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ear readers,

Few months have passed since the entire humanity was astonished by the evidence of water on Mars that NASA came up with. Whereas non­ professionals may have been sur­ prised by the totally new informa­ tion, the scientific community had eagerly expected such a discovery for a long time since the topic of presence of various molecules on Mars has been discussed for dec­ ades. Debates over various opin­ ions and their credibility have finally come to their conclusion. However, one unresolved issue has raised another uncertainties among public. Most of us probably seek the answers for newly brought up questions and wonder what is the underlying method leading NASA to the conclusion that there are water molecules on the planet that is 225 million kilo­ metres far from the Earth. Science, and specifically progress in the contemporary technology, provides the answer for such ques­ tions. Robotic rovers are useful in scanning the surface of planets; we have an access to a variety of telescopes and other technology that enables us to observe planets,

as well as other objects in the space. Technology, mathematics and sci­ ences in general provide us incred­ ible opportunity to explore such things. What the early mathemati­ cians and philosophers started, contemporary scientists have been continuing with and the incessant progress lead us to a conclusion that discoveries of the future can­ not be easily predicted nowadays. In science, as well as in any other field, it is not only important to continue with the progress on a purely academic level, but it is also necessary to introduce the new ideas to the wide public. The sec­ ond issue of the Open Science magazine that you are reading at the moment aims to introduce readers some of the most break­ through news from the world of science. Whether it is the discovery of water on Mars, 4D printing tech­ nology, Chemistry of touch or any other topic, all the articles give a reader unique opportunity to learn about the topics that contempo­ rary science deals with.

is fortunately not the case of Open Gate teachers. Teachers of human­ ities have actively competed in the first round of the Open Gate Teach­ er s Science Cup which you can read more about further on in this issue. The Open Science magazine also aims to provide an opportunity for students who would like to publish articles themselves. Especially to those of you considering writing a scientific article, I recommend to pay a special attention to the scien­ tific article dealing with the Chaos Theory, written by Michaela Mráz­ ková, which may serve as an inspi­ ration and template. Please, notice the QR codes next to the articles titles. They will lead you to the information about the authors of articles, their pictures and article references. Enjoy the reading!

Sometimes, people who are not directly involved in any area of sci­ ence do not realize the importance of staying in touch with it, and the more we are proud to say that this

If you have anything to ask to to tell us, don t hesitate to use our gmail adress: openscience.info@gmail.com

JAKUB KUNÈÁK Chief Editor

WORLD NEWS 4

CHEMISTRY 8

­ Harder than Diamond ­

­ Touch Chemistry ­

­ Gravitational Waves ­

­ Beware of Dihydrogen Monoxide ­

­ Beta cells made from skin cells ­ PHYSICS 10 TECHNOLOGY 5

­ Black Holes vs Worm Holes ­

­ Printing in 4D ­

­ Sweaty Mars ­ BIOLOGY 6

­ Neutrinos ­

­ Cow Heart Dissection ­ ­ Virtual Lab ­

TOP 10 13

­ New Target for MS Treatment ­

­ Strange & Interesting Phobias ­

ENTERTAINMENT 14 ­ Teaches Cup ­ Comics ­ SPECIAL INSERT 16 ­ Chaos Theory ­

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new form of solid carbon, called Q­carbon, was discov­ ered by Jay Narayan and his team this year. Syntesis of this materi­ al is done using a crystal ­ for example glass ­ which is coated with amor­ phous carbon, heated up with a strong laser beam, and then rapidly cooled down. The final structure ­ whether the result is a diamond or a Q­carbon, or both combined ­ depends on the rate of cooling; this can be

changed by length of laser pulse dura­ tion or by using different substances.

ust recently, one peer­reviewed paper giving the first evidence of the existence of gravitational waves was published by physicists from the Advanced Laser Interferom­ eter Gravitational­Wave Observatory (LIGO). If the measurement is repeat­ ed with similar results, the LIGO scien­ tists might easily receive the Nobel Prize for Physics. We know about grav­ itational waves for about a hundred years, but we were still waiting for confirmation that they are real. If the results prove to be correct, we will be reassured in our understanding of

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gravity, and our limited knowledge of the nature of spacetime itself could be expanded.

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pancreatic cell production and manu­ facture trillions of the target cells in a step­wise, controlled manner. This ac­ complishment opens the door for dis­ ease modelling and drug screening and brings personalized cell therapy for patients with diabetes a step clos­ er. "Our results demonstrate for the first time that human adult skin cells can be used to efficiently and rapidly generate functional pancreatic cells that behave similar to human beta cells," says Matthias Hebrok, PhD, di­ rector of the Diabetes Center at UCSF and a co­senior author on the study.

cientists at the Gladstone Insti­ tutes and the University of Cali­ fornia have successfully converted human skin cells into fully­ functional pancreatic cells. The new cells produced insulin in response to changes in glucose levels, and, when transplanted into mice, the cells pro­ tected the animals from developing diabetes in a mouse model of the dis­ ease. The new study, published in Na­ ture Communications, also presents significant advancements in cellular re­ programming technology, which will allow scientists to efficiently scale up

This process itself is rather inexpen­ sive ­ it can be done at normal atmos­ pheric pressure, and at room temperatures. More discoveries and assumptions have been made about this carbon phase. In natural state, it is found in the core of some planets ­ for example

Detecting gravitational waves is really difficult. The mechanism itself is quite simple, but requires extremely precise measurement. In LIGO, a laser is shot through space and divided into two identical lasers by a special optical de­ vice. The two created beams fly per­ pendicular to each other. They are reflected by a pair of mirrors, and re­ turn back to the initial position. If the light travels without any external fac­

Jupiter. It is highly ferromagnetic, and it glows when exposed to low energy levels. The strength of this phase and its ability to release electrons could be significant for development of new display technologies, or creating and processing nanomaterials. The re­ searchers are now working on finding a way to process this material, so that it can be used in the future for addi­ tional nanotechnology advances.

tors, the light beams should return aligned (as a single beam like it was first created). On the other hand, when one of the lasers meets a gravi­ tational wave, a change of the length of the light's path by either stretching, bending or shrinking the space might occur, which results in a different time of arrival of the light. However, since we cannot produce gravitational waves strong enough, and the light is very fast, the changes in the time of arrival are so insignificant, it was not possible to detect them before recent­ ly.

"This new cellular reprogramming and expansion paradigm is more sustaina­ ble and scalable than previous meth­ ods. Using this approach, cell production can be massively increased while maintaining quality control at multiple steps. This development en­ sures much greater regulation in the manufacturing process of new cells. Now we can generate virtually unlimit­ ed numbers of patient­matched insu­ lin­producing pancreatic cells."

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he technology of printing three­ dimensional objects from various materials has been known to the world for more than three decades. The first 3D printer was built in 1983 by Charles Hull, and the printing technology has been developing ever since, using different and more complex materials, varying from plastic and metals to even organic materials.

