“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.”

“The most beautiful system [the universe] could only proceed from the dominion of an intelligent and powerful Being.” (Isaac Newton)

The Newtonian physics, quantum mechanics, and the theory of relativity took the modern community to the boundary of the two realms of physical and metaphysical existence. Nevertheless, the nature of the fundamental physical dimensions still remains an open question resting on the related areas of science

The fundamental concepts of Newtonian physics are Time, Length, Mass, and Electric Charge by means of which all the other classical physical quantities such as velocity, force, momentum, energy, current, electric field, magnetic flux, etc. can be derived and expressed as their combinations. Classical physics stands on the assumption that material, having two basic intrinsic properties of mass and charge, and immaterial phenomena are all contained in an absolute space and an ever-flowing absolute time. These four physical dimensions, without asking the nature of them, provide a practical framework for a description of the gravitational and electromagnetic forces and thus a description of the physical world and an interpretation of the events occurring in it up to a certain degree. However, the Newtonian picture of the universe is neither adequate for a deeper understanding of the corporeal reality nor appropriate for linking that reality to the ones possessing higher degrees of the Universal Existence.

Starting from late 19th and early 20th centuries, the Newtonian picture of the world has been changed due to two revolutionary theories, which have been proved both experimentally and theoretically that they are superior to and not compatible with the classical descriptions and assumptions. They are the relativity theory and the quantum mechanics. In physics, a field is a physical quantity associated with each point of Space-Time. For example, the Newtonian gravitational field is a vector field specifying its value at a point in Space-Time, which requires three numbers, the components of the gravitational field vector at that point. Quantum field theory constructing quantum mechanical models of systems classically parameterized by an indefinitely big number of degrees of freedom, namely fields, is the natural and quantitative language of particle physics. The current set of fundamental fields and their dynamics are summarized in a theory called the Standard Model. All particles and their interactions observed to date can be described almost entirely by the Standard Model although most particle physicists believe that it is an incomplete description of nature, and that a more fundamental theory, the Theory of Everything, awaits discovery. Figure 1 represents an overview of the various families of elementary and composite particles, and the theories describing their interactions.

The relativistic quantum field theory of the subatomic world does not only include the strong and weak nuclear forces in addition to the electromagnetic and gravitational interactions of the classical picture, but also provokes some ideas about the nature of the fundamental concepts of the classical physics. Symmetry of a physical system is a physical or mathematical feature of the system that is preserved under some change. The Standard Model says, for instance, that the electric charge is the generator of the U(1) symmetry of electromagnetism. U(1), the unitary group of rank 1, is the simplest internal symmetry group of the Standard Model. It can be visualized as the rotational symmetry of a circle about a perpendicular axis passing through the center of the circle. It represents a continuous symmetry because a circle can be rotated by an angle and remains unchanged. It is an internal symmetry since this circle does not lie in the physical space but in the complex plane of mathematics. More abstractly and more generally, a charge is any generator of a continuous symmetry of the physical system under study. When a physical system has a symmetry of some sort, Noether’s theorem implies the existence of a conserved current. The thing that flows in the current is the charge; the charge is the generator of the symmetry group. This converts our classical concrete idea of electric charge into a mathematical abstraction. Conservation of energy and conservations of linear and angular momenta are nothing but the applications of Noether’s theorem to the translational symmetry in time and translational and rotational symmetries in space, respectively.

Classically, which is equivalent to macroscopically, mass is associated with matter and can be defined as a quantitative measure of an object’s resistance to the change of its speed. But in the Standard Model of the subatomic scale, the mass of the elementary particles are explained by the Higgs mechanism which refers specifically to the generation of masses for the W and Z bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN is currently searching for Higgs bosons, and attempting to understand the electroweak Higgs mechanism. The Higgs mechanism is the process that gives mass to elementary particles. In 1905, Einstein proposed mass-energy equivalence (E=mc2) in his paper entitled “Does the inertia of a body depend upon its energy-content?” In relativity, all of the energy that moves with an object (that is, all the energy which is present in the object’s rest frame) contributes to the total mass of the body, which measures how much it resists acceleration.

