Every human being asks the question “why,” though at different levels, to explain the unexplained. It follows a pattern of questions, like “Why did the financial crisis in the U.S. happen in August 2008?” “Why did the space shuttle Challenger explode?” “Why did the terrorists commit the September 11 attacks?” In statistical terms, such unexpected events are named “outliers,” however, they are part of the system and among the components constituting the overall system’s complex behavior. Thus, they need to be part of the explanation in order for the explanation to be complete. We are naturally tempted to come up with universal explanations of the complexity behind these major events so that we can be ready when a similar thing happens again. Though simple mathematical equations or relationships relate to us better and provide a universal explanation, they are typically practical only when the outliers are excluded from the system behavior. Statistics help us greatly in quantifying and characterizing the outliers, especially in the form of probabilistic expressions, such as “there is a 30% chance of a hurricane next week.”

Understanding the complexity around us involves the development of a model that is simple enough for us to comprehend but yet universal enough to capture most of the dynamics of the complexity. The simpler and the more universal the model, the more powerful it is. The universality of a model, however, is hindered by the potential inability to capture something unexpected. The tradeoff between simplicity and universality exists in all modeling efforts; and the models finding the delicate balance in this tradeoff are the most effective ones. A simple mathematical relationship known as “the power law” has been used extensively to characterize and model various natural and social phenomena.

What is the Power Law?

The “power law” does not refer to a misconception that “whoever has power will rule,” but rather it refers to a particular way of characterizing dependency between two quantities. When the number or frequency of an object or event varies as a power of some attribute of that object (e.g., its size), the number or frequency is said to follow a power law. In more general terms, there exists a power law relationship between x and y if y is growing or reducing polynomially when x is growing linearly (y _͌ x^–α). Mathematically speaking, this means that the relationship between y and x is mainly characterized by the exponent -a. An exponent is simply shorthand for multiplying that number of identical factors. So, 4³ is the same as 4x4x4; that is three identical factors of 4. As shown in Figure 1, a quantity with an exponent has three components: the base, the exponent, and the coefficient. So, for 4³, the base is 4, the exponent is 3, and the coefficient is an implicit 1.

Figure 1: Description of an exponent in a power law relationship.

Figure 2: Sample power law relationships between x and y, where y = x^–α.

The power law relationships are traditionally expressed with a negative exponent, which simply means the inverse of the quantity. That is, y _͌ x^–α is equivalent to y _͌ 1/x^α. For example, when a is 2, y will reduce from 1/4 (i.e. 0.25) to 1/9 (i.e. ~0.11) if x grows from 2 to 3. Likewise, when a is 0.5, y will reduce from 1/2 (i.e. 0.5) to 1/3 (i.e. 0.33) if x grows from 4 to 9. For those who enjoy graphs, Figure 2 illustrates these mathematical relationships in linear and logarithmic scales.

Power law on different scales: Atoms to planets

To start with, gravitation, acoustics, electrostatics, and light and electromagnetic radiation, all exhibit a form of power law in that physical quantity or strength that is inversely proportional to the square of the distance, which corresponds to a power exponent of 2. [1] Gravitational force between two particles, the electrostatic force of attraction between two electrically charged particles, the intensity of sound signals coming from a source, and finally the intensity of light or electromagnetic field coming from a source all follow a power law with respect to the distance.

What makes the power law relationships more interesting is their independence from scale or size of the measures being related to each other. This is why we sometimes call power law relationships as “scale-free” relationships or “scale-invariance.” For example, the gravitational force between two spherical particles decays with a power exponent of 2 regardless of the sizes of the particles though the actual force is certainly dependent on the particle sizes. So, the particles can be at nano scales (e.g. a group of atoms) or macro scales (e.g. a planet), but the relationship stays the same!

