The Proper Dodecagon Seam

 In my last post, “The Rhombus Twist”, I presented the topological model of forming bent, twisted, seam between two cylindrical segments, where both have been compressed to rounded rhombuses, both where one of the rhombuses has been twisted clockwise and the other one counterclockwise. However, in the post I confessed that while the model was topologically correct, it didn’t quite match the electron microscope image that was the reason why I felt compelled to make such a model in the first place. So, in today’s post I’m presenting a ‘Proper Dodecagon Seam’.

But before I present the model, I do have to say that I have a strong feeling that the complexity of the model is a strong indication that when lignin nanotubes (which are at the heart of the theory) collapse into an actual sphere, the pressure is so evenly distributed that these rhombus twist might never form. Rather, the formation of the tight corners seen in electron microscope images, might be just a function of the looseness of the structure, allowing the otherwise improbable shape to form.

Let’s recap what the rounded rhombus model is. In the “The Trypolite Crystal” post I presented that if you start with a closed pack layer of cylinders and compress these, you deform them into rounded rhombuses:

This allows the packed (hollow, despite the model above indicating otherwise) cylinders to take up less space. One might ask why rhombuses and not squares. This is actually a question I asked myself, but realized that the answer was isotropic pressure. When the collapse of short hollow nanotubules take place their arrangement is such that isotropic pressure produces rounded rhombuses. You might think that maintaining a rhombus shape takes more pressure than a cylindrical shape and shouldn’t the rhombuses relax to cylinders. The problem here is that the shape generated is six interlocked crystallites that would all deform in different directions were pressure to be relaxed. The only way for the shapes to relax would be to rearrange the crystallites. And I actually think this is a valid point: you probably need a bit of rearrangement for the shapes to be optimally stable. But this would require physical chemistry, not mathematics, so let’s assume this doesn’t happen in our case.

Next let’s compare the case where the cylindrical crystallites don’t fuse, but exist as separate crystallites. This is what we see in the left hexagonal ‘supercrystallite’ in the 3D model below. It is of not that there is always a cylinders at the seam of two crystallites that can’t be part of both, so in this model I’m adding the cylinders to the neighboring crystallite so that the shape forms counterclockwise twist. But if we apply pressure so that each of the crystallites are compressed against the neighboring crystallite, you’d get a bent cylinder at each of the seams. But I ignore this step and show what happens when the cylinders around the seam are compressed, so that half of the cylinders rotate clockwise and half counterclockwise. This way, the cylinders on opposite sides of the hexagon are (rather surprisingly) aligned towards the same direction when observed  from above, despite being twisted in the opposite directions, if viewed along the hexagonal torus. This  concept can be slightly hard to understand in words, but if you just look at the model closely, you see that the rhombuses at the opposite ends of the hexagon are indeed tilted towards the same directions, despite their opposite twist with their neighboring rhombuses.

https://video.wixstatic.com/video/ed0781_6c7632e9db464f4a9878b7dfa366e0eb/1080p/mp4/file.mp4

And while I still wouldn’t call the above model exceedingly pretty, one can imagine that if the resolution of the electron microscope image that served as the inspiration of the model was good enough, one might even observe some new details from this shape:

I was about to end this post with something a bit more trivial, but I realized that the twist that I’m talking about is probably very much linked with quantum pressure: a topic I’m also developing. Pressure is generating torque to the cylinders in the crystallite, generating the twist that is required to form the compact rounded rhombus shape. This means that at least indirectly my electron microscope image of lignin hexagons is the first evidence of the type of quantum pressure my theory presents.

The problem is that when I finally submit the manuscript for peer review, I’m expecting resistance from both editors and if I ever get to peer review, then from them as well. I’m sure they would want the evidence of quantum pressure to be something fancier. And conversely, reviewers of my theory of lignin crystallization will probably consider my theory of quantum pressure to be too fancy to explain something as mundane as the formation of lignin spheres. But I’ll have to try, even though it’s not going to be easy.

And a final comment about the name of this post. I was almost about to rename the post “The Proper Hexagon Seam”, as the final shape looks more like a hexagon than a dodecagon. But I decided to keep the original name. I think the fact that there is a dodecagon hidden behind a hexagon is a nice allegory on how looking at things with new eyes allows one to find new insights.