One of the latest and one of the most impressive achievements of the 3D print technology was printing out a protein a long chain of various different chemicals called amino acids. Shortly after this, it was followed by other organic tissues, such as bacteria, polymers, drugs or food materials. The machine capable of printing even single cells in chains has been released in June 2015 under the name Bio3D Explorer.

aligned next to each other thus creating a plain, the third one are these plains layered on each other to make space. The fourth dimension, consequently, is a set of infinitely many 3D spaces, put next to each other along the axis of time. They are all occurring at different times, and we can never observe any other but present space, even though we are partly capable of predicting upcoming events in spaces which are about to come. Each of these 3D spaces is slightly different than the previous ones, simply because objects change shapes or positions with time. If we watched different three­dimensional spaces in different times, we could observe changes, which we in our reality and with our limited physical abilities observe simply as movement.

The basic idea of printing in new dimensions is simply that the object changes its shape over time it moves and it is therefore different in every one of the 3D spaces found in fourth dimension, if certain conditions are satisfied.

So far we have been able to print in two­dimensional pictures on papers using ordinary office printers, and three dimensional objects from various materials using the technologies of 3D printing. A team, which consists mostly of MIT engineers, has recently announced a new invention, the technology of 4D printing. To understand what printing in 4D means, the concept of fourth dimension itself should be introduced. Other, lower dimensions are built up in following way. The first dimension is just a line, the second is an infinite number of lines

In case of the printing introduced by MIT engineers, the condition is that the straight straw­like object must be completely surrounded by water; then it slowly changes its shape to become, for example, a square.

As their technology advanced, an additional technique was released this time they managed to print a plain, which then folded itself into an almost perfect cube, or bent some of its parts to change shape.

By developing this technology and learning how to use it properly, we could build much more efficient 3D printers, which would, for example, take up less space. One can use a big printer to make a mug, or a much smaller printer to print something, which folds itself into the shape of a mug. Additionally, as scientists are now capable of printing proteins and other organic tissue, and are accustomed with the technology of self­folding shape­changing materials, a wide range of opportunities have appeared. For example, there are four levels of how protein is structured, beginning at simple order of amino acids, ending with complex pack of protein, which is already fully folded and ready to be used. The real challenge for researchers then is: are we able to print proteins which would fold themselves to create higher structures, even muscles, without any human help except fulfilling the conditions of folding?

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bsence makes the heart grow fonder, says one well­known proverb. As a young poet I fully agree, however, after the unforgetta­ ble experience with a cow s heart I am forced to make an exception. Al­ though the hearts had already been separated from their bodies for a cou­ ple of days, when they were given to us they were nothing but cold as ice. But us, brave and curious students,

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ractice makes perfect, right? But in terms of laboratory experiments at schools, there may not always be the possibility for practical work. It is a pity since many teachers and scientists would agree that you learn the most when implying your theoretical knowledge in real life. I definitely approve this idea, therefore I am grateful that at our school (Open Gate) we truly do have the opportunity to test ourselves while working with real biological material. For example, last month we performed a laboratory work, the dissection of a cow heart, about which you can read in the article above. However, not always are we so lucky to have a few cow hearts in a refrigerator, which are by the way about a size of a human head

were not afraid and, armed by our knowledge and kitchen knives, fought tirelessly with our bloodthirsty ene­ mies. The first main point of our strategy was to examine the external anatomy of the heart to figure out all of the opponent s weaknesses and strengths. This was not an easy task, nevertheless, we all successfully man­ aged to spot superior vena cava, aor­ ta, pulmonary artery and vein and left and right atriums and ventricles. Therefore, when the final clash of our blades and heart s unyielding tissue came, we were prepared. Directing our front lines on the apex and slowly but unmercifully cutting a straight line through our enemy we controlled the situation from the very beginning. But what we found inside was more than astonishing. We stared upon the de­ feated pieces which opened up their treasure to us. Perfectly carved right atrium leading through the tricuspid valve straight into the right ventricle. The chordae tendineae (also called the heart strings) proudly staying taut,

(sometimes even bigger). Therefore, there exists a good substitution, a virtual lab. Yes, even this is possible thanks to the internet and modern day´s technology. So, what is a virtual lab and how does it work? Basically, it is a simulation of real laboratory experiments carried out by professionals, which people from all around the world can benefit from and virtually try these specific scientific methods on their own. It helps people to learn basic laboratory techniques in all fields of study, therefore everyone will find their place. Usually, the users are advised on how to perform the lab either by some notes in the video or a talking guide , who reads the instructions for them. This way of practicing aims to

holding the muscle to the valves. Hard­ ly any of us could imagine that this dead beauty was once hiding in the body of a healthy cow, unflagging and strong, sending life through its veins with every single beat. But we quickly awakened from this astonishment to celebrate our victory. After describing the internal anatomy we even got a chance to explore this fascinating or­ gan under microscope. A tiny bit of the tissue was taken and observed so that we could see the structure of the cardiac muscle through which electric pulses are sent from the brain. Although this bloodshed was not an easy task, we proved that we are fight­ ers truly dedicated to our love of sci­ ence and showed that nothing is impossible. With sharp knives and smart brains we accepted the chal­ lenge of the dissection of the cow heart and, I believe, enjoyed the expe­ rience of holding such an organ in our own hands and being able to see all of the important parts of it that are so similar to our own hearts.

familiarise students and other non­ professionals with the work of real experts and the diversity that exists in the scientific career. The consequences of involving such methods into the teaching techniques are usually very positive and students show better understanding not only to the practical part but the theoretical one as well. To put it in a nutshell, if there is no possibility to carry out an experiment in real life just go online. Search for virtual labs, choose which laboratory work you would like to perform, follow the instructions in the video and simply enjoy! Hope you will have fun!

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ultiple sclerosis is a chronic disease which affects the nervous system. In the past decades, the ability to diagnose multiple sclerosis and to start treatments and therapies that can help modify the progress of this disease, has increased. The scientists understand it much better now; its pathogenesis and figuring out the genetic factors that are responsible for the disease may one day help to find the ultimate cure. A new study says that maybe there is a new way to reduce the cell nerve damage, becoming a new hope for the patients. Multiple sclerosis is a disease that affects people mostly between 20 to 40 years of age. The cause remains unknown as the scientists believe that numerous factors are responsible for the triggering of the disease. What we know so far is that myelin, a fatty substance that protects the nerve fibers, is destroyed and the communi­ cation between nerve cells, spinal cord and rest of the body is flawed. Thus the patients experience symptoms such as fatigue, muscle weakness, pain, tremors and problems with bal­ ance and coordination.

