“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.”

Observing Majorana fermions in the ferromagnetic atomic chains on a superconductor. Nadj-Perge et al. Science, October 2014.

In the universe, matter and antimatter particles are always produced as a pair and, if they come in contact, they destroy each other in a flash of energy. In 1937, an Italian theoretical physicist named Ettore Majorana had proposed that there can be unique exceptions to this rule: a stable particle could exist in nature that is both matter and antimatter. Scientists have been looking for that indefinable particle, also known as the “Majorana fermion,” for seventy years. A group of researchers recently reported that they were able to detect the Majorana particle which behaves simultaneously like matter and antimatter. Researchers designed an experimental system allowing them to observe an emergent particle inside a material. They first generated an extended chain of pre magnetic iron atoms on a superconductor made of lead. Then, they cooled the material to -272 C, just about one point above absolute zero, and monitored it using a giant two-story-tall scanning-tunneling microscope, which can track electrical signal changes with very high precision. Finally, they were able to capture a glowing image of an electrically neutral particle at the ends of atomically thin iron wires. The Majorana particle was surprisingly stable and the opposing properties make the particle neutral so that it interacts very weakly with its environment. The discovery of the Majorana particle has exciting implications for several areas of modern physics, engineering, and astrophysics. For example, Majorana particles are very similar to neutrinos, as they both have very weak interactions with the matter. Neutrinos are thought to make up most of the dark matter that fill the Cosmos. Perhaps, neutrinos are simply Majorana-like particles and Majorana particles are also a candidate for what dark matter is. As an industrial application, Majorana particles can be utilized in quantum computing which aims to create computers to handle incalculable systems. The current quantum computing technology uses electrons, but they are known to be very unstable due to high interaction rates with surrounding materials. However, since Majorana particles are neutral and highly stable, they can be engineered into a variety of materials to produce more reliable and powerful quantum computing applications.

The Bitter Side of Artificial Sweeteners

Artificial sweeteners induce glucose intolerance by altering the gut’s microbiota. Suez J. et al. Nature, September 2014.

There have been conflicting and confusing findings about the health effects of artificial sweeteners over the past several decades. Some studies found that they cause weight loss and others found the exact opposite. Some studies linked them to diabetes and other studies argued otherwise. A recent study provided a series of experimental evidences that artificial sweeteners disrupt the body’s ability to regulate blood sugar, and thus may cause metabolic diseases and diabetes. Researchers, using animal models and human studies, found that sweeteners significantly alter the gut’s microbiome – the collective name of bacterial colonies living in our intestines. The composition of our gut microflora plays a critical role protecting us from pathogenic bacteria, the metabolism of indigestible components of our diet, and modulating development and regulation of the immune system. Sweeteners – in the form of saccharin, sucralose, or aspartame – are found to alter the mix of microbes in our intestines and consequently change how our bodies metabolize glucose. Constant use of sweeteners in mice and human test groups caused typical glucose intolerance symptoms in which glucose levels rose higher after eating and declined more slowly than expected. Glucose intolerance can ultimately lead to serious illnesses like metabolic syndrome and Type 2 diabetes. Although this study will cause a lot of discussions and headaches in the food industry, the link identified between microbiome and glucose intolerance will definitely inspire novel therapeutic approaches to metabolic disorders such as diabetes.

Brainy Fingertips

Edge-orientation processing in first-order tactile neurons. Pruszynski JA and Johansson RS. Nature Neuroscience, August 2014

A new study found that neurons in human skin are able to perform advanced calculations that scientists thought only the brain was capable of performing. A group of sensory neurons that extend into the skin and record touch are called first-order neurons in the tactile system. Each nerve ending branches in the skin to form about 5mm2 elliptical receptive field, with up to 8 highly sensitive zones that are unevenly distributed within the field. It turns out that these neurons not only transmit information about when and how intensely an object is touched to the brain, but they also send complex information about the touched object’s shape. Researchers found that the sensitivity of individual neurons to the shape of an object depends on the layout of the neuron’s highly-sensitive zones in the skin. Computations that require untangling geometric shape information are classified as feature extraction computations in neuroscience and are typically attributed to the immensely complex circuits of the cerebral cortex. This study showed that neuronal populations localized outside of the brain, such as first-order tactile neurons, can have advanced processing capacity similar to brain neurons. These results can also potentially improve treatments for nerve injury and rehabilitation, as scientists previously assumed that the cerebral cortex was doing all the work.

While wandering near a creek, have you ever seen bugs walking on the surface of the water? Have you felt any resistance when you hit the surface of the sea with your palm? Have you ever thought about what causes these to happen?

The two examples of events given above happen to be related to “surface tension.” This situation is described as the force per distance unit that is generated in the opposite direction of the direction of expansion between two different surfaces. It can take place in between two different liquid layers, as well as among liquid-gas and liquid-solid layers. For example, the surface tension of a liquid forms in the transitional region where liquid and gas molecules make contact. The source of this force generated on the liquid’s surface is the intermolecular attractions that hold the liquid molecules together. Each molecule in the liquid is pulled via opposite but equal forces by neighboring molecules, thus no single force is acting on the molecules. However, the molecules on the surface are only surrounded by one side, therefore they are pulled inwards with a net force (Figure 1), causing a tension similar to an inflated balloon on the surface of the liquid.

When we look carefully to a stagnant pool of water in a container, the surface of the water seems to be covered with a thin layer of film, resembling a stretched membrane. In order for a substance to enter or leave the body of water successfully, it must puncture this membrane. In other words it has to overcome this intermolecular force. If a steel blade is laid horizontally on the surface of the water slowly, it floats despite that it is made of denser steel because it cannot overcome this surface tension. Surface tension is the principle responsible for the trampoline-like behavior of liquid surfaces. Many insect species created for aqueous habitats can maintain their lives on the water via their adapted leg parts. The best example of this is the water strider. This insect lives on water by taking advantage of water surface tension. Though the surface tension principle is a requirement to be on the water, it is also necessary that the strider not to stick to the surface. Therefore, this insect is also equipped with a paddle made of waxy hairs at the end of their legs (Figure 2).

Forces of cohesion and adhesion

The intermolecular force of a liquid among the same kind of molecules is called the “cohesion force,” and intermolecular attraction between different types of liquid molecules is called the “adhesion force.” These forces of adhesion and cohesion determine the behavior of a liquid in a container. If some mercury is put in a glass tube, because the cohesive forces among the mercury atoms is greater than the adhesive forces in between the glass container and the mercury, the mercury assumes a convex shape. Here, mercury has a tendency to reduce its contact with the glass and does not wet it. In contrast to mercury, when water is put inside the tube, the surface layer between the water and air takes an inward concave shape. This is caused by the greater adhesion force between the water and glass compared to the intermolecular cohesion forces of water. Water wets the glass since it has a tendency to spread towards the greatest surface possible (Figure 3).

When there is a thin layer of water or tea left in between a tea glass and its plate, the adhesion force glues the glass and plate together. Since the adhesion force is greater than the weight of the plate, the glass cup can be lifted together with the plate. Contact lenses also stay in position on the eyes without falling through the help of adhesion forces. Tears strongly pull both cornea and the contact lens together, holding it in place.

