Waterman TIE Fighter

In my previous post I presented the simple theory of the seeding and growth of truncated octahedral Waterman polyhedra. That is, how to make a sphere of spheres, where the core of the structure is a tiny octahedron of spheres, on top of which a second slightly larger octahedron of spheres is deposited, after which the next layers are truncated octahedra of very specific dimension. Like this: 

https://video.wixstatic.com/video/ed0781_b70c5768fb824a98b7f46ac520f5dd57/1080p/mp4/file.mp4

However, this model assumes that the structure grows gradually into a dented cuboctahedron, which isn’t really that spherical in comparison to the truncated octahedra seen above. In my previous post I mentioned that the microscope images of lignin crystallization into spheres indicates that the square layers, that would turn into pyramidal structures are actually obelisk-like instead.

This means that the simple growth model stops at some point (after making the pyramidal tip of the obelisk) and turns into one that produces a rectangular prism (the body of the obelisk). After some thinking and playing around with the model in Blender, an obvious solution presented itself. If instead of the growing the hexagonal and square planes at the same speed, if two square planes are grown for each hexagonal plane, this allows the hexagonal planes to grow from the hexa-hexa connections and the side being connected to the squares remaining the same length in each layer. I.e. the same as in the old model, but with the non-growing side flipped from the hexa-hexa connection to the hexa-quare connection. Below is the new model applied to the final step side-by-side with the old step. While the hexagon of the old step is not valid (or is embedded into the larger hexagon of the new step), the blue 3 x 3 inner square of spheres actually exists below the 2 x 2 inner square of the new model. This means that the obelisk is actually ‘serrated’, its thickness undulating by a single layer of spheres. This actually explains why the obelisks in the electron microscope image has something snaking around it: the extra half-layer of spheres is too loose to crystallize into the obelisk body, so it ends up winding around it.

If one looks at the microscope image, the obelisk is about 35 nanometers thick, meaning that with lignin nanotubules of 8 nanometers in diameter, the base is about  5 nanotubules in diameter, assuming that the hollow nanotubules don’t collapse into smaller shapes (such as hexagons). And as the nanotubules on the edge of the shapes will inevitably become part of the hexagonal crystallization, in lignin nanotube crystallization into colloidal spheres the pyramidal tip will grow quite a bit (probably to a square base of 10 x 10, or at least 9 x 9 (with the inside being 8 x 8 or 7 x 7), until the transitioning into the obelisk body stage.

To illustrate this obelisk-type crystallization I decided not to plot each and every sphere of the shape, but instead I only plotted three different planes at the boundaries of hexagonal crystallites. The spheres at the interface of the hexagonal crystallites are colored green, whereas those that eventually crystallize into obelisks are colored blue. And at the center of the shape, there is the pyramidal seed created for my previous post. Because of how this shape looks like, I’m calling it a Waterman TIE Fighter. The resemblance isn’t very close, but there’s definitely a spacecraft vibe to the model.

https://video.wixstatic.com/video/ed0781_8306be983a4d4616826a390bd5840c4a/1080p/mp4/file.mp4

One might imagine that this theory would be easy to publish. However, I have my doubts. The problem is that I have no idea how to publish papers in mathematics, because I’m a chemist. But this is too theoretical as a normal chemistry paper. So, the only reasonable way for me to publish this would be to introduce the theory into my already over-bleated “theory of everything” manuscript. Unless I’m never going to turn the preprint to an actual paper and instead prepare a completely separate quantum pressure paper and add these mathematics into that paper. This way I would actually have experimental data for the theory to explain. Of course, this poses the problem that reviewers always say there’s not enough data to warrant such a huge theoretical leap.

But perhaps I’ll just try my luck. Who knows, perhaps someday my theory will finally be published. As a curious side note, when I did an image search for my Waterman TIE Fighter, I did find an article in a Physical Chemistry journal with pretty much the exact shape, but not hollowed out (the exact picture is behind a paywall: google search shows it separately, but clicking the link shows the abstract page, with different pictures).