When we come to the remaining two fundamental concepts of Newtonian physics, we see that Time and Length, which we know instinctively, are no exceptions. The modern physics challenges our classical understandings of them too. Relativity theory argues that Time and Space are of equal ontological status; the reality is the 4-dimensional unity of Space-Time. Physics could no longer be understood as Space by itself, and Time by itself. It also states that simultaneity is relative, so there is no objective way to define a “Now” that would be the same for all states of motion which substantially affects the idea of causality. In addition, this Space-Time is not flat but rather curved due to the material and energy contained in it and not static but dynamic. Time and Space are neither uniform nor absolute.

The missing part of the so-called Theory of Everything is the quantum gravity, which attempts to develop scientific models that unify quantum mechanics describing three of the four known fundamental interactions with general relativity describing the fourth, gravity. The following quotation is from one of the leading quantum gravity researcher, Carlo Rovelli, stated in 1997:

“I believe that we are going through a period of profound confusion, in which we lack a general coherent picture of the physical world capable of embracing what or at least most of what, we have learned about it. The fundamental scientific view of the world of the present time is characterized by an astonishing amount of perplexity, and disagreement, about what time, space, matter, and causality are. But if a new synthesis is to be reached, I believe that philosophical thinking will be once more one of its ingredients. Due to the vastness of the problem involved, the generality and accuracy of philosophical thinking and its capacity to clarify conceptual premises are probably necessary to help physics out of a situation in which we have learned so much about the world, but no longer know what matter, time, space, and causality are.“

Lee Smolin, another theoretical physicist named as #21 on Foreign Policy Magazine’s 2008 list of Top 100 Public Intellectuals, stated the following in 2001:

“Atoms do fall, so the relationship between gravity and the quantum is not a problem for nature. If it is a problem for us, it must be because somewhere in our thinking there is at least one, and possibly several, wrong assumptions. At the very least, these assumptions involve our concept of space and time and the connection between the observer and the observed.”

It is true that quantum mechanics and the theory of relativity were born and are growing in the nontraditional atmosphere of the scientific enterprise. Thus, they can be considered as sharing the reductionist character of the Newtonian physics by having no direct reference to the hierarchy of physical and metaphysical existence. Nevertheless, we consider them as an improvement since they took the modern scientific community to the boundary of the two realms, by asking the old question of ancients about the nature of the fundamental physical dimensions. The mystery of them is still an open question resting, we believe, on the related areas of science and metaphysics.

The miracle of life appears in the form of a baby which develops from a fertilized ovule (zygote) following millions of other events. This series of events, which is almost always the same for every fetus, can be grouped as reproduction, differentiation, and development. The zygote completes its development in the womb; postnatal growth can continue until 20 years of age. Even though every event during the baby’s development seems to take place with chaotic reactions, harmony and order are there for us to discover. One of these astonishing events is the perfectly symmetric growth of the fetus/baby. Most organs in the human body appear in pairs and are symmetric. Babies are born with 300 bones; however, some bones later fuse with other bones, leaving only 208 bones in the adult human. It is still a mystery how long bones such as the humerus, radius, ulna, femur, and tibia are able to grow on both sides of the human body in a symmetrical manner.

Mechanisms that control growth in organs

In vertebrates, both internal developmental programs and the external factors which stimulate or inhibit growth play a role in the ultimate size of an organ. But the relative effects of these two mechanisms can vary significantly in different organs. When pieces of spleen from an embryo that is at a later stage of growth are transplanted to a newly developing embryo, each new piece grows, but not to the size of the original spleen. The total weight of all the transplanted spleen pieces is equal to a normal spleen’s weight. When the spleen reaches a certain weight, growth inhibiting factors are secreted, which stimulate negative feedback mechanisms that limit growth. When a spleen reaches a certain size, the density of the inhibiting factors increases simultaneously, halting growth. Growth in the liver is controlled by extracellular factors (various substances in the blood, hormones, vitamins, minerals, etc.). When a section is cut off of the liver, the section continues growing and developing until it reaches the size of the original liver. The thymus has a growth process that is executed by a cellular genetic program. When sections of a thymus taken from the embryonic period are injected into developing mouse embryos, every section grows until it reaches the ultimate size.

More evidence of cellular growth programs was acquired via an experiment that was carried out with the salamander genus Ambystoma. When the leg bud of the larger species was injected into the smaller species, it would at first grow slowly, but then it would reach the normal size of its own species (the larger species).