The capillary effect

A liquid inside a thin vertical tube is pulled upwards by the inner surface of the tube until the adhesion force becomes balanced with the liquid weight. This event is called the capillary effect or capillarity. Liquids naturally rise in narrow channels if there is sufficient adhesion force. This effect is enhanced in narrow tubes due to the smaller volume of the liquid, but reduced in wider tubes because of gravity. Therefore, there is an inverse ratio between the channel diameter and liquid height in capillarity.

The reason a sponge absorbs water effectively is the easy rise of water in the capillary openings of the sponge. In a similar fashion, there are small openings found in paper napkins and towels. When a napkin makes contact with a wet surface, water is pulled inside the small openings with capillary action, thus removing the water from the surface. This is because the adhesion force in between the napkin tissue and water is greater than the cohesion force of the water molecules. This principle is also utilized while getting blood samples with capillary tubes. In addition, the removal of continuously excreted tears by the capillary ocular ducts that extend into the nasal cavity is another example of this wise law.

Capillary action is also important for the transportation of water molecules from humid parts towards drier areas in soil, providing for the spread of water. The same principle is also vital to nourishment of trees. Every part of a tree encompasses capillary channels, all the way from the tips of the roots to very ends of the branches. Water molecules are transported to the leaves against gravity when they enter the tips of these capillary channels at the roots. Even though the adhesion forces between the water molecules and the root’s tissues win the war against gravity, at a certain height, this force becomes equal to the gravitational pull, thus not allowing water molecules to climb higher. This is the ultimate height a tree reaches. Capillarity also affects internal water pressure of a tree, leaf size, photosynthesis, and other factors. This is why the leaves of a tree are usually bigger on lower branches compared to higher ones (Figure 4).

The surface tension of water is the highest among the known values of other liquids and this has very significant biological effects. If the surface tension of water was to be lower, like other liquids, it would not be able reach the higher parts of plants through capillary action, thus preventing the survival of taller plants. The vegetation waits patiently as nourishment is delivered to its roots. Water has been assigned a vital role in this service.

The water-dependent survival of plants is made possible through the capillarity and surface tension. Could this amazing phenomenon, in which the capillarity is on duty to water the leaves on the highest branches of the tallest trees to ensure the maintenance of life, take place via blind atomic interactions or accidental occurrences?

How do liquid droplets get their shape?

Objects with a wider surface will have a greater surface tension. Since the force of surface tension, acting on per unit distance, is equal to the surface energy per surface area, a wider surface requires greater accumulation of energy on the surface. All the matter in the universe tends to stay at a lowered energy level. Therefore, it is ideal for objects to reduce their surface area. When the surface area to volume ratio of the known geometric shapes is investigated, the smallest ratio is found to belong to a sphere. A small value of this ratio means the most reduced surface area per volume. Among enclosed containers of equal volume, a sphere is also the one with the smallest surface area. When two equal volume watermelons of spherical and cubical shape are peeled, the spherical one will produce the least amount of rinds.

Because of the reasons mentioned above, liquids take a droplet shape immediately when they fall, reducing their surface area. That is why a water droplet dripping from a faucet, a falling rain drop, and a droplet on a leaf are all in the shape of a sphere (Figure 5). It is the same principle that makes planets and other heavenly bodies resemble a globular form. This indeed points to an Almighty Power who plans the motions, positions, and assignments of all the objects, from particles to giants, managing and dispatching them as The Self-Existent One holding everything together.

Factors affecting surface tension

Temperature increase is directly proportional to a decrease in the surface tension in most liquids. When the temperature of a liquid rises, so does the kinetic energy of the particles in it, making these particles move faster. This leads to a weakened intermolecular attraction that binds molecules together. Since this change affects the particles at the surface, it decreases the tension. Improved soaking of hands and laundry can be achieved with warm water during cleaning because heat reduces the surface tension. This helps with better cleaning results in a shorter amount of time.

In a similar fashion, soap and detergents also reduce the surface tension of water. If a small soap bubble is placed on a water droplet, the droplet spreads away instantly. This indeed tells us that the soap bubble reduces surface tension.

If a substance dissolves in a pure material, surface tension is found to change depending on the solute and the solvent structure. For example, salt decreases the surface tension of water. Salt weakens the intermolecular bonds of the water molecules, and therefore reduces the cohesion and surface tension. That’s why sea waves foam when they hit shore.

Can surface tension be associated with the ability of unconscious and primitive atoms as the principle behind many functions and tasks in the lives of plants and animals? Do such wondrous events happen by chance? Isn’t this principle such a blessing of the One who easily provides what is necessary to all living things, nourishing them in time according to their needs?

Gómez-Valdés et al. Lack of Support for the Association between Facial Shape and Aggression: A Reappraisal Based on a Worldwide Population Genetics Perspective. PLoS ONE, 2013; 8 (1)

It is a common misconception that some people are profiled to be angry or aggressive because of their physical appearances, particularly their craniofacial shapes. In addition, there have been some studies suggesting that men with certain facial traits (round-shaped faces) are more likely to develop aggressive and unethical behavior. A new study using a sample of around 5000 individuals from 94 different countries has found no correlation between facial shape and aggressive/criminal behaviors. Researchers analyzed fWHRs (facial width-to-height ratio) and 2D/3D craniofacial landmark coordinates to estimate any possible correlation between skull shape and aggressive behaviors in men. First, they utilized the famous skull collection in Hallstatt/Austria to investigate any potential correlation between skull features and life history parameters of individuals, such as their overall fitness. Second, they analyzed the male prisoners convicted of crimes like inter-personal aggression (homicide, robbery etc.) from Mexico City Federal Penitentiary to see whether there is any relation between skull shape traits and aggressive crimes. Analyses of both databases have found no significant correlation between skull shape traits either with the fitness of males or with their aggressiveness. This study has very important social and political implications in today’s societies, as we unfortunately see many ethnical, racial and even physical prejudices. This comprehensive study has undoubtedly showed once more that physical traits cannot be a reliable predictor of complex human behaviors, which are mostly shaped by external factors such as education and socio-cultural practices.

Biggest Structure in the Universe Discovered

Clowe et al. A structure in the early Universe at z ∼ 1.3 that exceeds the homogeneity scale of the R-W concordance cosmology. Monthly Notices of the Royal Astronomical Society, January 11, 2013

Throughout history, mankind has been trying to answer the questions of “how big” or “how far,” when looking into the vast expanse of the universe. As new technologies are developed, bigger discoveries and consequently bigger numbers are brought to light. An international team of astronomers recently discovered a collection of 73 quasars which form a single structure; the largest structure ever observed in the entire universe. A quasar, short for quasi-stellar object, is the luminous center of a galaxy that surrounds a super massive black hole. The distance of these newly large quasar groups to the earth is about 9 billion light years (1 light year is approximately 9.5 trillion kilometers). The size of these structures is simply mind-blowing. Even if we have a spacecraft that travels at the speed of light, it would still take about 4 billion years to cross. If we put this overwhelming size into perspective, the Milky Way—earth’s home galaxy—is only about 100,000 light-years wide and our neighbor galaxy Andromeda is only 2.5 million light-years away from the Milky Way. So these quasars are 1600 times larger than the distance from the Milky Way to Andromeda. This discovery seriously challenges the size calculations based on the widely accepted Cosmological Principle which assumes that the universe is essentially homogeneous when viewed at a sufficiently large scale. Cosmological Principle predicts that there should not be any structure in the universe larger than 1.2 billion light-years. A four billion light-years wide structure would obviously be an outlier when compared to other structures in the universe and it might contradict with the homogeneity of the universe. However, scientists think that such contradiction would not necessarily falsify the Cosmological Principle originally established by Albert Einstein. It might only change the assumptions of the theory that define at which scale the universe can sufficiently be viewed.