Presently, there is no known cure for this disorder, but there are drugs that are able to relieve the symptoms and slow down its progression. Only re­ cently, the last research led by Dr. Fang Liu in Toronto, Canada, discovers a new target for multiple sclerosis treatment, using MS animal models and deceased MS patients. The focus of her investigation was a spinal cord change that involved a pro­ tein. This protein binds to a specific cell receptor for the glutamate neuro­ transmitter the most outstanding neurotransmitter in our brain. Dr. Liu s team found that the spinal cord altera­ tion that involved this glutamate pro­ tein was present at higher levels in the animal models and deceased patients. Hence, the team developed a new peptide with the help of the tech­ niques developed in their lab. This small piece of protein was supposed to try and disrupt the spinal cord alter­ ation in MS animal models. And the results were astonishing. What the team found was that the peptide effectively stopped the previ­ ously identified protein binding to glutamate receptor. This improved the neurological functioning in the animal

models. The peptide was reported to reduce nerve cell death because it was able to protect myelin from damage. It also increased the survival of myelin­ producing cells. And importantly, the peptide did not suppress the immune response system directly and did not impair neuron transmission in the brain, which, as Dr. Liu notes, is a com­ mon side effect of drugs that target glutamate system. Commenting on the team's findings, she says: "We've identified a new biological tar­ get for MS therapy. Our priority now would be to extend this research and determine how this discovery can be translated into treatment for pa­ tients." While it still remains unclear whether the population will see a new and so far the only cure for the MS, one thing we know for sure this new target for the research is giving people who suf­ fer MS hope that one day they will overcome their disorder and be in con­ trol of their body again.

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ore than 1.8 billion users of smartphones worldwide. But what makes the smartphones so smart? Chemistry has the answer to this question! 84% of all stable elements, 62 different metals in average, 17 rare earth metals in various smartphones we have almost run out of the elements in the periodic table, and yet this is only a part of the recipe for the mysterious tool we use on a daily basis. One of the smartphone s smartest aspects is the display that a vast number of people consider as the most important in the construction of the overall impression of the phone. Touch screen enables you to control the phone. And if something can help us understand the principle of it, it is chemistry.

The screen of a smartphone is more than just a tough piece of glass. It responds to your touch and gives you a personal connection to your phone. There are various types of touch screens, but the most widely used is called capacitive. When a finger presses down a capacitive touchscreen, a very small

electrical charge is transferred to the finger, creating a voltage drop on that point of the screen. A controller within the smartphone processes the location of this voltage drop and orders appropriate action.

The most obvious component of touch screens is glass. However, the glass the touch screens are made of is not the same substance you associate with the word glass. In order to get an adequately tough and strong material, a substance called Gorilla glass is made of a combination of glass and ceramics. The reason for this combination is based on the characteristics of each of these materials. While glass lacks crystalline structure and is called an amorphous solid, ceramics tends to be crystalline and is characterized by ionic bonds between positive and negative ions.

to slip past another. Ceramics resists compression, but can break when it is bent.

Glass

Ceramics

Glass­ceramics is formed by overheating the glass, so a portion of its structure is transformed into a fine­ grained crystalline material. Glass­ ceramics is at least 50% crystalline, and in some cases, they are more than 95% crystalline. It is so strong that its name is Gorilla Glass. Gorilla glass is composed of an oxide of silicon and aluminium, and is also being called aluminosilicate glass. However, this is still not the final product. Gorilla Glass needs to go through one final step in which the glass is chemically strengthened and which gives it its tremendous strength. The glass is put into a molten bath of potassium salt, usually potassium nitrate, at 300 °C. Because the potassium ions are more reactive than sodium ions, they displace them. Potassium atoms are bigger than sodium atoms, and the same holds true for their ions potassium ions are much larger than sodium ions. Therefore, these potassium ions take up more space in the glass than do sodium ions. Cramming larger ions into the spaces formerly occupied by smaller ions results in a compression of the glass.

Glass does not contain planes of atoms that can slip past one another, so there is no way to relieve stress, and therefore, the excessive stress is responsible for a crack formation. On the other hand, ceramics is brittle since when it forms crystals, the strong force of attraction between ions of opposite charges in the planes of ions makes it difficult for one plane

In the same way, as the larger potassium ions push against each other, the glass is compressed, gaining strength. As a result, a lot of elastic potential energy is stored in the glass, much like the elastic potential energy that you might find in a compressed spring. »

enters your finger not enough you to feel, but enough for the screen to detect. As this electrical charge leaves the screen and enters your finger, the screen registers a voltage drop, the location of which is processed by the software, which orders the resulting action.

» The touch screen would not work without other components as well. The glass must be coated with a thin transparent layer of a conductive substance, usually indium tin oxide, which is laid out in crisscrossing thin strips to form a grid pattern. It is

because glass is an insulator and does not conduct electricity.

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properties. Additionally ­ and this is an alarming information in deep under­ standing of DHMO dangers ­ this chemical has higher pH than any acid so far discovered.

ow is the best time to reveal certain shocking results of a recently conducted set of ex­ periments regarding public health and safety. Those have acknowledged and proven that particles of Dihydrogen Monoxide are traceable in almost eve­ ry tap or other sanitary device used in households, companies, schools and even hospitals. The problem with the chemical called Dihydrogen Monoxide (shortly ­ DH­ MO) is the danger it represents to­ wards the human civilisation and welfare of all people. It possesses abil­ ities to mutate DNA, denature pro­ teins, disrupt cell membranes and chemically alter critical neurotransmit­ ters, up to a global level, and even worse: this chemical in its liquid state plays the key role in floods or the well known El Nino effect, caused partly by variation of its temperature. The sub­ stance itself has been detected in vari­ ous acids, poisons, explosives or toxic materials, and it is a crucial part of them ­ when taken away, the chemicals often cease their dangerousness, or adapt very different, harmless

The conductive grid acts as a capacitor, storing very small electrical charges. When you touch the screen, a tiny bit of this stored electrical charge

There are also unexpected correla­ tions one should not oversee when interested in the subject of DHMO avoidance. For instance, it has been proved that every person who has ev­ er absorbed DHMO into their body died, or is eventually going to die in near future. Ingesting too much of this

This tiny bit of electrical current enters your finger because your skin is an electrical conductor primarily due to the combination of salt and moisture on your fingertips, creating an ionic solution. Your body actually becomes part of the circuit, as a tiny bit of electricity flows through you every time you use the touchscreen on your phone. Is it smart, isn t it? That s why it is called a smartphone!

compound can have serious impacts on one's health ­ in the past, this knowledge has been also widely used for torture and interrogation ­ besides DHMO, fire and sharp tools were the most common and most favourite in­ terrogation devices; this can expose the true nature of the liquid, as it was used along with other deadly tools for malign purposes. Besides, it was used for killing people as well; dying, being emerged in DHMO is recognized as very painful and very unpleasant way of ending one's life. And now, it was announced that this dangerous substance can sometimes leak from water taps and other water sources we use on a daily basis. Possi­ ble impacts of this information leak are yet unpredictable, but sooner or later they will get more evident. Until some kind of solution for this currently revealed crisis is made, please be aware that whenever you drink from the water tap, you might unconscious­ ly consume the very dangerous Dihy­ drogen Monoxide, the potentially harmful substance with an enormous variety of possible impacts on any­ thing it gets in contact with.