Power law in frequency: Wealth, terror, and earthquakes

A common usage of power law relationships has been to model and understand frequency of a varying measure. A power law typically very well represents the distribution of wealth in a society. [2] According to a recent study, the distribution of wealth in China during the years 2003–2005 follows a power law with an exponent ranging from 1.758 to 2.285. If we consider an average exponent of 2 for Chinese wealth distribution, this means that if there are 1 million Chinese people who owned $1000 there were 1000 that owned $1M. Thus, the power law essentially expresses how skewed the distribution of a frequency is (see Figure 2). The larger the power exponent, the more skewed the distribution. In this case, a larger power exponent means a more imbalanced wealth distribution while a power exponent of 1 refers to an evenly distributed wealth.

Many other social patterns exhibit power law. A recent study showed that it exists even in terror events! The number of casualties per insurgent event and the number of insurgent events per day follow a power law. [3] Historical data for the last two centuries show further that the number of casualties per war or a terror attack follows a power law distribution. What is even more interesting is that the number of casualties and the number of attacks within an insurgent conflict both follow power law. That is, when only a particular conflict between two countries or ethnic groups is considered, the number of casualties per insurgent event and the number of insurgent events per day follow the power law. This suggests a “self-similar” pattern. Likewise, traffic measurements for many systems show power law distributions of size. For instance, if one observes the data traffic on an Internet connection and counts the number of bytes being transmitted per hour over that connection, a power law distribution of the count of bytes will emerge. Further, if this counting is done per minute instead of per hour, a similar distribution will still emerge – again showing a self-similar pattern. [4]

The power law has been observed in several natural phenomena as well. The frequency of earthquake magnitudes follows a power law. [5] This refers to the intuitive notion that the number of earthquakes with small magnitudes (which humans do not even feel) is much larger than the number of earthquakes with large magnitudes, (which can kill many humans). Small earthquakes are the norm while large ones the outliers. However, without the outliers, there is no power law distribution! Thus, the power law distribution of a quantity comes with an interesting observation: If a quantity is indeed following a power law distribution, then the likelihood of an outlier event increases as the time goes by without an outlier event. This is why geoscientists would make comments like “The region X is due for a major earthquake!” indicating that the region X has not been receiving a major earthquake (i.e. an outlier) for several years. The issue, though, is determining the threshold for an outlier is typically ambiguous and may require many years of measurements and data, which may be impractical.

Power law in growth: Rich get richer

Growth of systems also exhibit power law in various ways. Social growth follows power law due to the well-known “rich get richer” rule, which refers to the intuition that “important” people in the society attract more of the attention of newcomers. This dynamic situation is observed, for example, in the growth of the Internet. Several studies [6] showed that the connections between Internet Service Providers (ISPs) (e.g., AOL, Yahoo!, AT&T, Sprint) follow a power law distribution in that the number of connections per ISP (which shows how well an ISP is connected to the rest of the world) is represented by power law. In other words, there are few ISPs with many connections to other ISPs while most ISPs have a few connections to the others. This is believed to be due to the “rich get richer” rule since an existing ISP with many connections is more likely to gain the business of a new ISP who is joining to the Internet. So, it is somewhat an economic pattern too.

If economics (or the money) is taken out of the picture, social growth still exhibits power law. Online social networks such as Facebook, LinkedIn, and Flickr are clearly following a power law distribution. It is found that the power exponents are in the range of 2.5 to 3.7, indicating a highly imbalanced social growth pattern where few people are at the “center” of the social network with hundreds or thousands of friends, and many people have only one or two friends. [7] Again, the typical explanation for this growth pattern has been the “rich get richer” rule, but “richness” refers to the number of existing friends in this context rather than money.

Physical growth shows power law too in many ways. For instance, roughness of a growing surface as time goes by follows a power law distribution with an exponent ranging between 0 and 1 where an exponent of 0 refers to a smooth growth and 1 refers to a stiff growth. The surface roughness is measured by the variance of heights of surface locations. [8]

Does it really exist? Why does it exist?

