All human beings have similar organs, but each organ has qualities unique to itself and to the person it belongs to. Research has revealed that even some of the genes of monozygotic twins are different [1]. As opposed to the previous assumption that “when it comes to DNA, every cell in the body is essentially identical to every other cell,” it has been found that each cell has a DNA of its own [2]. This uniqueness originates from the fact that the order and number of molecules and proteins are different, which is valid for atoms and sub-atom particles as well. Having spent countless hours on microscopes and spectroscopes, a university professor of mine once said, “I witnessed different structures of iron atom each time I observed one. We even need to produce a separate periodic table for the iron element alone.”

We also witness that particles have been created uniquely, just like human beings are. From the micro universe (one millionth of a meter) down to femto universe (one quadrillionth of a meter), each particle is called a boson or fermion. When particles are examined to find out in what conditions they resemble each other in the femto universe, doors of a new world will be opened in terms of knowledge and contemplation. According to this new perspective, while the things and events are sometimes explained through mathematical equations and formulations, they are sometimes explained through experiments of thought trying to attain the truth behind them. Pauli exclusion principle is one such experiment.

Pauli Principle

There are around 380 sub-atomic particles [3]. In atomic and subatomic dimensions, two fermions (e.g. two electrons) of an atom cannot have the same set of quantum numbers. Wolfgang Pauli first explained this principle in 1925 and was awarded the Nobel Prize in Physics in 1945 after experiments proved that the theory was correct. This theory has gone down in history as the Pauli Exclusion Principle.

The motion equations of subatomic particles are expressed by their magnetic states. Pauli said that none of the electrons of an atom can share the same quantum state at the same time. In other words, two electrons in the same atom cannot share the physical features such as magnetism, motion, position and velocity in the same state [2].

In addition to the three quantum numbers (principal, subordinate, and magnetic quantum numbers) described so far, Pauli defined a quantum number for the spin of a fermion (e.g. electron). The spin quantum number is related to the rotation of the electron around its axis because an electron rotates around its own axis as it rotates around the atomic nucleus. According to the Pauli Principle, if an orbit has an electron with a spin of 1/2, an electron with a spin of -1/2 can be placed in the same orbit. If we assume that the spin of an electron rotating clockwise is 1/2 the spin of an electron rotating counterclockwise will be -1/2. Thus, with Pauli’s contribution, four quantum numbers were defined for subatomic particles and it was stated that even if the other three quantum numbers were the same, electrons with different spin quantum numbers could circulate in the same orbit.

The Pauli Principle shows that atoms are not identical. This principle also applies to solid crystals, that is, the materials we use daily. The Principle is also able to explain the fact that the ores that make up the element are different, and the fact that there are different elements in the periodic table. With this principle, important properties of superconductors have been discovered.

In 1927, based on the Pauli Principle, German physicist Karl Werner Heisenberg showed that it is not possible to measure the physical properties of a particle, such as the position and momentum, at the same time and that these results can only be expressed with uncertain probability and statistics [5].

This uniqueness also manifests itself in the space we call the macro world. Stars consisting of only neutrons are called neutron stars. A neutron star is very dense: its mass is 2–3 times that of the Sun, but only about 10 km in diameter. A neutron star should theoretically collapse into a black hole, but this does not happen. The reason for this, it has been discovered, is that neutrons that cannot share the same position prevent collapse and that neutron stars remain in balance [6].

The universe has been created with balance, and everything, from the minutest particles to the stars in the outer space, follows their particular courses to maintain the order.

References

1. scientificamerican.com/article/identical-twins-exhibit-d/
2. https://www.sciencedaily.com/releases/2009/07/090715131449.htm
3. lbl.gov/2017/listings/contents_listings.html
4. wikipedia.org/wiki/Wolfgang_Pauli
5. phy-astr.gsu.edu/hbase/uncer.html
6. phy-astr.gsu.edu/hbase/pauli.html

It was finally the weekend! After my long mathematics presentation, I came home to watch my favorite tv show, Person of Interest, to de-stress. Surprisingly, the episode was about the most famous mathematical constant, pi (π) which is equal to the ratio of a circle’s circumference to its diameter, commonly approximated as 3.14159. Mr. Finch (the main character) was acting as a substitute teacher and wrote on the chalkboard 3.1415926535. Then he asked the students, “What does this mean?” I answered the question in my mind, thinking, “If I have a bicycle tire with a diameter of 1, then one full revolution of the bicycle tire would travel the distance pi.” However, in the show, nobody answered. Then Mr. Finch answered the question himself, saying:

Person of Interest, Season 2 Episode 11 “2 Pi R”

“Pi, the ratio of the circumference of a circle to its diameter — 3.1415926535 — is just the beginning. It keeps going forever without ever repeating, which means that contained within this string of decimals is every other number; your birth date, the combination to your locker, your social security number, etc. It’s all in there somewhere. And if you convert these decimals into letters you would have every word that ever existed in every possible combination; the first syllable you spoke as a baby, the name of your latest crush, your entire life story from beginning to end, and everything we ever say or do. All of the world’s infinite possibilities rest within this one simple circle. Now what will you do with that information; what it’s good for? Well, that would be up to you…”

Although that scene was actually inaccurate, I loved it. This scene is beautiful because most teachers in the world struggle to be as good and as interesting of a teacher as Mr. Finch is here. His knowledge about the subject expands the discussion beyond the textbooks and keeps the students focused throughout the lecture.

Wikipedia, gif, “Pi Unrolled”

We have all been taught that pi is the ratio of a circle’s circumference to its diameter.

Unfortunately, this is wrong because mathematicians have not proved that pi has the characteristic of “normality” yet. In other words, mathematicians are not sure if pi contains all the finitely long permutations of digits from 0 to 9. They are not sure if every digit continues to be used after a certain amount of time or an unlimited number of times in pi’s decimal representation. Nobody knows what we will find in the digits of pi if we keep going. For instance, when we check the first billion digits of pi, we see that the digit 7 occurs almost 100 million times. This makes pi a nice random number generator. However, after some points, pi may not contain the digit 7 and might instead have a non-repeating number with just two or three digits such as 010203112233000111222333…

For instance, after the first 761 digits of pi, there is a famous mathematical coincidence where six nines occur in a row which is called the Feynman point.

Twitter, Feynman’s Library, “Feynman Point in Pi”

But we are sure that the digits of pi keep going on forever and in a random order. This makes pi interesting because the value of pi is finite; however, its decimal value is infinitely long. This is not a contradiction. Pi is a constant number because it is the ratio of the circumference of a circle and its diameter, which are finite values. Still, we need an approximate value for pi.

In 1768, Johann Lambert proved that the value of pi is an irrational number and it cannot be written as a rational simple fraction. 22/7 is a commonly used approximation but does not contain all of the digits of pi. This is because irrational numbers cannot be written as a ratio of two numbers, such as , because they continue on to infinity and do not follow a pattern. In 1882, Ferdinand Lindemann proved that pi is a transcendental number because it is not algebraic; it is not a non-constant polynomial equation with rational coefficients (“Transcendental number”, Wikipedia).

We can safely say that pi is transcendental because the mathematician Yasumasa Kanada found that the first trillion digits of pi appear to be statistically random. If you check the table below, you see that the event of each digit occurring is independent, and the probability of it is one-tenth of the time (“Kanada Laboratory,” Super Computing)

After many years, Emma Haruko Iwao found 34.1 trillion digits of pi in 2019. It took 121 days for Haruko and his computer, because calculating pi requires a lot of power, even for a computer. You can picture it in your mind like this; if you were to print a billion decimal values of Pi in normal sized, ordinary font, it would stretch from New York to Kansas.

