“Appearances are a glimpse of the unseen.”
Anaxagoras

Throughout the ages, human beings have looked at the sky with admiration. The star-studded sky on summer nights reminds us how enormous our universe is and how small we are in that immensity. NASA (National Aeronautics and Space Administration) has been scanning the universe with giant telescopes for decades. The fact that the starlight we see in the sky are their images from four years ago (that is, the stars are so far away that their light takes four years to reach us), should give us an idea of the vast size of the universe. The question, then, comes to mind: how can we have information about the places these telescopes cannot observe? The answer is that we get precise information about these galaxies, the planets, and even their moons, which are beyond the reach of our telescopes thanks to Mathematics and Mathematical thinking. These tools have been used for centuries and will continue to be used in the future. With mathematics, we can obtain accurate results with heuristic methods without directly seeing or observing the object.

Mathematics enables us to understand invisible structures with high-level thinking, allowing a meaningful system to emerge from paths that previously seemed random. Using our minds to “see” when we lack physical vision fosters abstract thinking, leading to extraordinary advances in human civilization. Let me clarify this by analyzing the three-prisoner problem.

In ancient times, three prisoners were brought to a room, blindfolded. The prisoners were told to line up behind each other, the tallest in the back, the shortest in the front, all facing the wall. Then, the guards put a hat on each of the prisoner’s head. They told them that the hats were selected from a total of five hats, three white and two black, but they did not tell them which color the one on their head was. The prisoners were expected to guess the color.

When the blindfolds are removed, the tallest man in the back could see the hats of the other two, the one in the middle could see the one on the shortest man, but the shortest man could see nothing but the wall. None of the prisoners could see the hat on his own head. The remaining two hats were hidden away.

The tallest man was asked to guess first. If he guessed the color of the hat correctly, he would be free. If not, he would die. Another option was to say, “I have no guess,” but for that he would have to live the rest of his life in the dungeon. The tallest man said, “No answer! I have no guess!” The man in the middle was asked if he knew; he also said, “No answer!” But the man at the front said he solved the puzzle and guessed correctly the color of the hat on his head: white. How was this possible although all he could see was nothing but the wall?

What’s lovely about this problem is that the person with zero visual information at the start becomes the first person to know their own hat color.

Let’s begin by looking at the possible combinations of hats for the prisoners. It can be solved using the process of elimination.

There are only seven possible combinations of hats on the three people:

• WWW
• WWB
• WBW
• WBB
• BWW
• BWB
• BBW

[W= white, B= black, and the order from last to first.]

The tallest man could have deduced his hat’s color if the other two had black hats – since there were a total of two black hats, the tallest one would surely have a white one. Since he chose not to guess, we can eliminate WBB. The knowledge that WBB has been eliminated is now part of the common knowledge held by all three people.

The prisoner in the middle also chose not to answer. If he saw a black hat in front of him, he could have deduced he was NOT wearing a black hat, since if he did have a black hat, then the person on the back would have guessed that his own hat was white. Since this did not happen, then we can eliminate all combinations with a B in the third place. We are left with WWW, WBW, BWW, and BBW. In all these combinations, the first person (the one in the front) is wearing a white hat. This is how he deduces the right color. The other prisoners could not know what color their hat was, for the remaining options included the chances that it could be either black or white.

This ancient three-prisoner problem is a fascinating example of mathematical thinking in action, showcasing the power of abstract thinking to unravel complex puzzles. Just as blindfolded prisoners use logic and deduction to determine the colors of their hats, we can strongly argue for the existence and qualities of many physical entities beyond our perceptions. For instance, most of the exoplanets (any planet beyond our solar system) have been discovered not with direct sighting, but through indirect methods, like the transit photometry by which we measure the dimming of a star’s light when a planet orbiting it partially blocks it. Another indirect method is the radial velocity which describes a planet’s gravity pull of the star either toward us or away from us and thus shifting the wavelength of its light (Doppler effect). And a third method is the microlensing method, when a star passes by another star, its gravity bends the light, making the other star brighter. If the passing star has a planet, this bend is even bigger, and it increases the star’s brightness.