Verifying existence of a power law distribution is not easy and requires enough number of samples to show the “tail” of the distribution. The tail of the distribution refers to the samples with large (or rare) values. For example, for the power law distributions in Figure 2, the portion of the distribution when x is greater than 10 (i.e. x>10) roughly corresponds to the “tail.” The tail corresponds to the rare samples. Though statistical theory calls those rare samples “outliers,” the distribution will not be a power law distribution without them. They are strictly parts of pieces that constitute a power law relationship, and observing them typically requires long periods or large numbers of measurements. Due to this difficulty, the existence of the power law is questioned for many real systems. Most of the time, claims of the existence of the power law typically come with an error factor indicating the confidence of the claim. The bottom-line is to observe trends in the samples and thus establish sufficient confidence (e.g., more than 95%) that the power law distribution does exist in the samples.

For those systems with clear exhibition of power law, the root causes of it have been of high interest. The “rich get richer” rule is intuitively one of the root causes, and it is intuitively a natural dynamic to get attracted by a rich member rather than a poor one. Growth certainly naturally follows the “rich get richer” rule, but we have system components slowing their growth, flattening, and then deteriorating. So, not everything is growing, and actually, we have as many things deteriorating as growing. For instance, participants join or leave the Internet or the social networks, and likewise, people join (i.e. birth) or leave (i.e. death) society. How does the power law stay in such systems then?

Due to the “rich get richer” intuition, the power law is considered to be the signature of “self-organization.” The fact that so many natural or synthetic systems are exhibiting this signature deserves the question: “Is it really self-organization?” Maintaining a global power law distribution for a system requires either (i) every member joining or leaving the system according to the “rich get richer” rule and having global knowledge of the whole system or (ii) somebody who knows everything about the system and gives explicit direct orders to each member when they are joining or leaving. Which one is more likely?

Murat Yuksel is an Assistant Professor at the CSE Department of The University of Nevada – Reno (UNR), Reno, NV.

References

[1] Wikipedia, “Inverse-square law,” http://en.wikipedia.org/wiki/Inverse-square_law

[2] M. A. Santos, R. Coelho, G. Hegyi, Z. Néda, and J. Ramasco. 2007. “Wealth distribution in modern and medieval societies,” The European Physical Journal, Volume 143, Number 1, pages 81-85.

[3] J. C. Bohorquez, S. Gourley, A. R. Dixon, M. Spagat, and N. F. Johnson. 2009. “Common ecology quantifies human insurgency,” Nature, Volume 462, December, pages 911-914.

[4] T. Karagiannis, M. Molle, and M. Faloutsos. 2004. “Long-Range Dependence: Ten Years of Internet Traffic Modeling,” IEEE Internet Computing, September/October, pages 57-64.

[5] T. Lay and T. Wallace. 1995. Modern Global Seismology, Academic Press, San Diego, CA.

[6] M. Faloutsos, P. Faloutsos, and C. Faloutsos. 1999. “On power-law relationships of the Internet topology,” ACM Computer Communication Review, Volume 29, Issue 4.

[7] R. Kumar, J. Novak, and A. Tomkins. 2006. “Structure and evolution of online social networks,” Proceedings of ACM SIGKDD, pages 611-617.

[8] A. L. Barabasi and H. E. Stanley. 1995. Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, England.

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.”

In this path of loving, how can it possibly be That we see the world through You, and yet we don’t see You?
Rumi

What is the universe? The search for the answer to this fundamental question has been the starting point of philosophy and science. However, believers, mostly turn to religion for an ultimate answer. Most Muslim thinkers have considered the Divine Names as the primary things to be comprehended through which we can gain knowledge of the cosmos. And growing numbers of people have suggested that the universe is a hologram which makes us images of a higher reality. The first question that comes to mind is this: Can God’s manifestation, as explained in Sufi texts, promote to a holographic universe? The article, however, does not aim to answer this question, but rather to show that holographic model of the universe can feed our imaginations to perceive self-disclosure (tajalli) of God in everything. For this purpose, we will first concentrate on the main characteristics of a hologram, explore what is meant by a holographic universe, and then use these as metaphors to understand the cosmos in terms of God’s names.