However, 34.1 trillion digits is still not enough to prove whether pi is normal or not (“Pi in the Sky, Google Cloud Blog). Supercomputers are still crunching the numbers. If you check the graph below, you will see the number of known digits of pi, by year, since 250 B.C.

FiveThirtyEight, Graph, “Even After 31 Trillion Digits, We’re Still No Closer To The End Of Pi”

Going back to Mr. Finch, we see that he is not 100% wrong. We can find our birthdays in pi easily. If you go to mypiday.com and type your birthday, it will give you the decimal place in pi. For example, my birthday occurs at the 675,097th decimal place.

If pi is a normal number, then we can say that our whole destiny is encoded in pi. The pictures we are going to take in the future, will be in pi because there are binary numbers behind images. All digital products are in pi. Even this article has been in pi for thousands of years. Furthermore, the DNA of every creature is in pi. Mr. Finch was actually right.

There is an interesting and artistic way to show the randomness of pi. Some scientists might be happy with their tedious scatter plots, but there are some artists who use colors for data visualization to communicate with the public. Martin Krzywinski is one such artist, who found beauty and artistry in the randomness of Pi. He took the digits of pi and gave each digit a different color. For instance, he gave 3 the color orange, 1 as red, 4 as yellow, and so on. Then he made a beautiful poster. And if you look at it carefully, you do not see any particular pattern to the colors.

Science Art by Martin Krzywinski

Should we stop working on pi? Or should we continue looking for a better approximation? Is assuming pi as equal to 3.14 good enough? Or is it enough to use 40 digits of pi to find the circumference of the Milky Way galaxy to an error less than the size of a proton (JPL NASA)? Are the first 152 digits enough to find the circumference of the observable universe at 93 billion light-years (WIRED)? There are hundreds of mathematicians who have been trying to figure out more digits of pi for years. It is like trying to get to the moon and then to the next planet, and so on…

But why? Why do mathematicians bother calculating any more digits? Why aren’t 34.1 trillion digits of pi enough? Is it because pi lurks in every circle?

The logical reason seems cryptic: is it because pi is a beautiful source to generate random numbers? Or is it that countries can show off their technology to other countries, because calculating trillions of digits of pi requires a very powerful computer? For instance, in the Star Trek episode “Wolf in the Fold,” Spock foils the evil computer by commanding it to “compute to last digit of the value of pi.” So asking a computer to compute pi is called “a stress test” and may make it crash.

On the other hand, we humans are just weird. Staying at home and drinking tea is a beautiful activity, but when we get bored, we try to climb the highest of mountains, befriend a tiger, or try to memorize the digits of pi, like Chao Lu, who correctly memorized the first 67,890 digits of pi. We will keep doing these things because we like to understand the world around us.

We are inevitably connected to the past, and pi is a thread that’s gone through all of human history. That’s why we can say that as long as there are people, there’s always going to be somebody who wonders what’s next. And I assure you that somewhere in the world there is a mathematician or scientist using pi for something important, because pi is still the mysterious constant of nature.

Finding Pi

The previous statement is utterly true: there has always been someone who works on pi. Math is as old as civilization. Pi has been studied by the human race for almost 4000 years. When the last mammoths were going extinct, people were studying Pi. As far as we know, Archimedes was one of the first humans who calculated pi. He was most likely helping wheel makers. But how did he estimate the value of pi?

Firstly, he saw that all polygons are a circle. According to Archimedes, if you keep increasing the number of the sides of a polygon, you get closer to the perfect circle. In other words, a pentagon is more circle than a square, but a hexagon is more circle than a pentagon, and so on… Thus, more than two thousand years ago, he defined a circle as a regular polygon with an extremely large number of sides.

His definition is useful because measuring a curved surface was hard to do accurately. He found a way to find the circumference of a circle. First, he drew a square with its corners touching the perimeter of a circle and found the perimeter of the inscribed square. Secondly, he drew another square with its sides also touching the perimeter of the circle and found the perimeter of the circumscribed square. He came to the conclusion that the circumference of the circle had to lie somewhere between the value of those two perimeters of squares.

Using this method, however, the difference between those two values was pretty big. So, he drew pentagons to see the upper and lower bounds of the circumference of the circle. This gave him a smaller range of bounds. He kept increasing the number of faces of the polygon that he was drawing inside and outside the circle. Each time he did this, his estimation was getting more accurate. Archimedes got up to a 96 sided regular polygon [called an enneacontahexagon] until he grew exhausted. The lower and upper bound that he found were 3.1408 and 3.1429. Thus, he calculated π to two decimal places.

Archimedes’ method needed improvement because his life span was not going to be long enough to find the other digits of pi by hand. Mathematicians needed to discover more efficient formulas and new techniques.

Before they could do this, they needed to discover algebra. Its discovery by the great mathematicians inspired a whole new way of looking at the world.

The next great jump in calculating pi was the invention of calculus. After that, mathematicians started working on infinite series. An infinite series is an expression with numbers added together until infinity; sometimes these infinite series converge to a particular value.

There are many methods available now to calculate pi. Gottfried Leibniz found pi in infinity. James Gregory was working on one of the astonishing infinite series for the inverse tangent function below. He added infinitely many small numbers together and found pi.

He put x = 1 into the inverse-tangent series. He showed us the further we go, the closer to the estimation of pi we get. However, in order to get 10 digits of pi, we need to write about 5 billion fractions.

After that, another great mathematician, Leonhard Euler – who officially adopted the Greek letter “π” as a symbol to represent the value – found a more efficient equation for pi, when he was only 28. The symbol became iconic. Euler’s Pi equation calculates an infinite sum. The Basel Problem was named after him.

Euler also used pi to write another beautiful equation, Euler’s Identity.

Thanks to the Indian mathematician Ramanujan’s obsession with pi, we have many new formulas to find pi. When he arrived at Cambridge from India, he brought with him a notebook in which there were 400 pages of formulas to find pi.

After the invention of mechanical computers, mathematicians used Leibniz’s, Euler’s, and Ramanujan’s infinite series to calculate a trillion decimal digits of pi (Stanford Cryptography Group). Without a supercomputer, finding this many digits of pi would be difficult. For example, the mathematician William Shanks managed to calculate the first 707 digits of pi by hand but unfortunately, he had made a mistake after the 527th place.

Pi is everywhere

Children start learning about pi when they are in 7th grade and use it until they graduate from college. Even after that, most people use pi again when their children go to school. Pi appears everywhere in the universe. It is literally woven into our universe: the orbits of planets, electromagnetic waves, rivers, the colors of auroras, the structure of DNA, the Great Pyramid of Giza… If a scientist wants to describe the structure of the universe or find the relationship between planets, he/she definitely needs to use pi: anything involving a circle or a sphere is about pi. Circles appear throughout the natural world, whether they’re a soap bubble or the moon in the night sky.

 A Gif showing the clever play on the letters “pi”

Sinuosity of rivers

Pi has a direct relationship with rivers. But how? To figure this out, we need to measure the length of a river in two different ways. Assume that we know the starting and ending point of the river. First, we need the actual length to see how bendy the river is. In other words, the distance that you need to swim from the beginning point to the ending point. This whole length will be “L”. Second, we need to find a straight length. In other words, this time we need to fly from the beginning to the end. And this direct route will be a lowercase “l”. Now we can write the formula for the sinuosity by dividing L by l. The sinuosity is a ratio and measures how bendy the river is.

