“There was a footpath leading across the fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more about mathematics.”
-Bertrand Russell, Nobel Laureate and Mathematician

It is mystical when you step outside on a snowy morning. Snowflakes are swirling around the vast sky and falling and blanketing the ground. If a snowflake lands on you, it is like a winter angel. There are no flowers around, for they cannot survive the cold; yet what lies before your eyes is an incredible beauty. And it’s remarkable, you come to realize, that no two snowflakes are alike. It is as if the uniqueness of a snowflake is controlled by a divine force. The individuality of a snowflake’s structure draws a parallel to human life. Like snowflakes, everyone has a unique story to tell.

I am not the only one who ponders about snowflakes; many mathematicians do the same. Actually, they think about the characteristics of snowflakes because they are particularly important for three basic mathematical principles: pattern, symmetry, and symmetry breaking.

A little-known scientist, Wilson Bentley, a.k.a. “the Snowflake Man” took pictures of snowflakes almost every day and observed them until he died. You can buy his book about his work on Amazon. If you wish to know why he did it, read about it at snowflakebentley.com.

“Under the microscope, I found that snowflakes were miracles of nature; 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. That beauty was gone, without leaving any record behind.”
-Wilson Bentley

When I checked the Oxford dictionary, there were 3 definitions for the word “pattern.” Two of these definitions [listed below] are important for this article.

Pattern: 1. A repeated decorative design; 2. An example for others to follow.

When we check the pictures and delve deeper into each snowflake, we will see that the structures of the snowflakes are totally different. However, they have something in common: symmetry and a hexagonal structure.

These perfect ice crystals are genuine, even though it is hard to believe they are not fake.

When I take a close look at a snowflake, the beauty of the combination of ice molecules fascinates me every time; each flake is unique. However, uniqueness is not the point here. The things that make snowflakes important objects for mathematicians are their symmetry and their hexagonal structure. Math-loving people have a lot of interest in transformations. They love moving objects. And, surprisingly, if an object is symmetric, transformations are not even noticed by many.

To be more precise, when you have a hexagonal symmetric snowflake, or any other symmetrical object, when you rotate it in any direction, 60°, 120°, 180°, 240°, 300°, or 360°, people watching you wouldn’t realize it. If you check the images below, you will see rotated shapes but no difference. It appears to be the same shape in exactly the same place.

1- Counterclockwise rotation by 120°

2- Reflection through a vertical axis

3- Reflection axes of a snowflake[1]

Snowflakes also possess reflectional symmetry. If we stand in front of a mirror, our reflection looks exactly the same. Hence, if we put a mirror in the middle of a snowflake, there will be a reflection. For a snowflake, we can put a mirror 6 different ways. Thus, we can say that a snowflake has 12 symmetries: 6 from reflections, and 6 from rotations.[2]

Now we can define symmetry as a transformation that leaves things unchanged. We can also claim that a combination of any of the transformations will give us exactly the same shape. For instance, we can rotate our snowflake 60° two or three times in a row and flip it over, and it will remain unchanged.

At this point, you might ask the question: “You have all these fancy symmetries for this particular snowflake. But, does every snowflake possess the same symmetries?”

Snow is a molecular structure of an ice crystal. And ice is a structured substance. It is a different form of water. When the water cools down, the molecules move more slowly, and this begins to impact how the molecules line up. Hydrogen atoms of one water molecule bond with two oxygen atoms. As the water freezes, the molecules arrange into hexagonal patterns. They prefer to stay as far away from each other as possible, and that makes them take up more space. The large space affects density. The density of ice becomes less dense than water. This is why ice floats. Almost all other liquids have a higher density when they freeze.

When we examine an ice crystal carefully under normal conditions, we always see a combination of molecules with six-fold symmetry. Snowflake molecules make a honeycomb structure. This results in an inordinate amount of hexagonal symmetry in these molecular three-dimensional structures.

Okay, we saw the structure of a snowflake under normal conditions. But, what if we changed those conditions? Johannes Kepler answered this question after his experiments and wrote a book about snowflakes, particularly The Six-Cornered Snowflake.

There are two key elements which affect the structure of a snowflake: temperature and moisture. Each time the temperature or the amount of moisture change, the structure of a snowflake changes. If you check the snow crystal morphology diagram below, you will see that when the temperature nears 0° and humidity is high, the structure of a snowflake will be flowery. Flowery structures are called dendrites. When you make it a little bit colder, the structure will be fancy hexagonal plates. We can apply many combinations and get varying structures.

