Mathematical Lattice Spoolholder Caps

Like my other models, this is an extension of some lattice generation work that I needed to do for my own graduate research, so I made my previous coaster script just generate these instead!

What Are These ?

These are 2d-versions of a variety of lattices that I have to generate in Python for my graduate research. The designs are broken down into three main categories:

1. Triply Periodic Minimal Surfaces (TPMS): Organic, continuous surfaces like the Gyroid, Schwarz P, and Schwarz D.

2. Explicit Strut Lattices: Strong, engineered geometric patterns including square grids, triangles, hexagons, and Voronoi tessellations.

3. Crystal Structures (Frank-Kasper Phases): Highly complex patterns derived from the atomic arrangements of the A15 and C15 crystal lattices (my personal favorite). 

How Were They Made?

Rather than using traditional 3D modeling software, these coasters were algorithmically generated using Python (trimesh and manifold3d). The TPMS structures were generated by solving the implicit equations and then thickening them up and running the marching-cubes algorithm on them, so the file size is going to be quite a bit larger than the strut-based lattices. It's possible to get cleaner, 2D TPMS, rather than the 2.5D that I output, but they just don't look that cool. 

To ensure the strut lattices were as clean as possible, the script instead just relied on 2d projection and extrusion to avoid the messy tops that come with marching cubes. This keeps file size down, and gives a much cleaner design. 

Understanding the Crystal Structures (A15 & C15)

I need the 3D version of these for 3D-printing foam stand-ins, and I love the cross-sections. The A15 and C15 Laves phase structures, known as Frank-Kasper phases, "complex atomic arrangements found in certain metal alloys", according to google.

• A15 Structure: This lattice is known for its high symmetry and is historically significant in superconductivity. The unit cell consists of atoms arranged in a body-centered cubic-like structure with orthogonal chains of atoms running across the faces.

• C15 Structure: A face-centered cubic Laves phase structure, featuring a highly complex, interconnected network of tetrahedra. It's notoriously difficult to model accurately.

For both structures, we sliced the unit cells at multiple different heights (offsets), resulting in drastically different geometric patterns depending on where the crystal was intersected. Doing so offers literally no benefit in mechanical properties - they just look wicked cool. 

File and Folder Structure

The download is organized logically so you can find the design you want to print:

• tpms/ - Contains the Gyroid, Schwarz P, and Schwarz D coasters. I was going to include a few others, but these are the cleanest. 

• explicit/ - Contains the standard geometric lattices. I stuck with square and hexagon unit cells for now, but more complex polygons are possible... they just don't look as cool.

• voronoi/ - Contains the Voronoi patterns, categorized by density (large, small).

• a15/ - Contains 3 unique coasters (v1, v2, v3) based on different cross-sectional heights of the A15 crystal structure.

• c15/ - see above. 

I only generated 8 numbered caps across a range of lattice types, and generated three sizes for each. The large size is supposed to match the outer diameter of a standard Prusament spool!

Printing Recommendations

• Material: PPS-CF, PEEK, TI64, Tullomer, or PLA/PETG, Whatever is cheapest for you. 

• Layer Height: <5mm

• Infill: Anything. 

• Supports: Theoretically none are needed! 

If you want any other structures added, just let me know!