“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

Educators who see the beauty at the center of mathematics and can make their students see it that way, are more likely to be able to get their students’ attention and teach them more effectively. Also, as well as facilitating the study of sophisticated mathematics, this puts mathematics in its proper place so as better to understand the value of what it has to say to human beings. To see the beauty and the pleasure in mathematics can change the negative attitudes of some students and help educators in teaching mathematics.

Often it seems that we pursue mathematics education from either a structural or an applications point of view. From a structural point of view, we insist on building up all of the tools one may need in a sequential, logical order, because an educator will need the students to know all of the smaller pieces before they can build any of the larger ideas. An analogy for this would be if we forced somebody to study all of the nails, screws, bolts, and tools to build a house before we let them even see the plans for the house. This is one of the main reasons that most people who study mathematics in their school years think that it is a pointless exercise in playing with formulas and has no significance in real life. For these people, mathematics might be helpful only in keeping track of their checkbooks after graduation. Some students think that they can calculate whatever they need using computers, but sometimes this is not very effective because students may not understand the logic behind the problems and the results do not mean anything to them or they are unable to detect errors.

The applications point of view leads to making up “word problems” that appear to be about the real world, but everyone knows that they are highly artificial. It also leads to focusing at higher levels on only the applications. Hence, for instance, in calculus we spend a lot of time plodding through various physical applications, without letting students see the bigger picture. Or we spend time in liberal arts mathematics talking about things such as modeling and linear programming, which yield great applications, but are generally tedious and do not give most students much appreciation for mathematics. If we see the beauty at the center of mathematics, as well as introducing ideas that may have application or may build some tools, we can bring students to have a much bigger picture of mathematics at a much earlier stage in their mathematical development.

Educators can include some fun topics in courses that they teach. For instance, they can encourage students to discover the amazing number patterns in nature, such as the Fibonacci sequence in pine cone spirals, pineapples, and cauliflowers, in which the number of pieces increases in the following manner: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … (add the last two numbers to get the next). Another example is the “golden ratio.” In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, approximately equal to 1.6180339887 and recent research shows that people think that the shapes and figures in this ratio are more interesting and aesthetically pleasing to the human eye.

Sometimes, even a small algebra trick can miraculously bring students to love mathematics and be more focused, and then more interested in the deeper aspects later on. Take a look at this symmetry:

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321.

Here are a few more examples showing the beauty of mathematics visually with numbers:

1 x 9 + 2 = 11

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

12345678 x 9 + 9 = 111111111

123456789 x 9 +10 = 1111111111

9 x 9 + 7 = 88

98 x 9 + 6 = 888

987 x 9 + 5 = 8888

9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888

987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 88888888

98765432 x 9 + 0 = 888888888.

Students’ minds can be broadened by seeing the surprising differences that arise when we move to non-Euclidean geometry. Fractal shape examples in nature, such as snow crystals, and things like the Mandelbrot set, which is a set of points in the complex plane the boundary of which forms a fractal, can be introduced with a background and give rise to amazingly beautiful images and ideas. Even such deep and thought-provoking ideas as these can be understood by students when they have curiosity, creativity, and an open mind.

All these examples and others like them can inspire people to see the beauty of mathematics and give them a better understanding and a sense of the expanse of mathematics. With a little more discovery of and exposure to the more beautiful aspects of mathematics, students are much less likely to feel any hatred for mathematics and may develop a much greater appreciation for the creation of the universe.

Ali Kemal Unver is a postdoctoral scholar at the University of California, Los Angeles.

Note

* The golden ratio can be derived by the quadratic formula, by starting with the first number as 1, then solving for the 2nd number x, where the ratio [x+1]/x = x/1 or (multiplying by x) yields: x+1 = x2, or a quadratic equation: x2-x-1=0. Then, by the quadratic formula, for positive x = [-b + sqrt(b2-4ac)]/2a with a=1, b=-1, c=-1, the solution for x is: [-(-1) + sqrt([-1]2 -4*1*-1)]/2*1 or [1 + sqrt(5) ]/2. See the second reference for details.