More Twists on Double Helix

Biffi et al. Quantitative visualization of DNA G-quadruplex structures in human cells. Nature Chemistry, 20 January 2013.

About 60 years ago, on April 25th 1953, James Watson and Francis Crick published a one-page paper where they described the “double helix” structure of the DNA, the molecule that carries genetic information from parent to offspring. This discovery not only revolutionized the biological sciences and medicine but also dramatically changed the way we perceive life, nature and most importantly ourselves. Yet, new findings on DNA structure keep surprising us. Scientists from Cambridge University discovered the first quadruple helix—a four-stranded DNA structure in human cells which they named “G-quadruplex.” These structures were previously observed in test tubes but they were never found in cells. The building blocks of DNA molecules consist of four different bases: Adenine (A), Guanine (G), Cytosine (C) and Thymine (T). G-quadruplexes (G stands for Guanine) are formed by four guanine bases that forms a square DNA helix. Researchers found that these structures are enriched in rapidly-dividing cancer cells, specifically at the ends of chromosomes called telomeres. When researchers targeted and trapped these quadruple DNA structures with synthetic molecules, they found that DNA replication slows down and cell division is blocked. Researchers suspect that these quadruple DNA structures in telomeres of cancer cells could explain why cancer cells rapidly proliferate and divide. It is still not clear whether G-quadruplexes exist in healthy cells but targeting these structures in cancerous cells with pharmacology seems to be a promising method to stop the spread of cancer.

Dear Peter! I, your muscular system, would like to talk to you today; I allow you to walk and do all kinds of movements very easily. In the most recent essay of this department, the skeletal system, which works together with me, discussed how it protected your body and enabled you to stand straight and firm. Sure, this is true. However, as you know, huge rocks and trees can stand straight, too; but they do not have what you do-the ability to move. They are rigid and inflexible because they do not have a system that allows them to move.

You on the other hand, as the most splendid creation in the entire universe, have mobility. All animals have mobility at different levels, thanks to the muscular tissues, which work dynamically behind all the moving organs. However, you, human beings are like neither the animals nor the plants. You have not been created to live like a tree that is pegged down in earth or like an animal that unconsciously tries to meet only its biological needs. Our Designer, God, has made you and your descendants the most important of all creation. He has given you qualities that help you to discover the world, learn, invent and establish new civilizations. To realize such potential and to carry out such duties, you need to first have the freedom to make changes in your small immediate world, which can only be done through motion. In order to enable you with this ability to move, my Creator has put me at your service. I am a system that comprises hundreds of muscles and millions of packed cells.

My most important feature is the cells that move by burning sugar, like a motor consuming fuel in order to work. My cells can shorten and lengthen thanks to the intracellular fibrils (myofilaments) that contract and expand. As a result of each contraction, I pull the bone or the organ that I am attached to and cause it to move or change shape. With the exception of your heart, your bones and organs cannot move on their own. Their ability to move depends on the nature of the muscle they are attached to.

Indeed, I can be called both an organ and a form of tissue. I can use my contraction ability not only as muscle tissue, but also as an organ and a system which runs throughout your body. That is the reason why I appear in so many different types and shapes of muscle bundles. Let me give you an example to help you better understand what a muscle consists of: Let us imagine that a thin thread is like a basic muscular cell. Let us now bring together a great number of threads and make string out of them. Then, let us bring together those strings and make a clothes line. Next, let us bring together a great number of clothes lines and make a very thick rope. Now, imagine this thick rope as a muscle and an organ. Yet, this example is too basic compared to the sophisticated muscle.

Very thin cotton fibrils make up the thin threads. In the muscle fiber, like those cotton fibrils, there are two filaments formed by the two types of protein molecules called actin and myosin, which help in the contraction function. Those little filaments are placed facing one another and they slide past each other during the contraction, which causes the muscle fibers to shorten. That is how the contraction and relaxation of a muscle occurs.

Peter, do you think that coincidence plays a role in this complex and wonderful mechanism and the incredible structure which I have attempted to simplify with an example? Not even a simple thread can be produced without a thread-maker or a machine. Each of your muscles comprises billions of fibers, wrapped all around your bones and giving shape to your body. Can such a complex and delicately intricate structure exist on its own and be positioned in the best place it could possibly be?

My muscles consist of bundles that are made of thousands of muscle fibrils; the size and shape of each muscle depends on which bone it is attached to and what function it does. For example, the muscles that move the bones in your arms and legs are long and spindle-shaped; whereas the ones that are attached to your body can be circular, or triangular, or spread over a broad area. Whatever shape they have, the red skeletal muscles, which are attached to your bones, are very strong and they are voluntary muscles, which mean you can control their movement. When you walk, run, do something with your hands, lie down or stand up, you always use my red striated muscles. The strias (stripes) can only be seen under a microscope because of the histological structure of these skeletal muscles that make up a great part of your body.

My smooth muscles are involuntary muscles that work without your control. Their movements are slow and their contractions last longer, which is the reason why they do not tire easily. The smooth muscles lie in the walls of digestive system, blood vessels, and urinary tract, but I will not talk much about them since each system has referred to the smooth muscles within itself and in detail in previous talks. Because they are not attached to your skeleton to work, the smooth muscles do not play a role in your movements, such as walking around; they only work for the movements of your inner organs.

The third type of muscle belongs to your heart (cardiac muscle) and although the heart has a little striated muscle tissue, it, too works involuntarily. Therefore, you should be aware of the fact that it is the striated muscles which work for the movement of the skeleton and do the major job, and that it is this that we refer to when we say “muscle.”

A great number of bones have been created in order to support your body, and joints have been placed between those bones for them to take the proper shape according to every movement. However, none of those joints have the ability to move by itself. A door or a window, no matter how good it is, cannot be opened or closed without an outside force to pull or push it. In the same way, a joint needs a force to move it and that force is produced by your muscular system. There are around 340 muscles included in your muscular system. It has been estimated that all the muscles in your body perform 510 different functions! While some of those functions are bone movements in your joints, other muscles can perform movements without moving a bone at all. Muscles that are placed in your forehead, face, eye lids and abdomen are those kinds of muscles. They can help you look worried by wrinkling your forehead or grimacing when you are disgusted by something.