Distinguishing growth and symmetry from one another

Both the arms and legs have long bones. A long bone consists of two parts (diaphysis and epiphysis). The diaphysis is the middle (core) part of the long bone. It consists of hard bone tissue, and is like a tube. The hyaline cartilage-covered joint forms the epiphysis of the long bone. In a growing bone, there is a growth plate (epiphysis plaque) made of hyaline cartilage; this is located between the diaphysis and the epiphysis. The epiphysis plaque causes the bone to grow longer; when growth is complete, the epiphysis plaque ossifies (becomes bone). In other words, growth stops. There are some clues that show the existence of positive feedback mechanisms which control the symmetric and balanced development of the arms and legs while the fetus is still growing. The arms and legs grow due to the development and growth of the plaques located at opposite ends of the long bone. The ultimate size of the arms and legs are proportional to the size of the finger bones (phalanx) and the metacarpus. According to current knowledge, growth in our arms and legs is only controlled by internal growth programs and the active growth of the plaques. We do not yet know the mechanism through which how much the bone must grow and symmetrically with the organ (the other arm or leg) on the other side of the body. But even if this is discovered in the future, we will continue to appreciate the perfect and miraculous aspect of this phenomenon.

In addition, in growth-plaque transplant experiments, the development of the transplanted growth plaque is dependent only on the age and size of the donor. Growth plaques cause the bone to grow, but the plaques themselves remain the same size for years. The cartilage cells they produce (chondrocytes) exchange places with the bone cells (osteocytes) in harmony and without destroying the length of the bone. Cells from different areas of the growth plaque act differently. Stem cells are found on the upper section, near the epiphysis. Immediately above them is an area where cells reproduce very quickly. At the bottom of the epiphysis, the cartilage cells grow up to 4 to 10 times larger than their normal size (hypertrophy). Cell reproduction here is mostly due to hypertrophic chondrocytes. The chondrocytes die and break up, then change places with the bone tissue. The dynamic process of these events in the growth plaque repels it from the bone area, and as a result, the bone grows longer.

Sustained symmetry despite cell sequence and speed of reproduction

The rapid growth rate in the legs and arms during the embryonic period continues to increase until the child is three years of age. This growth rate slows down until the individual reaches adolescence. During the fastest growth period, which is from adolescence to the early 20s, the growth rate rapidly increases. For example, most people who grow between 30 and 37.5 cm during the first two years of life can grow between another 7.5 and 10 cm every year during adolescence. At the onset of adolescence, rapid growth due to a sudden change in the volume of cells is observed. After adolescence a sudden falling off in the speed of growth can be observed due to the effect of hormones on the growth plaques in the spine and other long bones. The growth plaque now fuses with the neighboring cells and growth stops. However, the fusing of the growth plaque is the result of the cessation of growth, not the cause. After growth stops, the growth plaques begin to disappear. When the reproduction potential of the cartilage cells in the growth plaque has been exhausted, the growth plaque begins to disappear.

Growth plaques in different bones can trigger growth at various rates; these rates can differ as much as seven times. In fact, growth plaques on different ends of a bone can have different growth rates, provided that this rate is consistent with the genetic program. The number of cells on the growth line is 40 times more than in other areas. The number of cells produced here can exceed 10,000 cells per day. For symmetric growth between the arms and legs to be sustained, the number of cells in the growth plaque must be the same or very close. Experiments carried out on rats show that eight cartilage cells leave the growth plaque to exchange places with cells above them every day. It can be said that the growth of the bone is caused by the increase of cells in the growth plaque (which sustains its size). The growth rate caused by the growth plaque can be calculated by multiplying the growth plaque’s cell production rate by the average length of all of its cells. Different growth plaques provide different growth rates. This difference can be caused by the difference in the size of the growth plaques, the difference in cell production rates, and/or the difference in the hypertrophy (growth) rate of every cell. The upper growth plaque in the tibia of mice generates 16,400 cells every day; the average life span of these cells is around 30 hours. Can such harmonious, symmetric, and equivalent growth in the arms and legs-despite the large number and variety of cells-be the work of pure coincidence, mindless nature, or unconscious molecules?

Do hormones play a role?

