);

But nothing's working. The error says z must be a matrix, and I'm a bit lost.

Answer 1

mesh cannot work because you are plotting a curve. Mesh wants matrices as input. To plot a curve you can use plot3. Try this:

figure;
plot3(x, y, z);
grid on

Answer 2

3D printing requires the use of 3D file formats, such as stl (most common), stp, amf, obj, or paramaterized toolpaths (Gcode). you can export the parametric form of this to control the motion of a 3D printer, but you won't actually print anything. If you want to export an stl, you must create a large number of facets (triangles in 3D space, with normal vectors calculated by a right-handed cross product), and output them in the following format:

solid keyminus.stl
 facet normal -0.676239558096412 0.628228242471799 0.384745806249222
  outer loop
   vertex 0 0 107.76627426733
   vertex 1.5625 1.5625 107.961254120119
   vertex 1.5625 0 110.512566214802
  endloop
 endfacet
 facet normal -0.678155426589519 0.63117863312972 0.37645550928056
  outer loop
   vertex 0 0 107.76627426733
   vertex 0 1.5625 105.146531455889
   vertex 1.5625 1.5625 107.961254120119
  endloop
 endfacet

.... (39,000 more facets here)

 facet normal 0 -1 0
  outer loop
   vertex 198.4375 0 0
   vertex 198.4375 0 110.512566214802
   vertex 200 0 107.76627426733
  endloop
 endfacet
 facet normal 0 -1 0
  outer loop
   vertex 198.4375 0 0
   vertex 200 0 107.76627426733
   vertex 200 0 0
  endloop
 endfacet
endsolid