Keeping with tradition, the muscles that are included in my system have been named based on the function they perform. For example, the muscle which moves an organ part towards another part is called an abductor, while the muscle that straightens a joint is an extensor, and the muscle that bends a joint is a flexor; the muscle that raises a part of the skeleton is a levator, that which the muscle makes a part of an organ prone is a pronator, the muscle that rotates a part of an organ is a rotator and that which brings an organ into a supine position is called a supinator.

In order for you to make all the movements that your body needs, my muscle components have to be both very strong and flexible. The most important feature of my muscle components is that they can be trained and strengthened with a systematic workout. The main goal of all sportsmen who compete is to increase the strength and the endurance of their muscles. As a result of intense exercises with weight and speed, the number, the diameter and the length of my muscle fibers will increase. Thus, I can gain more power to be able to do more work and also gain the ability to contract faster.

However, in addition to all this training and exercise, genetic factors also play a role in my health. For this reason, not everybody who works out can become a good sportsman; but if the person has innate muscular and skeletal capacity, with good exercise this capacity can certainly be enhanced and developed. However, if a person does not have the proper muscular structure for a particular sport, it would be unfair to expect them to be a champion! Although my muscles always seem to be of the same type at first sight, I might show different behavior depending on the distribution of the special fibers inside them. Some of my fibers twitch fast and tire easily, some of them twitch slowly and tire later. Depending on the distribution of these different fibers, the movements and sport that every person can do differ from person to person. In this case, an athlete who can run only 100m and an athlete who can run 10,000m do not have the same development of muscles; they have different amounts and distributions of special muscle fibers.

The contraction of any of my muscles can occur in two different ways: If the pressure put on my muscle is stronger than the resistance of the tissue, the tension remains constant and the muscle shortens. This is called an isotonic contraction. If the pressure put on my muscle is equal to the resistance, the tension of my muscle increases and its length does not change. That is called an isometric contraction. The amount of force that occurs during the contraction of my muscle depends on its length and the amount of the stimulus.

In order to produce muscle contraction, an electrical signal is sent through a motor neuron to the synaptic gap, which is positioned between the muscle cell membrane and the nerve cell. As a result, a chemical reaction occurs, which, in a very short time, causes the actin and myosin proteins in the muscle fibril to slide past each other and thus shorten the fibril, contracting the muscular cells. During this reaction, the temperature also rises a little and the total heat generated by all the muscles determines your body temperature. For this reason, in cold weather, my muscles vibrate, increasing your body temperature and trying to maintain it. You may now understand why moving the parts of the body in cold weather helps people avoid from getting sick or freezing. As you can see, every act of my Creator is quite purposeful. He can create two or even more functions within one task: Through your muscles, He not only provides you with the ability to move freely, but heat is also produced and you are protected from getting cold.

When a muscle fiber contracts frequently as a result of successive electrical impulses from a nerve fiber, it becomes tired after a while and needs rest. In this case, other muscle fibers which have not contracted for a while will take over and continue the job. However, if the electrical impulses from the nerve come too frequently and my muscle fibers do not have an opportunity to rest, a condition of constant contraction, which is known as physiologic tetanus, occurs.

The tension receptors that are placed on my muscles help maintain the harmony and coordination of all your movements including walking and running, bouncing and sitting down. They do this by constantly signaling the nerve system and providing feedback about the condition of my muscles, and about the speed and the intensity of contraction. Thus, through these receptors which control and coordinate my muscle activities, the well-being of my system is ensured. It is this that prevents you from wobbling when you walk, or helps you to take a spoonful of soup to your mouth without spilling it.

Like any other tissue or system, I, too, have some special disorders. The most common disorders are: weakness, malformation, muscles that develop and move involuntarily and habitually, especially in your face (tic), infected muscles (myosite), muscle dystrophy, muscle rigidity (the Stiffman Syndrome), benign or malignant muscle tumors (leiomyom, rhabdomyoma, or Rhabdomyosarcoma). These disorders differ in their degree of severity and risk.

Dear Peter! You have now seen that each muscle helps your organs to move, holding your bones and giving shape and function to your body, making you a beautiful model and an inspiration for sculptors. You may have understood that this is a work of knowledge and might; there is no way that the myofibril in my one cell could form by itself as a result of a coincidence.

Irfan Yilmaz is a professor of biology at Dokuz Eylul University, Izmir, Turkey.

One category of galaxies is the spiral; this is dependent on the galaxies’ appearance. The magnetic field of the Sun is also a spiral. Among many things that have a spiral form are the cochlea inside our ears, our navel cord, our fingerprints, the teeth of mammoths, elephant trunks, some spider webs, the horns of some goats, cluster of sunflowers, thousands of types of mollusks, the pattern in which subatomic particles move, plus many more examples. Grapevine shoots, ivy, some microorganisms, and the positioning of some leaves around their branches are in the form of a helix. Nature displays brilliant examples of spiral and helix forms over a wide spectrum, ranging from fossils to galaxies. Below we will discuss some of them:

The Archimedean spiral

Named after its discoverer, this spiral is the geometrical location of a point which moves across a line turning around a fixed point at the speed of q and with a straight angle (Figure 1). The equation for the polar coordinates is p=aq. The distances between the curves are equal. A good example of this type of spiral is the spider web constructed with equal distances from the center.

The Equiangular (Logarithmic) spiral

This spiral type was defined by Descartes in 1638. In an equiangular spiral, any line that crosses the center cuts through all coils of the curve (Figure 2). The equation for polar coordinates is Inr=a.q or r=ea.q. Sea shells and the shells of snails are formed with this spiral.

Fibonacci Numbers and the Golden Ratio

The following numbers, the sequence of which is made by adding the last two numbers together, are known as Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … In other words, each number is the sum of the preceding two numbers. Let us divide each number with the preceding one and write down the quotients:

1/1=1; 2/1=2; 3/2=1.5; 5/3=1.666…; 8/5=1.6; 13/8=1.625; 21/13=1.615…; 34/21=1.619…; 55/34=1.6176…; 89/55=1.618…

If we continue to divide in this way, we will reach a mathematical constant, i.e., 1,618034, which is known as the golden ratio (φ).

Let us now draw a new geometrical shape with the Fibonacci numbers. Next to a 1-unit side square put another square that has equal dimensions. Then add another square, this time equaling the sum of the sides of the previous two (2 units). As we continue to add new squares with double the units of the previous two we get what is called the Fibonacci or golden rectangle. When we draw an arc from one corner of this rectangle to an opposite corner and continue drawing through neighboring squares, as in Figure 3, we will get a spiral. A good example of this is the nautilus shell. The golden rectangle and the spiral is frequently used in fine arts, architecture, and technology.

The Helix

The space curves that coil around a cylinder and cut through its main axis at a right angle is called a cylindrical helix (Figure 4). An ivy plant climbs a tree in a helix, and a helix is the shortest distance to a certain height. The Selimiye Mosque, Edirne, Turkey, features one of the best examples of helices in architecture. The architect Sinan designed the minarets of this mosque with three balconies, which are reached via different stairs that have no connections between them.