The main molecular players that organize longitudinal growth in bones during childhood are the growth hormone, the thyroid hormone, and corticoids. The sex hormones (androgens and estrogens) are programmed to influence growth during adolescence. Estrogen is the main determiner of characteristics related to increased height and an increase in bone quality, as well as adolescent-related physiology. These hormones are in charge of coordinating growth throughout the body. It is for this reason for women, after the menopause, the production in estrogen decreases and osteoporosis and brittle bones can occur. According to the current view, cartilage cells have a certain genetic reproduction potential, and when this potential finishes, growth stops. The growth rate during the embryonic period is 20 times higher than that of mid-childhood. The growth rate drops greatly during mid-childhood. If we exclude the noticeable increase during adolescence, the cells responsible for growth have begun to age. The bones on opposite sides of the body stay about the same size, despite all of these changes in growth rates. Circulating hormones and neuroendocrinal factors are believed to play important roles in maintaining symmetric growth. But there is no conclusive evidence to support this belief. Even though one can think of factors such as pressure, tension, and sports as helping control harmonious and symmetric growth of bones, no proof has been attained from controlled experiments. As a person ages, a gradual decrease in growth can be observed. Even if a growth plaque is placed into another organism, be it young or old, the growth rate of the bone does not change. This shows that symmetric growth in long bones is controlled by a program that is operated by internal factors, which is also compatible with the genetic program. When chemical-based medication is given to postpone growth, after the medication has been eliminated, the growth plaques grow faster for a short period to compensate for the lost time. These findings show that timing and the location and circumstances of the cell are critical parameters for reproduction. If the cartilage stem cells in the growth plaque have a certain reproduction potential, then it is clear that cartilage cell reproduction stops when growth comes to an end. If growth inhibiting factors slowly accumulate in the growth plaque, this might cause a deceleration of growth over time. Another possibility is some sort of “meter” in the unconscious and mindless stem cells, which keeps track of the number of cell divisions and thus controls aging. The estrogen in our body has a duty of closing down the growth plaques and speeding up the aging of cells. However, we should not forget that estrogen plays the special role of closing down all of the growth plaques at the same time. Estrogen is one of the visible causes of fertility, growth and development, and resilience. Estrogen also represents femininity and fertility at all levels.

When the signals from unconscious cells in the growth plaques and the quite sophisticated interactions among all the factors that influence growth, all of which require an all-encompassing knowledge to be executed, are taken into account, the impeccable genetic programs of different growth plaques on the two sides of the body that leads to the formation of the arms and legs, as if they have been molded in a factory, is absolutely amazing for anyone who reflects upon it.

References

• Wolpert L. (2010).”Unsolved Mystery: Arms and the Man: The Problem of Symmetric Growth.” PLoS Biology. 2010 Vol. 8(9). pp 1-3
• Extremity Development during the Embryonic Period (www.visembryo.com)

Be they physical or chemical, many attributes of beings are dependent on mathematical laws and their appearances are also shaped along mathematical principles. When we observe creation from this standpoint, we can perceive the perfection as well as the spectacular beauty that is inherent in every being. As reflections of the Attributes of the Names of God Almighty, Jamil (The Owner of Beauty), Bari (The One Who Creates from nothing), Sani (The Maker of All) and Musawwir (The Designer), this beauty found in the external appearance of beings is dependent on more than one factor coinciding. The most important factor here is “symmetry,” which is described as “an exact correspondence and beautiful balance among the parts of an object.” Beings are created with various symmetrical attributes and with great artistic beauty.

The most common symmetry type is the bilateral symmetry; this creates a mirror effect which is an exact correspondence between the right and left sides. An object forms an exact symmetry with its reflection in the mirror. A perfect symmetry that is very similar to the mirror effect can be found in the human body. The left and right sides of our body are symmetrically corresponding. Imagine a dividing line that passes from the middle of the forehead, through nose, chin and down the chest, we can see a perfect symmetry on both sides of the body. Our arms, legs, eyes, ears, nose and lips are designed with a bilateral symmetry. The same symmetrical structures can also be seen in most other creatures. All mammals, reptiles and birds are symmetrically created.

Another type of symmetry is rotational (radial) symmetry. Imagine a metal object that is in the shape of an equilateral triangular placed on the sand. If we will rotate this object 120o around an axis that passes through its center, the new position of the object will fit exactly into its original mark left on the sand. The reason for this is that the radial symmetry for equilateral triangles is 120 degrees. In the same way, a square has a radial symmetry of 90^o and a regular polygon with n number of sides has a radial symmetry of 360/n degrees.