What is important here is there’s no limit to how high sinuosity can be. Rivers can be really bendy. However, Hans-Henrik Stølum proved that theaverage sinuosity of rivers around the world is pi. If you find the sinuosity of all the rivers and take the average sinuosity of them, you should get pi (Meandering River).

There is another interesting fact about sinuosity. Rivers can be very bendy at some points. But suddenly, those rivers become straight and make the sinuosity around pi. So, it is hard to find the sinuosity of a river equal to 7 because of fluid dynamics. Mathematicians found the highest sinuosity to be around 3.5 and the lowest sinuosity around 2.7.

At the extremely bendy point, rivers cut off after the bend point and make a shortcut to become straight again. This phenomenon is known as an oxbow lake, which controls the sinuosity of rivers. This keeps the sinuosity of a river around Pi.

Pi in Space

There is a mathematical order inherent in our universe. For instance, to understand our solar system, we need pi. We know that our planet moves in front of its host star. And the light comes from host stars. To talk about that light, we need to know how big the host star is. In other words, we need the surface area of the host star. The formula for the surface area of a sphere is 4πr², with r being the star’s radius. The size of a planet also helps scientists to guess whether it is habitable or not.

Another good example to show the relationship between pi and the universe is electrostatic force, which is the force between two electric charges. An electron exerts a force in all directions and forms a sphere field. Electrons also interact with each other on an electric field. To figure out that interaction, we need to find the surface area of spheres, where again pi comes up.

There is also a connection between pi and gravity. If you have had a chance to see Einstein’s field equations, you might notice that pi is there also:

The formula above calculates how objects with a large mass, such as stars and galaxies, can curve space and time with their gravity. Einstein said that, just like a ball sitting on a bedsheet, any form of momentum and energy can also curve space-time around it. In words, the formula is saying:

Gravity = 8 x π x Energy & Momentum

Lastly, if you take the square root of Earth’s gravity, you almost get pi.

Pi Day

After so many years studying pi, people decided to organize an official celebration of pi on March 14th. Since 1988, people have celebrated this magical constant. Coincidentally, Albert Einstein was born on pi day – March 14, 1879. Einstein also published his theory of general relativity on pi day.

The Google logos for Pi Day.

“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

A new Artificial Intelligence (AI) approach to translate brain waves into speech

Akbari et al. Towards reconstructing intelligible speech from the human auditory cortex. Scientific Reports January, 2019.

In a recent study, researchers demonstrated that brain signals from the human auditory cortex can be reconstructed into intelligible speech with a high rate of success. In the study, the brain signals from the auditory cortexes of five participants (suffering from epilepsy) were recorded while they were listening to stories and numbers being read. Researchers specifically chose epilepsy patients as they were already undergoing a clinical procedure involving the placement of electrodes into the temporal lobe of their brains. This procedure is called invasive electrocorticography. The researchers then spoke 10 numbers into the ears of the test subjects: zero to nine. Each number produced a unique response in the brain. The sound produced by the vocoder in response to those brain signals was analyzed and cleaned up by neural networks, a type of artificial intelligence that mimics the structure of neurons in the biological brain. Remarkably, the robotic translations were successfully understood by 75% of people enlisted to test the technology. These results suggest a hopeful future for the Brain-Computer Interface. In particular, this new technology has great potential at giving a voice to people who are paralyzed or severely injured. Despite the potential, the path for a practical device is still long. To go beyond simple numbers and access the thoughts that go with speaking, the researchers will have to probe a different part of the brain known as the Broca area. It is found in the frontal lobe and linked to speech production and language processing.

Milky Way is warped and twisted, not flat

Chen, et al. An intuitive 3D map of the Galactic warp’s precession traced by classical Cepheids. Nature Astronomy February, 2019.

Astronomers have been studying the Earth’s home galaxy, the Milky Way, for centuries in efforts to get a better understanding of its size, structure, and place in the universe. While modern instruments have provided invaluable details about our galaxy and others, a truly accurate model of our galaxy has been missing. Researchers have recently built, for the first time, an accurate 3D map of the Milky Way. We often think of spiral galaxies as a flat thin disk of stars that orbit around a central region, where billions of stars provide the massive gravitational force to hold everything together. However, a new study showed that the Milky Way’s gas disk is no longer confined to a thin plane, but is instead organized in an S-like, warped appearance. The researchers determined the Milky Way’s exact shape at high accuracy by measuring the distances to so-called Cepheid variable stars. Cepheids are young stars that can be up to 20 times as massive and 100,000 times as bright as our Sun. These high stellar masses live fast and die young, sometimes after only a few million years. They also show day-to month-long pulsations, which are observed as changes in their brightness. Combined with a Cepheid’s observed brightness, its pulsation period can be used to obtain highly reliable distances.  By using the data of 1,339 Cepheid stars as benchmarks, researchers were able to map out accurate distances between objects in the Milky Way. Interestingly, in the Milky Way’s outer regions, they found that the S-like stellar disk is warped in a progressively twisted spiral pattern. Researchers suggested that the Milky Way’s warped spiral pattern is most likely caused as a result of rotational forcing or “torques” by the massive inner disk. The further stars are from the center of the galaxy, the weaker their gravitational pull is. This new morphology is expected to provide a crucial updated map of our galaxy’s stellar motions and the origins of the Milky Way’s galaxy.

Smart fabric self-cools or self-insulates depending on conditions

Zhang et al. Dynamic gating of infrared radiation in a textile. Science, February 2019

Decades of innovation in fabrics have yielded high-tech materials that react to hot and cold conditions, allowing them to keep marathon runners cool or alpine hikers warm. However, there was never a material that could change the insulating properties of fabric in response to the environment. A recently developed fabric is the first to automatically warm wearers up or cool them down as needed. Infrared radiation is the primary way the body releases heat and is the focus of this new technology. When the condition is warm or moist, like when a body sweats, the fabric allows the infrared radiations to pass through. Conversely, in cool and dry conditions, the fabric traps the surrounding heat that escapes. The fibers are made of two different synthetic materials coated in carbon nanotubes – one absorbs water, and the other repels it. When the material gets hot and wet during sweating, the strands twist and warp. This distortion brings the strands of yarn closer together, which opens the pores in the fabric to allow heat to escape. When a wearer is too cool, this mechanism is blocked, trapping heat close to the skin. The reaction takes place almost instantly. The garment cools people down before they realize that they are getting hot. Athletes perhaps will be the first people to wear clothes made from the material, but it could be useful for infants, people with disabilities, and the elderly. Researchers say it could be in production within months as the base material is easily available and the carbon coating can be effortlessly added during the standard dying process.

Concepts like the number zero, negative numbers, complex numbers, matrices, and spatial geometry are inventions of mathematicians which were studied earlier and presented especially for the use of physicists. When mathematicians theorized the “Group Theory,” they were reported to have said, “Finally we have developed something that physicists cannot use.” However, this theory, which stemmed from intangible algebra, was found to be useful in investigating the symmetry of physical systems and had serious applications in particle physics.

If there were no mathematical advances, would physics and other sciences have developed as far as they have? What does it mean that sciences are found in such an interrelated state, and thus support each other?

The perfect relationship between physics and mathematics, as in the expression of many physical laws via simple mathematical equations, is truly amazing. The laws that describe the physical world – from equations expressing the laws of motion (like X=V.t, V=a.t, F=m.a h= (gt2)/2), to basic electrical equations (like V=I.R, P=I.V, E=V/d), going all the way to the equations that define gravitational forces and the expansion of the universe (like F=G.m1.m2/d2, V=H.d) – can easily be expressed through mathematics. Furthermore, this simplicity and plainness in creation of the universe fascinates many scientists. Einstein expressed this fascination when he said, “The most incomprehensible thing about the universe is that it is comprehensible.”