The 3D Archimedean spiral and the Logarithmic spiral (Helico spirals)

Conical helices are the space curves that coil around a right cone and cut through its main axis at a right angle. Sea snails, or limpets, have this spiral shape (Figure 5).

Galaxies and hurricanes

Galaxies and hurricanes are also spiral in shape and they have some similar features. Sharing the Stamp of Unity, the law of which governs the entire universe, both galaxies and hurricanes are affected by major forces, like the force of gravity, angular momentum or rotation.

Spiral galaxies are divided into two categories: elliptical and barred spiral galaxies. Barred spiral galaxies have arms that extend away from the main core (Figure 6).

(As evidence for a people open to belief) We have assuredly set in the heaven great constellations, and We have made it (the heaven) beautiful for those beholding. (Hijr 15:16)

The Nautilus: A wonder of creation

The hard shell of the nautilus has a beautiful logarithmic spiral shape. Each coil is at a distance from the next at an increasing proportional distance, each coil is multiplied by a constant. The chambers in the shell are similar, but they widen in a geometric sequence. It is amazing that calcium carbonate, the material that makes up the shell, can accumulate in such a way so as to comply with this geometrical pattern. In this pattern, the nautilus occupies the least space that is possible, thus losing as little heat as possible. Architects have been inspired by the nautilus to produce designs to use the smallest possible space to contain the most possible room.

The Cochlea

The cochlea in our ears is like a double-ramp tunnel coiled upon itself. Etymologically, the word cochlea comes from a Greek word that means snail. The spiral shape of the cochlea reminds one of sea shells.

Horns

Horns of the sheep and goats have the shape logarithmic spiral; they grow in the form of helicoids, as if coiling around a cone.

The Rose

The leaves of a rose are lined up and shoot out in a spiral shape.

Spirals open for us gateways to thought in our efforts to explore the wisdom and beauty that have been set in motion in the universe and are constantly maintained. Spirals, like other living or non-living objects or beings around us, are exquisite works of art that point to the fact that nothing exists from coincidence. Looking through a telescope to a marvelous galaxy in outer space or examining a sea shell on the beach or holding a rose in the spring may become a rewarding act if we contemplate on their Fashioner, for such “contemplation for an hour is worth voluntary prayer for a year.”

The Earth’s shape is becoming rounder as a result of the construction of projects like the Three Gorges reservoir. The weight decrease due to the melting icecaps has played a major role in these changes.

From time immemorial, humanity has wondered about the shape of the Earth. Over the centuries countless studies exploring the Earth and its shape have been conducted, and they still continue today. With advancements in technology, the methods and measuring devices have constantly changed. In earlier periods the Earth was believed to be flat; nevertheless from around the fifth century bc there were varying opinions suggesting that the Earth was actually round and calculations were conducted to measure its radius. Particularly from the seventh century ce onwards, the number of studies regarding what the Earth really looked like have increased tremendously.¹

In later years, with the advent of Islam and its open encouragement of Muslims to explore the universe and make advances in science, Muslim scholars made huge progress in astronomical research. Historians who have studied these advancements in astronomical science agree that the era between the eighth and fourteenth centuries can aptly be designated as a period of Islamic astronomy. In the years following the sixteenth century, significant research was undertaken in relation to the shape of the Earth and measurements of its radius in both the Islamic world and the West.

In the eighteenth century astronomic research and the advancement of technology proved not only that the Earth was round, but that it had a distinctive shape. According to calculations, the Earth was bulging around the equator and flattened at the poles. The maps produced by satellite systems show that the Earth is not completely round or smooth, but rather it has protrusions, creating an uneven surface that resembles a face with spots.2

“Geoid” is the term scientists prefer to use when referring to the physical depiction of the Earth’s surface. The shape of the Earth is not a perfect ellipsoid, thus, scientists use this representative surface that is thought to be most approximate to sea level, in order to identify departures from the ellipsoid shape. Due to the events of nature and human-related factors the geoid constantly changes, and this is why a precise mathematical account of the geoid has not yet been possible.

The Earth is known to be a geologically active planet. Just as everything else in the universe, from atoms to galaxies, has not been left to their own fate, the Earth is also constantly being transformed, thus making our magnificent ecosystem possible. The continuous geological process of changes in the Earth’s crust is related to a variety of factors: the varying density of the rock layers which form the Earth’s crust, the activity of the tectonic plates, as well as the movement of the continents, shifts in the center of gravity, tidal activity, hydro-spherical and atmospheric phenomena, and human intervention in some regions.

Researching the variations in the gravity of the Earth with satellite systems is a relatively new method of recording the changes in geoid elevations. The distance between the center of the Earth and its surface is constant (the tallest mountains will rise or decrease 1-2cm per year at most); if we take into consideration that the Earth’s physical body does not vary much and disregard the other forces, then we can say that the main reason for these infinitesimal changes in gravity on the surface of the Earth is due to differences in mass. While there is a decrease in weight in specific regions from melting glaciers, in other areas there is an increase in weight due to melting water flowing into reservoirs; both these phenomena play a significant role in the variations of the Earth’s gravity.

Even a minor variation in mass can be detected by measuring gravity. The change of mass location on the Earth’s surface results in gravity variations in the same region; in brief, today the commonly used gravity measurements are the most important source for detecting and identifying variations in geoid elevations, as well as determining the actual reasons for these changes. Gravity measurements are conducted via satellite systems; these indicate changes in mass location by detecting an increase or decrease in weight. The most modern technological satellite systems that can detect gravity change and allow us to follow the variations in masses on the Earth’s surface are the satellite used by the European Space Agency, called GOCE, and NASA’s satellite, called GRACE. GOCE has been designed to perform accurate studies of the Earth’s gravity field as it progresses into orbit. As the satellite passes over the regions where gravity is intense or weak, it measures the variations in gravity with signals that have been conveyed by a device called a gradiometer. GRACE is a pair of identical satellites that are flying in the same orbit, 136 miles apart; they orbit the Earth at a distance of 300 miles. These satellites can measure distances with microwave signals, and can detect changes of less than 1% the thickness of human hair; thus the twin satellites are able to accurately measure the distance to the surface of the Earth. The measurements provided by this satellite system make it possible for changes in gravity to be calculated. The GRACE satellite data is 1,000 times more accurate than other gravity field detection systems.

The enormous waves that occurred on the sea surface as a result of the Sumatra Island earthquake, which measured 9 on the Richter scale, caused a level ridge, measuring about six meters in height, to form on the shore. According to data produced by GOCE, such changes in the mass of the Earth’s surface caused a variation of 18 mm to occur on the geoid; this is recognized as a relatively high degree of change.

