“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”
Albert Einstein

I have always been pretty sure that mathematics and science are the best languages to explain the natural phenomena. While for many mathematics is too abstract, for me, it is the beautiful language of the universe.

There is no upper limit to the numerical abilities of humans. First, we discovered fire, to get warm. Then, we needed light, so we invented electricity. When we needed to talk to someone 10,000 miles away, we invented the internet. Behind all these inventions, was mathematics.

Of course, the universe cannot speak or think. However, we, the people, can read the universe. There are many scientifically and mathematically inclined people who can read the universe and find answers and then describe them. We, the normal people, can also use our imagination as an apparatus to read the universe and nature. If we can read, hence, something is written. In order to write, a language is always needed. So, the universe should have a language. The letters are circles, triangles, hexagons, etc.

Everything in life has mathematical patterns. Think of the wild animals with stripes or patterns for the purposes of camouflage. But why does a leopard or cheetah or tiger have a particular design?

The Enigma codebreaker, Alan Turing, had a mathematical theory about leopard’s spots. Turing suggested in his paper “The Chemical Basis of Morphogenesis” (published in 1952) “a mathematical schema for the formation of the patterns found in animals and plants.” This was 60 years ago [1].

Stars have patterns. Astrologists have been looking at the outer space searching for patterns to better understand life. Whatever it may be that they find, it is always about mathematics.

Seasons have patterns. They come and go. And they influence nature: the climate changes, animals migrate north or south, rain comes, snow melts, the earth changes color, etc.… Of course, seasons cannot make these miracles. They can only have mathematical patterns.

Einstein had pondered for years on how mathematics works so perfectly. He knew that mathematics is the bridge or the language that connects humans with the universe. And being a connection between us and the universe makes mathematics the greatest achievement of mankind.

If you take a closer look at the patterns of our world, you will witness the language of mathematics. Let me give you some specific examples.

Fibonacci, the golden ratio, spiral, cabbage…

Our universe is filled with spiral designs. Spirals can be found in the shapes of the DNA double helix, flowers, elephant tusks, sunflowers, hurricanes, draining water, animal horns, a nautilus shell, a snail shell, a pinecone, a cabbage, a fingerprint, algae, galaxies… the list goes on and on. Tons of lifeless and living things have spiral designs. And they are not random spirals. They have something in common: the golden ratio! And surprisingly, “there is a strong case that this so-called ‘Golden Ratio’ (1.61803…) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time” [2].

All these spirals in nature tell us there are numbers all around us. Let’s observe the numbers of petals on some flowers. When you count the number of petals of the flowers in your garden, you will get the numbers 3, 5, 8, 13, 21, 34, or 55. These numbers are not random numbers. These are very unique numbers; they are part of a sequence developed by Fibonacci, a 13th century mathematician, by adding up the last two numbers starting from 1:

1+1= 2, 1+2= 3, 2+3= 5, 3+5=8 …

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

F_n= F_n-1+F_n-2, F₁=1, F₂=1

But, why are those Fibonacci numbers so important? The key is, the relationship between the progression of growth and the proportion. There is a harmonic proportion hidden in the Fibonacci sequence.

A fact: If you divide one number in the sequence by the previous number, the answers result in or come closer to phi:

For example: 5/3 = 1.6666;

13/8 = 1.6250; 377/233 = 1.61802575; 317811/196418 = 1.618033399

Definition: In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities [3].

These numbers can be demonstrated with the spiral of the florets in a sunflower. The florets in a sunflower head also form two spirals. If you count the clockwise and counterclockwise spirals that reach the outer edge, you’ll usually find a pair of numbers from the sequence: 34 and 55. If it is a very large sunflower, you will get 89 and 144 [4].

These spirals are not only in sunflowers. You can see them if you look at a pine cone or a daisy. If you mark the spirals and count them, you will always get a number from the Fibonacci sequence. And if you count in the other direction, this time you will find an adjacent Fibonacci number.

The nautilus shell, the golden mean

What makes the nautilus shell so special for mathematicians? Having the Golden Mean. But how do we know that the nautilus shell has the Golden Mean?

First of all, we will start with drawing a small, one unit square. Then we will draw another square which is larger than the previous one. We need to add in a counterclockwise direction. The length of each square has a value from the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … Then we can draw spirals, starting with the smallest one, outward through the largest one. Then the Golden Mean will appear.

Flowers, plants, or objects have no idea about mathematics, yet they manifest the best of mathematical patterns. This marvelous mathematical art has been placed in their nature for us not to be fascinated only but also to explore the mysteries behind it.

Note

1. https://www.bbcearth.com/blog/?article=the-maths-behind-a-leopards-spots
2. https://www.sajs.co.za/article/view/4033
3. http://mathworld.wolfram.com/GoldenRatio.html
4. http://www.sciencemag.org/news/2016/05/sunflowers-show-complex-fibonacci-sequences

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 article, we will delve a little deeper into this magnificent miracle of God: the mathematics of nature.

