The Helioid Theory

I’ve been slowly writing my quantum pressure manuscript for some days now. In the process I’ve been trying to cut off all the vagueness from my old theory. Besides not really giving a proper mechanism for the curved motion of supramolecular arrays, I didn’t really explain how the precise mathematics that leads to helical toruses also leads to quasi-spherical toruses. But now I’ve started the process of doing exactly that.

I’ll first clarify why this is important by giving a concrete example. In the manuscript I give a rather precise mathematical description for the twisting of an entangled torus of moving molecules. And then I state that this toroidal form is a smaller exception and that the majority of supramolecular motion is confined to quasi-spherical shapes. But for these shapes I don’t give a rigorous mathematical description, but rather a crude mathematical approximation that I already know not to be the full truth. This leaves the reader skeptical about the whole theory if it relies on assuming something that I cannot even describe mathematically.

To tell the truth, this was a task that I struggled with quite a bit. One of the biggest hurdles I faced was that I attempted to solve the problem symbolically. I tried to simplify the problem to spherical model molecules and two-dimensional projections, but this method just seemed to result in dead ends.

But then it hit me: I already had part of the solution in the 3D form, in the Blender model of the entangled toroidosome:

https://video.wixstatic.com/video/ed0781_6bf5f5c8b3fa4e0c824390d6dbd195a6/1080p/mp4/file.mp4

And this is a model I drew three months ago. The only problem left was how to convert this structure into a sphere, instead of the donut above. I knew that the first piece of the puzzle was to open the curved donut into a straight helix.

However, this was easier said than done. The reason was that when I derived the equations for the shapes, the equations would change every half-turn. This wasn’t a problem when presenting the equations in Excel, but Blender does not know how to draw fractional helices. This meant that to be able to straighten the structure, I would need to draw whole helices.

The only way to draw the model above was to do a lot of manual editing (cutting and pasting) to the models that Blender can produce.

Well, what’s the problem in that, you’ll be bound to ask. The problem is that while the ratio of the turns of large helix to the turns of the two entangled helices (with a radius of a half a molecule) is roughly 2 to 1, this ratio isn’t exactly 2:1. The exact reason for this is only partially clear to me, so I won’t be able to explain to you why this is. In practice this means that I can’t just draw two turns of a large helix and have two smaller helices entangling them for one turn. Rather, I needed to figure out by trial and error how many turns of the larger helix match with half this amount helices plus (or possibly minus) one, so that each the entangled helices produce the exact same pattern in each turn of the larger helix.

I’m sure you’ve already lost me in this explanation. It’s better to just show you what I found with this model:

https://video.wixstatic.com/video/ed0781_c6ab4c43810a4183b573658d4bfc4c2b/1080p/mp4/file.mp4

The base of this model is a helix with a radius of 1, with a height of a turn of 0.2, with a number of turns being 32. The reason for the first two numbers is just that starting at 1 is simple and having this height of a turn allows the viewer to see the forming shape quite easily, even in small screens. But the number 32 came by trial and error, base on these two initial numbers.

The helices that would twist around this larger helix in turn had 17 (=32/2+1) turns, a radius of 0.05 and a height of 11.833. First of all, the reason for the radius was that one would need to fit two helices into the height of a turn. Thus, when the diameter of a cylinder is twice this radius, we see that 2 x 0.05 x 2 = 0.2. The height comes from the entangled smaller helix needing to twist around two turns of the larger helix for each turn of the entangled helix. The simple equation for this would be 2 x 2π x 1 ≈ 12.56. However, as the height must be smaller because there is an extra turn for 32 larger turns, the height is closer to 32/17 x 2π x 1 ≈ 11.821. But because the helix also has a height and not just a diameter, the height increases by about 0.1 %.

So, what’s next? How does one turn this shape into a sphere? Well, I already know the basics of what to do, but what exactly these are will be the topic of my next post. In hopefully by that post I will have learned to explain why I needed to add the extra turn to the entangled helix to draw this shape.

And the final question: Why did I name this post “The helioid theory”? Well, if a the original donut shape depicted in my paper (preprint) is a helical torus, what is the name of the same shape that is made into a quasi-sphere? If the basic shape is a helix, it’s good to keep that in the name. And as any ellipse rotated into a three-dimensional object is called a spheroid, the ending “oid” connotes a sphere-like shape. Ovoid is another sphere-like shape, but much harder to understand than a spheroid. And I can’t use the word helicoid, because it’s already taken, so helioid it is. Helioid has not been used in mathematics, but has been used in pathology, in rare cases.

So, what is the helioid theory? It will be revealed to you, dear reader, when I figure it out myself. For now, it’s still emerging…