The Snow Crystal Morphology Diagram. Source: Snow Crystals – http://www.snowcrystals.com/science/science.html

Professor of physics Kenneth G. Libbrecht is the owner of the diagram below. In a PBS interview, he said, “It’s a mystery as to why snowflake shapes go from plates to columns to plates to columns as the temperature lowers. That’s one of the things I’ve been trying to understand. It has been a mystery for about 75 years, and it’s still unsolved.”[3]

The Shapes of Snowflakes | Source

In the end, although the structure of (almost) all snowflakes are the same, some of them are not completely hexagonal. For instance, there are some snowflakes that have tree structures. Some snowflakes have branches, and each branch has tiny branches.

But, why is the structure of some snowflakes not hexagonal?

So far, we have talked about pictures which were taken at a particular instant. We have seen the pictures of the motion of the snowflakes for the smallest amount of time that can be measured. However, a snowflake never stops spinning in the air. They tend to oscillate. That means the shape of the snowflake is changing all the time. But how? When you see a snowflake in the air, it changes its place after a second because it would be whirled about, and it will be under different conditions at that time. This process will occur up until the snowflake lands on the ground. We know from the diagram that the temperature and amount of moisture always affect the shape of a snowflake. While small-scale conditions are almost the same, on a larger-time scale, conditions will differ. And these differences will change every corner of a hexagonal snowflake, resulting in a different structure. This is the main reason behind the variety of snowflake structures and uniqueness.

In conclusion, we can say that a snowflake can preserve its six-fold symmetry at all times. I think we have another reason to love mathematics! I want to finish my piece with Hermann Hankel’s words:

“In most sciences one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.”

Carbon atom, which composes nearly 0.2% of the earth’s crust, has a very special place among elements. All living beings are formed from carbon-based compounds. In addition, 94% percent of compounds—that is, more than 4 million—contain carbon atoms. Certain carbon compounds form nearly 18% of the matter in living beings. The rest is mostly water. These compounds are used as building blocks in cell formation. Only carbon has the ability of compounding with other elements in sufficient variety and complexities in order to carry out the main functions that life is based on. As carbon atoms easily form chains by making chemical bonds, no other element is given such a quality. These chains formed in a line can separate into branches and can connect to form rings. These rings are actually polygons formed by three, four, five, six, or more carbon atoms. Due to this quality, carbon is the element that forms the basis of many things from the foods we eat, to the clothes we wear, from the fuels we use, to the furniture we have.

The heat level that makes carbon compounds possible is between -20 and +120 0C. Carbon compounds begin freezing at -20 0C and they begin breaking at 120 0C. In space, where extreme heat and freezing cold exists, the only heat range that makes carbon compounds possible is found on earth and this is a very sensitive heat range. Temperatures in our neighboring planets give a better idea: as hot as 450 0C in Venus, and as cold as -53 0C in Mars. Under these temperatures, it is impossible for carbon element to form compounds and thus living beings. Therefore, the earth is the only planet created with the suitable conditions that make life possible.

Crystal structures of carbon atom

Particular repeating arrangement of atoms in three dimensional space to form a certain geometric shape is known as a crystal structure.

Different crystal structures of the same substance are named “allotrope.” Carbon has three different allotropes found in nature: amorphous carbon (coal), graphite, diamond. In addition to these, an artificially produced allotrope is fullerene.

Amorphous carbon (coal)

Amorphous structure is one without a definite crystal structure; that is, one where atoms take their places in free order. A mass of carbon atoms of amorphous structure is known as coal. After plants die, they undergo chemical transformation with the activities of microorganisms. If dead plants collect in a suitable wetland and are buried into the ground with a geological process, the carbon amount in their body increases and they begin transforming into coal. Types of coal are categorized according to the carbon percentage they contain. Geologically, this transformation process takes a period of 15 to 345 million years. Coal is one of the most commonly used forms of energy.

Graphite

In graphite, carbon atoms are found in a hexagonal crystal structure. These sheets, resembling the surface of a honeycomb, pile up and form graphite. As the sheets are not connected with firm bonds, they easily shift when some force is applied. This is why graphite is used for eliminating friction at machine industry. The black substance in pencils is graphite hardened by adding some clay. Graphite can resist very high temperatures. Therefore it is used within the steel industry and melting metals. In addition, it is a very good conductor of electricity. For this reason, the brushes of the electric engines in household machines such as a washing machine and a vacuum cleaner are made of graphite. In recent years, graphite has been used as heat shields of space shuttles.