FLOWERS:

For example, look at the pictures given below. For 1-petalled flowers, we offer white calla lilies; for 2-petalled flowers, we offer the very rare euphorbia; and for 3-petalled flowers trilliums, lilies, and irises.

<cellpadding=”15″ cellspacing=”15″>White Calla Lilly

Trilliums

Did you ever wonder why 4-petalled flowers are so rare, and why everyone gets excited when they find a 4-leaf clover? The reason for this is because such flowers are very rare, for 4 is not a Fibonacci number. Some violets and bluets also have 4 petals.

Flowers with 5 petals are rather common. Among them are buttercups, wild roses, larkspurs, and columbines.

Examples of 8-petalled flowers are bloodroots and delphiniums. Examples of 13-petalled flowers are ragworts, corn marigolds, and cinerarias; those with 21 petals are daisies, asters, and chicories; and those with 34 petals are oxeye daisies, sunflowers, plantains, and pyrethrums.

Some families of daisies, such as the michaelmas daisies from the asteraceae family, have 55 and 89 petals.

Seed and flower heads

The echinacea purpura is a member of the daisy family native to the Illinois prairie. You can see in Figure 1 that the orange œpetals seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiraling to the right as you go outwards, you will notice that there are 55 spirals (a Fibonacci number). A little further toward the center, you can count 34 spirals (another Fibonacci number). If you count the spirals going the other way, you will see that the pair of numbers (counting the spirals curving toward the left and toward the right) are neighbors in the Fibonacci number series.

The same happens in many seeds and flower heads, among them sunflower seeds, daisies, pineapples, and pine cones. The reason for this is that such an arrangement packs the optimal number of seeds so that no matter how large the seed head is, the seeds are always packed uniformly at any stage. As they are the same size in any given area, there is no crowding in the center and no scarcity at the edges.

The spirals form a pattern: The œcurvier ones

appear near the center, while the flatter ones, which are more numerous, appear the further out you go. Thus the number of spirals we see in either direction differs according to the size of the flower’s head. On a large flower head, we see more spirals further out than we do near the center. The numbers of spirals in each direction are (almost always) neighboring Fibonacci numbers!

Let’s observe the spirals in these beautiful arts of the Infinite Artist. Consider the daisy. In the close-packed arrangement of tiny florets in the daisy blossom’s core, you can see the phenomenon in an almost two-dimensional form. As shown in Figure 2 there are 21 (a Fibonacci number) counterclockwise spirals and 34 (another Fibonacci number) logarithmic or equiangular spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms in the Fibonacci sequence.

The Math Behind It

Botanists have shown that plants grow from a single tiny group of cells right at the tip of any branch or twig belonging to a growing plant or tree. This tip is called the meristem. They grow in size after their formation, but new cells are formed only at such growing points. Cells further down the stem expand, and thus cause the growing point to rise. This means that a growing plant produces seeds at the flower’s center, and that those seeds then push the other seeds outward. Each seed settles into a location that turns out to have a specific constant angle of rotation relative to the previous seed. This constant rotation forms the spirals.

Consider the following case. There are n seeds in the arrangement. The nearest seed is seed 1, the next one is seed 2, and so on. If each seed has an area of 1, we have a total area of n and a circle with a radius of from the area formula (Area= ). Given this, the distance from the center to each seed is proportional to the square root of its seed number. If we call this angle alpha, the angle of seed k will be alpha*k. So we can easily describe the location of any seed in polar coordinates with and theta=k*alpha.

If we chose an angle of 0.15 (54A) the result will be far better. But we will be done again after 20 seeds, for 20*0.15=3, so after 3 complete turns we will be back on the same line. If we choose 0.48, which is a little bit better than 0.15, we will come back to the original line in 25 rotations.

Therefore, speaking logically, if we choose an irrational number we will not return to the original line. Of course we will get close to it, since every irrational number has some kind of rational approximation. In nature, we mostly observe the so-called golden ratio, an irrational (1.618 = ), which is the root of the equation , which is the limit of the ratio of two consecutive Fibonacci numbers.

With this angle of rotation, each seed is rotated approximately 1.618 revolutions from the previous seed (i.e., 0.618 revolutions or 0.618*360=222.5A). Notice how well distributed the seeds appear; for there is no clumping and very little wasted space. Although the pattern grows quite large, the distances between neighboring seeds appear to stay constant. In nature, you can see that plants grow their seeds simply where there is the most room.