This means that although we do not have a direct image of most of the exoplanets, we still know they exist. This falsifies the argument “I do not believe what I do not see.”

Rational methods and logic enable humans to see the invisible and know the unknowable. When multiple ways are used together, we can learn the vital statistics of whole planetary systems without directly imagining the planets themselves. The best example is the TRAPPIST-1 system about 40 light-years away, where seven roughly Earth-sized planets orbit a small, red star. Think that 1 light year is 9,461 trillion kilometers, then 40 light years are 3784.4 trillion kilometers away from the Earth. In addition, an astronomical unit (AU) is the equivalent of 150 million km / 93 million mi, and the Sun is 1 AU away from Earth. In light-years, the Sun is 0.00001581 light-years away, while in light minutes, the Sun is 8.20 light minutes away, or 500 light-seconds away from Earth. Just to give an idea about how far this system is, it would take 6,912 years of non-stop flight for the Parker Solar Probe, the fastest spacecraft ever built, at the speed of 430,000 mi / 692,017 km per hour, to arrive at the TRAPPIST-1. The seven known exoplanets that revolve around the TRAPPIST-1 planets have been examined with ground and space telescopes. These observations revealed their diameters and the subtle gravitational influence these closely packed planets have upon each other. From this, scientists determined each planet’s mass. We also know how much energy their star radiates onto these planets’ surfaces, allowing scientists to estimate their temperatures. We can even make reasonable estimates of the light level and guess the color of the sky if we were standing on one of them. And while much remains unknown about these seven worlds, including whether they possess atmospheres or oceans, ice sheets or glaciers, they have become the best-known solar system apart from our own.

“Appearances are a glimpse of the unseen,” says Anaxagoras (d. 428 BC), one of the earliest philosophers of natural science. One of my math teachers once said that what we know is perhaps the size of a small island in an ocean of unknowns.

If you think of what we know as a circle, the more you know, the wider the circle around it and the greater its relationship with the unknown. It is like the more you know, the more you do not. The larger the island of the known in the ocean of the unexplored, the longer the coast of the unknown. “Ignorance is bliss,” some say, perhaps in contrast to the fear of being lost in this ever-growing ocean of the unknown. The seen are the forerunners of the unseen, just like an iceberg. Much more of an iceberg stays underwater than what meets the eye. We only see one-thirtieth of the iceberg. Canny inferences can be more accurate than physical observations. Mathematical thinking shows us all aspects of the iceberg.

As we continue to push the boundaries of knowledge, fueled by curiosity and guided by mathematics, we unlock new insights into the universe’s mysteries. By expanding our understanding of the cosmos through abstract reasoning, we embark on a journey of exploration that promises to reveal the secrets of existence and inspire future generations to reach for the stars.

 “Those who believe in the unseen…” (al-Baqarah 2:3)

Throughout life, people seek the elusive formula for happiness. The myth of King Midas serves as a poignant illustration that controlling external circumstances does not necessarily lead to a fulfilled existence. Like many individuals, King Midas believed that amassing immense wealth would guarantee his happiness. Thus, he struck a deal with the “gods,” eventually securing the ability to turn everything he touched into gold. It seemed like an extraordinary bargain that would ensure his status as the richest—and, consequently, the happiest—person in the world.

However, poor Midas soon discovered the cruel irony of his wish. As he attempted to eat, the food in his mouth transformed into inedible golden morsels, and the water he tried to drink instantly solidified into unquenchable gold. Trapped in a gilded prison of his own making, King Midas met his untimely demise, surrounded by abundant golden plates and cups. This tale serves as a cautionary reminder that pursuing wealth alone does not lead to genuine happiness or fulfillment.

Unfortunately, for some individuals, the formula for happiness appears deceptively simple: acquiring wealth. The belief that financial prosperity automatically equates to happiness suggests that the key to an easy life lies in a substantial bank account or a stroke of inheritance. It is understandable why such thoughts persist, as many people have contemplated this idea at some point.

In a legend attributed to the ancient Brahma priests, a promise was made to bestow golden discs upon a fortunate individual, enticing people to visit their temples and learn about their religion. The tale goes that within the Brahma temple, there were 64 golden discs arranged on a stake. Adjacent to this pillar, two additional posts stood empty.