Diamond

Diamond is the hardest natural substance we know. In spite of being a transparent substance and having no color of its own, it can be found in pastel colors such as yellow, brown, or even dim black, owing to being mixed with other minerals. Diamond is a perfect electric isolator and is the substance with highest heat conductivity. For this reason, it can be cut without being deformed. In diamond, carbon atoms are found in a pattern to form a cubical crystal structure. Extraordinary resistance of carbon-carbon bond and its hard and integrated structure prevents its reacting with other things around. It burns at a heat of 850 0C. In a piece of diamond, there can be other atoms that cause impurity and decrease the value. In good quality natural diamonds, there is only 1 alien atom versus 100,000 carbon atoms. In addition to jewelry, diamonds are widely used at industrial products such as drills, glass cutters and the like. 75-80 % of diamond production is used in this industry.

Formation of coal, graphite, or diamond from carbon

Carbon based organic compounds were buried underground as a result of movements by the earth’s crust millions of years ago. Physical and chemical changes occurred with those organic masses through heat and pressure. Gradually, water, carbon dioxide, oxygen, and—in the highest phases—hydrogen leaves these masses. This organic matter called “turba” (first transforms into lignite, then to sub bituminous coal, then to bituminous coal, and then to anthracite). If the conditions allow, it transforms into graphite. Coal is the first type of substance formed by carbon atoms on their journey to become diamond. As lower values of heat, pressure, and time suffice for coal formation, graphite requires much higher values. Diamond is formed in the mantle layer of the earth at about 150-200 depth. This valuable substance is later carried to the surface of the earth by volcanic rocks such as lamproite and kimberlite. In order for diamond to form, an atmospheric pressure of 50,000 atm, 2,400 0C of heat, and a period of 3 billion years are required. Without this immense pressure and long time, the substance to be formed by carbon will simply be graphite. It is possible to transform graphite into diamond artificially; however, according to calculations, a minimum pressure of 10,000 atm is required. In 1955, for the first time artificial diamond was obtained under 100,000 atm, 2,500 0C heat, and by using chrome as catalyzer. However, the pieces of diamond obtained were small and black, most of them did not classify as jewels. In another attempt made in 1962, graphite turned into diamond under 200,000 atm, 5,000 0C heat, without using any catalyzer.

The similarity between carbon and human beings

As carbon atoms’ properties change according to the crystal structure, people’s lifestyle and view of life depends on the community they live in and their position in that community. In order to become diamond, the highest level of its kind, a person needs to undergo hard conditions. If carbon atoms were to say, “this is more than we can bear, we prefer easier conditions,” then they can be nothing more than graphite. If the conditions for graphite are avoided as well, then one cannot go beyond the level of coal. Diamonds are kept in safes and worn in most important occasions and graphite has different kinds of practical use as an industrial material. As for coal, it ends up in fire. The situation of human beings in a way resembles carbon atoms. Every person is made of the same elements biologically. Their value will naturally be different, according to the processes they underwent and the behaviors they presented. Some show patience in the face of misfortune, put their sincere trust in God, and become the diamonds of humanity. Some others, whom we can compare to graphite, attain a desirable level even if they cannot become diamonds. Those who choose to assume the lowliest form are likely to face a similar fate with the coal.

References

• H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl ve R. E. Smalley. 1985. “C60: Buckminsterfullerene.” Nature 318. DOI:10.1038/318162a0.
• L. Vlasov, D. Trifonov. 2005. 107 Stories about Chemistry, TUBÝTAK., Ankara.

Wilson Bentley took the first photomicrographs of a snowflake, thereby initiating extensive research on the snowflake. When he was 15 years old, his mother gave him a microscope as a gift. He began studying several things under the microscope, among them raindrops and snowflakes. Later on, he discovered how to photograph this delicate ice formation and presented it to the world. His enthusiasm for photographing snowflakes continued until his death 47 years later. Through his photomicro graph collection, we can see just what a complex and wonderful creation each snowflake is and why it has been the subject of such debate over the years.

The average snowflake is made up of 2 to 200 separate snow crystals; much larger ones can contain as many as 1,000 separate snow crystals. These snow crystals begin to form around tiny dust particles that have been carried up high into the atmosphere. When the temperature drops below freezing at these high altitudes, water vapor clings to these dust particles. Interestingly, the water vapor skips the liquid state and turns directly into ice, a process called sublimation. When the air contains enough moisture and the ice crystals accumulate, these crystal formations begin to fall as snow.