Changes in the polar glaciers also cause variations in the geoid; data provided from satellite GRACE shows that the layers of ice in Greenland and the Antarctica are melting at a higher rate than previously expected. The melting icebergs are causing a rise in sea levels of up to 0.41 mm every year and the weight of water produced from the melting icecaps is causing changes to the shape of the Earth’s surface.3

One of the interesting facts attained by GRACE is the changes in the Earth’s gravity field that have been caused by Three Gorges in China, the largest reservoir ever built. The lake region of the reservoir that is being built measures around 372 miles long, 70 miles wide and approximately 574 feet deep; when the casing of the reservoir is completed it will house an amazing 39.3 billion m3 (9.4 cu mi) of water. The area that this reservoir will cover once the project is completed is so great that it will make an estimated 1.5 million people homeless. It has been observed that the enormous accumulation of water in the completed sections of the reservoir has increased the gravity level in that region, which in turn has caused changes in the geoid structure.4

Scientists have confirmed that the Earth’s shape is becoming rounder as a result of the construction of projects like the Three Gorges reservoir. It is also estimated that the weight decrease due to the melting icecaps has played a major role in these changes. In some regions of Scandinavia and Canada the ground is rising 1 cm every year due to the melting glaciers. The water produced from the melting glaciers is forcing the currents in the Atlantic Ocean towards the equator, while the decrease of mass at the poles and the increase of weight in the equator region have caused significant changes in the shape of the Earth.

Many scientists claim that changes in the Earth’s surface have been caused by changes in the climate. Unfortunately, according to a report published by the UN Intergovernmental Panel on Climate Change (IPCC), humans are responsible for 90% of global warming. As a result of these vast variations, the geoid shape of the Earth is becoming rounder and its radius is increasing annually by 0.4–0.8 mm. The reasons for these changes are being closely monitored by scientists. According to scientists, the variation of the geoid that has been caused by changes in mass location is having an effect on the Earth’s dynamics, with the transfer of mass demonstrated by the changes in gravity causing a reduction in the speed of the Earth’s rotation around its axis; this is expected to result in variations in the daily time-zone.

Abdullah Sancak is pursuing an academic career in engineering in Turkey.

Notes

1. For more information on this topic see James R. Smith, Introduction to Geodesy, The history and Concepts of Modern Geodesy, John Wiley and Sons, Inc., 1997.
2. There are interesting images that show how the earth looks in the following link: University of Texas Center for Space Research and NASA (7 Haziran 2005) http://www.csr.utexas.edu/grace/gallery/gravity/
3. G. Ramillien, A. Lombard, A. Cazenave, E. R. Ivins, M. Llubes, F. Remy, R. Biancale, Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE, Global and Platenary Change 53 (2006) 198-208.
4. San Shaoan, Institute of seismology, CEA, Wuhan, China, Gravity change before and after the first water impoundment in Three Gorges Project. http://www.sgg.whu.edu.cn/icct/html/icct_ppt/S1/sun.s.A___fourth.pdf

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.

The human learning process begins with birth. The person who starts to understand their body, organs, environment, and the world in this first stage enhances their skills by practicing them within the limits of the capabilities that God has given. With the help of the senses, such as “hearing,” “seeing,” “touching,” and “smelling” the amount of knowledge proliferates rapidly. Every warning and message that reaches us is an important factor in helping learning, developing consciousness, and becoming aware of life. While the development of the intelligence and mental activity of children who lack sufficient warnings and messages becomes slower, the development of the ones who are overloaded is dispersed. It must be kept in mind that the fastest growing system in a child’s body during the first three years of life is the nervous system. Therefore, appropriate messages must be provided very early on, even from the very beginning of the pregnancy.

Messages that are received through the sense of hearing have a significant importance in child education. The mental development of children who have received ample and appropriate messages is more significant and positive. It is a well-known fact that even in the earliest stages of pregnancy the baby can distinguish the mother’s voice, and is relaxed via the sense of hearing; this sense starts to develop at the very beginning of pregnancy. Unfortunately, some parents turn the television or the radio on in the baby’s room, frequently letting them listen to it. However, what is best for the baby is to have suitable audio incentives that can be easily understood. On the other hand, to remain silent when with a baby for a long time will negatively affect the language learning process. A conversation between parents in a relaxed and gentle tone, talking about pleasant things, for example, a beautiful poem, the sound of someone reading the Qur’an, or the sound of the adhan coming from the minarets, are all useful messages for babies.

Messages received through the sense of sight are as important as those received through hearing. Humans recognize nature and themselves by seeing, touching, and trying. Although the visual capabilities of a baby develop in the womb, it takes some six months for sight to reach an appropriate level. Babies accept everything that they see during this time as training material. Everything that is seen has an important role in forming the baby’s personality. Parents must always be good models for their children, acting in commendable ways. It is a well-known fact that babies pay attention to their parents’ behavior. Since imitation is one of the earliest learning techniques; parents should remember to behave in a way that they want their children to behave. Helping each other, forgiveness, responsibility, hard-work, empathy, and correct behavior are qualities that parents should demonstrate. Seeing a parent working hard, reading, and praying will help the conscience of the child to develop. Fighting, noise, and angry words will negatively affect the development.

Messages sent through meta-communication are more effective. For example, in an advertisement of a certain product, the fact that people using that product are smiling is more important than the properties of the product. This is because such behavior attaches the message “you will be happier when you use it.” For children, living in a positive environment helps them to observe correct models. Since they use the very first form of learning-imitation-providing good examples of behavior is extremely important. In the learning process, sight is one of the most frequently used methods. Therefore, parents must display appropriate behavior to help their children to learn. Displaying correct behavior is more effective than explaining what correct behavior is.

The environment in which we live has a great importance in the education of children. The school that is chosen for children must be considered from different aspects. Most people never forget their first teacher; this is one of the milestones in our lives. The teacher starts to shape our personalities at an early age. Therefore, choosing the right teacher, someone who is not only capable, but is also virtuous and pleasant, is of great importance. The teacher’s style of dressing, way of talking, their reaction to stimulus and their behavior tells us more than the teacher can; these are also the most effective means of sending messages to our children.

The children’s group of friends is another environment that needs to be monitored carefully. Having the right friends positively affects the development of the personality of the child, while having inappropriate friends will have negative effects. Many parents cannot attain positive results, despite all their efforts in education, merely because they have forgotten the importance of the peer group on the child. Starting form the ages 6-7, the effect of friends increases gradually, peaking in the teenage years.

The effect of the media on children is indisputable, especially in today’s world. We can see violence, fear, and a lack of moral values in our children; these are all the effects of media. It is known that in cases where babies spend much time watching television these children fail to develop their linguistic skills and become withdrawn from society. The massive visual or audio input from television can cause significant problems in the psychological development of the child. Some parents allow their children to watch advertisements, as they are unaware of this fact. In child education, especially in the early stages, parents must be very careful to combat the inappropriate stimuli of the television and computer. During that period the baby needs the parents’ interest, love, conversation, and physical closeness more than they need the picture on the television or computer. Parents must remember that every stimulus during these stages has a long-lasting effect throughout the child’s life. Therefore, the stimuli given to the child must be appropriate.

As a drop of water can shape a rock by falling on it everyday, so to can appropriate or inappropriate messages shape the personality of a child over time. All the messages must be suitable to the age; they must contribute to the child’s development. They must be neither too difficult, nor incomplete and insufficient. They must be in close relation with environment. They must not contain opposing messages, and they must not impose fear or despair.

As a result, in the education of the child, parents, educators and teachers must provide appropriate, sufficient, and positive warnings and messages for children.