The discs varied in size, with the largest positioned at the bottom and each subsequent disc decreasing in size, culminating with the smallest at the top. The smallest disc weighed one kilogram, while the largest weighed 64 kilograms. If one were to possess not just one or two, but the entirety of these golden discs, the cumulative wealth would be staggering—two tons of gold. Imagine the prosperity that could be calculated from such a collection, symbolizing substantial wealth and abundance.

Another perspective could liken it to winning a lottery. Brahma priests had a game. They give the winner of this game a chance to take away these golden discs. The rules of the game were seemingly simple: move the gold discs from the first pole to the third or even the second pole. However, there was a catch—a tiny stipulation with a value potentially worth billions of dollars. When transferring the discs from the first pole to the third, no larger disc could rest on a smaller one. The poles positioned two meters apart. A total of 2080 kilograms of gold discs awaited transfer from the first pole to the other posts. The one who could carry out this task without violating the condition would be the fortunate possessor of the gold. It might sound like a tale from Aladdin’s lamp for those unfamiliar with the mathematics behind it.

In reality, the priests had no intention of parting with the gold. The task at hand was not a simple one; it would take generations to complete. Even if descendants continued the mission, moving 64 gold discs to another pole would take an unfathomable amount of time.

Don’t be mistaken in assuming these discs can be easily transferred within a short distance. The minimum number of moves to be made for a legal carry is only 15 for three discs and 1023 for ten discs. No problem so far, but that would suddenly be 1,048,575 for twenty discs. For 64 discs, the number skyrockets to a staggering 18,446,744,073,709,551,615. Even the most optimistic estimate of one second per move, the time required for this task is calculated to be 600 billion years. To put this into perspective, the entire universe is approximately 13.7 billion years old, underscoring the staggering magnitude of time involved.

What the priests might be implying is the profound extent of human greed for wealth, a desire that won’t even endure for a billionth of the time required to complete the replacement of the 64 discs. Alternatively, could they be suggesting that life is fleeting and should not be preoccupied with trivial pursuits? It’s unclear, perhaps even a ploy to entice people into their temples. However, one undeniable fact remains: mathematical applications permeate various aspects of life, and a knowledge of mathematics proves essential.

Think about it: would someone with the ability to play the game by the rules and calculate from the outset that taking all the gold and leaving is impossible ever visit such a place? Perhaps, driven by curiosity. Nevertheless, this game, which is called the Tower of Hanoi, came to be recognized as a mathematical problem towards the end of the 19th century, and since then has expanded the horizons of mathematicians, offering insights into both gold and the formula for happiness. In the Tower of Hanoi problem, transferring three discs from the first stick to the third necessitates at least seven moves. The number of moves increases to 15 for four discs and 31 for five discs, following the pattern. This leads to the formula for the minimum transfer number: 2ⁿ–1, where “n” is the number of discs. Applying this formula to the 64 gold discs mentioned in the legend, the number of moves is calculated as 2⁶⁴–1. Interestingly, the same formula is employed to determine the number of subsets for clusters at the middle-high school level, showcasing the ubiquity and versatility of mathematical principles.

One of the most notable contributions of this problem was its role in discovering the largest known prime number until 1951. At that time, the record-holder was the number 2¹²⁷–1. Before computers took center stage in scientific endeavors, it made sense to tackle this challenge manually, moving 127 discs to calculate the number of moves required to unveil the most significant known prime number! Beyond this, there’s a surprising revelation when we articulate the movement of 5 discs: from the smallest disc to the largest one, it becomes an intriguing formulation, and a sequence appears.