What is a Hologram?

Holography is an imaging technique much like photography. However, in the case of holography, the image of the object is three-dimensional. The main instrument behind this dimensionality is the laser light. Let’s see how we can obtain a hologram, for example, of a flower. To achieve this, a beam of laser light is separated into two by a beam-splitter. One beam falls directly onto light-sensitive film. The other beam is reflected from the flower and then shines onto the same film. When these two beams overlap, they form an interference pattern on the film which is called a “hologram.” The pattern, which looks like the ripples formed by rain drops on a pond, can be seen once the film is developed. A three-dimensional image of the flower is produced as soon as the hologram is illuminated by another laser light. That is because the whole message of all the visual aspects of the flower is enfolded on the two-dimensional surface of the film. Actually, what happens is the revelation of the flower information, as recorded in the interference pattern. The startling feature is that if we cut the hologram in half and then illuminate one piece by a laser, we will still be able to produce the entire image. Even, if we keep cutting the film into smaller and smaller pieces, every single piece will still possess the whole information of the flower. Furthermore, we can see the different sides of the flower when we look at its image from different angles, giving us a convincing illusion of seeing a three-dimensional object.

The Holographic Universe

The idea that universe can be a hologram is brought forward by the fact that not only light waves, but also matter waves, can display the behavior of interference. A deeper level of interpretation for a holographic universe arose through the theoretical models that try to explain the coordinated action of the four fundamental forces in nature. These theories suggest 11 dimensions of space-time which actually appear four-dimensional on a human scale. This would, in a way, imply that the information of reality in eleven dimensions is projected onto the four-dimensional space-time substrate that we live in. However, it is with Gerard’t Hooft’s “Holographic Principle” that the holographic nature of all physical systems became apparent, for it explains how a three-dimensional physical system can be described by a theory based on a system’s two-dimensional surface area. Thus, the idea of a “holographic universe” was on stage once this principle had been applied to universe, the biggest physical system we know of. With the holographic model of universe comes the realization that each point in the universe contains the whole universe in itself: every grain of sand is connected to every planet in the cosmos, just as every subatomic particle comprises a web of interconnections by which it becomes intertwined with the human cell. David Bohm is one of the 20th century scientists known for recognizing this wholeness in nature. Bohm’s interpretation of the universe as a hologram came as an explanation to a 1982 experiment, performed by Alain Aspect and his team, which revealed how fast sub-atomic particles communicate with each other. Actually, the communication is found to be so fast, even faster than speed of light, as if particles “knew each others’ fates.” Bohm suggested that particles do not need any signal to communicate with each other simply because the separation between them is an illusion. The explicit separateness is a projection from a higher level of reality, that he calls Implicate Order, where everything is connected. The particle acts as if it knows the other particle’s fate because it has the other particle’s information within itself. Another scientist who saw the holography in action is neurophysiologist Karl Pribram. He was trying to find which specific locations in the brain are assigned to store our memories. His experiments revealed there is no such localization. Pribram claims that the pattern formed by the interference of electrical signals from each nerve cell in the brain is where, in fact, the memories are “stored.”

God’s Manifestation

Many Muslim thinkers, from Ghazzali to Ibn al Arabi, have presented God as Light, and all entities in the cosmos as the dim reflections of that Light. The Qur’an affirms this approach with its many verses, one of the best known being that “God is the light (nur) of the heavens and the earth” (24:35). However, since God has neither resemblance nor similarity to any of His creatures, this kind of description should not be considered as a likeness to the light we know of, but as a metaphor. The symbolism of light is used mainly to explain the relationship between God and creation. But first, one should know about Divine Names to understand this relationship.