The beautiful symmetry of snow flakes, with their regular hexagonal shape are a beautiful natural phenomenon. In addition to these there are shapes in nature that have a three-dimensional radial symmetry. The most significant of these shapes are regular polyhedrons. An example of such polyhedrons is the salt crystalline elements that have cubical structures. Until recently, the fact that there is a creature in nature that has a regular polyhedral shape, consisting of twenty sides, was unknown. However, when a type of adenovirus that causes infections and hepatitis in dogs was discovered, it was found that there is a creature with twenty regular sides in nature.

One of the most beautiful samples of radial symmetry in nature is the daisy. Symmetrical structures do not only exist in the normal world and in the micro worlds, but also can be found in the macro world, like all the huge celestial objects, the Sun, the Moon, galaxies, star clusters in the sky . . . . All planets move around the Sun in a symmetrical manner, whereas galaxies have a spiral symmetry. It is interesting that the symmetrical structure of living beings is overwhelmingly apparent externally, rather than internally. For example, the internal organs in the human body, like the lungs, liver, stomach and intestines are not symmetrical and we have only one heart in one side of our chest cavity. Moreover, the lobes of the brain are not symmetrical either. However, all the metabolic processes in human body function properly. Does this mean that the mathematical beauty found in our external appearance is merely for aesthetical reasons? God does not create things for only one reason or purpose, on the contrary, He creates them to serve many motives and in relation with many functions. For example, if we did not have two eyes and if they were not symmetrically placed on our faces, we would not be able to see objects three-dimensionally. In the same way, if our ears were not symmetrically placed on our heads, then we would have great difficulty in determining the direction and source of sounds. If we did not have symmetrical feet and legs, we would not be able to walk well, and if our arms were not symmetrical, we would not be able to balance our body’s center of gravity while walking. If birds did not have symmetrical wings, they would not be able to fly, and if the fins of fishes were not symmetrical, they would not be able to swim smoothly.

Symmetry is also closely related to physical and mental robustness. According to one study, women who suffer from an infectious disease during pregnancy are more likely to have babies with asymmetrical features. The same study claims that asymmetrical babies are more susceptible to heart disease than symmetrical babies.

Another study shows that people with asymmetrical teeth are more likely to have more harmful microorganisms in their mouth than those who have symmetrical teeth. It is interesting that there tends to be a greater difference between the fingerprints on the left and right hands of schizophrenic people than on those of normal people.

Symmetry is a phenomenon that is used by animals and insects. For example, an experiment showed that bees prefer flowers that are symmetrical. Actually, flowers with perfectly symmetrical shapes produce more nectar than those that are asymmetrical. In one investigation, a symmetrical flower was made asymmetrical with a pair of scissors. The flower had been attractive to bees before its shape was changed; after made asymmetrical, the flower became unattractive to bees, even though it had just the same amount of nectar as before.

All these facts reveal that there is much wisdom and beauty hidden within the symmetry that the Almighty Designer uses to shape all beings. We take symmetry for granted. To have two eyes placed equidistance and two ears on each side of the head is the norm. Anything else strikes us as strange. But if we just take a few moments to think about why our eyes are where they are, and why our ears are placed on the sides of our heads, the answer is obvious. God’s mercy is infinite; in even the simplest example of symmetry there is a reason. We should not take this world for granted, but rather use every opportunity to dwell upon and be thankful for the wonderful world that has been created for us. 

References

• Stewart, I. & M. Golubitsky, Fearful Symmetry, Blackwell, 1992.
• Rosen, J., Symmetry Discovered, Cambridge University Press, 1975.
• Tarasov, L., This Amazingly Symmetrical World, Mir Publishers, Moscow: 1986.

What Is Symmetry?

Physicists usually define symmetry as the invariance under transformation. Simply put: Remaining unchanged after a change. The basic example is time“translation invariance, which states that the homogeneity of time symmetry leads to the invariance principle that nature’s laws always remain the same Thus, they are independent of when we measure them.

Let’s assume that we measure and record a pendulum’s movement today. If we do this tomorrow or even next week under the same conditions, we always will obtain the same result. Without symmetry’s principles, for instance the time“translation invariance, a computer built today most probably would not work tomorrow. We could not even build a computer, a calculator, or a watch, for nature’s laws would be in a state of random change at every moment. Nor could we guarantee that gravity would not turn into a repulsive force in the next moment.

