The answer is as simple as ... 1, 2, 3. If you guessed the first, second and third patterns ... you were right !
The reason is that those and only those have hexagonal symmetry, a sixfold axis. This means that if we rotate the object 1/6th of a whole turn as if it was a wheel, we would end up with a new orientation which would look identical to the original (aside from the little deffects added to make them look more real).
Drawings number 4 and 5 have diferent symmetries. Number 4 has an eightfold axis (octogonal). Number 5 a fivefold axis (pentagonal).
The fact is that snow crystals always grow forming hexagonal patterns. The shapes are infinite, they have endless variations, but the symmetry deep within each little speck is the same.
There was an american naturalist, a self-educated farmer in Jericho, Vermont, named Wilson A. Bentley (1865-1931) who spent 40 winters of his life studying the tiny little crystals that flock together to form snow flakes. He examined and photographed more than 5000 of them by adapting a microscope to a bellows camera (a true an extraordinary pioneer of photomicrography). He was struck by the beauty of those fragile patterns and recorded their shapes. He never found two of those intricate crystals exactly identical.
"Under the microscope, I found that snowflakes
were miracles of beauty; and it seemed a
shame that this beauty should not be seen and
appreciated by others. Every crystal was a
masterpiece of design and no one design was
ever repeated., When a snowflake melted, that
design was forever lost. Just that much beauty
was gone, without leaving any record behind."
Wilson "Snowflake" Bentley 1925
Modern day electron microscopes let us look at things like snow crystals with an amazing detail. The following are images of real snow crystals taken by the people at the Electron Microscopy Unit of the Beltsville Agricultural Research Center, Beltsville, MD. See below
The reason for the complicated shapes of many snow crystals is to be found in their process of growth. Under certain atmospheric conditions water will freeze and will quickly grow at the right speed to produce plenty of those little "branches" shown in crystal 2 (dendrites). With slower growing rates snow crystals might have simpler shapes like that of crystal 1.
The answer is to be found deep inside the crystals. We need to look at the atoms and molecules that form our snow crystals to understand. But that is so deep in fact that no microscope can be used to just look and see the answer.That is the reason why snow flake crystals are always hexagonal. The crystal structure at the atomic level marks the way these crystals will look. And this is not an isolated example !. The way things are at the atomic and molecular level is the kernel for the structure, toughness and properties of materials in general.
It's all right. Scientist have devised ways to "look" at atoms, not just with microscopes but also with many other tricks. There is for instance a way to get pictures of ordered arrays of atoms in crystals by shining X-rays on the crystals and recording the reflected rays. A suitable mathematical treatment of the measured data leads to information on the arrangement of atoms. This technique is known by scientists with the fancy name of "X-ray crystallography". We will call it "X-ray Vision", and it lets us "see" how Oxygen and Hydrogen atoms are bonded to form water molecules and how those molecules interact with each other in solid ice and snow crystals.
Look at this X-ray vision image of ice !.............
it is equivalent to a x100000000 magnification !
That´s right ! it's got the Hexagonal symmetry too!
In this image Oxygen atoms are drawn as red balls and Hydrogen atoms as smaller cyan balls.
All the bonds between them are drawn as white sticks in order to show the honeycomb network formed by all those molecules of H2O.
In liquid water there are H2O molecules too, but they are hotter than in ice. That means that they move freely and that is the reason why liquid water has no shape.
But when water freezes the forces of interaction between H2O molecules win over their (now reduced) thermal motion and they form a most stable arrangement... precisely with HEXAGONAL symmetry.
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If you want to learn more about snow flakes, find out other interesting facts and see more images of snow crystals I recommend you to visit the Snow Crystal Research page (Electron Microscopy Unit Snow Page, BARC) at www.lpsi.barc.usda.gov/emusnow/