In point of fact, in the Qur’an, the verses generally end with a mention of some of the Divine Names-The Life Giver, The Slayer, The Forgiver, The All-Provider, The All-Knowing, The Creator, etc. Actually, we name Him “Creator” after witnessing the effect of this name on creation. For example, we witness mercy on creation and so we call Him “The Merciful.” Everything from physical beings to the sciences manifests God’s names in some mode or another. All types of hearing originate from Him being “The All-Hearing.” “The All-Just” shines in the way the planets are placed in their orbits, while “The All-Provider” can be seen in how each and every animal is taken care of. Medicine reflects “The All-Healing,” while geometry reflects “The All-Shaping.” To explain God’s manifestation (zohur), Said Nursi once gave the example of the sun and its light in his Sixteenth Word: We can think the light from the sun as God’s light and its attributes as God’s attributes. For example, the heat of the white light can be thought as God’s Power, and brightness as God’s Knowledge. The moment the sun reflects, let’s say on a mirror, its heat and brightness are also there in addition to its image. In a similar manner, God manifests Himself in all beings with all His attributes. However there is ranking in this manifestation which depends upon the being’s abilities and quality. We can use the analogy of light to see how different qualities give rise to different manifestations. A flat mirror has the ability to produce the image of the object with its original size and shape, while mirrors in fun houses can cause distortions depending on the curvature. Actually, in Sufism, the fact that the Names are various and are being manifested to varying degrees is given as the reason for variety in the universe as well as in human beings. And thus, human beings, created with the most complex abilities, have the highest place in the ranking, while the heart, the subtlest faculty of a man is considered as the center of manifestation.

Taking the analogy of light further, and using the allegory of a hologram, we can better visualize the relationship between the creation and God and the idea of ranking in His manifestation. In our analogy, laser light represents God’s attributes, the hologram (film) represents four-dimensional space-time, and the interference pattern corresponds to interfering with God’s names. Therefore, everything in the cosmos can be viewed as a pattern of Divine Names enfolded throughout space and time. As a result, one can conclude that the

Divine Names are not so much ontological entities but simply the effect of such interference, like two lights overlapping. God’s attributes give birth to all existent things just as laser light interference gives birth to a hologram. This analogy is consistent with the Sufi texts that draw the distinction between God’s Names (asma) and Attributes (sifat). Imam Rabbani, a well known Islamic scholar, emphasizes this distinction in his Letters. Divine attributes (such as Existence, Having No Beginning, Eternal Permanence, Being Unlike the Created, Self-Subsistence, Life, Knowledge, Power, Speech, Will, Hearing, Seeing, and Creating) are the features that cannot be separated from God Himself (Dhat). But once God discloses Himself, the effects of the Attributes are manifested and we call these effects Divine Names. In a nutshell, this means, the Attributes are the sources of Names, just as the light is the source of interference.

Furthermore, with the analogy of space-time as a hologram, if we cut the space-time into an infinite number of pieces, the whole universe is present at every location. Mahmud Shabstari expresses perfectly such an interconnection and unity in all creation in his Gulshan-i Raz (The Mystic Rose Garden):

Know the world is a mirror from head to foot,

In every atom a hundred blazing suns.

If you cleave the heart of one drop of water,

A hundred pure oceans emerge from it.

If you examine closely each grain of sand,

A thousand Adams may be seen in it.

In its members a gnat is like an elephant;

In its qualities a drop of rain is like the Nile.

The heart of a barley-corn equals a hundred harvests,

A world dwells in the heart of a millet seed.

In the wing of a gnat is the ocean of the life,

In the pupil of the eye a heaven;

What though the grain of the heart be small,

It is a station for the Lord of both worlds to dwell therein. (Translated by E. H. Whinfield)