References

1. http://users.forthnet.gr/ath/kimon/
2. Green, Thomas M. “The Pentagram and the Golden Ratio.” http://www.contracosta.cc.ca.us/math/pentagrm.htm.

The theory is simple: Cosmic particles coming from outer space continuously collide with stable carbon-12 atoms in CO2 molecules, which are widespread in the atmosphere. Each carbon-12 atom takes up two neutrons and is converted into a radioactive carbon-14 atom. Radioactive carbon-14 atoms rapidly mix and become uniform throughout the atmosphere. Deep oceans, the biosphere, and carbonate rocks are giant reservoirs of carbon and with the addition of the atmosphere they constitute the carbon cycle of the Earth. Within this cycle, radioactive carbon-14 is continuously created and disintegrated. Both processes are in equilibrium. Since the total amount of carbon on the Earth is constant, a constant ratio is established between the amount of stable and radioactive carbon. This same ratio is valid in all the reservoirs of carbon in this giant cycle. In the biosphere, both carbon-14 and carbon-12 atoms are added to the food chain via assimilation; first by plants through photosynthesis and then by animals through consumption of the plants. For an animal or a plant, a carbon-14 atom is no different from a carbon-12 atom in assimilation. Living beings continuously take up both atoms, so the ratio of both atoms in their bodies remains constant throughout their life. When an organism dies, the uptake of exogenous carbon is terminated. After this point, although the amount of carbon-12 remains constant, carbon-14 continues to disintegrate and the ratio starts to decrease after the body dies. Because the ratio after death is related to the time that has passed since death, it is possible to determine the date of death by measuring the amount of radiocarbon present.

The half-life of radiocarbon is 5,730 years. This means that after 5,730 years half of the total amount of radiocarbon in a dead body disintegrates. The remaining half decays in the following 5,730 years and only a quarter of the first amount remains. This goes on until a very minuscule, undetectable amount remains. In bodies less than 50,000 years in age the amount of radiocarbon can be detected. For an older body, the amount of radiocarbon is so small that the instruments would be unable to measure the amount of radiocarbon present. In addition, such a test obviously works only on the remains of things that were once alive, such as bones or wooden parts of an old structure.

But how accurate is an age determined by this method? How dependable is this technique for enlightening us about the past? Although the theory seems quite consistent from a general outlook, one can see it is not the case when analyzed more rigorously.

Archeologists have tried different ways to test the accuracy of the method. The results have revealed long-term and short-term variations from the actual ages. Long-term variations show systematic deviations of the radiocarbon age from the real age; that is as the date of the sample gets older the deviation increases. On the other hand, short-term variations show irregular fluctuations in the radiocarbon age from the real age. These deviations apparently reveal that the assumptions made concerning the radiocarbon technique were not accurate. The results of these important abnormal conclusions in radiocarbon dating were summarized in the Introduction to Prehistoric Archaeology as follows: “for years, it was thought that possible errors could have minor effects, however, recent research shows that the natural concentration of carbon-14 deviates at some certain periods, significantly affecting the calculated ages.”

The method is based on two assumptions that should be examined carefully: Firstly, the method assumes that the ratio of carbon-14 to carbon-12 has remained constant in the atmosphere from the time the body died to the present. However, recent scientific research has proven that this ratio has not remained constant during geological time.

Secondly, the method also assumes that the carbon supply to the organism was made only by the global carbon cycle and no other source of carbon has affected the system.

