Monthly Archives: July 2017

Creating a 3D-printable Lorenz attractor

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The Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz in the 1960’s. It is notable for having chaotic solutions for certain parameter values and initial conditions. 

In Winter 2015, my colleague Professor Greg Dresden used the Lorenz attractor as an example in his course on Partial Differential Equations. At that time, I worked with Dave Pfaff in the IQ center at W&L to find a way to 3D print a physical model of the solution curves. Dave also created model of the solution curves that could be viewed with 3D glasses in our stereo 3D lab. To create the models we followed three steps:

Step 1:  Create points along the solution path (the Lorenz curve) using Mathematica. Here is the Mathematica code Greg developed for this purpose.

(* Here are the differential equations *)
leqns = {  x'[t] & == -3 (x[t] – y[t]),
y'[t] & == -x[t] z[t] + 28 x[t] – y[t],
z'[t] & == x[t] y[t] – z[t] };
(* Here, I define two paths, p1 and p2, which start at slightly-different initial values. *)
p1 = NDSolveValue[{leqns, x[0] == z[0] == 0, y[0] == 1}, Function[Evaluate[{x[\#], y[\#], z[\#]}]], {t, 0, 30}];
p2 = NDSolveValue[{leqns, x[0] == z[0] == 0.03, y[0] == 1},  Function[Evaluate[{x[\#], y[\#], z[\#]}]], {t, 0, 30}];
(* Let’s look at a plot of these two paths, to verify that they seem correct *)
pic1 = ParametricPlot3D[p1[t], {t, 0, 30}, AxesLabel -> {x, y, z}, PlotStyle -> {Red, Opacity [0.5]}];
pic2 = ParametricPlot3D[p2[t], {t, 0, 30}, AxesLabel -> {x, y, z},
PlotStyle -> {Black, Opacity [0.5]}];
(* Finally, we export them as two separate files *)
tableout1 = Table[p1[t], {t, 0, 30, 0.01}];
tableout2 = Table[p2[t], {t, 0, 30, 0.01}];
Export[“TableOut1.xls”, tableout1]
Export[“TableOut2.xls”, tableout2]

Step 2:  Use Excel to tweak the data into a form we can use.

  • Open the files in Excel.
  • Insert a column before the three columns of numbers, this is column A. Make the first row 1, and the second row 2 in column A. 
  • Highlight these two entries in column A, then drag the box down to row 3001. The numbers in column A will automatically fill in the correct numbers.
  • Highlight all of the numbers in all four columns. Click on Save As, then save the file as a Tab Delimited Text (.txt) file. 

Step 3:  Create the thickened Lorenz curve in Cinema 4D.

  • Open Cinema 4D
  • Construct the spline with the Lorenz data.
    • Left click and hold on the Spline button on the top row, then select the Linear (spline) button. Click once on the view-port to add one point.
    • Go to the Object Manager. On the right side, click on the Structure button. You’ll see the Point 0, with X, Y and Z coordinates. 
    • Click on the File button above the Object Manager. Click on Import ASCII Data… Open the .txt file created above.
    • Click on the 0th point (the one you added initially) and Delete it. The remaining points are from the Lorenz data (see figure.)
    • Click on the Objects button on the right side of the Object Manager. 
  • Thicken the spline for 3D printing.
    • Left click and hold on the Subdivision Surface button on the top menu, then select the Sweep button.
    • Left click and hold on the Spline button on the top row, then select the Circle button. Under the Attribute Manager, make the radius of the circle 3mm (0.3cm).
    • In the Object Manager, move the Circle and the Spline under the Sweep. The Circle should be above the Spline (see figure). The thickened spline is now complete, and the Lorenz curve can now be exported (as a .stl file) and then 3D-printed.

A thickened Lorenz curve in Cinema 4D is shown below on the left. Two curves can be added to the same plot in Cinema 4D and given different colors.  Dave Pfaff printed two such curves using the Project 260 3D printer at WLU. (This printer uses a gypsum-like powder hardened via a laser then finished with superglue.)  The Project 3D printer can print in color (using inkjet cartridges). Dave also designed a stand for the curves. This gives the beautiful model shown below on the right. Cinema 4D also allows the user to animate the curves. Dave created such an animation, which nicely shows the chaotic nature of the Lorenz attractor. The initial points of the two curves were very close, but in the long term, the curves diverge.

 

 

Five Intersecting Tetrahedra

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One of my favorite mathematical models is the Five Intersecting Tetrhadra model. I first became aware of this model from Thomas Hull, a mathematician at Western New England University. He has written extensively about mathematics and origami (check out his book Project Origami). In particular he has written fantastic instructions for creating this model using modular origami.

dodec1tetThe five intersecting tetrahedra model is based on the dodecahedron. This one of the five classic regular polyhedra consisting of 12 pentagonal faces and 20 vertices. Take 4 vertices in the dodecahedron which are the same distance apart. These form the 4 vertices of a regular tetrahedron, as shown on the right (figure from Tom). Since the dodecahedron has 20 vertices, we can inscribe 5 such tetrahedra inside the dodecahedron.

The origami version comes about by making a thin frame for each tetrahedron. Provided the frames are thin enough, they won’t intersect each other. Instead they will form a marvelous interwoven pattern. As shown on the left, I have successfully followed Tom’s instructions for making the origami version of this model. 

The next natural question is whether or not the model can be 3D-printed. The answer, is of course, yes! I used crsfdr’s model Interlocking Pyramids  from Thingiverse to print the model on the UPrint SE printer at W&L.  The photo at the top shows the finished version. The one to the right shows the model just out of the printer before the supports have been removed. Since the supports are made of a material that dissolves, the UPrint was the perfect printer for the job.

My colleague Professor Michael Bush has used the Five Intersecting Tetrahedra model when teaching introductory Group Theory to undergraduate students. Indeed, this was the motivation for 3D-printing the model as the origami version is not really robust enough to use in a classroom setting. The model is a great tool for discussing the rotational symmetries of the dodecahedron (or its dual the icosahedron). Roughly speaking, the rotational symmetries of the dodecahedron act in a natural way on the five tetrahedra giving a permutation representation of the symmetry group. After some noodling around this allows one to see that this group is the alternating group \(A_5\). (Michael usually suppresses details about the faithfulness of the representation at this level.)