For n=1, S1=1

For n=2, S2= 1, 2, 1

For n=3, S3=1, 2, 1, 3, 1, 2, 1

For n=4, S4= 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1

For n=5, S5=1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1

The sequence here is remarkable. The journey to the largest disc entails an incremental progression from 1, 2, to 1, followed by a return in the same sequence, concluding with a decrease by one. The first move is significant in the Tower of Hanoi. The algorithmic structure of the tower of Hanoi is used as explanatory material in computer coding lessons. The right start is the most significant factor in completing the game with minimum moves. However, the conclusion of the legend is not particularly optimistic. Even if all 64 discs are successfully transferred, the outcome is not a joyous one, as the passage of time heralds the approaching end of the world. Whosoever places the final gold disc on the third pillar will not benefit from the wealth either, as it is believed that chaos will ensue. According to the Brahma priests, the world’s end is expected after approximately six hundred billion seconds, assuming one second is spent on each disc. What the Brahmins mean is that happiness should not be delayed. Terms do not bind it. It is not with money. Happiness is found in the present moment, in working for a meaningful life and being fully engaged in it, right here and right now. Also, studies show that happiness is a function of reality minus expectations. While some studies show income is correlated with well-being and happiness, other research, such as the one conducted by Nobel-prize-winning psychologist Daniel Kahneman, shows that above a certain threshold, somewhere between $75K and $90K per year, minor adjustments may be needed to account for variations in location or other factors [1].

On the other hand, additional income beyond this threshold is not correlated with increased happiness or well-being. Emotional well-being appears to peak after the $75K mark. In a study conducted across 164 countries, involving 1.7 million participants, individuals were questioned about their purchasing power, life satisfaction, and overall well-being. The findings revealed that people within the income range of $75,000 to $95,000 generally reported the highest levels of happiness.

Contrary to popular belief, those who earn more than $95K experience varying levels of happiness. Exploring the mathematical relationship between wealth and happiness, it appears to exhibit an exponential and parabolic structure. Money can’t always buy happiness. If we visualize a curved design, with money on the X-axis and happiness on the Y-axis, and the graph starting from the origin point (0,0), we observe an interesting relationship. Initially, the graph suggests a correlation between money and happiness, echoing the common belief that without money, there is no happiness—a notion applicable in many contexts but not universally true, such as in the village of Piraha in the Amazon jungle.

As depicted in the chart, the association between money and happiness is not a straight line; it follows a curvilinear pattern. The more money one has, the higher the reported happiness, but this increase is not constant. Instead, it peaks and then starts to decline after a certain point. Research has identified the peak of this curve as an annual income of $95,000. It’s noteworthy that the level of happiness experiences only marginal improvement beyond $65,000, and once the maximum point is reached, further increases in income can lead to a decrease in happiness.

This phenomenon underscores the idea that beyond a certain threshold, additional money does not contribute positively to happiness; instead, it may have the opposite effect. This aligns with the saying, “Do you have money? Then you have a big problem!” as the graph reveals a diminishing return on happiness after reaching a specific income level.

However, a recent 2021 study of over one million participants challenges this notion, indicating that there might not be a definitive inflection point beyond which increased wealth does not equate to heightened happiness, at least not until an annual salary of $500,000. This study employed a more comprehensive well-being assessment, focusing on participants’ current emotional state rather than relying on recollections of their past experiences over weeks, months, or years. According to this real-time evaluation, individuals with significantly higher incomes reported experiencing elevated levels of happiness [2].

On the flip side, the geographical location where even lottery winners reside can significantly influence the long-term effects of their newfound wealth on psychological well-being. Factors such as local culture, social support networks, access to resources and services, and community norms all play a role in shaping how individuals perceive and experience their lottery windfall. Moreover, the cost of living, economic opportunities, and lifestyle expectations in different locations contribute to the overall impact of lottery winnings on psychological well-being over time.

A notable study from Sweden delves into this phenomenon, examining how lottery wins affect individuals’ lives. Remarkably, even years after winning, the study reveals that lottery winners consistently report higher life satisfaction and mental well-being. They also demonstrate an enhanced ability to navigate challenges such as divorce, illness, and solitude when compared to individuals who did not win the lottery. The possession of a substantial sum of money significantly alleviated the difficulties associated with these life events for the winners [3].