Ancient people, considering it very important to determine Earth’shape, derived two important clues from the night skies. According to Aristotle (384-322 bce), these were lunar eclipses and the North Star. Lunar eclipses occur when the sun, Earth, and the moon line up in such a way that Earth temporarily blocks the sun’s light from reaching the moon while its circular shadow gradually crosses the moon’s face. The North Star appears lower in the sky the further south we go: at the Equator it lies directly on the horizon, at a latitude of 45 it is 45 above the horizon; and at the North Pole it is directly overhead. However, it is not visible south of the Equator.(1) As both of these indicate a spherical Earth, the scholars of that time discarded the idea of a flat Earth.

The more challenging question was how to determine Earth’s size. Eratosthenes of Alexandria (third century bce) had a simple yet brilliant idea: insert a gnomon (a vertical stick) into a level piece of ground. This enabled him to determine noon’s exact time (when the shadow was the shortest). It was also used as a compass, for in the Northern Hemisphere the gnomon’s shadow points north.

But can such a simple device determine Earth’s size? Aswan, located about 500 miles south of Alexandria, sits on the Tropic of Cancer. So, at noon of June 21 (the summer solstice), a gnomon inserted there has no shadow. By doing just that in Alexandria, Eratosthenes found that the angle was 1/50 of a circle’s circumference (i.e., 2p/50). In other words, the angle at Earth’s center corresponding to the arc between Aswan and Alexandria on Earth’s surface is 1/50 of a circle’s circumference. Since the distance between Alexandria and Aswan is 500 miles, Earth’s circumference should be 25,000 miles, which is its actual circumference.(2) Thus, Earth’s size and shape was pretty well established over 2,000 years ago.

This knowledge was lost to Europe when the ancient civilizations crumbled. But Islamic civilization and culture, which was rising at roughly the same time as the West was declining, produced scholars and scientists who translated and refined quite a bit of this ancient knowledge. For example, in 1424 al-Kashi used Archimedes’ method of computing to determine its values to 16 decimal places. Ulug Beg compiled the greatest star catalog known at that time. During al-Ma’mun reign (813-833), al-Khwarizmi measured one degree of latitude on Earth’s surface and obtained the result of 57 miles. This means that Earth’s circumference is 360×57 = 20,520 miles.(3) Thus, in the ninth century, Muslim scientists knew that Earth was spherical and had a good idea of its size. Most Europeans at that time, believed that Earth was flat and the universe impenetrable.

The Qur’an describes Earth’s geographical shape and change in that shape: Do they not see how We gradually shrink the land from its outlying borders? Is it then they who will be victors? (21:44).(4) The reference to shrinking could relate to the now-known fact that Earth is compressed at the poles.

At a time when people generally believed that Earth was flat and stationary, the Qur’an explicitly and implicitly revealed that it is round. More unexpectedly still, it also says that its precise shape is more like an ostrich egg than a sphere: After than He shaped Earth like an egg, whence He caused to spring forth the water thereof, and the pasture thereof (79: 30-32).

The verb daha’ means “to shape like an egg,” and its derived noun da’hia is still used to mean “an egg.” As this may have appeared incorrect to pre-modern scientists, some interpreters misunderstood the word’s meaning as “stretched out,” perhaps fearing that its literal meaning would only confuse people. Modern scientific instruments recently established that Earth is shaped more like an egg than a perfect sphere, and that there is a slight flattening around the poles and a slight curving around the Equator.

The West receives “lost” knowledge

An enduring Western myth is that Columbus had to overcome a pervasive belief that he would sail off the edge of a flat Earth by sailing west to Asia. This myth stems in part from compressing the past and conflating the early Middle Ages, when Europe’s belief in a flat Earth was widespread, with the late Middle Ages, when Europe’s knowledge had caught up with and partially surpassed that of ancient Greece and medieval Islam.

During the Renaissance, Europe came into contact with “lost” knowledge by translating Greek and Arabic works. One important book was Ptolemy’s Geography, which accepts Earth’s spherical shape. Geography once more became available in the original Greek, which was not widely known in the thirteenth century. This book was translated into Latin in the late fifteenth century and became widely known. Columbus owned a copy printed in 1479.

By the time of Columbus, the idea of a spherical Earth was widely accepted in theory. Columbus believed this and wanted to sail west to the eastern shores of Asia. Earth’s size was the real issue. Ptolemy’s estimate was as much as 20% too low. Also, he vastly overestimated Asia’s size. The resulting map depicted an Earth with oceans between Europe’s western tip and Asia’s eastern tip, which was well within range of the provisions that ships of that time could carry. Columbus’ estimate of the distance to Asia was wrong, as was his assumption that there was no land between Europe and Asia. Fortunately for him, these two “wrongs” made a “right,” with all of its attendant fame and glory.

The Shape of the Universe

So far, we have given external information (i.e., lunar eclipses and the North Star) about Earth’s spherical shape based upon its position in the universe. If we use this method to determine the universe’s shape, we must observe it in an external manner. As this is not possible, let’s reconsider the question of Earth’s shape with a slight change: Can we determine Earth’s shape by using measurements and observations done only on its surface, and thereby acquire intrinsic information that can inform us of the universe’s shape?

Karl Gauss (1777-1855) answered this question positively by inventing “curvature,” which measures a given surface’s “bumpiness” at a specific point. A flat piece of paper has no bumps and so its curvature is zero. But if we look at a sphere at each point, we see some bumpiness. Gauss called such bumpiness “positive curvature.” Another kind of bumpiness is “saddle-shaped.” We can think of positive curvature at a point as follows: If we put a piece of flat paper on a surface at that point, the surface lies totally on one side of the paper. But in negatively curved space, this cannot happen.

To describe this concept formally (minus some technicalities), assume constant curvatures on the shapes in question. In other words, the shape is totally symmetric and every point has the same amount of bumpiness. There are several ways to describe curvature. Gauss’s formulation for curvature is brilliant. But before that, let’s look at his intrinsic proof for a spherical Earth. Imagine an orchard so large that any deviation from flatness is perceptible. First plant trees on the Equator every 100 kms (the approximate distance between two meridians on the Equator). Then plant another tree 100 kms (the approximate distance between two parallels) north of each tree, and do this several times. If Earth is flat, the distance between them would be same. But since the distance between the two consecutive trees (on the same parallel) decreases, Earth is spherical.

Having seen that an intuitively positive curvature implies a spherical shape, we want to follow this method to get an idea about the universe’s shape. Georg Riemann (1826-66), trying to do just that, invented “curved space” and explained how to compute its curvature. We could launch six probes at equally spaced points along the Equator, and have each of them continually monitor the distance to the two adjacent probes. If space is flat, the distances at any point in its journey would equal the distance from the probe to Earth’s center (an equilateral triangle). For negative curvature, the distance between probes would grow faster than the distance the probe had traveled from Earth; in positively curved space, the distance between probes would grow slower than the distance covered by the probes since leaving Earth.