One of the most important relations between mathematics and physics is that the independent study of intangible works of mathematics unexpectedly became one of the best tools to describe the physical world. Numbers, for example, are one of the greatest inventions of humanity. Humans have used them to quantify their properties. For thousands of years, people used numbers, but then the concept of negative numbers developed for seemingly no reason. For centuries, negative numbers were seen as nonsense. Of course there is not a square with a negative side length, a circle with a negative value area, or a classroom with negative number of students. However, negative numbers were found to be applicable in many areas of physics; they are now accepted as being as real as positive numbers. For instance, it is almost impossible to graph position-time, speed-time, acceleration-time, and the momentum-speed relation of objects without negative numbers.

Ellipses, parabolas, and hyperbolas (plane sections of cones cut in different shapes) were studied by Apollonius (BC 262-200), who was a contemporary of Archimedes. Interestingly these shapes were one day used by Kepler and Newton to describe the orbits of heavenly bodies like planets. Three dimensional pentagonal and hexagonal patterns, like those on the surface of a soccer ball, were also investigated by Archimedes. This shape has been found to be in exact configuration of a special carbon molecule composed of 60 atoms.

Likewise, the number zero, which was introduced by Muhammad bin Ahmad, in 967, led to many innovations in mathematics, as well as physics. Did al-Khwarizmi (780-850) know, when he found and utilized 1st and 2nd degree equations, that he was working on something mathematicians and physicists could one day never do without?

Imaginary numbers, as proposed against the main principles of arithmetic, also provides a very good example for this topic. We cannot think of a number whose square is negative in normal conditions. In other words, when a number is multiplied with itself, the resulting number is always a positive number. But mathematicians thought of a number that is negative when squared and continued various studies accordingly. Again, these studies have proven to be an important tool, especially in understanding electrical circuits by physicists.

Let’s finish our examples with ones from modern physics. Riemann (1826) was a mathematician who studied spatial geometry and proposed the concept of space curves. Mathematical equations designed by Riemann, pertaining to spatial geometry, were used by Einstein in 1908 to describe and formulate the concept of general relativity. Einstein also used Minkowski’s four dimensional geo-spatial continuum when developing his theory of general relativity.

There are many more examples. The famous Russian mathematician Friedman established a mathematical model that allows the expansion of the universe by improving Einstein’s model of universal geometry. This model was also later confirmed by the physicist De Sitter in discovering universal expansion and by Hubble in formulating the expansion. In addition, well before the discovery of quantum mechanics, Davit Hilbert proposed the complex vector space with a very different mathematical purpose known as “Hilbert Space.” This concept of space with an infinite number of dimensions is today used by quantum mechanics.

Sometimes physical realities can be foreseen via these invented equations. For example, Dirac proposed the existence of a particle known as the positron (or as we call it, the twin of the electron; it just differs by the charge) through his mathematical equation that he wrote in 1928. Four years later this particle was discovered by Carl D. Anderson, as predicted. To name, James Clerk Maxwell (1831-1879), a famous physicist and mathematician, predicted the presence and speed of electromagnetic waves mathematically via his own equations. Later, these waves were detected by Hertz (1886) through experiments. Again, Maxwell calculated the speed of electromagnetic waves via his equations, and by revealing that it was equal to the speed of light, it was understood that light was also a type of electromagnetic wave. Nowadays, the particle called the “graviton,” which is supposed to be in charge of gravitational forces, and the “Higgs” particle, that theoretically fills space according to quantum theory, are waiting to be discovered.

The book of nature is written in such a way that it is expressible by mathematical language. Famous physicist Sir James Jeans (d. 1946) expressed this situation as follows: “From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.” Yes, the level of knowledge that is at play in the universe encompasses both physics and mathematics. The overall interconnectedness of sciences and the interdisciplinary character physics and mathematics point to an owner of this knowledge.

As a conclusion we can deduce that physics and mathematics, just like material and non-material worlds, are in fact intertwined with each other’s various dimensions. The physical face of the universe is the place where records are kept and concepts of matter like heavy or light, big or small, and soft or hard, exist. The mathematical face of the universe (as if spiritual) is the unseen side of events or materials that are hidden and intangible. In a way, this relation between physics and mathematics is a display of the material and spiritual sides of universe.

Nuri Balta is the Head of Physics department at Samanyolu Schools in Turkey. He is also pursuing a PhD degree in Physics at Middle Eastern Technical University, Ankara, Turkey.

Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. He argued that an object in motion can never pass from one position to another, because between the two there is always an “infinity” of other positions, however close, that must be successively traversed in the course of the motion, and this “infinity” can never be exhausted. David Hilbert is a German mathematician who postulated a hypothetical hotel with “countably infinitely” many rooms, all of which are occupied. Since the hotel has “infinitely” many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, and so on, and fit a newcomer into room 1. By repeating this procedure, it may be argued that it is possible to make room for any finite number of new guests, although every room of this hotel initially contains a guest.

The following two quotations from two contemporary authors may provide more substance about the nature of the problem:

On the other hand, involvement with the infinite brings with it a huge range of difficulties. In particular, there are the many puzzles and paradoxes that have been outlined in the pages of this book. Moreover, there are the many quite fundamental problems that arise for such apparently simple notions as counting, adding, maximizing, and so forth. Because we are so firmly wedded to limit notions-“best,” “first,” “greatest,” “maximum,” and so forth-that do not sit easily with the infinite, it is very hard to see how we can make our peace with the infinite.

The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation.

In the present article, we attempt to summarize Rene Guenon’s (1886-1951) alternative way of thinking on the idea of infinite from the perspective of the traditional metaphysical science. Much more detailed presentation of his perspective can be found in his (1886-1951) valuable study The Metaphysical Principles of the Infinitesimal Calculus from which we will extensively quote here.

Guenon considered mathematics as providing a particularly proper symbolism for the expression of metaphysical truths to the extent that they are expressible. However, he states that “in order for this to be so it is above all necessary that these sciences be rid of the various errors and confusions that have been introduced by the false views of the moderns.” To follow Guenon’s paradigm it is necessary to start with the metaphysical notion of the universal-All which comprehends all possibilities, the non-manifested as well as the manifested Universe, that is, the cosmos. The universal All leaves outside itself only the impossible that is a pure nothing. A determination is to define a certain domain of possibilities in relation to all the rest which is expressed by Spinoza (1632-1677) as omnis determinatio negatio est (all determination is a negation). The first of all determinations is Being itself. “Number is only a mode of quantity, and quantity itself only a category or special mode of being, not coextensive with it, or more precisely still, quantity is only a condition proper to one certain state of existence in the totality of universal existence.” Number, space, and time are all determined conditions.

The Infinite, understood in its true, metaphysical sense, has no limits since its opposite, finite is synonymous with limited. Therefore, according to Guenon,

… one cannot correctly apply this term to anything other than that which has absolutely no limits, that is to say the universal All. Furthermore, there can obviously be only one Infinite, for two supposedly distinct infinities would limit and therefore inevitably exclude one another.

He further states, “The Infinite, in its true sense, can have neither opposite nor complementarity.” The scholastic distinction between “the infinite in a certain respect” and “the absolute infinite” cannot be accepted. If a thing is not limited in a certain sense or in a certain respect than one can legitimately conclude that it is limited in no way at all, and since a determined thing does not include every possibility, as such it can only be finite.

