“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.”

The appeal of the Golden Ratio to the human eye and brain has been scientifically tested. When subjects are presented with a range of rectangles, people invariably pick out as most pleasing ones those whose sides are of the Golden Ratio. This golden number is the basic number at work in aesthetics; there are even claims that the Golden Ratio was used by Leonardo da Vinci when painting the Mona Lisa, and by the Greeks in building the Parthenon. But the surprising thing is that a number deemed aesthetically pleasing by human beings also crops up in nature and science. In a newly published article in Physical Review B, it is stated that the Golden Ratio appears in the structures of some metals. The Golden Ratio is seen in the arrangement of seeds on flower heads, in the spirals of sea shells and galaxies, even in black holes. This ratio can be found almost everywhere in the universe.

Although the Greek mathematician Euclid first defined the Golden Ratio in around 300 BC, the followers of Pythagoras probably knew of it two centuries earlier. Euclid defined it as a line that can be divided into two unequal parts (Figure 1), where the ratio of the smaller part of the line to the longer part is the same as the ratio of the longer part to the whole. This ratio is 1.6180339887…, the Golden Number.

What makes the Golden Ratio special is the number of mathematical properties it possesses. The Golden Ratio is the only number whose square can be produced simply by adding 1 and the reciprocal of which can be arrived at by subtracting 1. If you take a Golden Rectangle – that is a rectangle where the length-to-breadth ratio is equal to the Golden Ratio, and take out a square, what remains is another, smaller Golden Rectangle. Also, think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. If you start with 7 and 11, then what you have is:

7, 11, 18, 29, 47, 76… When you have written down approximately 20 numbers, calculate the ratio of the last to the penultimate: the answer should approximate the Golden Number.

In mathematical terminology it is (an/an-1) equals to the Golden Ratio.

Another example of the Golden Ratio is the Fibonacci Numbers, that is a number series where each number is simply the sum of the previous two numbers. These numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… The ratio between any two successive Fibonacci numbers approaches the Golden Ratio as the numbers get larger. We can find the Fibonacci series particularly in the spirals of sea shells and in the arrangement of seeds on sunflower heads.

It was the elusive nature of the Golden Ratio that led the Italian friar and mathematician Luca Pacioli to equate it with the incomprehensibility of God. In the 15th century, he wrote a three-volume treatise, Divina Proportione (Divine Proportion), that was crucial in the dissemination of the Golden Ratio beyond the world of mathematics. After him, many artists, architects, and musicians used the Golden Ratio in their works; for example, musicians such as Debussy and Bartok and the architect Le Corbusier.

Let’s examine the arrangement of leaves on the stem of the plant Phyllotaxis. As each new leaf grows, it does so at an angle offset from the leaf below. The most common angle between successive leaves is 137.5 degrees – the Golden Angle; 137.5=3d 360-360/G, where G is the Golden Ratio. Why does the Golden Ratio play a role in the arrangement of leaves? It is all down to the irrationality of the number. A new leaf must collect sunlight without throwing too much of a shadow on the leaves below. A plant must arrange its leaves in such a way that the greatest number can spiral around the stem before a new leaf can sprout immediately above a lower one – that is at 360 degrees. If the leaves were arranged at an angle of 120 degrees, then the leaves would grow as 3 separate columns with large gaps between them. This would effectively block out the sunlight to the lower leaves. If the angle were 50 degrees, then there would be spaces between the leaves. But, with an angle of 137.5, the maximum amount of leaves can be arranged with a minimum of space being left between the leaves.

The Golden Ratio also crops up in hard sciences. Let’s take a look at the growth of “quasi-crystals.” These maintain a five-fold symmetry, which means that they make a pattern that looks the same when rotated by multiples of one-fifth of 360 degrees. Since the time when these crystals were discovered in 1984, many physicists have been researching their properties. In Brookhaven National Lab in New York State, Tanhong Cai imaged the microscopic terrain of the surface of such crystals made from alloys of aluminum-copper-iron and aluminum-palladium-manganese. It is found that flat terraces are punctuated by abrupt vertical steps. The steps come in two predominant sizes, with the ratio of the heights of these two steps being the Golden Ratio. This fact was discovered in 2002.

The most surprising place where the Golden Ratio appears is in black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies, such as the sun, have a negative specific heat. This means that they get hotter as they lose heat. In a spinning black hole there is an outward centrifugal force acting to prevent any shrinkage of the hole. The force depends on how fast the black hole is spinning. It turns out that at a critical value of the spin (when the ratio between the square root of the mass value and the square root of the spinning parameter is equal to the golden ratio), a black hole flips from having negative to positive specific heat. In other words, the Golden Ratio determines the character of the black hole.

Figure 2 shows the finger bones of a hand. The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the Golden Ratio. Pine-cones show the Golden Ratio spiral clearly (Figure 3). If one looks carefully at an ordinary cauliflower, one can see a center point where the florets are smallest, and the florets are organized in spirals around this center in both directions (Figure 4). The flower, Echinacea Purpura, has the same spirals (Figure 5).

With the help of developing science, new examples of the Golden Ratio are waiting to be discovered in the universe. The latest discoveries demonstrate that by using this ratio in technology, new products which make our lives easier will soon be available to all. This mystery that is spread throughout the universe may be an opportunity to renew and change our points of view on life.

Reference

• Chown, Marcus, “Why Should Nature Have a Favorite Number,” New Scientist, 21-28 December 2002, 55-56.

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).