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very child or adult who is even a little interested in the universe will sooner or later come across the terms black hole and worm hole . It is very easy to find the basic information about black holes. They are extremely dense and heavy. They attract all objects so strongly that even light cannot escape their gravita­ tional pull, if it gets close enough. Most of us know that. Worm holes are somewhat trickier but usually, they are used for interstellar travels in sci­ ence fiction movies. Even though they are both named holes, there is no real connection between them.

accelerating you toward the black hole. Now, you might start to think that we should see it growing as we get closer. But the truth is you can't see it at all. What you see is quite a disturbing phe­ nomenon which gives the full meaning to the term black hole . You literally see a black object. But forget about the black we can normally see around us. If you have been somewhere deep underground with your light turned off for a while, you know that there is nothing really black on the surface. However, there is still some light un­

When you get very close, you experi­ ence significant time dilation you and your surroundings are much slower than everything further away from the black hole. This is due to the effect described by the Theory of Relativity great mass and energy can bend both time and space. After some time, you reach a border called the event horizon , which is a very self­explanatory. It is the border, from which nothing, not even light, can escape. Also, gravity behind this border is so strong that time and space are too compressed and stop making any sense . In reality, it means that anything behind this line is so slow that from our point of view, that nothing really moves there. The tem­ perature and density inside the black hole are so extreme that no­one is

sure what happens to the matter in­ side. The problem with observing black holes is that light doesn't escape from behind the event horizon. The only way we can empirically confirm black holes in the universe is by watch­ ing their gravitational pull on near ob­ jects. There are many much more interest­ ing aspects of black holes. Anyone who has the basic knowledge of the General Theory of Relativity could fig­ ure out that such massive object will distort the space­time quite signifi­ cantly.

To show the most interesting features in a coherent way, let's pretend you are floating next to a black hole; near enough to experience the gravitation­ al force. The first interaction is not re­ ally surprising the gravity starts

der the ground, just too little for you to see. Black holes take this to a whole new level there is not even a single photon coming from the black hole into your eyes. Therefore, you basical­ ly see nothing growing in front of you. As you get even closer, you should start rotating around the black hole, just as a planet orbiting a star, except that you would be moving in a spiral converging into the black hole. The orbiting speed increases as you are nearing the center due to the increas­ ing gravitational force acting upon you.

The state of space­time inside the event horizon is called singularity. It means that the space is so bent that the trajectory of any particle ends in one point. That is the reason why nothing can escape. There is no way anything could be moving away from the center because the only direction is towards it. It is impossible to imag­ ine it but the mathematics describing this effect is clear. Black holes are consuming any­ thing around them this way. The more massive they are, the more they grow.

However, you don't have to fear they would once consume the entire uni­ verse. The famous physicist Stephen Hawking discovered something that was previously thought to be impossi­ ble particles escaping the black hole. » This is possible without breaking the law about singularity. Quantum me­

chanics together with the Theory of Relativity tell us that it is possible for random particle­antiparticle pair to oc­ cur in one place with opposite veloci­ ties without breaking the energy conservation law the sum of ener­ gies is still zero. This phenomenon oc­ cur relatively frequently near black holes. The particles with positive total energy move away from the black hole and their opposites with negative en­ ergy, and therefore negative mass (E = mc2) are absorbed by the black hole. Absorption of a particle with negative mass makes it lose its total mass. This phenomenon is more significant in tiny black holes which often quickly disap­ pear due to this process.

Worm holes are something complete­ ly different. Their real existence has not been proven but hypothetically, they are possible. I will not describe the process of travelling through a wormhole because that would be very boring as you would experience the same thing as going through empty space.

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ummer. ­23°C. Tiny dark lines of water tumbling down the steep crater walls on the surface of Mars. The image of Mars being once covered with oceans and rivers seems now more likely than ever before; however, over time, more of the liquid disappeared into the space due to the process of evaporation. Recently, scientists came up with the idea that water may still flow on the Red Planet today. Their observations show positive results, nevertheless one thing remains unclear; Where does this water come from? Two major suggestions have been made to answer this question. The first possibility is the melting of a frozen subsurface saltwater, seeping onto the surface and trickling down the slopes. The second possibility which seems to be the most favoured by sci­

entists, is that the water might be of atmospheric origin. If the humidity in the Martian atmosphere gets high enough, perchlorate salts will absorb the atmospheric water until the salt dissolves and forms a liquid solution, says Mary Beth Wilhelm of NASA s Ames Research Centre.

Worm holes' basic mechanism is also based on the Theory of Relativity. Space is bent due to mass and energy. Worm hole is a place where two differ­ ent positions of the space­time meet at the same point due to the warp. To explain it very simply, instead of travelling somewhere far, you just move the location closer to yourself by bending the space. However, this is easier said than done. Such warp would require tremendous amount of energy, much more than we are capable of creating. Also, such curvature of space is behaving unpre­ dictably and a wormhole would be too unstable to be used in practice. Time and space are connected, so if you can theoretically bend space as you want, can you also travel back­

Despite our unawareness of the source of water on Mars´ surface, there is no doubt about its existence. The meaning of it is still a mystery but scientists hope that it will help them to find something even more valuable in the future, such as life, either past or present.

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f you watched the news, you could not have missed the Nobel Prize awards. As they are taken very seri­ ously, it is convenient to know the win­ ners and their achievments. Doctors Kajita and McDonald have received Nobel Prize for Physics in 2015 for proving that neutrinos have mass and can oscillate change flavors. Now, these terms might sound very exotic to anyone who has not studied parti­ cle physics. Despite the complexity of the research, the basic principle can be explained simply enough for everyone to understand. And for others, it is not clear why this discovery is so impor­ tant. However, as it belongs among the most influential ones, everyone should know what it means for phys­ ics perception of reality. Neutrinos are a group of elementary (indivisible) particles that are charac­ teristic by being almost undetectable. They do not have charge, and until recently were thought to have no mass. They do not interact with most matter. In fact, there are billions of neutrinos passing through our bodies every second, there is just no way for us to notice. Neutrinos occur mostly as byproducts of nuclear fission which takes place in stars. We have devised a few mecha­ nisms to detect some of the neutrinos for example huge tanks of a chlorine­ containing chemical usually deep un­ derground. The neutrinos passing through the liquid can interact with chlorine atoms if they collide in a par­ ticular way. Each chlorine then trans­ forms into a radioactive isotope of argon. We detect the amount of ar­ gons formed. From that number, we can estimate the number of neutrinos passing through the liquid. There are three types of neutrinos ­ electron, tau and muon neutrinos. The attribute that defines the three neutri­ no types is called flavor charge. To be clear, flavor charge is not a type of energy as electrical charge, it is just a term used to distinguish between dif­ ferent groups of particles.

