The Waterman-Lintinen Cylinder Crystal

In my last post I presented The Perfect Cylinder-Sphere, or a way to crystallize cylinders into a sphere. However, in the post I only presented the crust of such a crystal and led the reader imagine how the crystal structure continues to the core of the sphere. However, even when posting the model, I knew most people would be able to imagine the internal crystal structure. The biggest reason for knowing this is that despite knowing the structure of the crust, I couldn’t properly envision the crystal structure inside the sphere. But in today’s post, I take on the challenge and build it, one cylinder at a time.

Because I knew the structure of the surface and I knew the basic rules, the task of building the crystal wasn’t really a mathematical challenge, but just a task of labor. In my last post I describe the basic rules, but I can repeat them once more. I made a cubic array of spheres and then drew a larger (52 times) sphere located at the corner of the cube. Then I deleted all of the spheres that wasn’t primarily located within the sphere. Partially located spheres, with more than 50 % of the sphere located within the larger sphere were kept in. At this point one of the square faces of spheres was pointing up. Then I rotated the eighth of a sphere of spheres by 54.736°, so that a hexagonal face was pointing up. The exact reason for this is a bit too mathematical for this, with it allowed me to split the structure in ‘butterfly wedges’, represented by this shape:

After that, the only thing left was tedious work. I would only learn any other rules helping in the process after creating the shape in the most tedious manner imaginable. But now that I have the final shape, I won’t me remaking it with the extremely inefficient original manner. After seeing the final shape I found all manner of tricks to make the process much easier. But I’ll probably leave this tricks/rules to the eventual manuscript.

But this wasn’t the whole method. Once I had generated the basic shape with tiny spheres, I replaced most of them with cylinders. I only left the spheres on the edges, so that the shape can be copied and rotated to create a uniform shape with complete hexagons and squares. After this process of ‘cylindrification’ the only thing left was to mirror the shape twice to form a zig-zagged cylinder that looks like this:

https://video.wixstatic.com/video/ed0781_6bc7fe1f96ad49d1b950816ddf596eec/1080p/mp4/file.mp4

The most notable features are that only the quasi-cylindrical edge exposes the sides of the (smaller) cylinders, while rest of the shape shows the ends of the cylinders (or spheres in this model). There’s a clear reason for this. The linear crystals end the surface of the zig-zag and face the tips of other zig-zagged cylindrical crystals, either at an angle of 60°, or at 90°, depending on the location of the crystal interface.

And this base cylindrical crystal is made into six identical copies, five of which are rotated along the three grooves of the zig-zag. One copy is rotated by 90° in the groove, making up the square crystallites. And two copies each are rotated by 120° and 240° along the two groove that make up the hexagonal crystallites. And by these five rotations the six zig-zag cylindrical crystals fill up a spherical volume. And the maximum surface roughness of this cylinder-sphere for this 52 cylinder-diameter (26 radius) cylinder sphere is 100 % / 26 ≈ 3.8 %, because of the rule of cutting all of the small spheres/cylinders that extend more than 50 % beyond the surface of the sphere. And the average roughness is significantly below this.

And as a final point, you might ask why I use two different colors in making this shape. The answer is pretty simple: with only one color it would be extremely hard to visually follow where one plane of cylinders ends and another one begins. Two colors reveal thee internal structure of the shape much better than just one.