De Econometrist neemt een statistische kijk op de wereld.
It’s cold outside and the last couple of days it has been snowing heavily. Some people hate the snow and others love it, but I think we can all agree that a thick layer of snow gives some extra brightness to the world, especially in these dark winter months. This beautifulness of snow does not only apply to the new world it creates, it also holds for just a single snowflake. Most of us have seen a picture of a snowflake (or equivalently, a snow crystal) and the mesmerizing shape it has. But why does a snow crystal has this six-fold symmetry pattern? Why doesn’t it have a shape with five sides or four? Or why just no specific shape at all? Furthermore, this six-folded shape seems to show up even more in nature. Think of beehives, the eyes of a fly or if you look closely to the pattern created by bubbles in foam. Is there some link between all these natural phenomena?
To find out why snow crystals have six sides, we must have a very close look at a snow crystal. Actually, the answer lies on the level of molecules and atoms. If you could zoom in on a single snow crystal, you would see that it consists of hexagonal rings. Such a ring consists of six water molecules and a single water molecule consists of an oxygen atom and two hydrogen atoms. In the figure below you can see a small section of a snow crystal. If such a ring is rotated by 1/6 of a turn, it still looks the same. Hence at this level we also have the six-fold symmetry like for the crystal in general. But why do these molecules form a hexagonal pattern? And is this always the case? We now have the exact same question as for the crystal, but on an even smaller level.
The common explanation for the hexagonal shape is that it is caused by the shape of a water molecule. The shape forces the molecules to link up into the six-sided rings. However, this answer does lead to two new questions.
Firstly, water can form into 14 different shapes of which only two shapes result in the hexagons, so why does it always result in the six-sided shape? The reason for this is that the conditions in which we live, are the ideal conditions in which the molecules favour to form the hexagonal rings. Neither very low temperatures or externally applied pressure is needed to get this shape.
Secondly, what are these forces that place the molecules into the rings? These are electromagnetic forces. Each water molecule has four poles: two positive poles, which are given by the hydrogens, and two negative poles. These negative poles are predicted by quantum mechanics, but for simplicity we will not go into any further detail on how this works. Positive poles of a molecule attract the negative poles of a different molecule, forming a bond. This happens for all four poles of a molecule. As a consequence, each molecule bonds to two other molecules in the ring, as shown in the figure below, as well as to a neighbouring ring and to a ring either above or below. Since the angle between two poles on an individual molecule is approximately equal to the corner needed for a hexagon (120º or 60º, depending on the way you look at it), we are left with the hexagonal structure. You could say that this arrangement of the molecules results in the most efficient structure.
So it’s clear that the water molecule’s shape produces the microscopic hexagons, but how can this result in the crystal having the six-sided shape, as it is enormous compared to just a few molecules.
In a structure of molecules, the molecules on the outside are pulled towards the inside. So the molecules on the surface of a crystal are pulled towards the inside of a crystal. The forces of this “pulling” depends on which parts of the hexagonal rings make contact with the surface. Places where only one molecule touches the surface (red dashed places in figure below), the surface grows relatively slowly compared to the places where two molecules of a ring touch the surface (green dashed places in figure below). So places with one molecule touching the surface experience a larger force when getting pulled to the inside. As a result of this, when the crystal starts growing, the places with two molecules touching the surface grow faster (like in situation B below) while the slow growing places stay on the outside. Eventually this leads to the hexagon shape as given in situation C in the figure below. Since only places of the rings with one molecule (which could be called the corner of a ring) touch the surface, we are left with the six-sided shape and not with any other kind of shape.
Finally, to obtain a snow crystal which has the typical “symmetrically” six-sided shape and which is large enough to see, three things need to happen. Firstly, the crystal must grow thin and broad, meaning that the top and bottom faces must grow much slower than the six side faces. Secondly, the corners of the crystal must grow faster than the rest of the six faces to create the branches. Lastly, all branches should grow at nearly the same rate. If these three things happen, you obtain the snow crystal with a hexagonal symmetric shape.
So, we now know why a snow crystal has its hexagonal shape, but how does this relate to the beehive, the eyes of a fly and the structure of the bubbles in foam? Do these other three natural phenomena have a similar reason for the presence of the hexagons? The answer is yes, they all have the same underlying reason why they have a hexagonal pattern.
The hexagonal shape in nature is a result of nature trying to be as efficient as possible. To explain what this implies, let us take the bubbles as an example. A single bubble is always a sphere, as this is the object which takes the least amount of surface. If you deform a bubble it will always return to the most efficient form it can take. If we have an enormous field of bubbles, then if all these bubbles are round, they do not cover all the needed space efficiently. Hence the bubbles take on another form to optimally cover the field it needs. This optimal form turns out to be a hexagon! With some slight alterations but in general the same principle, you can also explain why beehives and the eyes of a fly have a hexagonal structure. The underlying reason for all these things to form into the most efficient structure is the molecule composition of the phenomena, which was also exactly the reason why snow crystals are six-sided!
Jon Nelson, How the crystal got its “six”. https://www.storyofsnow.com/blog1.php/how-the-crystal-got-its-six
This article was written by Casper de Vries