The Rhombus Twist

In today’s post I’m finally presenting quite a literal twist to the trypolite crystal. Actually, at first I was planning on just presenting a visualization of the seam in the dodecagonal pyramidal stack, but as I was writing the post, I realized I could do something even better.

But before that, let´s recap. In “The Dodecagon Pyramid Stack of Cylinders” post I presented the model for the formation of pyramidal stack of rotating cylinder, that would eventually collapse into dedecagons. However, in that post, I didn’t show the collapsed state. Conversely in “The Trypolite Crystal” post I showed how hollow nanotubular cylinders compress into rounded rhombuses within their own crystallite. but I intentionally didn’t even try to illustrate the seams between the crystallites.

I think there are decent grounds to suggest that these nice seams don’t exist within the spherical LignoSphere crystals. But then again, I know that I can definitely see the crystallites fusing, when I intentionally disprupted the collapse (or the self-assembly) of the hollow lignin nanotubules into spheres:

So, I do feel obliged to offer at least an example of a dodecagonal seam. In this first attempt, I make things a bit easier for myself and stack rotating cubes into quasi-hexagons, instead of cylinders. The reason for this is that introducing a bend to rectangular prism is easier in Blender than it is for a cylinder. Don’t ask me to explain why: I learner this by trying both ways and properly succeeding only with rectangular prisms. Conveniently, the rhombus cylinder in the trypolite crystal resembles a rectangular prism as much as it does a cylinder, so this simplification seems somewhat grounded in reality. 

https://video.wixstatic.com/video/ed0781_62823f97f48a4ef58e74316e38371941/1080p/mp4/file.mp4

For this illustration I began with a hexagonal array of cubes, where the cubes form a loose dodecagon. Then I collapsed the cubes that I could into triangular wedges, leaving the remaining cubes loose at the seams. Then I made a trick, where I first stretched the loose cubes into rectangular prisms and used a simple deform modifier in Blender to the, bending them into 60-degree arcs. This produced a relatively convincing dodecagon, but I wasn’t 100 % happy with it. It looked a bit too rounded and not as seamless as the hexagonal plane that I saw in the electron microscopy image. My next attempt was making a hexagonal prism by making a cylinder with 6 vertices in Blender. I then cut a hexagonal hole into this with a smaller hexagonal prism, making this a perfect hexagonal four-sided torus with 4 minor segments. Then I did the same for the smaller hexagonal prism and to the next smaller, until the shape was hexagons toruses to the core (with just a small hexagonal prism). Then I applied a bevel modifier to each, creating a much smaller bend, more closely resembling the electron microscope image above.

And the above would have been today’s post, but for some reason I was reluctant to publish it. The reason for this reluctance is that the nanotubules that make up the rhombus crystallites at either side of the seam are quite a bit more complicated than the bevel seen above. Indeed, only when I started tinkering with the model of the seam, did I realize that the seams have a twist. You see, half of the rounded rhombuses are tilted towards the center of the hexagonal crystal and half are tilted away from it. This means that if the initial state is a rhombus pointing upward, the seam will have a twist where one side points inwards and one side outwards. And to my surprise, I was able to illustrate this twisted seam in Blender, but only after a lot of tinkering and asking ChatGPT for advice and grumbling when the advice was incorrect and reasking until I was able to create this model:

https://video.wixstatic.com/video/ed0781_4057b527f1874824baba56638e47aeb4/1080p/mp4/file.mp4

It is of note that the above seam is not the same in the neighboring seams. Rather, the seams at either ends of a rhombus crystallite have opposite twists.

So, you might ask: the seam above looks mighty big, when the seams in the electron microscope image above look very tidy. This is because the presented seam model is only topologically correct. It still needs fine tuning to match the experimental data. Who knows, the tweaking of the model might still throw a new curveball.

Also, one might ask, is the length of each hexagonal torus an integer multiple of a specific length x, that follow the theoretical model, where the innermost true hexagon has a circumference of 6x, the second one 12x, the third one 18x and so forth? My answer is “it might be, but it doesn’t have to be’. You see, the only microscope image that shows the clear 120-degree angle between linear faces, isn’t a true hexagon. Rather, the symmetry of the shape is quite badly broken. There’s a smaller hexagon on top of it that might be a better hexagon, but it is so occluded by the obelisk shape above that it is quite hard to say anything too conclusive about the shape.

However, as there is relatively clear experimental evidence of the seamed collapse, the above model can be considered as a rough topological illustration of the general phenomenon. I still wish to reach a point where the rough model matches the electron microscope image. If I had to guess, this could take a day or two, or only minutes/hours, instead of weeks or months. But until I have the solution, I can’t be absolutely sure.