Scientists believe that there are only four different types (shapes) of these six-sided snowflakes; hexagonal plates, stellar stars, stellar and plate combinations, and spatial dendrites. Hexagonal plates are thin, solid, or partly snow crystals. This pattern is made up of a variety of ridges and hollows, as well as thick and thin ice. The stellar star pattern is the one many know as the symbol of a snowflake. It assumes this pattern because ice crystals tend to cling together in “cottony” clumps and have the corners of a star, unlike hexagonal plates.

The stellar and plate combination pattern is formed when plate and stellar star characteristics unite. The resulting flake is considered the most exquisite of all crystals. The plate pattern is in the middle, and the stellar star branches out from the plate. Finally, the dendrite, another stellar type, is identified by small crystals that branch out, fern-like, along each of its six “rays.”

It is believed that the shape of the snow crystals forming these snowflakes depends on the temperature of the cloud in which it is formed. Ice crystal columns are formed in the highest clouds, which have the lowest temperatures. Dendrites and star-shaped crystals are formed in the slightly warmer middle clouds, and needle ice crystals are formed in the lower clouds. These temperature variations cause each snowflake to assume a specific shape. Different sources give slightly different temperature ranges and different explanations of a snowflake’s developmental stages. This may be to the fact that scientists do not have exact knowledge of the conditions and formation of these delicate crystals, for they base their assumptions on laboratory experiments that seek to create the same weather conditions. It must be pointed out, however, that all snowflakes “created” in laboratories are always deformed and do not resemble the perfectly symmetrical flakes found in nature.

The first wonder I would like to describe is the snowflake’s construction. Nuclear physicists and crystallographers are still trying to explain this complex bridgework of molecules that form the ice crystals into a snowflake. A brief explanation behind the construction mystery is that an average hexagonal-shaped crystal may contain 100 millon more water molecules. The ice crystal grows by adding more and more molecules. Its growth proceeds in a way that is both perfectly horizontal and perpendicular, thus building a broader and thicker crystal. Amazingly, this process is always carried out within the same hexagonal symmetry.

An ice crystal’s framework is a marvelous example of solid geometry, for it always presents an ingenious grouping of molecular parts. Not only does the ice of a snow crystal grow perpendicularly by interlocking pyramids, but at the same time its horizontal ice particles extend themselves in overlapping hexagonal patterns.

But not all of a snowflake’s beauty can be seen with the naked eye. Each crystal contains an invisible masterpiece of construction resembling an ongoing pattern that becomes smaller and smaller. Such a development is produced by the ice crystals themselves, which bond to each other and thereby increase the snowflake’s size. Over a period of 15 minutes, and under the conditions necessary for sublimation, a snow crystal gradually assumes the shape of the first stage. This baby crystal is unbelievably tiny, from .008 to .009 of an inch in diameter.

The average initial crystal may appear as hexagonal plates, sector plates, various stellar forms, or as capped columns. This later shows evidence of a skeletal structure, surface design, and pattern that subtly determine the snowflake’s final pattern. Later, this plain star will begin to develop either crystal twigs or fern-like plumes. As the final ice structure becomes heavy, it begins its journey to the ground.

An additional wonder is that no two snowflakes are alike! Each snowflake has a unique combination of ice crystals, which creates a unique snowflake. No two identical snowflakes have been found.

Another mystery is why each snowflake has six sides. Johannes Kepler, a physicist and mathematician, has studied this for years. In his The Six-Cornered Snowflake, he mentions some very important and thought-provoking questions and explanations. He also states that there must be an agent for such perfection and calculation, some definite reason why a snowflake’s initial form invariably displays the shape of a six-cornered starlet. Why always six? If this were the result of chance, should not some of them at least have five or seven corners?

In his search for a logical reason for the six sides, he asserts that if you ask geometers on what plan honeycombs are built, they will respond “on a six-cornered plan.” Each cell is surrounded by six others, each of which ends in an obtuse angle, pointing downwards, formed on three planes. The architecture causes each cell to share six walls with six cells surrounding it in a row, and also three plane surfaces with three other cells from the contrary row. Each bee, as a result, has nine neighbors.

Kepler also observed that the insides of such fruits as pomegranates and peas are squeezed into six sides. Why? One reason, perhaps, is that a plane surface can be covered without gaps by only three shapes: a triangle, a square, and a hexagon. Of these, the hexagon is the roomiest, and so has the most storage space, for example, for the honey produced by bees. Therefore, bees instinctively build their hives in this shape rather than others. Why and how?

Kepler concludes that this original, well-thought-out pattern can only have been imprinted on it by a Creator: our Creator. He concludes that the cause of each snowflake’s hexagonal shape is the same cause that shapes plants and numerical constants. Nothing happens without a reason, but rather with a reason guided by a Supreme Reason.