From the particle equations of fusion reactions that take place inside our Sun, we can calculate how many neu­ trinos we should be detecting here on Earth. The fusion reactions can pro­ duce only one type of neutrinos elec­ tron neutrinos. We have been detecting the electron neutrinos for a long time but each recollection of pre­ vious detections has shown an inter­ esting result. We detect approximately only a third of the neu­ trinos we expect. The Nobel Prize­winning theory ex­ plains that the neutrinos can in fact change their flavors as they travel through space. However, it was proved that neutrinos in vacuum move slower than light, which means they must have mass. Even though the mass is tiny, it tells us quite a lot about the Standard Model, the most used theory describing the behavior of dif­ ferent particles. Its first real issue is that it cannot explain why neutrinos have mass. The theory clearly says they should not. However, as this evidence directly contradicts the Standard Model, it can be concluded that the Model is incom­ plete or entirely wrong. This is a huge breakthrough because even though many scientists guessed the theory would be incomplete, we now have a concrete evidence. Some theorists claim that we need to add more spatial

dimensions or abandon the distinction between matter and anti­matter. The first solution has already been pro­ posed by String theorists through a completely different reasoning proc­ ess, so it might be the right way to go. However, matter and anti­matter is such a basic concept explaining so many phenomena that it would be complicated to change it. It is likely that the Standard Model will be aban­ doned completely, being replaced by different complex theories, for exam­ ple String Theory or other attempts to unify the different rules of the Stand­ ard Model and correcting mistakes such as the neutrino mass. The main point is that after a few dec­ ades, the dominating Standard Model has been proven imperfect. New mod­ els that are expected to appear or to be finished will most likely tell us something new about the Universe. Or perhaps a new, revised version of the Standard Model will be created. All we needed was a proper reason to persuade us that working on the cur­ rent Standard Model only is not enough to understand everything about our Universe. The fact that this discovery motivated many scientists to work on different explanations on how our Universe works, and there­ fore starting a new era of theoretical physics, is the reason this research gained a Nobel Prize.

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hobias, the compulsive fear of something for no apparent rea­ son, are incredibly common. In fact, one in five people on average have one. Despite being so ordinary, phobias haven t been discovered until the 18th century, where scientists started to diagnose deficits of the hu­ man psyche for the first time. They found the most common ones, like fear of dogs, spiders or heights. Little did they know how absurd and inter­ esting they can be. Here are my top ten picks

Marriage can be scary at times, but did you know an actual phobia of mar­ riage exists? Gamophobiacs fear any relationships, though, not just mar­ riage in general. This is social awk­ wardness taken to the extreme. Scientists treat it rather as a personali­ ty disorder, since it interferes with the behaviour of a person too much.

People with radiophobia are extreme­ ly scared of radiation. This is one of the phobias that is developed, often through lack of understanding of X­ rays. This phobia could be understand­ able these people are really scared of cancer and have no idea how rays work.

Fear of justice. It follows both justice and retribution as moral concepts. It is hard to diagnose and hard to treat, as it often links to schizophrenia and oth­ er serious disorders. It can be devel­ oped through crime, too. But as long as you are healthy and a good citizen, dikephobia should not be a problem.

Fear of beautiful women. This phobia is acquired through a tragic experi­ ence with a beautiful woman at some point in a person s life. One of the few phobias that troubles exclusive­

ly men. There must be some truth in the statement beauty is danger­ ous .

Is the extraordinary fear of garlic. Trig­ gered by a close proximity to garlic or even when mentioned. Scientists do not know where it comes from ­ it is possible that it is hereditary. Allium­ phobia can be cured, as the patient needs to learn to confront garlic. It is not true that people possessing this phobia are vampires.

Similar to the previous one, but the person is scared of cheese instead of garlic. The patient might fear one spe­ cific type of cheese or cheese alto­ gether. This is a great example of a phobia acquired during childhood, through a bad experience with cheese, as many children despise the taste and start eating cheese at a later age.

The fear of an atomic explosion or an atomic war. Victims of atomosophobia live constantly under the feeling that an atomic war can start at any mo­ ment, even if they are in a peaceful and neutral country. Signs of this pho­ bia were reported after the end of the Second World War. Symptoms are dis­ tress and panic attacks whenever at­ om weapons are mentioned in the media.

Also known as the fear of clouds. The origin is unknown, as it cannot be logi­

cally linked to any triggering events, but it can be understood. Clouds are pretty scary, as they are huge objects floating in the sky, and the mind of a nephophobiac has trouble confront­ ing that. It might trigger the feelings of being trapped. Although it seems bizarre, nephophobia complicates life a lot, as going outside is a problem.

This phobia is very different from the other ones in this list. Panphobiacs fear everything, easily put. It is more a fear of some unknown, vague, ab­ stract evil within, that is omnipresent and can never be destroyed. Panpho­ bia was one of the early ones to be studied, namely by Théodule­Armand Ribot in 1911. It is often grouped with generalized anxiety disorder, and not registered as a phobia in medical terms. Panphobia is very hard to treat as it doesn t fit in the standard treat­ ment criteria.

The winner of our top ten is not only a phobia, but also a paradox. Phobopho­ bia is the fear of phobias. The fear of fear. Whenever a phobophobiac is scared of something, they notice their fear and gradually become even more stressed. Phobophobia can also occur when a person finds out they are de­ veloping a new phobia; they become stressed because of it. When a patient has developed phobophobia, their condition must be diagnosed and treated as part of anxiety disorders, because it is too complex to treat as a phobia. Yes, something as strange as phobophobia can be cured today.

The first round is over! Have the eight nominated teachers done well? We bring you a report on that! The teachers had to deal with question of high standard which you can try to answer yourself:

How many types of forces exist in the universe? What is the mass of the universe? What compound makes plants green? What chemical is wood mostly made of? How can our brain create and transfer a thought? How much is a mole? How fast does light move (in metres per second)? Which of following options does NOT increase evaporation rate of a liquid? How much money does a winner of Nobel Prize earn?

Do these seem difficult? We have made it a bit easier and teachers had four options for each of these questions (except of the last one where they were expected to write an estimate). Our teachers haven t disappointed us in way and moreover, some of them were so enthusiastic about answering the questions that they have made sophisticated comments aside. We have decided to share some of these with you, read on! In the following table, you can see the final scores:

Although all of the teachers surprised us with their outstandingly good knowledge of science, some of them got higher score then others and from each of the competing pairs, we will see the winner in the next round! From these winners, we have randomly made pairs: PAIR ONE Mr. rùta vs. Mr. Ka par

PAIR TWO Miss Buffry vs. Miss Pospí il

By next time, we will know the two science gurus who will continue to the final round, so make sure you do not miss it! And no, this is this not all from us. The additional task that all teachers had to complete was drawing a neuron. (The pluses you have seen in the table are for the extra effort put into the drawing). Here you can some of the masterpieces we have received as well as the promised additional comments on the questions:

If you have anything to ask or to tell us, don t hesitate to use our gmail adress: openscience.info@gmail.com Also, use your QR code reader to gain more information about the authors and sources of our articles.