It is really amazing that a single fixed angle can produce the optimal design no matter how large the plant becomes. For example, once a leaf’s angle is fixed, that leaf will œdo its best not to obscure the leaves below and œdo its best not to be obscured by any leaves that will grow above it. Similarly, once a seed is positioned on a seed head, the new seeds continue to push the older ones out in a straight line. However, it retains the seed head’s original angle. The seeds will always be packed uniformly on the seed head regardless of the head’s size. The principle that a single angle produces uniform packing no matter how much growth appears thereafter was proved mathematically only in 1993 by the French mathematicians Douady and Couder.

We frequently observe the golden ration in nature. In addition, we can try flowers and flowers, at least mathematically and as models.

Conclusion

We look at nature and see God’s creation. Most people just look at the general design, but others also study the seeds and their designs. As the Qur’an’s first verse tells us to œRead, in the sense of reading the signs in nature (the Qur’an was revealed to a largely illiterate people who had no significant body of written literature), we should understand this as an indication to analyze nature.

As a mathematician, I see that God has arranged everything in nature according to a mathematical order. He puts the maximum number of seeds in a minimum area. Bees use hexagons to store the maximum amount of honey in a minimal space. This fact is even mentioned in Qur’an 16:68: Your Lord revealed to the bee. Those who believe in evolution say that the picture is very clear. But there is a great art in front of us, one which is very well balanced and in perfect accord with mathematical harmony. This shows that there is no luck or chance in the world, and that everything is based on the rules that God has laid down for His creation.

Footnote

1. Fibonacci (Leonardo da Pisa, c1175-1250): The son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. In 1200 he returned to Pisa and used the knowledge that he had gained on his travels to write Liber abaci, in which he introduced the Latin-speaking world to the decimal number system. Fibonacci numbers begin with 1,1 and the next term is the sum of the two previous terms. The first ones in the series are 1,1,2,3,5,8,13,21,34,55,89,

References

Mathematics Magazine 75:3 (June 2002): 163-73.

http://ccins.camosun.bc.ca/~jbritton/fibslide/jbfibslide.htm.

G. J. Mitchison, œPhyllotaxis and the Fibonacci Series, Science 1965, 270-75.

P. Stevens, Patterns in Nature (New York: Little Brown and Co.,1974).

There are two major theories about the origin or essence of mathematics. One of these theories is attributed to Plato, and the other to the so-called Formalist school. According to Plato, mathematics exists independently of man. What man does is to discover its objective reality, just as other ‘laws of nature’, which we tend to call ‘Divine laws of nature’, are discovered. The Formalist school by contrast asserts that mathematics is a product of human thinking. In order to understand the difference between these two schools, we may cite as an example their view of prime numbers (that is, numbers like 7, 17, 41 which can only be divided exactly by themselves and the number 1). Platonists argue that the prime numbers exist independently of us: before we discovered their existence, they existed in infinite number. Whereas, Formalists are of the opinion that the prime numbers exist because we have defined them as such, and it is meaningless to think about whether they are of infinite number or not.

The language of numbers

Formalists assert that numbers came into existence when human beings began to count. A well-known account of how this happened is that of a shepherd who used to put a stone in his bag for each of his sheep and by matching a stone with a sheep could find out whether any of his sheep had been lost or not. Later on, people began to call numbers each by a different name and since there were two fingers in the two hands, they found it easier to make calculations by the decimal system. This was followed by the operations of addition and subtraction.

According to the Formalists, even the simplest mathematical operations like the four basic ones consist in some logical rules based on certain axioms. They say that we do mathematics by expressing certain rules with certain symbols. That is, we take, say, 5 and 7, a couple of signs whose meaning in the physical world we do not know, and put between them the plus sign, a third sign whose meaning in the physical world we do not know, followed by an equals sign. And we know we must write 12 after the equals sign because that is a requirement of the axioms and rules of logic we are using. This is just what a calculating machine does, that is, it goes through the operation required of it without knowing what it is doing.

Let us suppose that an adding operation consists only in applying axioms or certain logical rules, and has nothing essential to do with the physical world. If we were to take our number signs and apply them to physical objects like stones and sheep, we should be surprised, amazed even, as if by a miracle, that 5 and 7 stones or sheep added together (according to the same rules as 5+7) make 12 stones or 12 sheep. We would come to know that the abstract, conceptual realities in our mind correspond to physical realities in the outer world. According to Paul Davies, the renowned physicist, if we lived in a universe where different physical realities prevailed, in a space where, for example, there were not any countable things, we would not be able to make most of the calculations we make today. David Deutsch claims that counting emerged as the result of experiences. According to him, we can do arithmetic because physical laws allow the existence of physical models convenient for arithmetics.

