The Perfect Cylinder-Sphere
- Kalle Lintinen
- Aug 6
- 3 min read
It’s been nearly two weeks since my last post on The LignoSphere Cylinder Crystal, where I presented the mathematically accurate way to crystallize cylinders into quasi-spheres. Ever since then I’ve attempted to improve the theory, so that I can more closely describe the electron microscopy images that have guided me in working on the theory in the first place. One of the biggest needs for improvement has been how to make the model to produce more spherical crystallites. And today I will show how I managed to do this.
First of all, I probably would have been much faster at coming up with the accurate model were it not for my memory/processor heavy way of testing the model. To be able to test the model with spheres roughly the size of a LignoSphere, I needed the model to be about 75 cylinders tall, wide and deep, which means a starting point of a cube with 75³, or 421875, cylinders, which I would carve to the right shape. But because of Blender, whenever I went past about 1 000 objects, Blender started slowing down considerably and already at 10 000 objects the program would freeze, and I couldn’t do anything anymore.
So, I had to learn all kinds of tricks to reduce the memory load. One of the earliest ways was to use very rough icospheres, with minimal number of subdivisions and only introduce cylinders when the shape looked right. That way I wouldn’t need to rotate any cylinders, as that’s a very memory/processor heavy task, when done for thousands of objects. And then I stopped trying to make the whole model, but instead looked at the FCC packing structure of a small Waterman polyhedron, and extrapolated the basic orientation of the cylinder/sphere lattices and the locations where the cylinders could be. Thus I could make a spherical/circular surface which would define the edge of the cylinders/spheres.
And next came the most important mathematical trick: I cut the sphere into ‘curved fish shapes’ that is defined by first dividing the sphere into a piece-of-eight, or an eighth of a sphere, and then cut this piece-of eight into thirds with tilted circles, located at exactly the right place, so that when three identical thirds are overlayed and rotated by 0°, 120° and 240°, they make up an eighth of a sphere.
While the original shape consisted of just spheres, the way I turned them into cylinder-crystals was by removing all but the spheres on the edges of linear arrays of spheres and replacing these with a cylinder of the same length. Or actually plus one, so that the cylinders reach the center of the edge-spheres. And this is what the pieces-of-eight as projections:
But this doesn’t look like a sphere, you might say. Well, I copied the shape into a half-circle, then a full circle and then copied and rotated by 90 ° into this shape, which I don’t even know how to name:
I guess someone could name it after me, but I think I can’t do it by myself.
And if I copy this shape twice and rotate the first copy around one axis by 90° and the second copy around another axis by 90°, these three shapes make up a non-overlapping spherical shape. Like this:
And in this shape, unlike in my previous post, there are loose cylinders, which are actually required, as my electron microscopy image clearly shows that the crystallization shape isn’t a clean hexagon, but a rather messy shape, just like in the model above.
So, the theory is ready. Also, the obelisk part that I talked about in my previous post, but I’ll leave it out of this post. Next, I should write this up into a manuscript. However, the million dollar question is whether it should be a separate paper, or whether I should include the theory of quantum pressure into the same manuscript, as the theory explains why biomolecules should form cylinders at all. However, it seems that adding too many new things into one paper will distract from the main point of the paper. Perhaps I’ll just write the theory into a preprint and publish it in ChemRxiv, just like my previous paper. And if I get it peer-reviewed, or better yet published, it’s just extra.
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