OPEN SCIENCE SPECIAL INSERT:

CHAOS THEORY By Michaela Mrรกzkovรก

What is the relation between the chaos theory, complex numbers and fractal geometry? Abstract The chaos theory is the field of study in mathematics that studies the behaviour and condition of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Through abstraction of a hypothetical case that I present, readers have the opportunity to comprehend the mathematical derivation of the logistic equation which results in chaos when solved in the set of real numbers. The article also connects the chaos theory with complex numbers and fractal geometry since it demonstrates how the logistic equation can be solved in the set of complex numbers, generating fractals.

Content: 1. 2. 3. 4.

Meaning of the word chaos History Application of chaos Mathematical ways to chaos 5. Complex numbers and fractals

Introduction Since its discovery at the end of 19th century, the chaos theory has played a crucial role in answering many issues of science and even of the practical life of all of us. At the very beginning, the whole theory appeared to be mysterious and people who were engaged in it, were labelled as insane. Some scientific magazines even used to reject articles pursuing this topic. Mathematicians stayed away from this theory and considered it a field of physics which was destroying their simplified theories and appeared to be too abstract. However, soon, this approach started to change and scientific institutions began to compete with each other in employing experts on the chaos theory which became an upward trend that time. Nowadays, the awareness of this theory has been gradually increasing, not only among the expert scientific community, which leads to a vast number of books and papers being published. In the current sources of literature, some of which I cite in this article, a reader may find complex mathematical analysis or treatise about specific applications of the chaos theory. However, despite the amount of materials I have read through, I have not come across any comprehensive text that would summarize the essential information and provide an overview approachable for readers who have never heard about the chaos theory. The purpose of this article is therefore to introduce this topic to such readers, present an overview of possible applications of the theory, clarify the essential way to generate chaos through the logistic equation and expound the miraculous relation between deterministic chaos and structured fractals.

1.

Meaning of the word chaos

Usually, we perceive chaos as something unarranged and confused. In the Greek mythology, Chaos is considered the beginning of the existence (“In the beginning, there was only Chaos…”) Nowadays, expert community associates the word chaos most commonly with the deterministic chaos. Although this phrase may seem as an oxymoron, it is in fact not. When we talk about the deterministic chaos, we put aside random chaos (unpredictable, unrepeatable) and focus on the chaotic behaviour that even though in the relation to time behaves irregularly, is still deterministic, which means that the same initial conditions lead to the same outcome. This article will deal with this type of chaos.

2. History For a long period of time, people had been convinced that small deflections and errors in the input parameters of deterministic processes were negligible. However, this presumption could be applied only on a very small scales, in very simple systems. Towards the end of the 19th century, scientists began to pay attention to more complicated processes in which they could not use the simplified physical laws. It was discovered that in more complicated (non-linear) systems, there were critical points when even the smallest deflection could radically influence the result. Those critical points, however, could not be analysed in detail without the use of a huge number of computational operations. This was the reason why the chaos theory had not started to develop until the first high-performance computers were introduced. Lorenz’s discoveries have been considered a turning point in the history of the chaos theory. Back then, the first computers began to appear, although when compared to today’s modern technology, they were rather primitive. The team lead by Lorenz launched a computation of weather forecast. Input values were for simplicity rounded to 3 decimal places, despite the fact that the accuracy of the computational operations was stated to be 6 decimal places. They assumed that difference in tenthousands and less was negligible. Despite that, they decided to make the results even more accurate. Lorenz made the input data precise on 6 decimal places and gave himself a break. When he returned, he found out that the result differed significantly from the preceding values. As it was discovered when both computations were compared in detail, initially, the differences were really negligible. However, as time progressed, the deflections were increasing until they developed into chaos (analogous to the scheme below).

Picture 1

Lorenz, meteorologist and mathematician, decided to dedicate himself to the issue of chaos and as a result, he presented a study about the so called Butterfly effect. In the Butterfly effect study, Lorenz indicated that a flap of the butterfly’s wings in Brazil could cause a tornado in Texas. Another way Lorenz contributed to the development of the chaos theory was the description of chaos by set of three non-linear differential equations. Initially, only few scientists besides Lorenz engaged in the chaos theory because “meteorologist not only despised predictions, but in 1960s most of serious scientists didn’t trust computers neither.” 1 Gradually, the theory started to be accepted as a new significant field of research. Scientists began to discover new principles of deterministic chaos not only in physics, but also in mathematics and meteorology as well as in other fields. In the course of research, many definitions of the chaos theory were formulated. Nowadays, the most commonly stated is the Devaney’s definition. Devaney defined deterministic chaos by three conditions (dependence on initial conditions, topological transitivity, periodic points dense in V). Since understanding of these definitions is not necessary for understanding of this article, I do not include their explanation.

3. Application of chaos In the field of medicine, knowledge of the chaos theory can enrich exploration of epilepsy, heart beats, thermoregulation of body, genetics, cancer, neurology, gerontology, anthropology or endocrinology. If economics is the field of your interest, you might find interesting the relation of the chaos theory and fluctuations of prices on the stock market. In ecology, this theory has been used in modelling of population growth (e.g. Ricker’s model). There are even programs that process musical parameters which the users can enrich by adding chaotic elements. If you have a background of programming, music and the chaos theory, you can create such a program yourself. Even managers should not forget about this theory. It helps them to realize that “company is not only a mechanism working on a principle that from clear inputs, we get clear outputs. It also warns about underestimation of social or psychological factors in the functioning of the company and about the fact that in planning, factors that might seem insignificant should be considered, as they might have a significant impact on the future.” 2

If you are interested in psychology, you should realize there is an important connection between your character and chaotic attractors (= phase diagrams of chaotic movement). Principle of deterministic chaos is also used in cryptography and is used mostly in the form of chaotic maps (such as Baker’s map). In chemistry, this theory might be useful in an attempt to supress the chaotic oscillations that are used for stabilization of reactions. Chaos can also fasten the process of combustion. If you are interested, I recommend you to read publications about the Belousov-Zhabotinsky reaction.

