The Tricolor Spherical Crystal

This past week I’ve been slowly polishing the visual representations of my theory, and in the process I was learning a bit more about how the structure is grown. I had already illustrated the shape in my post “The Perfect Cylinder-Sphere”, but I thought I could make the shape ‘pop out’ more. In the post “The Waterman-Lintinen Cylinder Crystal” I showed the inner structure of the crystal. However, I wasn’t fully happy with the way the coloring expressed the crystal layer. That model made it appear as if crystallization began at only one spot and continued in a single way throughout the shape. Then in “Exploded Waterman-Lintinen Hexa-Crystal” I took a different path on this quest and illustrated how the cylinder crystals form a spherical crystal from 36 almost independent crystal wedges. In today’s post I show how “The Perfect Cylinder-Sphere” is layered from tree slightly different types of layers to make the colors make structural sense.

Before showing the three-color sphere, I’ll first show the three-color cutout of the truncated octahedral shape. To make it, I first made the whole truncated octahedral shape and then selected exactly half of the spheres in the shape and moved them into their own folder(s) in Blender, allowing me to put the other half into their own folder(s). It is of note that the contents of the two folders weren’t identical, because a significant fraction of the spheres were on the y-z plane, which meant these central spheres would only be located in one folder.

Thus, when I hid the second folder, I could see the cutout of the shape by viewing it exactly along the x axis:

In the above shape there is a lot to see. First of all, at the core you see there is no sphere at the origin. Rather, there are blue spheres located along the y and z axes by a distance of √2r, where r = radius of the sphere. If the x-axis was visible, you would see all of the six spheres along the three axes. This shape is a regular octahedron. Overlaid on top of the blue spheres are black spheres that appear to form an octagon in the y-z projection, but when viewed in 3D, these form a truncated octahedron. As this is the first truncated octahedron of the shape, it also defines the angle where the hexagonal faces change to square faces. Looking carefully at the shape above, this angle is tan¯¹ 0.5 ≈ 26.57°.  However, only one third of the overlaid truncated octahedra have their edge along this line, which might be a bit confusing to the uninitiated.

After this easily defined black crust, there is a slightly more complicated yellow crust of sphere. The edge length of the planar spheres on the yellow crust are identical in size with the edge length of the planar black spheres, but the edge length of the non-planar square crystallites increases with each step. Though this might be hard to see from the cutout above, you’ll just have to trust me on this. After the yellow crust, I’ve built the next (fourth) crust again from blue spheres, because their basic rules repeat after three crusts. If you look at the windmill (or x) shape in the above picture, at its center each of the four ‘windmill blades’ comprise of three different colored pairs of spheres, after which the blade is widened into three different colored quartets of spheres, and then three different colored sextets of spheres and so on, until the surface of the truncated octahedron is reached.

While these windmill blades made of spheres along the y-z plane (each one being an edge of a hexagon) are easily visible, the military cross shape of the squar crystals are a bit tougher to see, as they are at a 45 degree angle towards the x-axis. Specifically at 45 degrees along the x-y plane for the vertical shape and at 45 degrees along the x-z plane for the horizontal shape. In the military cross shape, there are actually four planes for each three planes of the windmill blade, as the step where the windmill blade widens creates gap. And no, this isn’t just an optical illusion. There are truly a third more layers in the  square faces compared to the hexagonal faces.

And that is how to build a truncated octahedral crystal of spheres. But that isn’t the whole story. After building the truncated octahedral core, you also need the spherical caps to make the shape into a quasi-sphere. In the 3D model below toy see a yellow truncated octahedral core, as well as eight stacked hexagonal caps and six square caps that fill the full spherical volume. You can actually see that the hexagonal caps are thicker, with four layers, while the square caps only have two. Using three colors instead of two emphasizes the difference quite well, as the square caps have no yellow, while there is both an additional yellow and a blue layer.

https://video.wixstatic.com/video/ed0781_45962fefef0741ee8a539f04fc0c557d/1080p/mp4/file.mp4

It took quite a bit of ingenuity and advice from ChatGPT to be get my puny laptop to generate the shape, as it is quite complicated. That’s why there are a few holes in the shape. I did notice them once the model was ready and I had rendered the video (with Powerpoint), but knowing how my computer is already on the brink of freezing, I decided to leave them be. Even with the holes the shape should convey quite clearly how the shape is formed.

I can’t claim too much originality on the basic shape, as there are very similar examples online. However, at least ChatGPT says it can find no instances where each plane is colored with one color, with the neighboring plane having a different color. So this might be the first illustration of this kind. Which I would say is quite cool.