This way of looking at everything may lead a believer to embrace the doctrine of wahdat-al wujud, the interpretation of oneness of existence. This results in the denial of the existence of the universe and because of the belief that the only thing that exists is God. One remark necessary at this point is that this might sound a lot like pantheism. However, a critical distinction is that in pantheism, what is denied is God, not the universe. Actually, many people, in trying to integrate the perspective of holographic universe with Sufism, adopt the Sufi belief “hama ost,” meaning “All is He,” another name for wahdat-al wujud. But the holographic metaphor, as presented above, may better serve in our attempt to understand God’s Unity (tawhid) without denying the universe. Thinking all beings as an interference of God’s names is consistent with the faith that every entity is a Divine location. However, note that we drew the distinction between Divine Names and Divine Attributes. Every being is kept in existence through God’s names. Things exist not because God abides in them with his infinite Being. In other words, and most importantly, it is not that God Himself is everywhere in the cosmos, but rather that His Names are present at every point. He does not dwell in cosmos, but is, instead, the source of cosmos. He is eternal-while cosmos ceases to exist once its connection to Him is cut. This standpoint is referred in Sufism as “hama az ost,” meaning “All is from Him.” In this view, the existence of universe is not denied but its level of existence can be questioned. If the true existence is defined as God’s Absolute existence, the universe does not have a true existence but is still in a mode of existence which is temporal and contingent. This existential challenge is explained by Islamic thinkers through the example of a shadow. The relationship of creation to the Creator is like that between a shadow and the actual shape. Thus, the likeness witnessed in a shadow and a hologram is such that we can conceptualize God’s manifestation-but He is not limited by these dimensions.

In fact, the similitude of hologram can also help us to conceive the spiritual unveiling. Interference patterns originating from different objects can be recorded on the same film by projecting the laser beams at different and varying angles. In such a case, depending on the direction and frequency of the beam that you send through the film, a different three-dimensional image will appear. Two points to emphasize are that the use of laser light is necessary to see the higher dimensionality, and that one can see different realities. In a similar fashion, in a spiritual journey, a traveler needs to have radiance from God’s light to be able to witness further dimensions. With the light of God, spiritual unveiling occurs and a Sufi may see different instances of a reality depending upon His station, which Divine name He is reflecting on or reciting, etc. Thus, the complex information that every element in the universe can reveal about every other entity depends on the fact that God’s names denote each other. We can further use the notion of interference to delineate the concept of ranking in God’s manifestation. In an interference pattern, not all points have the same brightness. The degree of brightness depends on the location and the way the two lights overlap. There are even points where no light can be observed. This, however, does not mean there is no light there; it simply cannot be seen due to destructive interference. Extrapolating this into God’s manifestation, every entity is a Divine location, even if it cannot manifest Him.

To end this train of thought, we should find an answer to a mysterious question. If everything in four-dimensional space-time comes into existence once a higher reality is projected on the void of darkness by God’s Light, what, then, is that higher multidimensional reality? It is a reality which can only be witnessed by those who see behind the appearance of things-by those who free themselves from the material world and carnal desires. It is, perhaps, the Preserved Tablet (Lawh al-Mahfuz) mentioned in chapter, The Constellations (Buruj). Ultimately, this is a question that can only be answered by those who have purified their hearts and can see beyond.

In a book that deals with questions of logic, when we see numbers written in succession, such as 5, 15, 25; then we are able to discover the relation between these numbers, to predict the next number and to realize that this pattern was made by somebody. However, if we were to be told that these figures indicate the distance covered in equal segments of time by a pebble that has been dropped from a certain height, most of us would not think about the One Who made this general rule.

Or for example, the equation 11.111.111 x 111.111.111 = 12.345.678.987.654.321 may be as amazing to some people as the verses of a beautiful poem, whereas it will leave others cold.

Likewise, the famous mysterious symbols of mathematics, e, i, , are nothing more than some alphabetical signs for most of us, nevertheless they must have meant a lot to the famous physicist, Richard Feyman, since he wrote the following equation in his diary, noting that he admired it: ei+1=0

If we write f (z) = Z2 + c;, this will not be a meaningful sentence for most people. But that will not change the fact that it is an incredibly simple expression of biological and physical reality in an expression which concerns our lives (as in fractal logic). And the picture you see here (Figure 1) is nothing but the analytical projection of this equation on a computer screen.