There are two common misconceptions about the curvature of space. The first one is that curvature is a rather vague or qualitative concept. In reality, it is quite precise and assigns to each point in space and each direction at that point an exact number determined by the shape of the space near the specific location. The second one is that to describe curved space, one must think of it as “curving” into a fourth dimension. This can be useful in visualizing curved space for people familiar with four-dimensional Euclidean space (four-dimensional coordinate space). Unfortunately, science popularizers and science fiction writers often lace this concept with mystical overtones. This is more likely to confuse average people. In other words, measurements made in ordinary three-dimensional space may disagree with the results embodied in Euclidean geometry, for curvature measures the degree and kind of deviation from the Euclidean model.

Riemann also proposed a radically different (non-Euclidian) model for the universe: “spherical space.” This would be the case if space had a constant positive curvature. Based on this, he said that the universe should be a hypersphere (a three-dimensional sphere). The usual sphere is two-dimensional and lives in three-dimensional Euclidean space. In general, n-dimensional sphere is described as in the (n+1)-dimensional Euclidean space, and the set of points whose distance from origin (the point 0) is 1.

The more intuitive way to describe hypersphere comes from the usual sphere. Starting from a point in the sphere called the South Pole, and as we go in a direction in the sphere, we see concentric circles becoming larger until we reach the Equator, after which they become smaller and we finally reach North Pole. The situation is similar in hypersphere. Start from a point in the sphere called the South Pole, and as we go in a direction in the sphere, the concentric “spheres” become larger until we reach the Equator, after which they become smaller until we reach the North Pole. We can generalize this concept for any sphere of any dimension.

Earlier philosophers speculated that the universe was infinite in extent; others (e.g., Plato, Aristotle, Newton, and Leibniz) rejected this as implausible. But the alternative seemed equally dubious: If it did not go on forever, then “like the flat Earth” it had to end somewhere. And, what was beyond that? This model solved the Euclidean paradox of the universe’s “edge,” for if the universe is positively curved, it can be finite in extent and still not have any “edge.” In Riemann’s model, every part of the universe looks just like every other part, as far as shapes and measurements go.

Qur’an 51:47-48 mentions the universe’s spreading out or expansion in space: And the firmament: We constructed it with power and skill, and We are spreading it. This verse reveals that the distance between celestial bodies is increasing, which means that the universe is expanding.

Hubbel’s law

The most surprising discovery of the twentieth century was made by Edwin Hubble in 1929: The universe is not static, but is in a state of rapid expansion. Based upon his observations, he stated Hubble’s Law: Other galaxies are receding from us, the rate at which they recede depends upon their distance, and there is a constant ratio (the Hubble constant) between their velocity and their distance from us.

This law’s most dramatic consequence is what it tells about how we got to where we are now. If distances between galaxies increase as we look toward the future, they must decrease as we go back in time. Each ring of galaxies must have been closer to us in the past; the further away (or back in time) we go, the closer they would have been, and the faster they appear to be moving toward us.

Hubbel’s evidence was limited to a few relatively nearby galaxies. Over the years, however, thousands of observations extended and refined the measurements, and confirmed the general correctness of the velocity-distance relation. Current best estimates are that those galaxies are a billion light-years away (a light-year is roughly 6 trillion miles). Assuming that light always travels at the same speed, those galaxies must have been 1/20 of a light-year closer to us each year in the past. To have ended up a billion light years away, they must have started at exactly the same point as we did “the Big Bang” some 20 billion years ago.

Let’s start by using concentric rings of galaxies at intervals of a billion light-years. Then there are 20 rings, because five rings from us represents galaxies 5 billion light-years distant from us. To see them, we need to see their light that has been traveling for 5 billion years. Thus, we now see their position 5 billion years ago. As there was nothing 20 billion years ago, the outmost ring must the twentieth ring. This might sound paradoxical, as the circles of galaxies seem to grow larger as they move further away from us. However, the paradox is only apparent. Assuming Earth is in the South Pole and that the rings are a sphere’s latitudes, the rings become larger by the Equator and then become smaller until, in the twentieth ring, we reach the North Pole. This time, the rings are spheres and thus fit in the hypersphere. So Hubble’s Law supports our model of hypersphere for the universe.

But how can an expanding universe fit into our picture? In the sphere, the whole surface is expanding, just like inflating a balloon. So the distance from us (at Earth) to the Big Bang is increasing in all directions. In other words, any two points in the universe recede from each other, just as any two points on the balloon recede from each other during inflation.

The issue of time

So far, we have considered the universe’s shape at a fixed time. But, in physics, it is useful to consider space and time together. After Einstein’s brilliant publications about special relativity, Hermann Minkowski (1864-1909) proposed a very useful four-dimensional space-time model as the fabric of the physical universe. In a global picture, each fixed time represents a thin slice of space-time. Like an onion, each layer (assuming there are infinite very thin layers) corresponds to the universe at different fixed times. Given that each fixed time is a hypersphere, the layers are hyperspheres. According to Hubble’s Law, the hypersphere becomes larger as time passes. Just like an onion, the inner layers are smaller and the outer layers are larger.

In an ordinary onion, the layers are usually spheres; in the universe, the layers are hyperspheres. Assuming that the outer-most slice represents the universe at this time, the space-time “so far” is a four-dimensional onion, with layers of the universe at different times. The center of this “onion” corresponds to the Big Bang. We receive the picture of space-time until “this time.” Now, let mathematics predict the future of the space-time, just as Newton’s laws allowed a detailed description of the solar system’s future course. As time evolves, the universe expands, distances between galaxies grow, and gravity weakens. Thus, the space-time curvature diminishes and successive hyperspheres grow at a slower rate.

Two possibilities emerge: The hyperspheres continue to grow indefinitely, although at an ever-decreasing rate, or reach a maximum size and start to contract, in exactly the same fashion as the parallels of latitude on Earth: starting at the North Pole, growing until they reach their maximum size at the Equator, and then begin contracting toward the South Pole. If the universe contracts, distances between galaxies would decrease, gravity and curvature would increase, and the successive hyperspheres would shrink ever faster, eventually contracting into a single point: the “Big Crunch.”

We could then draw a map of the universe as a succession of hyperspheres growing in size for during the first half of its life and contracting during the second half. All space-time would then form a kind of super-hypersphere a four-dimensional object known as a “four-dimensional sphere.”

Conclusion

At a fixed time, the universe should be a hypersphere. When the time dimension is added, space-time should be super-hypersphere, with a “Big Bang” like the South Pole, each fixed time of the universe corresponding to the parallels, and finally a “Big Crunch” corresponding to the North Pole in our space-time model.

Almost everybody has heard that time is the fourth dimension. Even though this concept is easy to imagine, people find it hard to understand because of its mystification by science popularizers. We live in three-dimensional space. This means that I can parameterize the universe such that I can describe any point in it by using just three letters (a, b, c).

Footnotes

1. All of these statements are only approximately true. They would be exactly true if the North Star was precisely overhead at the North Pole, instead of being off center by about 1.
2. The real estimate might not be 25,000 miles, as we do not know the exact correspondence between ancient and current measurements. However, this was a very good estimate for that time.
3. Despite the potential errors mentioned in footnote 2, this also was a good estimate for that time.