While the impact of lottery winnings on psychological well-being is influenced by various factors, it’s crucial to recognize that happiness is a complex and multifaceted concept. Beyond financial considerations, individual satisfaction is shaped by the broader context of one’s life and surroundings. The Sweden study on lottery winners serves as a compelling illustration. Despite the evident positive effects on life satisfaction and mental well-being among winners, it underscores that true happiness encompasses more than just monetary gains. The ability to navigate life’s challenges, build meaningful connections, and find fulfillment in personal pursuits are integral components of a contented life. Therefore, while financial prosperity can certainly alleviate certain hardships, the broader understanding of happiness extends beyond the confines of wealth.

In summary, the findings in these studies show that happiness is not necessarily linked to worldly, materialistic wealth. Rather, the financial means at one’s disposal can yield varying levels of satisfaction. Interestingly, the results also reveal that individuals in lower-income groups tend to excel at discovering ways to derive meaning from life, find joy, and cultivate happiness even within the constraints of their limited conditions. This suggests that the pursuit and attainment of happiness are complex and can be achieved through diverse avenues, transcending the mere accumulation of wealth.

References

1. High income improves evaluation of life but not emotional well-being, Daniel Kahneman and Angus Deaton, September 7, 2010,107 (38) 16489-16493, https://doi.org/10.1073/pnas.1011492107
2. “Experienced well-being rises with income, even above $75,000 per year.” Matthew A. Killingsworth, PNAS, January 18, 2021, 118 (4) e2016976118 https://doi.org/10.1073/pnas.2016976118
3. “Long-Run Effects of Lottery Wealth on Psychological Well-Being,” Erik Lindqvist, Robert Östling, David Cesarini,The Review of Economic Studies, Volume 87, Issue 6, November 2020, Pages 2703–2726, https://doi.org/10.1093/restud/rdaa006

Among the captivating sciences that humanity has explored, mathematics holds a prominent place. Mathematics has always been an essential part of our daily life, from the fundamentals of arithmetic and algebra to the complexity of calculus and beyond. Math is a critical element of our problem-solving process, whether we’re buying groceries, calculating taxes, or analyzing data. But, besides being a vital skill, mathematics has numerous benefits for our brain and overall health. While it may seem like learning math is a daunting and insurmountable task, even small efforts to improve our math skills can have significant benefits for our brains. This article will examine how learning math can help us improve our cognitive capacities, promote healthy aging, and increase our overall well-being.

Mathematics promotes neuroplasticity

The acquisition of arithmetic skills fosters neuroplasticity, highlighting one of its major cognitive benefits. Neuroplasticity is the ability of the brain to change through time, allowing us to learn, adapt, and grow. According to research, engaging in mathematical activities such as problem-solving, mental calculations, and arithmetic practice can enhance neural connections and promote the formation of new brain cells. The underlying principle allows for learning and memory, which is critical for our total cognitive health. Mathematics includes both tangible and abstract thinking, which can aid in the development of brain muscles. When we perform mathematical tasks, neurons are activated together, strengthening their connections. As a result, brain plasticity increases, and cognitive function improves. The more we practice arithmetic, the stronger our neural connections become, which leads to improved memory, reasoning, and problem-solving abilities.

The neuro-scientific theory that explains how neural connections are formed in the brain is called the Hebbian principle. According to this principle, when two neurons are consistently activated simultaneously their relationship becomes more robust. In other words, the more frequently we use a particular neural pathway, the stronger it becomes. This is commonly phrased as “neurons that fire together wire together.” An example of the Hebbian principle in action is London taxi drivers. It has been found that many taxi drivers in London have a significantly larger hippocampus, the part of the brain responsible for spatial navigation and memory. This is believed to be in relation to the fact that these drivers study and practice London’s complex network of streets and landmarks extensively so they can navigate entirely from memory. They are even required to take a knowledge test for this, which is famously challenging. The test, which is called the Knowledge, involves memorizing over 25,000 streets and 20,000 landmarks within a six-mile radius of Charing Cross. In addition to street names and landmarks, the test includes questions about the most efficient routes between locations, traffic regulations, and historical and cultural information about London. The amount of information that needs to be memorized for the Knowledge test is significant and requires extensive training and practice to master. By navigating the city’s streets and landmarks repeatedly, taxi drivers activate the same neural pathways in their hippocampus frequently. This consistent activation strengthens the connections between the neurons in their hippocampus, resulting in a more extensive and efficient neural network. Similarly, learning mathematics requires us to consistently activate specific neural pathways in the brain, strengthening the connections between neurons and promoting neuroplasticity. By engaging in mathematical activities regularly, we can increase our capacity for abstract and concrete reasoning and improve our problem-solving skills and capacity for systemic thinking. These skills strengthen our mathematical abilities, enhance our cognitive abilities, and promote brain health.

