Lorenz attractor

The following post is by Henry Delgado as part of George Mason University Math 493, Mathematics Through 3D Printing.

Figure 1

The Lorenz attractor is a system of ordinary differential equations (ODE), named after Edward Norton Lorenz.  Mr. Lorenz was an American mathematician and meteorologist (1917 - 2008).  As a meteorologist Lorenz looked further than linear modeling and started to take an approach in chaos theory (also known as the butterfly effect) in which resembles weather patterns.  Figure 1 Lorenz attractor resembles a figure eight and figure 2 contains the Mathematica calculation used to create the design.

The Lorenz equation consists of three ODE.  Figure 3 displays the Lorenz equation with variables 
σ, ρ, and  β

Modifications to each variable develops a different chaotic design.  For example, calculation on Figure 4 has been changed to (σ, ρ, β) = (-3, 26.5 and 10, 1) .

Calculation on Figure 5 has been changed to  (σ, ρ, β) = (-1 and -3,26.5 and 5, 1).

Calculation on Figure 6 has been changed to  (σ, ρ, β) = -3,100, 1).

 

The below design was created off of the MakerBot 3D 5x printer.  This model took roughly 1 hour to generate.  Calculation on Figure 7 has been changed to  (σ, ρ, β) = (-13 and -3, 26.5 and 10, 1).