Laura Taalman / mathgrrl

Knots are embeddings of circles in 3-dimensional space, but they are typically studied in terms of their projections into 2-dimensional space. We can use 3D printed models to investigate knots in a more 3-dimensional way. In this series, we present nine Stainless Steel 3D printed knot conformations: Tritangentless Trefoil, Figure Eight Stick Knot, Cinquefoil Lattice, 5_2 Lissajous, 6_2 Petal Knot, two (7,2) Torus Knots, 8_19 Hyperboloid Stick Knot, and a Midway Perko Knot. Knot conformations were chosen based on key properties of each knot from the literature. The knots were scaled before construction to form a consistent series while preserving thickness. Designed with OpenSCAD as well as Mathematica, Blender, and Knotplot.

In collaboration with mathematician Steve Lucas, we recently developed a computationally reasonable method for creating 3D-printable models of chaotic attractors. This collection of five designs provide three-dimensional physical, tangible versions of the beautiful two-dimensional renders created by artist and photographer Phillipe Put. Our method takes as input a set of differential equations and initial values, produces a sequence of datapoints in MATLAB spaced by curvature constraints, and “sweeps” that data into a curve-surrounding polyhedron in OpenSCAD. The image shows our 3D printed model of the Aizawa Attractor. Four other models are currently in production and will be completed by August 2018.

This model illustrates a continuous morphing between the two knot conformations known as the Perko Pair. The red knot is "Perko A", the knot known as 10_161 in the Rolfsen table. The purple knot is "Perko B", the Knot Formerly Known As 10_162. The remaining knots are snapshots of a continuous morphing from Perko A to Perko B that shows that they are, in fact, the same knot. Each knot in the morph was extracted from KnotPlot, Laplacian thinned in Meshlab, Catmull-Clark remeshed in TopMod, and then arranged, scaled, and painted with color in Meshmixer.

This knot conformation is halfway between “Perko A” and “Perko B”, the two knots in the Rolfsen knot table that were famously shown to be equivalent by Kenneth Perko in 1973. Extrusions were added to each mesh face in the model to create spikes that highlight the path of the knot through three-dimensional space. Designed with KnotPlot and TopMod, optimized for 3D printing with Cura. The spikes make the model extremely challenging to print on a desktop 3d printer. After many unsuccessful print attempts, the final model took over six days to print and an additional week to clean up breakaway support material and for post-processing.