Richard Feynman, regarded as the greatest physicist after Einstein, says about mathematics that the problem of existence is a very interesting and difficult problem. When you take the third power of certain numbers and then add them with each other, you obtain interesting results. For example, the third power of I is 1, of 2 is 8, and of 3 is 27. The addition of these numbers gives the result of 36. The addition of 1, 2 and 3 is 6 and the second power of 6 is also 36. When you add to this the third power of 4, which is 64, the result is 100. The addition of 6 and 4 is 10 and the second power of 10 is also 100. Added to this the third number of 5, which is 125, the result is 225. 225 is the second number of 10 plus 5, i.e. 15. And so on. According to Feynman, we may not have known this typical characteristic of numbers before but when we do come to know such characteristics of numbers, we feel that they exist independently of us, and that they existed before we discovered them. However, we cannot determine a certain space for their existence. We feel their existence as conceptions only.

Let us take another example. Ibrahim Haqqi of Erzurum, a Turkish Sufi, religious scholar and scientist of the 18th century, discovered a way of checking the correctness of an operation of addition which may still be unknown to modern mathematicians. In order to check or prove the addition, we first add up the digits of each of the two numbers we are going to add up. Let us say, we are going to add 154 to 275, for which we get the answer 429. Adding the digits of each of the first two numbers, we get 1+5+4 = 10 and 2+7+5 = 14. The next step is to subtract 9 from each of these two sums, giving us 1 and 5 respectively. The third step is to add these two results together, 1+5 = 6. Now we do the same thing with the digits of the answer we are wanting to check, namely 429, and again subtract 9: 4+2+9 = 15, 15-9 = 6. The fact that we end up with the same number (i.e.6) means that our addition was correct. This way of checking an addition exists independently of us. We did not create it, we discovered it.

As water had the force of lifting objects of certain weight before Archimedes discovered it and, again, objects thrown into air or a fruit disconnected from its branch fell before Newton discovered the law of gravity so also numbers have many characteristics only some of which have been discovered.

Heinrich Herzt, a physicist, says that we cannot help but feel that the mathematical formulas discovered so far exist out there independently of us. We know that these formulas existed before we discovered them but we cannot determine a space for them. Rudy Rucker, a mathematician, is of the opinion that there is, besides the physical space, a space of mind, which he calls ‘mindspace’ and it is that that mathematician study.

Most of the distinguished mathematicians follow the view of Plato. Kurt Godel is one of them. Before Godel, it was almost a generally accepted view that mathematics is a function of the working of mans brain consisting in the collection of the logical rules which we establish between the symbols of two sets. Godel persuasively argued that there have always been correct mathematical expressions even though their correctness cannot always been proved. Another Platonist mathematician, Roger Penrose, believes that beyond the thoughts of mathematicians there are profound truths or realities in mathematical conceptions. Human thought is directed to extend into these eternal realities and they are there to be discovered as mathematical facts by any one of us. Penrose mentions complex numbers as an example for his argument. According to him, there is a profound, timeless truth in complex numbers. Penrose cites the set of Mandelbrot as another example to prove his argument. The reality this set reveals is the fact that even the lines, twists and shapes of mountains and clouds were or are formed according to certain mathematical formulas. 

What flowers reveal

Almost everyone has heard of the series of Fibonacci. This series, named after the famous mathematician, Leonardo Fibonacci, progresses as 1,1, 2,3,5,8,13,21,34,55,89,144, and so on, each term being equal to the addition of the previous two. That is, I and I make 2, and I and 2 make 3, and 2 and 3 make 5, and 3 and 5 make 8, and so on. This is the series found in nature. For example, when we count the spirals formed of the seeds in a sunflower, we find that those arranged clockwise are 55 and the others arranged anti-clockwise are 89. Both of these figures are among the consecutive terms in the Fibonacci series. These figures may vary according to the size of the sunflower: we may find the figures of 34 and 55 in a relatively small flower, and 55 and 89 in a normal sized one, but the arrangement is always as consecutive numbers in the Fibonacci series. The spirals are arranged in pine cones in 5 to 8. We may encounter the same figures in the arrangement of tobacco leaves. Another extremely interesting characteristic is found in the numbers of petals of flowers. A lily has 3 petals, while a buttercup has 5, a velvet 13, a dahlia 21, and a daisy 34 or 55 or 89, varying according to its family. It is impossible to attribute this miraculous arrangement to chance or ignorant nature. If the DNA of a sunflower or a pine cone determines random numbers for its petals or spirals, how can you explain their correspondence with the terms of the series of Fibonacci? The ratio between the consecutive terms in the series of Fibonacci is quite near what is called the golden ratio’ and known in classical art as the ratio most pleasing to human eye. In order to explain the origin of this miraculous reality, you have to either accept that flowers know what is most pleasing to human eye or that the ‘Hand’ of One, the All- Knowing, the All-Wise and the All-Beautiful, is working in nature.

In short, what Fibonacci did is to discover this characteristic in nature. This means that the universe has a mathematical order or mathematics is the branch of science studying the miraculous order of the universe, the order which the Absolute Orderer and Determiner, One Who determines a certain measure for everything, has established.