Given any number, one can form the next by adding a unit gives the sequence of numbers to us. Therefore, we cannot actually reach its limits. However, the impossibility of reaching the limits of certain things in the manifested Universe should not cause the illusion that these determined things have no limits at all. In order to replace the false notion of “determined infinite,” Guenon introduces, the idea of the indefinite, which is precisely the idea of a development of possibilities the limits of which we cannot actually reach; and this is why we (Guenon) regard this distinction between the Infinite and the indefinite as fundamental to all questions in which the so-called mathematical infinite appears.

According to Descartes (1596-1650), the indefinite is that of which we do not perceive the limits, and which in reality could be infinite. On the contrary, Guenon affirms that

(T)he indefinite cannot be infinite because it always implies a certain determination, whether it is a question of extension, duration, divisibility, or some other possibility; in a word, whatever the indefinite may be, and according to whatever aspect it is considered, it is still of the finite and can only be of the finite.

The idea of an “infinite number” understood as “the greatest of all numbers,” or “the number of all numbers” is contradictory. The impossibility of an “infinite number” can be established by various arguments:

(T)o every whole number (integer) there corresponds another number equal to its double, such that one can make the two sequences correspond term by term, with the result that the number of terms must be the same in both; but there are obviously twice as many whole numbers as there are even, since even numbers alternate by twos in the sequence of whole numbers; one thus ends up with a manifested contradiction.

Guenon insists that number, despite its indefinitude, is by no means applicable to all that exists and the multitude of all numbers cannot constitute a number, which, moreover, is finally only an application of the incontestable truth that what limits a certain order of possibilities must necessarily be beyond and outside that which it limits.

On the other hand, the idea of multitude, contrary to that of number, is applicable to all that exists which allows one to speak of the multitude of divine attributes for example, or again of the multitude of angels, that is, of beings belonging to states that are not subject to quantity, where, consequently, there can be no question of number.

Number itself can also be regarded as a species of multitude, but on the added condition that it be a “multitude measured by the unit” according to the expression of Saint Thomas Aquinas (1225-1274).

The term “indefinite” consists of something unfinished. The “non-measured” is that which has not yet been defined, which is only incompletely realized within manifestation. The multitude of all numbers is “innumerable” or “non-measured,” which is not to say they are infinite, but merely that they are indefinite.

Guenon calls whole number as true number or pure number. He accepts that the numbers other than whole numbers can be considered as the extensions or generalizations of the idea of number. However, he adds that these extensions are also distortions. According to Guenon, numerical quantity has a discontinuous character, whereas spatial or temporal magnitudes, for example, are continuous quantities. “Between these two modes of quantity is a difference of nature such that a correspondence between the two cannot be perfectly established.” He distinguishes the arithmetical unit from the “units of measurement,” which are magnitudes of another sort than number, notably geometric magnitudes. He defines a continuous quantity as an extension-however small it might be that will always remain indefinitely divisible.

Guenon is against atomism, which necessarily implies the discontinuity of all things. He argues extension cannot be composed of indivisible elements, for these elements would have to be extensionless to be truly indivisible, and a sum of elements with no extension can no more constitute an extension than a sum of zeros can constitute a number, that is why points are not the elements or parts of a line; the true linear elements are always distances between points, which latter are only their extremities. Points multiplied by any quantity at all can never produce length, since, rigorously speaking, they are null with respect to length; the true elements of a magnitude must always be of the same nature as the magnitude, although incomparably less: this leaves no room for indivisibles.

Further,

The point, which, being indivisible, is by that very fact without extension, that is, spatially null, but which, as we (Guenon) have explained elsewhere, is nonetheless the very principle of all extension.

For Guenon, Zeno of Elea’s arguments are against atomism and indeed, they prove that without continuity there would be no possible motion.

It is this very conception of motion that is in error, for it amounts in short to regarding the continuous as if it were composed of points, or of final, indivisible elements, like the notion according to which bodies are composed of atoms; and this would amount to saying that in reality there is no continuity, for whether it is a question of points or atoms, these final elements can only be discontinuous.

And, “The possibility of motion presupposes the union, or rather the combination, of both temporal and spatial continuity.”

We consider Guenon as an important and prominent example of thinkers who tried to remind people of the traditional metaphysical ideas. This metaphysical perspective does not share the modern tendency to attribute more importance to the practical applications of science than to science itself. This perspective attempts to link science back to principles of a higher order so that a particular science can be used as a support for elevating oneself to a higher knowledge.

How about quantum computers? Quantum computers make use of a quantum mechanical phenomenon, so-called quantum superposition (being in different states simultaneously). Classically, voltage across a circuit element can be either positive or negative when measured by a voltmeter, but not simultaneously negative and positive. Could it somehow be possible to be in both states simultaneously?

Quantum superposition

For electrical circuits, the answer is obviously no. In microscopic scales of single atoms, or photons (i.e., single quantized packets that constitutes the light beam), however, the answer is yes. Consider an optical component, for instance, that splits an incoming light beam into two beams of equal intensity. In optics, such a device is called a 50/50-beam splitter. You can ask what happens when a single photon is sent to such a beam splitter. Since a single photon cannot be split in this simple experiment, you might expect that it would either be transmitted or reflected with equal probability . Experiments, however, show that this is not actually true in the single photon level. The single photon is indeed simultaneously reflected and transmitted.

Once microscopic quantum superposition is brought into our macroscopic world, we can imagine many interesting phenomena. Simultaneously occupying many different places and being dead and alive at the same time are only two of them. Of course such technology, especially applied to humans is highly science fiction, given current experimental and theoretical challenges. Nevertheless, quantum superposition has a strikingly interesting similarity with the spiritual states already achievable by saints, such that they can be available in more than one place at a given time or become dead and alive, in the sense that they live both in the future and in the past.

It is not known exactly why quantum superposition exists, but what we know is that it is a necessary ingredient for our complex universe to perform its vital functions in a finite amount of time. Quantum superposition principle reflects the great wisdom and power of the Omnipotent. Similar to the single photon example above, with this principle, God gives the underlying particles of the universe an immense power to achieve many tasks simultaneously. Otherwise, regarding the finite age of the universe (about 15 billion years), our physical universe and the events taking place all around us would not come into existence. The Quantum superposition principle has also inspired researchers to build unprecedentedly fast computers to solve the problems that are intractable with any classical computing method. In this article, we introduce this new strategy to computing.

Quantum computing with superposition

Having provided some background about the quantum superposition, we ask the question “How could we exploit quantum superposition for fast computing?” Below we will give a glimpse of that power. Consider a three-bit register. It can only store one out of eight numbers in the set, {0, 1, 2, 3, 4, 5, 6, 7}, in a given moment of time. For example, number 5 is stored in a three-bit register as “101.” Now suppose that these three bits are replaced by their quantum cousins, so-called qubits (short for quantum bit). You can imagine, for example, a quantum register consisting of three rubidium (Rb) atoms. These individual atoms can be prepared in the 0 and 1 logical states simultaneously by shining a laser beam for a certain amount of time. Then it is possible for three atoms combined to be prepared in a superposition of eight numbers, which is impossible classically. In other words all those eight guys physically exist in the same room, although it doesn’t allow more than one guy to fit classically. If we want to make operations on all of these numbers, we don’t need to perform serially; instead, we can achieve that in only one computational step on a single hardware. Thus, quantum superposition leads to a massive parallelism, which renders the computational complexity (i.e., a measure of how efficiently a given problem could be solved) highly reduced for various difficult problems in computer science.

