Exploded Waterman-Lintinen Hexa-Crystal

Note: I posted a version of this post earlier today, but because I wasn’t happy with how the 3D model looked, I rewrote the text a bit and made a recolored model.

In my last post I showed how to cut a cubic array of spheres into quasi-cylindrical zig-zag crystal. At the end of the post, I talked about how the shape can be multiplied into six copies that can be rotated to fill a circular volume with no gaps or overlaps. But it’s one thing to talk about something and another to actually show how it is done. To do this, I decided not to use the rather complicated actual crystal of spheres/cylinders, but to simplify it by cutting identical wedges out of a solid sphere. Note that this produces a ‘too perfect’ a shape, but for this illustration I don’t mind producing a perfect sphere.

So, first of all I made an identical zig-zag (in blue) cut as I did in my last post and aligned it so that it the groove with 45° slopes was aligned along the x axis. But unlike in my last post, I recolored the six wedge segments with three colors: the wedge with two 30° slopes I colored red and two of the four wedges with one slope 30° and the other slope 45° I colored blue and the other two of the four I colored yellow. This way no neighboring wedge would have the same color. Then I rotated it counterclockwise by 45°, so that one slope was horizontal and one slope was vertical. Then I copied the shape, colored it yellow and rotated it around the z axis by 180°, so that the two shapes would form a simple blue and yellow checkerboard-shape on either side. Then I copied the pair and rotated it around the y axis by 90°, at which point the shape would already fit without overlaps, but to align the colors I rotated the shape around the z axis by 90°. Then I copied the original pair and rotated around the z axis by 90° and around the y axis by 90°  (notice the symmetry) to obtain a full sphere, comprised of the six crystals of six wedges that fit perfectly, with no overlaps and no gaps:

https://video.wixstatic.com/video/ed0781_20b3072a776040d28580c9aa6f07ec9d/1080p/mp4/file.mp4

But the above model doesn’t just show the end result. Because from the final shape it’s difficult to see the individual components, I decided to turn it into an exploded-view drawing. Or actually a hybrid of the assembled shape and the exploded-view. First of all, I took each of the crystals and pulled them away from the origin, along the axis they crystallized. This allows the viewer to see each component crystal both in its correct orientation.

Also, as an added benefit, as I mentioned before the drawing shows that there aren’t just six crystals in the shape, but actually six times six, or 36 wedges, as each wedge is basically disconnected from its neighbor in the crystal of the same orientation. And in the same vein, this exploded-view diagram shows how both the square and hexagonal cylinder crystallites are formed. However, perhaps showing the striped surface used in my previous post would emphasize this effect.

I can’t believe how anyone, after seeing this exploded-view diagram and the cylinder crystal of my previous post, would have the audacity to say that there’s no evidence of such shapes, especially when this is paired with an actual electron microscopy image clearly showing signs of this shape. But knowing the history of science, the evidence necessary to change paradigms is extremely large, as evident in the story of Ignaz Semmelweis. I actually wrote a long post on the topic almost exactly three years ago. I predicted that the paper can be rejected outright, because the editor or peer-reviewer says “that the manuscript takes too large hypothetical steps”. In the post I said that this phrase is the academic equivalent of “have you stopped beating your wife?” The phrase might sound neutral, but it really assumes that what is said is wrong without offering any proof of this. It is left to the author to guess what they could do to convince the reviewer, but the reviewer cannot be bothered to state which sentence or equation the reviewer objects to.

The last time I talked about the topic, I wasn’t actually talking about the theory of the structure of lignin, but about the theory of the elementary particle of energy. Back then I was so correct about the impossibility of publishing the theory that even after three years, I’m no closer to getting the theory published. The main reason for this is that unlike with lignin, I have no compelling experimental evidence that could satisfy peer reviewers. The problem always was that the only self-collected evidence I had related to the structure of lignin and my theory on its structure. And everything else was deduced from already published experiments. And what I have learned is that chemist are highly skeptical of theory without experiments and physicists are highly skeptical of theories created by non-physicists, using non-physicist terms.

But this time there is at least some hope. The manuscript that I’m about to write will deal with an industry already worth half a trillion dollars. So there is significant benefit in knowing what the structure of lignin is. On the other hand, I still have to be realistic and embrace resistance and cries of “Not enough evidence!” and “Too hypothetical!”