Mathematics improves our cognitive abilities

Learning mathematics has been shown to improve cognitive abilities across our life span. Working memory stands as a crucial cognitive skill imperative for achieving proficiency in mathematics. Our cognitive ability to retain information in our minds is closely linked to mathematical performance. Basic math processes, such as counting apples in a basket, are simple examples of working memory. In contrast, calculating statistical significance and computing derivatives are complex examples. Mathematics is a problem-solving process that requires a lot of focus on the problem. When we have good selective attention, we can quickly discover the irrelevant parts of a complex issue. This process of particular attention allows us to focus our mental resources on the most relevant aspects of the problem, leading to improved problem-solving abilities. Furthermore, it has been demonstrated that engaging in mathematics enhances our attention and inhibition capabilities, crucial for effective cognitive control. Cognitive control involves maintaining attention and focus, inhibiting irrelevant information, and switching between tasks efficiently. These abilities are crucial for academic success and overall mental health.

Engaging in mathematics may promote healthy aging of the brain

Despite the decline in cognitive abilities that accompanies aging, research indicates that mathematicians retain an advantage in preserving their cognitive functions, particularly in tasks related to executive functioning and short-term memory. Older individuals practicing mathematics not only experience these benefits but also receive a recommendation from neurology scientists to hone their mathematical skills. Increasing evidence suggests that individuals who engage their brains through activities such as mathematics are less susceptible to the development of Alzheimer’s and dementia. Consequently, the study of mathematics not only pushes individuals beyond their comfort zones but also provides a stimulating challenge for the brain, potentially mitigating the risk of this irreversible cognitive disorder. There have been several studies that suggest that engaging in math may promote the healthy aging of the brain. For example, a study published in the Journal of Neuroscience in 2013 found that older adults who engaged in cognitive activities such as playing games, reading books, and doing puzzles had less brain atrophy than those who did not. The study also found that the participants who engaged in more complex cognitive activities, such as learning a new language or doing challenging math problems, had even more excellent protection against brain atrophy. Another study published in the journal Neurology in 2017 found that people who engaged in more intellectually stimulating activities, including reading, writing, and doing puzzles, had a slower rate of cognitive decline than those who did not engage in these activities. The study found that this effect was powerful for people who played chess, took classes, or did math problems. A 2018 study published in the Journal of Alzheimer’s Disease found that older adults who engaged in more intellectual activities, including reading, writing, and doing math problems, had superior cognitive function compared to those who did not engage in these activities. The study found that participants who engaged in more complex mental activities, such as learning a new language or completing challenging math problems, had even greater cognitive function.

Mathematics can contribute to healthy aging by enhancing overall brain health. Engaging in mathematical activities encourages the growth of new brain cells and strengthens existing neural connections. This can help prevent age-related cognitive decline and diseases like Alzheimer’s and dementia. Moreover, studies have revealed that engaging in mathematical training can augment the size and activity of brain regions associated with problem-solving, spatial reasoning, and cognitive flexibility. With age, these regions typically diminish and decrease in activity, potentially leading to cognitive decline. However, consistent involvement in mathematical activities can serve to safeguard these critical brain functions, thereby fostering healthy aging. In addition to promoting brain health and cognitive abilities, learning mathematics can also benefit physical and psychological health. Participating in group math activities has been shown to have a positive effect on overall well-being and may even reduce the risk of developing dementia.

Mathematics serves as a means to comprehend and interpret the intricacies of the world. From deciphering natural patterns to unraveling the complexities of the universe, mathematics equips us with the tools to grasp and analyze the phenomena that shape our existence.