For example, let’s consider RSA, a well-known algorithm (i.e., a set of instructions to solve a problem on a computer) for secure communication that was invented by Rivest, Shamir, and Adelman, hence the name, in 1977 at MIT. It is widely used in electronic commerce protocols. The details of RSA are out of scope in this article (See the FAQ section of the RSA Laboratories’ web site in Ref. [1] for a brief introduction to RSA). Here, we only want to mention its vulnerability to quantum computers if they were to exist. The security of the RSA cryptosystem relies on the difficulty of factoring large numbers, which is intractable with classical computers. Factorization for small numbers, say 15, is quite simple. When the number of digits increase up to a few 100s, for example, enormous computational resource is required. RSA Laboratories publish the RSA challenge numbers (see Ref. [2] for the list of challenge numbers and the prize) on their web site to test the security of their algorithm at various key lengths. The largest integer, RSA-640, which has 193 decimal digits (640 bits), was factorized recently by F. Bahr, et al. The next challenge number in turn is RSA-704, and the prize is $30,000. Imagine factorizing a 1000-digit number. You would probably be a considerably rich person in just a few minutes, if you had a moderate quantum computer and the RSA Laboratories kept feeding you with new challenge numbers, because the factorization of such a large number with current computational resources takes forever, perhaps even more than the estimated age of the universe. Of course, the RSA Laboratories will not let you be very rich, by simply quitting posting new challenge numbers. They would be interested in your quantum computer, though.

How does the quantum computer crack the world’s most secure cryptosystems with little effort? One can construct new algorithms for quantum computers based on above described principle of superposition. These algorithms can take the outcome of previous calculations and input them as a superposition to the next stage of the instructions, which results in a highly efficient form of computing (please consult Ref [3] to have for a simple explanation of quantum superposition for fast computation). In 1994, Peter Shor from AT&T’s Bell Labs in New Jersey just did that. He developed the world’s first quantum algorithm, which efficiently performs factorization. In 1996, Lov Grover also at Bell Labs invented the unstructured database (i.e., a disordered list such as a list of city names not in alphabetical order) search algorithm for quantum computers, so-called Grover’s algorithm.

Suppose that there is a basket with ten balls in it. You are now asked to find a specific one with your eyes closed, say red. It is known, however, beforehand that there is only one red ball in the basket. All you, or your smart digital friend, can do is just pick one randomly and see if it is red. If you are lucky enough, the first ball you pick might be red. In the worst case, however, you will be successful at your last choice. So, classically you have to repeat the process on average at half times the number of balls. If you made a quantum friend rather than classical, however, your life would be smoother. You would be able to find and manage your stuff easily, no matter how messy you are. Quantum computers speed up such unsorted database searches quadratically. You can find, say your favorite socks, in a number of trials that is about the square root of the total number of your stuff. You may think that you don’t have that much stuff. But consider identifying a specific element in a considerably large pool of unsorted data. As the number of elements in the set increases, it quickly becomes intractable to find what exactly you are looking for. In that case the significance of quadratic boost cannot be denied.

Motivated by the above mentioned factorization and unsorted database search algorithms, the power of quantum computing has inspired great attention, since their invention, among many disciplines including physicists, computer scientists, mathematicians, engineers, and material scientists.

Quantum computer today

Despite promising developments in theory, progress in the physical realization of quantum circuits, algorithms, and communication systems have been extremely challenging to date. There are many approaches for quantum information processing. Major model physical systems include nuclear spins, ions, neutral atoms, solid state nanostructures, superconductors, and optical circuits. In optics, for example, the qubit can be represented by the polarization (i.e., direction of oscillation of electric field) state of a single photon. So that the instructions described by the algorithm could be implemented by manipulating the polarization states of single photons. Unfortunately, all the models for quantum computing have their own drawbacks besides their advantages.

Given the trends, nobody knows whether or not a sufficiently scalable (i.e., large enough to harvest its potential power) quantum computer would be available in the decades to come. Nonetheless, D-Wave Systems, Inc., The Quantum Computing Company, was eager enough to unveil the “world’s first commercially viable quantum computer” (see Figure 1, and Ref [4] for the story.). D-Waves’ 16-qubit quantum computer makes use of superconducting element niobium, which operates at an extremely low temperature. It can search for molecular structures that match a target molecule, create a complicated seating plan, and fill in Sudoku puzzles. Although the device is very slow compared to an inexpensive PC, D-Wave intends to develop a 1000-qubit quantum computer.* The goal is to scale the quantum computer to about 10 thousand qubits to solve the most challenging problems outright, which are simply intractable with classical computers. The researchers, however, are not very optimistic. Prof. Lloyd of Massachusetts of Institute of Technology, a pioneering scientist in superconducting approach for quantum computing that underlies the D-Wave’s quantum computer, says “It’s too good to be true.”

Once quantum computers of reasonable power are built, the world will be unimaginably exciting and perhaps scary too. When the first commercial computer, Universal Atomic Computer I (UNIVAC I) (see Figure 2), was shipped to the United States Air Force in 1952, nobody was indeed aware of what this fat guy would lead to in our social, economical, political, and psychological life. Its descendants, however, are now inevitable parts of our lives. They are helping us in many aspects of daily life. Controlling machines, sending electronic mail, scheduling our plane tickets, communicating with our best friends, playing games, making our payments are only some of them.

In this article we only sketched the quantum superposition principle as an important ingredient for quantum computation. This is certainly not the whole story. “Entanglement” [6], for example, is another complementary resource for quantum computing and communications, as well as quantum mechanics to test its foundations.

Contrary to its classical counterparts, the power of quantum computers indeed comes directly from our granted capability of tailoring and mimicking the amazing design hidden in the microscopic world of atoms, photons or other quantum particles. Quantum computers sooner or later will bring the most science-fiction into reality. They will play a significant role especially in the development of ultra-intelligent machines and robots superior to classical ones, and communication systems whose ultimate security is guarantied by the nature’s architecture which was lay down by God. Quantum computers will reveal to us the deepest secrets of our Creator embedded in our universe, which cannot be explored using conventional computers. That day, the future will only be lacked by our limited imagination.

Acknowledgment

This article was produced in MERGEOUS [7], an online article and project development service for authors and publishers dedicated to the advancement of technologies in the merging realm of science and religion.

Omer D. Ikramoglu is a freelance writer in optics and quantum physics.

References

1. RSA Laboratories, http://www.rsa.com/rsalabs/

2. RSA Challenge Numbers, http://www.rsa.com/rsalabs/node.asp?id=2093

3. A short introduction to quantum computation by A. Barenco, A.Ekert, A. Sanpera and C.Machiavello from La Recherche, November 1996. http://cam.qubit.org/articles/intros/comp.php

4. J. R. Minkel, “First “Commercial” Quantum Computer Solves Sudoku Puzzles”, Scientific American, Feb 13 (2007).

5. UNIVAC I, http://en.wikipedia.org/wiki/UNIVAC_I

6. S. Candaroglu, “Quantum Entanglement: Illusion or Reality?” Fountain, Issue 61 (January-February 2008).

7. http://www.mergeous.com/

* At the time of writing D-Wave Systems had only 16-qubit quantum chip and they were intending to develop a 1000-qubit quantum computer by the end of 2008. Although they couldn’t meet their goal, they now have a design of a 128-qubit most powerful ever quantum chip which awaits the tests (see http://www.dwavesys.com for up to date information).

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 infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. David Hilbert

The concept of infinity has always fascinated humankind. The excitement and awe that are affiliated with the very idea of this concept inspire us. Poets, philosophers, artists, astronomers, mathematicians and ordinary people have pondered the questions related to infinity over the centuries. Is the universe finite or infinite? Is matter or space infinitely divisible? Is there eternal life? Where does a finite being get the idea of infinity from?