The wonder of the snowflake is seen not only in its wealth of variety and form, which is perfectly constructed with complete beauty and perfection, but also in how these six-sided crystals are formed with perfect symmetry by various processes in the clouds.

As I mentioned earlier, such findings are merely realizations of a fact. The term realization is used because discoveries like these have been in existence since the beginning of time! They have always been around us and will continue to be. But now we can see some of these small miracles due to recent, rapid advances in technology. This displays to us more about the reality, existence, and attributes of the Creator. Yet even a tiny snowflake shows us, through sight and reasoning, the attributes of our Creator. Among these attributes are artistry in creation, finality in creation, countenance, and divine teaching and directing.(1)

Artistry in creation: “The whole of creation exhibits an overwhelming artistry of dazzling worth. Yet it is brought into being, as we see it, easily and in a very short time. Furthermore, creation is divided into countless families, genera and species and even smaller groups, and each of these exists in great abundance. Despite the variety and abundance, we see only orderliness and art and ease in creation. This shows the existence of one with an absolute power and knowledge, who is God.”

For example, each snowflake is literally a masterpiece of art, with a perfectly eye-appealing design. It is remarkable that such a masterpiece is created within minutes in the clouds, yet has perfect symmetry and a complex pattern. This is the first attribute of our Creator that we see when we look at a photo of a snowflake. Hence, our Creator is a master in artistry.

Finality in creation: “Nothing in the universe is for nothing, pointless. As ecology in particular shows, everything in creation, no matter how apparently insignificant has a very significant role in existence and serves a certain purpose…. There are many purposes for every thing, every activity, and every event in it. Since this requires a wise one who pursues certain purposes in creation, and since nothing in the world-except for man-has the consciousness to pursue those purposes, the wisdom and purposiveness in creation necessarily point to God.”

An example is the snowflake. The causes for it and its six sides, which point to certain reasons, are still being examined. If something cannot be explained at this point in time, all it means is that we have yet to understand its complexity.

Countenances: “[Uncountable] human beings have lived since man’s first appearance on the earth. Despite their common origin-a sperm and ovum, which are formed from the same sort of foods taken by the parents-and although they have all been composed of the same kind of structures or elements or organisms, every human being has an individual countenance distinguishing him or her from the others…. This obviously shows one with an absolutely free choice, and all-encompassing knowledge, and He is God.”

Just as each snowflake is different in pattern and type, no snowflake has been identified as identical with another, even though uncountable snowflakes fall each year. This truly points to a Creator with an unbelievable attribute of countenance.

Divine teaching and directing: “For man to direct himself in life and distinguish between what is good or bad for him needs a minimum of around fifteen years. However, many animals can do this very soon after they come into the world. A duckling, for example, can swim as soon as it hatches. Ants start to dig nests into the earth when they get out of their cocoons. It does not need a long time for bees and spiders to learn how to make their honeycombs and webs, respectively, while each are marvels of handiwork beyond the capacity of man…. How can you explain all these astounding facts otherwise than by attributing them to the teaching or directing of one who knows everything and has arranged the universe with all creatures in it in a way that enables every creature, big or small, to direct its life?”

The snowflake is an example of such directing. How can a snowflake, which is completely devoid of intelligence and consciousness, create itself in such an absolutely perfectly manner within a matter of minutes? Such events can only be the result of Divine teaching and directing.

In conclusion, I would like to highlight two points. First, such astonishing facts exist all around us. Yet in order to appreciate them, first we must observe our surrounding environment and consider its complexity, purpose, and perfection. Second, acknowledging that we have limited senses and that our technology continues to advance rapidly, I am excited about all of the other wonders in this world that might be revealed to us, for all of them will enable us to better understand and marvel at our Creator.

FOOTNOTES

1. The four following attributes are taken from Fethullah Giilen, Understanding and Belief: The Essentials of Islamic Faith (Kaynak, Turkey: Kaynak, 1997), 4-8.

REFERENCES

• Bell, Corydon. Wonder of Snow. New York: Hill and Wang, n.d.
• Blanchard, Duncan. The Snowflake Man. Weatherwise: 1970.
• Gulen, Fethullah. Understanding and Belief: The Essentials of Islamic Faith. Kaynak, Turkey: Kaynak, 1997.
• “How Do Snowflakes Form?” Lansing State Journal. October 8, 1997.
• Kepler, Johannes. The Six-Cornered Snowflake. Clarendon Press: 1966.