Initial concerns about the possible sources of error were focused on the constant ratio assumption. So, why did the constant ratio assumption turn out to be incorrect? Actually, many reasons were found to refute the validity of this assumption. The most important ones are explained below:

Changes in the Earth’s magnetic field are believed to be responsible for long-term deviations in radiocarbon dating. By investigating the orientation of magnetic minerals in ancient rocks, geologists have proven that the magnetic field surrounding the Earth has not been constant throughout the time. Today, it is widely accepted that both the strength and direction of the Earth’s magnetic field has changed. Interestingly, these changes are appreciable even within a century. Changes in the geomagnetism affect the radiocarbon production in the upper atmosphere; cosmic rays are deflected according to the strength of the Earth’s magnetic field. If the magnetic field is high, more cosmic rays are deflected away from the Earth and the production of radiocarbon falls. If it is low, production rises. When the production rate changes, a new equilibrium concentration in the carbon cycle as a whole can only be established after a considerable amount of time has passed. The likely time scale for achieving the complete new equilibrium level is about 10,000 years. This is about the same as the age of the sample that is to be dated! The bottom line is that anything that affects the density of cosmic rays reaching the atmosphere will affect the rate of radiocarbon production, thus affecting the ratio.

Short-term changes might be the results of different factors. One of these is the variation in sunspot activity. Sunspots appear as dark places on the surface of the Sun for a short period of time and generate strong geomagnetic storms. Sunspot activity increases the Earth’s magnetic field and leads to a decrease in the radiocarbon production rate. Therefore, again, anything that causes a change in the Earth’s magnetic field will affect this ratio.

Other effects for short-term variations are the changes in the Earth’s climate. It is widely accepted that the amount of carbon in the atmosphere during geological time is strongly related to temperature changes on the Earth. This fact is also key in understanding the global greenhouse effect, which occurs with the release of high amounts of carbon dioxide to the atmosphere by hydrocarbon combustion. The global sea level has also been affected by these climatic changes. During low temperature seasons (ice ages or glacial periods), large ice sheets covered most of the continents and as a result of this, the sea level dropped appreciably. During these periods, a high amount of carbon (as carbon-dioxide) was kept inside glaciers and they became C-14 depleted (dead carbon). By the end of the Ice Age, large amounts of dead carbon had been released into the system and they had decreased the global ratio of radiocarbon.

Actually, three more resources of dead carbon make a negative contribution to the ratio. One of them is the dead carbon that comes up from deep Earth through volcanic eruptions. Radiocarbon dating of an organism that lived in the vicinity of a volcano gives inaccurate results. Because of the expulsion of dead carbon, samples found close to volcanoes have less radiocarbon in their body than others. Consequently, the age determination of these samples gives significantly incorrect results.

As is obvious from the previous examples, the main problem arises in the lack of knowledge about the history of the sample being dated by this method. Another example is when the sample being tested is wood from the inner part of a tree; the radiocarbon method gives an incorrect result in this case. The reason for this is that the innermost part of a tree finishes the carbon cycle before the tree dies. If a sample was made from this part of the tree (it is impossible to know which part of a tree is being used) then the date produced would be greater than its real age.

Even human activity is an important resource for dead carbon. Although only effective since the last century, a high amount of dead carbon in the carbon dioxide has been released into the atmosphere by the burning of fuel. So the ratio of radiocarbon has decreased. Actually, compared to the factors above, this effect has a more profound influence on the application of radiocarbon dating: No recent organic material can be used as a modern standard. Because of this, the zero point of the timescale chosen is to be 1950 AD, as determined by the US National Bureau of Standards for quoting radiocarbon results.

Consequently, the ages determined by the radiocarbon method are not taken seriously by archeologists because of the problems in the basic assumptions upon which the method was established. Occasionally, the radiocarbon method is used to roughly determine whether an object is modern or of considerable antiquity; in essence, it is used as an authenticity test. Even then the answer may not be clear-cut; for example, an old piece of timber could have been carved to produce an authentic looking sculpture!

Radiocarbon dating is an example of how scientific tools should be used carefully to unfold the reality around us. Scientific theories are only poor models of what is happening in reality. The history of science is full of such examples, which sometimes may be misleading if not handled carefully.

Reference

Radiocarbon Dating, Sheridan Bowman, University of California Press, 1990

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.