Naturally, I cannot forget about the field in which the application of chaos is obvious, physics. The theory of deterministic chaos has been used primarily in modelling weather, in chaotic electric circuits, in the study of turbulences and in control of chaos in plasma. Although the chaos theory has various application across the whole spectrum of human activities, mostly mathematicians and physicists are engaged in it. This is great, but research in these fields does not bring huge advancements any more. If anybody working in a particular field thought about the connections of his field and the chaos theory, we could soon be witnesses of breakthrough discoveries. In many fields, this theory has already been recognized, but there is still space for advancement.

4. Mathematical ways to chaos In various sources of literature, I have found two basic ways of chaos generation with the usage of equations. The first way is the set of three non-linear differential equations. The another way is the generation of chaos through the logistic equation and this method I would like to elaborate in this article. The logistic equation is a generally known equation, which has been successfully applied, mostly in economics: đ?‘Ľđ?‘›+1 = đ?œ†đ?‘Ľđ?‘› (1 − đ?‘Ľđ?‘› ); đ?‘› = 1,2 ‌ 0 < đ?‘Ľđ?‘› < 1 Not only was I interested in the process of derivation of this equation, but also in the subsequent application in the chaos generation. For better understanding, I present a hypothetic case, from which it is possible, through abstraction, to get to the generator of chaos.

Hypothetical case Entomologist has 5 terrariums; in each of them he breeds the same type of insect. In the first terrarium he creates 100% conditions, which means there is enough light, moisture and nourishment. In the second terrarium, there are only 90% conditions (for example less light and moisture). There are only 80% conditions in the third one, 70% in the fourth and 60% in the fifth one. The entomologist then observes development of the population in reproduction cycles. Consider he chose such a type of insect where a parent dies before the offspring hatches. In the first generation (n=1), there is always just one fertilized female. It lays the eggs and dies, the new generation hatches and the cycle is repeated. Moreover, this all happens with the same coefficient of increase of population depending on the created conditions. Entomologist writes down the results of his observation of each terrarium into tables. Since we still consider the particular example with number of individuals, we are for now in the set of natural numbers (number of insect individuals according to the coefficient of increase is rounded). 1.TER 100% q=4 n 1 2 3 4

yn 1 4 16 64

2.TER 90% q=3 n 1 2 3 4

yn 1 3 9 27

3.TER 80% q=2,5 n 1 2 3 4

yn 1 3 6 16

4.TER 70% q=2 n 1 2 3 4

5.TER 60% q=1,2 yn 1 2 4 8

n 1 2 3 4

yn 1 1 1 2

n – order number of generation yn – number of individuals of insect in n-generation in particular terrarium q – quotient of geometric progression

Consider this worked ideally. Then we could call all of these cases geometrical progressions. “Geometrical progression is a kind of arithmetical progression where all components except of the first one are permanent multiples of components that precede them. These multiples are called quotients of geometrical progression (q) and for progressions with non-zero components they are equal to ratio of any component (except the first one) and component that precedes it.� For such a progression, this formula can be applied:

đ?‘Śđ?‘›+1 = đ?‘Śđ?‘› . đ?‘ž

Recurrence relation Relations in geometrical progression may be expressed in the form of recurrence relation, in which there is lambda coefficient as a ratio of increment and value of the preceding component: Îť=

yn+1 − yn yn

This equation can be modified: Îť. yn = yn+1 − yn yn+1 = yn . (Îť + 1)

We get the relation between q and Îť:

q =Îť+1

However, if q>1 or Îť>0, the values of such progression would be growing above all limits which contradicts our hypothetical example with limited capacity of terrariums. Let us therefore assume that the maximal capacity of each terrarium is K=300 individuals of insect. This value is our critical value and by approaching it, the conditions in terrariums are getting worse, so the rate of growth of population decreases. y

Initially, when( Kn → 0), this kind of deceleration of population growth is negligible, but as the number y

of generation increases, ( Kn → 1) the growth is gradually ceasing. We get a modified recurrence relation: yn+1 = yn . [Îť. (1 − After modification:

yn+1 = (Îť + 1). yn −

yn ) + 1] K

Îť.y2n K

This result is a rudiment of the logistic equation which is usually not stated in this form as it is still dependent on the specific value of K. Despite that, according to this equation it is still possible to alter calculations in our table so they would better fit the reality of growth in a limited space. It is also suitable to assume that at this moment we do not necessarily talk about specific terrarium or number of members of population, and therefore we do not need to continue to round the values of yn.

For better idea of progress of values of yn, I present a table and graphs (2 chosen values (q=4; q=2)). From the graphs, it is noticeable that the higher the value of q, the faster the values of yn reach the value of K.

In the table, you may notice that in some cases, the value of yn exceeds the value of K. On the example of the insect reproduction, it is explainable that a huge percentage of generation that exceeds a critical point dies instantly. (refers to the yellow mark in the table)

Substitution For the complete abstraction of the example, it is necessary to get such a form of the logistic equation which would not be dependent on a specific value of K. In literature, I have found suitable substitutions: 1.

Substitution

Substitution for q will be derived from the relation between q and λ. 𝑞 =λ+1 2. Substitution If we want to use the original form of logistic equation to gain a general equation independent of domain of values of yn , we make a substitution by which we convert relation between yn on relations between xn, with single domain 0 < 𝑥𝑛 < 1. 𝑦𝑛 =

1+𝜆 1+𝜆 . 𝐾. 𝑥𝑛 (i. e. 𝑦𝑛+1 = . 𝐾. 𝑥𝑛+1 ) 𝜆 𝜆

Subsequent substitution: đ?‘ž 2 đ?œ†. ( ) . đ??ž 2 . đ?‘Ľđ?‘›2 đ?‘ž đ?‘ž đ?œ† . đ??ž. đ?‘Ľđ?‘›+1 = đ?‘ž. . đ??ž. đ?‘Ľđ?‘› − đ?œ† đ?œ† đ??ž đ?‘Ľđ?‘›+1 đ?‘ž đ?‘ž = . đ?‘Ľđ?‘› − . đ?‘Ľ. đ?‘›2 đ?œ† đ?œ† đ?œ†

After modification we get the most widely used form of logistic equation: đ?‘Ľđ?‘›+1 = đ?œ†đ?‘Ľđ?‘› (1 − đ?‘Ľđ?‘› ); đ?‘› = 1,2 ‌ 0 < đ?‘Ľđ?‘› < 1

In the same way, we may transform obtained values of yn from the hypothetical case to abstract values xn. Note: With shortening time period between generations (n), it is possible to convert the logistic equation into differential equation. It is possible to use differential calculus and investigation of course of the function to mathematically prove the above written hypothesis of critical values of q. 3

Bifurcation diagram The final step suitable for visualization of the chaotic behaviour of logistic equations for � ≼ 3 is construction of so called bifurcation diagram. Bifurcation diagram is a visualization diagram which is being used for clearer notion of behaviour of onedimensional dynamical system. Bifurcation diagrams enable us to look at large number of equations as a complex and clearly demonstrate changes in behaviour of these equations depending on changes in parameters. In our specific case, we would have to add to our obtained values hundred thousands of columns with finer iteration q in interval 3 < � < 4. For specific line n, we would then obtain values of xn (vertical axis) depending on values of q (horizontal axis). The result would correspond with the diagram shown below. Also, it is acknowledged that bifurcation diagram of logistic equation is a simple fractal with repeating self-similar motives.