The spiral form (Figure 2) which can be seen in various things from cone shells to nebulas, has a very simple formula, r2=a2/A which is fascinating for those who spend some time to think about it. All these examples have a point in common. These numbers, which are abstract concepts, are as real as the concrete objects of the physical world. While other sciences make sense, more or less, for a layperson, mathematics can only be appreciated by people who know it well.

The mythological Princess Dido of Phoen-icia fled from the city where her husband (the King;s brother) had been slain by the King. She wanted to settle in Carthage, in North Africa. There the King only allowed her to buy as much land as could be covered by the skin of a cow. Dido decided to interpret the word ;cover; in a wider sense. She had her servants cut the skin in thin strips, connecting them to each other. In the end, she obtained a long cord, estimated to be somewhere from between 1,000 to 2,000 m. long. When it came to placing it on the ground, Dido wanted to find the shape that would cover the largest area. She found the right shape. She made a circle on the ground and she was able to encompass quite a large area of land. As a matter of fact, looking at some ancient castles, we can understand that they were built in this way in order to create the largest structure over the smallest possible area. This explains why the cross-section of a vessel tissue is circular, because it occupies minimal space in the body (Figure 9).

Did you know that mathematical reality applies in our body and in the universe? This fact was realized when scientists developed fractal geometry. The fractal structure we see in the roots and leaves of plants and in the human respiratory and vascular systems are very good examples of this fact. Such excellence indicates the All-Knowing Omnipotent One Who is behind these geometrical designs (Figures 4, 5, 6).

What is the invisible secret of this visible structure? Dido had to enclose the maximum area by using limited material, which she accomplished. Such optimization also exists in the human body and in other living things. The biological systems we have mentioned above have vessel systems designed as fractal networks, delivering the necessary substances to cells. Essentially, these systems are designed in such a way that the vessels occupy minimal space while serving all the cells within the system; this can only be realized through such a fractal structure.

The most striking proof supporting this idea is that if the human veins, which do not take up a great deal of space in the body, were all added together, they would reach a length that is three times the circumference of the world.

How can this be possible? This can be explained quite simply: find an equilateral triangle and carry out the following instructions. First, divide each side of the triangle into three equal sections and place another (smaller) equilateral triangle on the middle section of each side you have divided, facing outwards. If you repeat the same thing for each of these small triangles, you will create the pattern below (Figure 8).

In a fractal structure, as the number of branches near infinity, the shape of the structure resembles a circle more and more. The new area we will find cannot be bigger than the area of the circle, and the points of contact with the circle will approach the maximum value (Figure 9).

We can also explain this fact in the following way: take a circle with a radius of 3 cm. This will serve as the cross-section of a cylindrical object. Then draw seven smaller identical circles inside the first circle. You will see that the proportion between the sum of circumferences and the sum of areas is 5/7. Those who are interested in mathematics will see that as the value of the r (radius) decreases, the difference increases. This clearly indicates that fractal structures are always advantageous. So, what about organs like the brain or the lungs? They do not have a completely fractal structure, yet they really need to have a larger surface area than other objects of equal size. These organs have been enabled to have the largest possible surface area by being convoluted. Otherwise, man would be a strange creature, burdened with a huge mass on his back. Similarly, when we look at a map that shows the coastlines, we see that a coastline that has many capes and bays has a longer coast line than its counterparts, which run straight along the land. All these living things or organs (man, trees, brain etc) have such structures from the very moment they are created. This system or project (Figure 10) cannot have developed on its own, by chance, without there having been a Creator.

Sir James Jean says, ;The Creator must be a perfect Mathematician; in his book The Mysterious Universe. In so saying, he draws attention to mathematics, the mysterious pattern in the universe, and points to the Artist behind the ornamented beauty seen in Creation. The magical science of mathematics whispers its secrets to those who try to perceive it through the eyes of wisdom.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

FOOTNOTES

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

REFERENCES

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