Ball and stick components for rhombic dodecahedron packing
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Components for building a ball and stick frame of a rhombic dodecahedron packing, the Voronoi cells of a sphere packing.
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See https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres for a discussion of regular sphere packing and the lattices that result.
These components: cube and tetrahedral joints, along with equal-length struts, fit together into rhombic dodecahedron skeletons and can be continued into a lattice. Assembly is self-explanatory if you know the shapes you are building.
I put some effort into getting the struts to hold in place and maintain a rigid structure. They have a trapezoidal cross-section at each end, rotated to match the holes on adjacent lattice joints. Ideally, they should be triangular, but are flattened on the bottom for printing and on top to keep them symmetrical. The transition over the length of the strut is a smooth loft.
There is nothing special about the length of the struts. You should be able to scale them for larger or smaller constructions. The red and blue pictures at the end show 30mm pegs, which I stretched out lengthwise in the slicer before printing.
There's a tradeoff between getting the structs to fit and keeping them in place. I tuned them by trial and error for PLA in my printer. You might need to scale them a little. Because the size of the holes cannot be printed as uniformly as they appear, I have provided two optional tools. One is a tapered pin that can be pushed into a hole to widen it. Another is a cube that fits a strut on its diagonal, matching the joints. This provides extra leverage for struts that are difficult to insert.
There is a "deep" version of the pushing tool that I added after breaking struts while pushing. It holds about 15mm of the strut inside. The only disadvantage is it may get stuck when you first use it. I recommend printing and trying both.
This shows the parts and how they fit together.
I can assemble most of these without the tools, but they improve reliability. When struts are too loose, the result is fragile and hard to build on, though it is self-spacing and will hold its shape when complete. I recommend some experimentation.