Another basic example is the space“translation invariance. Space has the symmetries of homogeneity (being the same at every location) and isotropy (being the same in every direction). As a result, laws are independent of where we measure them, for whether we are in America or China, we use the same formulae: F= m.a, S= k.log W, E= m.c …

Why Does the Universe Behave the Same at Every Instant and at Every Point in Space“Time?

Materialists reply: Because laws under space“ time translations are invariable. This will bring another why, answered by: Because of homogeneity and isotropy. Such a pattern will continue, for each question is no more than an effect ready to be explained by a cause, and each answer (cause) will appear to be another cause’s effect.

Followers of Aristotle, Plato, Ibn-i Sina (Avicenna), al-Farabi (Al-Pharabius), or the Illuminist philosophers will urge you to keep asking why. The point at which you stop asking will be your destination, where you reach the initiator: God. All answers to your previous questions, when taken together, turn out to be attempts to attribute intermediaries or partners to God. If you ask those who believe in God’s Oneness and give causes no share in His Dominion, you will hear: The same laws, being seen at every point and at every instant, can be explained only if there is one God governing every point at every instant with His universal Will, absolute Power, and all-encompassing Knowledge. Thus all causes, bound to each other so that they can be understood, are bound directly to God, Who is not bound by space“time.

Why Is the Universe Accessible to Us through Laws?

While new symmetries were being revealed, skeptical scientists were wondering whether symmetry in nature’s laws on a macroscopic scale is valid on a microscopic scale.

Every symmetry is associated with a conservation law, the most important one being energy conservation. This implies a symmetry in time-translation. The validity of energy conservation has been proven on the microscopic level. One of the latest verifications is that in a perfect vacuum, particle and antiparticle pairs are created and annihilated constantly. In other words, microsized Big Bangs occur all the time out of the vacuum. This does not violate energy conservation or the laws of quantum mechanics. If they were somehow violated, and if we did not see the same concepts as the result of symmetries, the similar formalism, and the same techniques from micro- to macroscales, we could not comprehend the universe.

Consider Einstein’s amazement at this: The most incomprehensible thing about the universe is that it is comprehensible. Even the most atheistic scientist accepts that the universe is not absurd. But why is universe comprehensible and accessible to us?

Atheists and materialists believe that we come into being from nothing, by nothing, and for nothing, and so consider such questions meaningless. However, people who unify everything in God’s name can assert: God cannot be comprehended fully by humans, and so wills to be comprehended and known through His art and the system and order apparent in the form of laws. We are here to comprehend, and the universe lies open. In the words of Paul Davies, author of The Mind of God, we are meant to be here.

What Is the Consequence of the Matter“Antimatter Asymmetry?

Existing theory says that every particle has a corresponding antiparticle. Physicists believe that particle“antiparticle pairs behave symmetrically, like mirror reflections of each other (mirror-reflection symmetry). In 1964 at Doe’s Brookhaven Laboratory, when a slight but definite asymmetry between a subatomic particle and its antiparticle was noted, physicists saw the breaking of symmetry in nature. It was time to ask why.

It is now thought that this asymmetry may be responsible for matter’s dominance in the universe. Our universe appears to be made entirely of matter. If there were substantial amounts of antimatter on Earth, we would be annihilated as we react with our antiparticles, for particle“antiparticle pairs annihilate each other when reacting and leave behind electromagnetic radiation.

Any explanation offered for this asymmetry must account for the asymmetry in the universe’s earlier periods, beginning with the Big Bang. Astrophysicists think that certain massive particles, formed soon after the Big Bang, decayed in such a way that slightly more particles than antiparticles were created. Even though this asymmetry’s exact origin is unknown, we do know that if symmetry were not broken, we would not be alive. Amazingly, the laws of physics allow life to exist.

Does Time Flow from the Past to the Future?

Time-asymmetry is perhaps the most fascinating. Even though all successful physics equations are symmetric with respect to time, we perceive time as flowing from the past to the future: Newton sees the apple fall down, people grow old, we throw the stone and the window is broken. Newton used a time arrow to distinguish between time’s forward and backward directions. Einstein preferred a time river that meanders as it passes massive objects. But this sounded unscientific: past, present, and future are only illusions.(1)

Roger Penrose, a prominent mathematician and quantum theorist, suggests that our perception of time as flowing irreversibly from the past to the future should have something to do with quantum mechanics and consciousness. This is quite similar to Einstein”but more scientific. Along with many others, Penrose and Hawking have discussed the issue at great length. Hawking says that the observed difference between past and future must come from the universe’s boundary conditions. Since the Big Bang was the beginning of creation of matter and energy, and space and time, the reason should lie there. They disagree on the underlying reason for time-asymmetry, but agree that the quantum theory of gravity is needed to describe the complete nature of space-time.