Even small efforts to improve our math skills can have significant benefits for our brains. Embracing mathematics represents a holistic mental exercise that fosters neuroplasticity, advances cognitive faculties, supports healthy aging, and uplifts overall well-being. By embracing math and challenging ourselves to learn new concepts, we can improve our brain health and become more well-rounded individuals. Mastering mathematics transcends the realm of equations and formulas; it offers an invitation to individuals of all ages and walks of life to embark on a journey of intellectual enrichment. Whether you’re a student, a professional, or a retiree, there is always an opportune moment to commence your mathematical exploration and savor its myriad benefits.

My father used to tell me a story in which a young boy was searching for a place where nothing like math existed. I am unsure if he crafted it for me or read it elsewhere. However, I still liken those with a fear of math to that child. Individuals with “math anxiety” spend their lives trying to avoid mathematics and its formulas, which often haunt them.

Numbers are ubiquitous in our lives, and a world without them is highly improbable. Consequently, a life devoid of mathematics is unimaginable. In an episode of the once-legendary TV show “Twilight Zone,” a man wakes up to discover that nobody understands him. Imagine waking up one morning and realizing that your loved ones no longer comprehend your language. It is a nightmarish scenario where those who understood you yesterday fail to grasp your words today—your spouse, children, neighbors, and relatives. It highlights how the modern world can isolate us from one another, rendering us incomprehensible to those closest to us. Similarly, picture waking up in a place where numbers are not utilized. Is such a place possible? Can we find a location where numbers are absent?

Surprisingly, such a place exists in the Amazon jungle. The Pirahã tribe, an Amazon-native community, has no concept of mathematical numbers in their language. This tribe represents one of the rare communities globally that lack numerical words and mathematical concepts in their linguistic structure. Comprising approximately 700 to 1000 individuals living in scattered villages of 10 to 15 adults each, it takes around ten days to travel from one village to another by canoe. The Pirahã possess a unique language, referred to by anthropologists as “anumeric.” Essentially, their language lacks expressions that indicate quantity. After significant effort, researchers have only identified three words related to the concept of numbers: “Hoi” means “one” or “a small quantity,” “Hooi” signifies “two” or “a slightly larger amount,” and “baa gisa” represents a collective or “many.” Thus, their numerical vocabulary is limited to “one,” “two,” and “many.”

The first encounter between the Pirahã tribe and the modern world occurred in 1977, bringing this community to the attention of the wider population. Professor Daniel Everett played a pivotal role in introducing the world to the Pirahã people. Initially, Everett went to the tribe as a missionary, teaching them about Christianity and the Bible over several years. However, he found himself learning a great deal from them and eventually reconsidered his own beliefs. The experience was so profound that he not only stopped his missionary work but also resigned from being a missionary altogether. When Everett first discovered the Pirahã, he was filled with the same excitement as any missionary embarking on a new mission. In his own narrative, he expected this supposedly primitive tribe to enthusiastically embrace his faith. However, their own set of beliefs rendered his efforts fruitless during his first three years.

The tribe refers to themselves as Pirahã, meaning “straightforward” or “only human.” Everett’s initial observation was their sincerity. Despite the challenges they faced, the Pirahã people never complained about their conditions and always found reasons to be grateful and content. Everett even considered them to be the happiest people in the world. Their livelihood primarily relied on hunting and fishing, and all food obtained was shared among the villagers. Other Amazonian tribes exist alongside the Pirahã. However, what sets the Pirahã apart is their contentment with what they have and their lack of envy towards the possessions of others. Unlike other tribes, they do not yearn for the belongings of Brazilians, such as guns or horses. When they cannot acquire similar possessions, it does not break their spirits or make them feel bad. The Pirahã people take pride solely in being Pirahã; their self-esteem is exceptionally high, and they have no regrets.