In this article, we first consider the concept of infinity from a mathematical point of view and then try to explain how that is connected to Divinity and the Attributes of God. An incorrect understanding of infinity in mathematics leads to seemingly irresolvable paradoxes; likewise, an incorrect understanding of the Attributes of God leads to an incomplete or incorrect understanding of creation.

Mathematical infinity

Any serious consideration of infinity inevitably leads to mathematics, where the concept has its deepest roots. Mathematicians use the symbol ∞ to denote infinity. To many people infinity is a sort of number that is larger than all other numbers. However, that is not an adequate description of infinity. Strange things can happen in the realm of infinity. Many intuitive ideas fail when infinity enters into the picture. This leads to apparent paradoxes. Here is one example that is attributed to David Hilbert (1862–1943), the famous mathematician whose quotation appears above.¹ Consider a hypothetical hotel with infinitely many rooms. On a stormy night all the rooms of the hotel are occupied, and a wet and miserable couple shows up asking for a room. The concierge comes up with this clever idea to make room for these people. He says, “The hotel is full but here is what I can do for you. I can move the guests from room 1 into room 2, the guests from room 2 into room 3, and so on. The guests in room n will move into the room n+1, for each n = 1, 2, 3, …”.

So, if we can have a hotel with infinitely many rooms it is possible to accommodate an incoming guest even if the hotel is totally full. But then we see that by a similar reasoning we can accommodate any number of guests, and in fact infinitely many more guests! Clearly, this is not possible in ordinary hotels with finitely many rooms that we are familiar with.

To get more sense of what might happen in the realm of infinity let us consider a couple of other examples. You would have no trouble accepting that two sets have the same number of elements if there is a one-to-one correspondence between them. For example, the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} has seven elements because it is in a one-to-one correspondence with the set {1, 2, 3, 4, 5, 6, 7} given by Monday1, Tuesday2,Wednesday3, Thursday4, Friday5, Saturday6, Sunday7. Note that there are other possible ways of setting up a one-to-one correspondence between these two sets but what matters is the existence of one such correspondence.

There is absolutely nothing surprising about this example. We all know that there are seven days in a week. However, if the concept of infinity pops up in a similar situation, counterintuitive conclusions are obtained. You may be surprised to find out that the set of all positive integers {1, 2, 3, 4, 5, 6,…} is in a one-to-one correspondence with the set of even positive integers {2, 4, 6,…}. One way of setting such a correspondence is

1 2 3 4 5

← ← ← ← ←

2 4 6 8 10 …

Therefore, there are as many even (positive) integers as there are of all (positive) integers (this is a mathematically correct statement), even though the set of even integers is a strictly smaller subset of all integers. Such a situation is not possible with finite sets.

There are other well-known examples of “paradoxes” involving infinity such as Zeno’s paradox, and Gabriel’s horn.²

Treating infinity as if it is like other numbers, even with the assumption that it is greater than all other numbers, is a major reason for these “paradoxes.” Their resolutions are obtained with a proper understanding of the nature of infinity and the concept of limit. For example, adding or subtracting one element, or any finite number of elements, from an infinite set does not change its size. In some cases adding or subtracting infinitely many elements from an infinite set does not change its size either!

Even mathematicians struggled with the idea of infinity a great deal before rigorously formulating the concept and the rules about it. The fundamental notion of modern calculus is limit, which is intimately connected to infinity, in terms of either infinitely small quantities referred to as “infinitesimal” or infinitely large quantities. It took mathematicians a long time to formulate the modern rigorous definition of limit, the rules involving infinity and infinite quantities, and sets of infinite size.

Infinity and understanding the Attributes of God

While the concept of infinity is essential in mathematics, it is not the only field where it is so important. Another field in which the concept is very important is theology. The concept of infinity is intimately connected to the concept of Divinity. In mathematics, compared to infinity all numbers are equal, and equally small. This fact is mathematically expressed as a/∞ for any number a, whether it is 7 or 17 million. This is parallel to the theological fact that compared to God we, and the rest of the creation, are infinitely small. Nobody can claim to be greater than any other human being or any other being in the universe, no matter how powerful or wealthy he or she might be. Therefore, we cannot attribute to anybody greatness to the degree of regarding him or her as divine. Nursi expresses this view in the following words:

“O human! It is a Qur’anic principle that you must not consider anything other than God Almighty as greater than you to the degree that you worship it. Nor must you consider yourself greater than anything and thus claim greatness before and dominion over it. For, just as all creatures are equally far from being the Object of Worship, they are also equal in being creatures.”³

We have seen that treating infinity like other numbers (even with the understanding that it is greater than all other numbers) leads to seemingly irresolvable paradoxes. Similarly, treating God like humans leads to an incomplete and incorrect understanding of God and the creation. God is not only greater than all that is created, He is also not bound by the kinds of restrictions that created beings are subject to. With His infinite power, creating one flower is as easy for Him as creating all the creatures on earth every spring. Creating all the trees in the world is no more difficult for God than creating one piece of apple. He can create countless things at the same moment. Creating a trillion things is no more difficult for Him than creating one thing.

Unfortunately, it is all too common to see the evidence of attributing incorrect characteristics and restrictions to the Almighty. For example, some “try to free God from the burden of special acts of creation”4 by attributing the development of species to evolution, natural selection, and random phenomena. They do not seem to understand that it is no “burden” for God to create anything, whether it be a tiny individual or billions of species. He can create a great number of things at once. He is not forgetful or incompetent. Nothing He does interferes with another one of His actions. He continues to create every moment everywhere in the universe and He has power to do so. He is closer to every one of us every moment than we are to ourselves: Assuredly, it is We Who have created human and We know what suggestions his soul makes to him, and We are nearer to him than his jugular vein (Qaf 50:16). Therefore, there is no burden for God in dealing with special acts of creation.

Since we are subject to many restrictions as humans, we may fall into the error of attributing similar restrictions to God. So, we need to be more careful when we consider the Attributes of God. One of the restrictions by which we are bound is time. Unlike us, God is above and beyond time. Many people have difficulty comprehending the idea of destiny (or qadar, or divine decree in Islamic literature). If we understand that God is not restricted by time, it becomes much easier to understand this concept. Since He is beyond time, He does not need to “wait” to see the consequences of events in the universe. While it is not our purpose to have a full discussion of this important article of faith here, we want to point out one of its connections with the topic. A detailed explanation of divine decree can be found in Said Nursi’s Twenty-sixth Word.5 Those who do not understand that God is not restricted by time ask questions like, “What was God doing before creating the universe?” This question is meaningless with a proper understanding of the Attributes of God.

Attributing to God deficiencies by which we are constrained, such as forgetfulness, tiredness, and incompetence, is a great mistake, one similar to treating ordinary numbers like infinity. It leads to paradoxes that we cannot resolve. It prevents us from having a proper understanding of God and the creation. We can only understand the mystery of the creation by a proper understanding of its Creator, and we need to be very careful when talking about the Attributes of this Creator.

Nuh Aydin is an associate professor of Mathematics at Kenyon College, in Ohio, USA.

Notes

1. Maor, E. To Infinity and Beyond: A Cultural History of Infinity, Birkhauser, 1987. See also http://www.brainyquote.com/quotes/quotes/d/davidhilbe181572.html; http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Hilbert.html; http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
2. Ibid.
3. Nursi, Bediüzzaman Said, The Gleams, Seventeenth Gleam, NJ: Tughra Books, p. 160.
4. Collins, F. S. Language of God, Free Press, 2006, p. 140.
5. Nursi, The Words, Twenty Sixth Word, NJ: Tughra Books, pp. 479–494.