What Is Supersymmetry?

Einstein spent his last days trying to synthesize quantum theory and gravity to find a quantum theory of gravity. Despite great advances, such a theory remains elusive. This has guided them to the so-called theory of everything, which combines the four fundamental forces: nuclear, electromagnetic, weak, and gravitational. Theories describing the unification of the nuclear, electromagnetic, and weak forces are called grand unified theories (GUTs). The unification of these forces require the unification of symmetries, known as supersymmetry.

However, a large problem arises when trying to add gravity. This is because gravity is located in the field of general relativity, the theory of galaxies, quasars, and black holes, while the other three fundamental forces are described by quantum theory, the theory of the very small, which accepts subatomic particles as point-like particles. Nevertheless, this theory of everything should combine general relativity and quantum mechanics.

Unfortunately, it is not that simple. Right now, the most elegant idea for such a theory is supersymmetric string theory, in which point-like particles in quantum mechanics are replaced with string-like entities. Different vibratory resonances on these strings correspond to different particles in the same way as different frequencies correspond to different notes on a violin string. These strings, however, are not connected to the ends of a cosmic violin, but float in space“time. When they float, they warp space“time as predicted by general relativity. Thus strings unify the quantum theory of particles and general relativity.

Once the theory of everything is completed with its supersymmetry, we might be able to explain many mysteries, among them the existence of atoms, molecules, and dark matter,2 and the asymmetry between matter and antimatter. If we succeed, one more step will have been taken toward time travel. Even DNA, whose double-helix structure is interpreted under X-ray crystallography, which is a fruit of crystal lattice symmetries, may reach a full description with sypersymmetric string theory.

In the near future, if supersymmetric string theory is completed both experimentally and theoretically, we will be able to express bravely that all forces in nature are different aspects of the same thing. Scientists and believers in God’s Oneness will adopt this theory, for such a comprehensive unification takes us to God’s Unity.

Conclusion

Symmetry, which is displayed in art, music, inorganic and organic nature, always has been fascinating to the human mind. Space“time, upon which all laws are displayed, also is constructed on the basis of symmetry. This construction brings an amazing result: the universe behaves the same at every instant and at every pont. Through symmetry, we witness unity, harmony, and perfection. Ironically, the breaking of symmetry (asymetery) is not the braking of prefection, for the universe’s perfection is completed with asymmetry, the outcome of which is human life. Today, we human beings are looking for a supersymmetic string theory that will reveal the underlying unity beneath the diverse creation.

Footnotes

1. Escape from the Quantum Whirlpool, New Scientist (April 1997). Included in Robert B. Leighton (contributor), et al., The Feymann Lectures in Physics, Symmetry in Physical Laws (Addison- Wesley: 1964).
2. Astronomers calculate the mass and velocity of galaxies or distant stars by detecting their spectrums and redshifts. However, they discovered that the high velocities of galaxy clusters cannot be explained fully by their calculated masses. As a result, they posit the existence of dark matter surrounding galaxies. As this dark matter does not radiate, it is not detected in the electromagnetic spectrum. It is expected that one of the supersymmetric particles (the neutralino) might make up the missing dark matter in the universe.

References

• Glashow, Sheldon L. The Charm of Physics. Springer Verlag: 1991.
• Gribbin, John R. The Search for Superstrings, Symmetry, and the Theory of Everything. Little Brown & Company: 1998.
• Hawking, Stephen W., and Roger Penrose. The Nature of Space and Time. Princeton University Press: 1996.
• Icke, Vincent. The Force of Symmetry. Cambridge University Press: 1995.
• Krauss, Lawrence M. Fear of Physics. Basic Books: 1993
• Leighton, Robert B. (contributor), et al. The Feymann Lectures in Physics, Symmetry in Physical Laws. Addison-Wesley: 1964.
• Penrose, Roger. The Emperor’s New Mind. Oxford University Press: 1989.
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