Picture 2

5. Complex numbers and fractals Logistic equation, which I have already introduced, “is advantageous to solve in the set of complex numbers. Notice, that some fractal discoveries in the set of complex numbers are described using the same equations as chaos in the real variable.� 4

Complex numbers In this section, I would like to introduce complex numbers to readers who have never heard about it. “Complex numbers extend the concept of the one-dimensional number line to the twodimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. Complex number can be expressed in the form đ?‘Ľ + đ?‘Śđ?‘– where x and y are real numbers and i is the imaginary unit, that satisfies the equation đ?‘– 2 = −1.â€? Complex numbers have a form: đ?‘§ = đ?‘Ľ + đ?‘Śđ?‘–; where đ?‘Ľ, đ?‘Ś ∈ đ?‘…

(“Number đ?‘Ľ ∈ đ?‘… is called the real component of complex number; number đ?‘Ś ∈ đ?‘… is called the imaginary component of complex number.â€?) 5 Whereas we can use 1D number line for visualisation of real numbers,

Picture 3

for visualization of the complex numbers, we use 2D Cartesian coordinate system (Complex plane).

Picture 4

Every complex number has exactly one assigned point on the complex plane as its image, and reversely, every point on the complex plane has exactly one assigned complex number as its template. 5

Fractal geometry Although most of the objects that surround us are irregular, at school we usually learn to work with the geometry of regular shapes, so called Euclidean geometry. This geometry is full of perfect and simultaneously simple shapes (e.g. triangles, cones, spheres). It is very useful to know this geometry and be able to use it properly. However, I would like to introduce the fractal geometry which can be used to describe our real world in a more precise way. Quotation of excerpt from my favourite book about chaos: In the past two thousand years, there was no space for discontinuity or Cantor set and similar phenomena in the field of geometry. Shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. By their abstraction, powerful philosophy of platonic harmony was inspired. Based on these findings, Euclid constructed geometry that persisted for two millenniums and up to now, it is the only geometry that majority of people approaches. Artists saw ideals of beauty in those shapes and Ptolemaic astronomers based on them their theories of the structure of the universe. However, for understanding of complex and complicated phenomena, they are wrong types of abstraction. “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” 6 The new geometry is the reflection of our space that is uneven, not straight and blemished, not smooth. 7

Fractals Fractal is a geometrical object that has the following characteristics: it is self-similar – which means that if we observe the given object in whichever scale, we observe repeating motive or characteristic shape; at a first glance, it seems to be complicated shape, but it is generated by the application of simple rules. Fractals may seem to be the most complicated geometrical objects that current mathematics deals with, however, they often mathematical have very simple structure. 8 We may find fractals anywhere, in rainforests, in the world of wireless communication nets. Fractals surround us. Lungs, blood vessels, kidneys, flowers, climatic systems, heart rhythm, all of them now can be explored from a completely different point of view.

Picture 5

Conclusion: Let us come back to the original question raised in the title, what is the relation between the chaos theory, complex numbers and fractal geometry? The answer to this question has hopefully been clearly demonstrated to the readers. When we solve the logistic equation in the set of real numbers, the result is chaos. However, when we solve the same equation in the set of complex numbers, the result is a fractal. The purpose of this article was to introduce the deterministic chaos to people who have not heard about it before and provide them a mathematical way to chaos. Hopefully, this attempt was successful and every reader has the important basics that are crucial for further understanding of other papers dealing with the chaos theory and its specific applications. Personally, I consider the hugest contribution of this article to be the breakdown of the logistic equation which any high-school student with good mathematical background should be able to understand.

References: Picture 1: Lorenz's experiment [Digital image]. (n.d.). Retrieved April 10, 2015, from http://www.tokenrock.com/stock/chaostheory2.gif Picture 2: Bifurcation diagram [Digital image]. (2005, September 14). Retrieved June 5, 2015, from https://upload.wikimedia.org/wikipedia/commons/7/7d/LogisticMap_BifurcationDiagram.png Picture 3: Number line [Digital image]. (n.d.). Retrieved July 13, 2015, from https://s-media-cacheak0.pinimg.com/originals/3f/10/75/3f107509047090a1e9801486f9f27b26.gif Picture 4: Complex plane [Digital image]. (2007, May 29). Retrieved May 6, 2015, from http://pirate.shu.edu/~wachsmut/complex/numbers/graphics/plane.gif Picture 5: Fractal [Digital image]. (2010, October 9). Retrieved March 3, 2015, from http://www.wired.com/wpcontent/uploads/images_blogs/wiredscience/2010/09/fractal_10.jpg What are the complex numbers? (n.d.). Retrieved May 20, 2015, from http://stuleja.org/vscience/materialy/mandelbrot/B.htm Definition of differentiation. (n.d.). Retrieved May 30, 2015, from http://www.glouny.cz/matematika/mat_sem/index.htm Fractal Generator. (n.d.). Retrieved June 25, 2015, from http://www.fractalsciencekit.com/ Tarak. (2013, October 13). Fraktály – hon na skrytou dimenzi. [Video file]. Retrieved May 6, 2015, from https://www.youtube.com/watch?v=AfMQt1AJdRU 1, 7

Gleick, J. (1987). Chaos. Ando publishing.

Ing. Vojtěch Hordějčuk. (n.d.). Retrieved April 3, 2015, from http://voho.cz/wiki/matematika/relace/ Ing. Václav Rada, C. (březen 2012). TEORIE ROZHODOVACÍCH PROCESŮ . Retrieved from Fakulta stavební VUT v Brně: http://www.fce.vutbr.cz/tst/rada.v/ROZHPROC/w-cw05-rpr-pr17-ther.ppt 4

Křížek, M., Somer, L., & Šolcová, A. (2009). Kouzlo čísel, Od velkých objevů k aplikacím. Praha: Nakladatelství Academia.

6 Mandelbrot

Fractal Z^7. Retrieved May 26, 2015, from Deviant art: http://elbrujodelatribu.deviantart.com/art/MandelbrotFractal-Z-7-341695245 2

Podstata teorie chaosu a její přínosy pro obory ekonomie. (16. 6 2008). Retrieved May 24, 2015. From MANAGEMENT A MARKETING: http://managment-marketing.studentske.eu/2008/06/16-podstata-teorie-chaosu-jej-pnosy-pro.html 5

Polák, J. (2008). Přehled středoškolské matematiky. 3 For

further reading, I recommend p.15 of Bachelor’s study of Dagmar Plháková titled “Logistic differential equation”

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