As is the case in many scientific arenas, the first studies in the field of algorithms were conducted by Muslim scientists. The concept of algorithm was used for the first time by the worldwide renowned Muslim scientist Al-Khwarizmi, who lived between 780–850 AD in Baghdad. Al Khwarizmi put his name to historic studies in the fields of mathematics, astronomy, and geography. His work entitled Hisāb al-jabr wa-l-muqābala (Calculation by Completion and Balancing) constitutes the first collection of algorithms. The Latin translation attracted great attention in Europe. The Europeans used the term algoritmi, rendered from “Khwarizmi,” to refer to rules for solving arithmetical problems by using Arabic numerals.

Having first developed as a branch of mathematics, algorithms have been described as a sequence of instructions that is used to solve a certain problem. These finite number of steps,begin at a certain point and come to an end with a result. But, today when we talk about algorithms, what comes to mind are the methods pursued in the operations of sequencing in computer programming; though algorithms are used in many fields, including physics, chemistry, biology, and music.

When different algorithms are used for solving the same problem, this enables us to reach the same solution through different means. There are, for instance, a great number of sorting algorithms (selection sorting, merge sorting, quick sorting) that have been developed for the purpose of incrementally sorting a given set of numbers. The same also applies to the search algorithms which find a given number in a set of numbers. What does this imply for us? Different people can attempt to solve the same problem with different algorithms or approaches that are better, truer or more “beautiful”. Thus, it is always important not to reject an idea without a prior examination or understanding, to welcome proposals of different opinions and to try to benefit from better alternatives or solutions. Thus, we can understand from an algorithmic approach that establishing dialogue and showing mutual respect are essential moral virtues in transforming differences into wealth.

Computer algorithms solve certain problems according to pre-defined parameters. Thus, we should never forget that computers cannot carry out a spontaneous operation, but can only carry out functions that have previously been programmed into them by human beings. Although some researchers, by relying too much on recent developments in computer sciences, and particularly in fields like artificial intelligence, attribute human characteristics to computers. They even estimate that they can produce human-like things in the near future, which seems rather unfeasible when we consider countless physical and spiritual features of humans. What does this imply for us? Pursuing a systematic method when solving problems in both computer sciences and daily life would considerably increase one’s success rate in solving any problems encountered. The most critical stages in problem solving are obviously a correct diagnosis and full description of the problem. The algorithms are developed and the most coherent and suitable are then chosen.

After an algorithm is designed in a flowchart, it is translated into software and tested against verifiable data prior to its usage in real operations. This test is necessary to prevent probable flaws in the software. What does this imply for us? By utilizing all the talents and especially the intelligence granted to us, human beings are always able to find what is better or truer. Research has been conducted in scientific studies by keeping in mind that there is always a next step in any development that has been achieved; the experience of the history of mankind demonstrates that a method that has been developed today might prove to be inadequate tomorrow.

When analyzing a number of different algorithms that are used to solve the same problem, the cost is assessed, with the least costly and most suitable one being selected. The cost of an algorithm is determined according to the number of operations carried out in solving any problem. An algorithm which solves the same problem with a greater number of steps has a lesser efficiency and a lesser performance.

Sorting algorithm by insertion

Sorting algorithms are one of the most frequently used algorithms in computers. As sorted data is more easily processed and used, data are usually sorted first with a sorting algorithm before it is processed within more costly operations.

In insertion sort (i.e., sorting by insertion), numbers to be sorted are inserted into a new “sorted” set. The new sorted set is empty at the beginning, and the numbers are inserted one by one into the right position of this set. During each insertion, the number at hand is compared to the numbers of the new set from the smallest to the largest. During a comparison, if the term (number) at hand is smaller than the term in the new sorted set, then the term is inserted right before the term in the sorted set. As each term is generally compared with all the terms which precede it, sorting of n numbers is done by about n2 comparisons. Thus, the cost of such a sorting algorithm is O(n2).

The quick-sort algorithm

This algorithm is used to find the shortest way between two points. Let us suppose that we have a limited number of points, some of which are inter-related. During this type of calculation, the shortest way is systematically found by checking all the combinations that have been able to be established among given points. The web sources that are provided for planning routes and mapping services with PNDs (Portable Navigation Devices) for travelers can be given as examples; here the minutest details of every highway and road have been recorded to the virtual media. In accordance with certain criteria, by taking into consideration all the probable routes between the two addresses provided by the user, the optimum driving route, including road and street names, is suggested.

Use of algorithm in medicine

Any research or inquiry used in the diagnosis and treatment/curing of a disease is defined as a medical algorithm. The logic of a decision tree is used in diagnosing, curing and following up diseases with these algorithms. Guiding algorithms, for example, “if symptoms A, B, C are observed in patient, then, the patient is probably suffering from the disease D, so, use the treatment E” are used in medical expert systems. The purpose of such algorithms is to standardize the medical services provided and thus minimize potential uncertainties and errors that likely to arise. Moreover, such algorithms are also used for educational and training purposes and as a guide to approaching the patients, diagnosing and curing diseases and towards solving their problems with greater ease. A great number of medical information that has been published has been transformed into numerous algorithms, ranging from those that use simple calculations to those that make highly complicated decisions. For instance, in the algorithm that was developed to measure the Body Mass Index (BMI), age, sex, height and weight are among the questions that are asked of the patient. The data obtained from the patient is later applied to a formula developed from the experiences of medical science, and the said index can thus be calculated. However, doctors should compare the results obtained from such algorithms with their existing knowledge, for even such simple algorithms as BMI become undependable in different cases, for example an athlete or an expectant mother. Algorithms only provide guidance for physicians, but the final decision should be taken by the physician after having individually assessed the status of each and every patient.

Algorithms and music

The algorithmic composition has been used in music for centuries to produce music with a methodological approach. Canon, for example, is a form of music produced with an algorithm in which a melody is replicated by one or more imitations of it played after a given duration.

Human brain, nature, and algorithms

Although the secrets of brain have not yet been fully explained, vital information has been compiled about its biological structure and functioning, thus proving that different sections of the brain fulfill different roles. Such sophisticated tasks as controlling the bodily organs, mental discernment, functions corresponding to the use of tongue and other sensory activities, the perception of colors, calculation of actions and the perception of physical conditions, are all carried out by use of brain mechanisms and faculties. All these tasks are accomplished by algorithmic processes which have not yet been fully explained. The issue of the unification of the data or images received by the eyes is, for instance, one of the matters that scientists do not yet fully understand. Although many have tried to imitate the human brain, such simple cerebral functions as walking and communicating have, to date, not been convincingly imitated.

What algorithms imply for us

As science and technology develop more and more, new inventions are being made, algorithms are being proposed for unsolved problems, while algorithms of already solved problems are being further developed. For example, many ciphers, which in the 1970s were believed to be unsolvable within an acceptable span of time, are being easily solved today thanks to the algorithms that have been developed and the ever-increasing operational power of computers. Consequently, developments in algorithms suggest to us that we should always search for that which is better and more effective; also we should remember that there are a number of different ways that lead to the very same target. Thus, we should always be open to the ideas of others and willing to consult with them before making a decision about any subject.

Ahmet Isik has a PhD in mathematics and is a freelance writer.

References

• Cormen, T. H., C. E. Leiserson, R. L. Rivest, Clifford Stein. Introduction to Algorithms, MIT Press, 24th Edition, 2000.
• http://www.nist.gov/dads/termsType.html#P