The Octahedral Waterman Seed

In today’s post I’m taking a mathematical detour. In most of my posts I’m talking about topic relating to quantum gravity, and lately more specifically quantum pressure. But today, I’ll return back to Waterman polyhedra.

I think I’ve mentioned Waterman polyhedra a ton of times in my previous posts, but I can’t remember most of them. I think the most important post and possibly the first one was in the post “Lignin vs Water”, where I describe the crystallization of lignin nanotubules into spherical particles. However, that post was more about lignin and less about Waterman polyhedra.

So, what is a Waterman polyhedron? According to Wikipedia:

A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing (CCP), also known as the face-centered cubic (fcc) packing, then sweeping away the spheres that are farther from the center than a defined radius then creating the convex hull of the sphere centers.

While this is the accurate description of a Waterman polyhedron, this isn’t exactly the way I use the term. You see, both Steve Waterman and I came up with the idea when thinking about how to describe spheres using smaller spheres. But where this idea is used is quite different between the two of us. Waterman used the description to draw accurate maps, whereas I use it to describe the crystallization of lignin into spherical particles. This means that the original definition of a Waterman polyhedron is looser, where all of the geometric shapes created in the sweeping of a spherical hull are allowed. This video shows shapes from triangle at least up to dodecagon. However, in the case of lignin crystallization, not all shapes are equal.

To be specific, my original theory, and later electron microscopy (below) says that the short hollow lignin nanotubules crystallize into long linear nanotubules, meaning that the angle between neighboring crystal units is 120°. This means that if the shape is not defined by its convex hull, but rather by a simple seed polyhedron, with a tightly hugging hull of the same polyhedral shape on top of it, with each consecutive hull increasing in size, like a nesting doll.

Based on the Wikipedia site for Waterman polyhedron, I knew the general shape of this nesting doll to be a truncated octahedron, but at least for me it was difficult to figure out the exact rule for how this nesting shape without making it myself. So that is exactly what I did. I went to Blender and created a huge closely packed array of spheres and started slicing extra spheres off until I had created a truncated octahedron. But this looked a bit off, so I generated a sphere inside this octahedron, where the whole sphere fit inside the shape and I realized that while the shape was a truncated octahedron, there were extra planes, making the shape very non-spherical. So, I removed the planes that were totally outside of the spherical core and I got a shape made of 6 x 6 squares and alternating equiangular hexagons, with side lengths of 4 and 6. From here we see that the sides of a square are always attached to a hexagon, where the length of the side is 6, but the hexagons are also connected to other hexagons, with a side length of 4 spheres.

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

If we peel the truncated octahedral hull off, a nearly identical truncated octahedron is revealed, comprising of 5 x 5 squares and hexagons with side lengths of 4 and 5. That is, the size of the square is reduced by one as is the length of the sides in the hexagon attached to the squares, but the side length between hexagons remains four. And this is repeated in the next level truncated octahedron, with 4 x 4 squares and equilateral hexagons with all sides of four spheres. From here we can observe two rules. Rule 1°: each successive square, where the initial sides are n x n, becomes (n – 1) x (n – 1) in the hull below. Rule 2°: each successive hexagon, where the initial sides are n x 4, becomes (n – 1) x 4 in the hull below. This rule is followed up until the final shape is a square of 1 x 1 and hexagons with side lengths of 4 and 1. Only, at this point, while the mathematics holds, the shape no longer looks like a truncated octahedron. The shape has become a true octahedron. The reason for this is actually pretty simple, while the length from one sphere to another is one unit (simplified to 1), half of this length in the corners of the truncated octahedron lies outside of it. This means that for truncated octahedron with n x n sphere squares and n & 4 sided hexagons, the actual length of the sides is (n – 1) and 4 – 1 = 3.

But this isn’t all. After the truncated octahedron has become an octahedron, the shape has turned from six squares and eight hexagons into no squares and eight triangles, because an alternating equiangular hexagon with a side lengths of 4 and 0 is just a triangle with a side length of 4. However, we haven’t yet reached the core of the shape. You can still peel this hull and reveal and octahedron with each side having a length of two spheres, or one unit, because 2 – 2 x ½ = 1. And at the center of this octahedron is nothing, because the spheres at the edges of the octahedron fill the edges of the center.

Then, the next obvious question is over what happens when n becomes a very large number, or when more truncated octahedral hulls are added to the shape above? Well, the square sides increase in size predictably, keeping the exact square proportions. But the seed of the alternating equiangular hexagon is a has side lengths of 4 and 0, or more commonly just a triangle. While three of the sides remain 4 in each consecutive hull, the other three size grow relentlessly, first to and equilateral hexagon with all sides of 4 spheres, but then to shapes that begin to resemble dented triangles, where the second sides are much larger than 4. If there was nothing preventing this type of growth, already at n = 100 the shape would resemble a slightly dented cuboctahedron.

But here is where pure mathematics breaks down and physics kicks in. While this simple mathematical logic points to lignin crystallizing into a cuboctahedral shape, this is not visible in electron microscopy. Rather, it doesn’t seem that the square sides grow uncontrollably to form a cuboctahedral shape. Rather, it seems that there comes a point where the square sides stop growing, producing an obelisk shape, with a pointed tip and a rectangular prism base. This seems to be caused by adjacent lignin nanotubules fusing with each other, if the angle between the two is 60° (as seen in the microscope image), but not when the angle is 90°. Rather, when the angle would be 90 °, the lignin nanotubules rather crystallize with the nanotubules in neighboring planes, producing the obelisk shapes visible in the microscope image.

The funny thing is that art (or cartography) imitates nature, maps drawn with waterman projections follow this same logic. The base of the map is a truncated octahedron with the square sides rather small and the alternating equiangular hexagons large in comparison, resembling the shape with n = 2, or n = 3.

But how this more complex structure is described mathematically, will be a topic of another post. This should have introduced a sufficiently large number of new ideas already.