Printing The 3D Fractal Future

2D line-division type fractals are pretty easy to create on the MakerBot as it happens. Recursive algorithms can neatly define the act of repeatedly inserting a sequence in between copies of itself to generate the Coch Curve and other related shapes. Further, a wealth of algorithms have long been known for generating fractals for pen plotters, and shapes like the above Coch Snowflake are just a matter of repeatedly performing those very commands at different elevations of the tool head.

3D fractals, like the Serpinski sponge, aren’t as easy to create. The Serpinski sponge in particular has overhangs that will make it poorly suited for printing on a non-support printer, and although the self-similar nature of fractals may make it plausible to make 3D fractals which by construction do not need supports, the added complexity of the formulas, coupled with their lack of innate suitability to the plotting method will cause some trouble. Also likely to cause trouble is the lack of study in this field– 3D fractal research has been done with polygon modeling or point clouds more often than with 2D slicing.

However, I think this is a really important area of study for a number of reasons, one of which is that the biological world consists more or less entirely of 3D printed fractals.

Of course the 3D printing that goes on in our cells runs on vastly different hardware: 1D extruders called ribosomes spray out chains of amino acids which electrostatically contract into useful objects like cellulose molecules, enzymes, and other ribosomes, (mother nature’s repraps are miniaturized!) but at each step of the way, you’ll find self-similar patterns, from the repeatedly-coiled helices of our DNA to clusters of hair follicles, all of which are defined at the bottom level by extruder modules, positioning themselves in three dimensions using relative coordinate systems based on chemical gradients and other signals to create macroscopic machines of dazzling complexity, such as cats.

3D printed fractals made of ABS plastic probably won’t do things like replace lungs, but the research we in the open-source 3D printing community do to make pretty trees and self-similar LED fixtures may inform future efforts in printing with PLA and later more advanced biomaterials to create support structures for re-growing organs or optimize surface area to volume ratios for biological experiments.

There is a revolution in molecular assembly coming, and regardless of whether the biologists and chemists or the engineers and machinists are the first to arrive at a general assembler, I expect that getting there will require fractal geometry, not the Cartesian stuff we’re used to. And learning how to map between the two domains is something we really can learn by designing 3D printed fractals.