The Pirahã tribe holds firm to their own values and beliefs. They believe in the existence of invisible beings known as “fast mouths” that reside in the forest. According to their beliefs, these beings are similar to the Pirahã people but speak in a different way. The Pirahã only believe in what they see, except for the “fast mouths.” Whenever Everett spoke to them about his faith, he faced the same question repeatedly: “Have you ever seen Jesus? Why do you continue to talk about things you have never seen?” It took Everett three years to gain acceptance among the tribe and learn their language. They refer to anyone who does not know their language as a foreigner and never truly consider them one of their own. Despite occasionally being called “brother,” this did not contribute to Everett’s missionary work. One day, a Pirahã individual approached Everett and said, “We know why you are here—to tell us about Jesus. We appreciate your friendship. You are a good person, but we do not wish to hear about Jesus either. Remember, we are not Americans.” It was then that Everett began to question his own actions. How justified is it to impose one’s beliefs on others? Eventually, these and similar thoughts led Everett to give up his missionary work. He underwent a personal transformation, realizing that these people were not superstitious and did not discuss things they had not seen or heard. According to Everett, the Pirahã people were the happiest individuals he had encountered in the world. Moreover, their lives were free from the complexities of mathematics, numbers, taxes, and mortgages.

Although the Pirahã people lack mathematical knowledge and do not use quantities as we do, it would be a mistake to assume they lack intelligence. They possess extensive knowledge about the plants in the forest and readily recognize which plants can be used to cure various ailments. They have the ability to train wild animals, such as raising a baby eagle from an early age and then using it for hunting. They compare their hardworking nature to that of ants. Their environmental knowledge is astonishing, even though they may be considered “primitive.” According to the Pirahã people, Pirahã individuals work day and night, just as ants do. If they choose to, they can interact with people in the big city. Everett, who lived among the Pirahã people for seven years, recounts the story of a girl from the tribe who went to work in the city. She worked as a cashier in a supermarket. However, after two years, she returned to her village, expressing that she found city life highly stressful and missed the tranquility of her village. This example alone demonstrates that the Pirahã people possess cognitive abilities akin to mathematics, although they prefer not to use them in the conventional sense. Their language includes concepts related to objects, colors, and intonation, but it lacks numbers and the concept of time. They divide time into morning, noon, and evening. If a tone is not heard, communication can become challenging. Due to the absence of numbers in their language, questions related to quantities can be perplexing for them. For instance, a simple question like “How many children do you have?” holds no meaning for them. They find such questions strange and wonder why these outsiders ask them. Understanding the unique dynamics of tribal life provides insight into why this question, in particular, seems odd, irrespective of their lack of numerical concepts. Within the Pirahã community, children belong to the tribe, and everyone collectively takes care of them. The same sense of communal responsibility extends to the sick, the needy, and the elderly. Money is not used within the tribe, as they adhere to a culture of exchange. Although Brazilian merchants attempted to introduce money to the tribe, their efforts were unsuccessful.

The Pirahã people, an indigenous tribe living in the Amazon rainforest, have a unique perspective on mathematics and life. Their numerical system is limited to the concepts of one, two, and many. While their understanding of mathematics may differ from other cultures, these concepts hold significance in their worldview. From a philosophical and religious standpoint, one can interpret these concepts as representing different aspects of existence. One can symbolize the Creator, who is believed to have the power to create everything from nothing. The Creator is seen as unequalled and knowing Him is considered valuable. For the Pirahã, life feels incomplete and meaningless without this knowledge. The concept of two represents humanity. The union of a male and a female, symbolized by Adam and Eve in various belief systems, is seen as the origin of humankind. Despite their differences, the unity of these two parts is considered essential for giving meaning to life. The continuation of human life relies on the existence of couples, which the Pirahã attribute to the world order established by a higher power. Lastly, the concept of many represents the diversity found in the universe. The Pirahã observe the multitude of life forms, such as animals, insects, plants, mountains, and stones. The Quranic verse from Surah Nisa, verse 1, emphasizes the scattering of men and women from a single human self and its mate, highlighting the abundance and diversity of human beings.

The Pirahã’s limited numerical system of one, two, and many may be rooted in their cultural perception that the abundance and diversity of the universe reflect the Creator’s design. It’s important to note that the Pirahã culture and worldview are unique, and their understanding of these concepts may differ from interpretations in other cultures and belief systems. The perspective presented here reflects a specific religious and philosophical viewpoint. Different individuals, cultures, and belief systems may have alternative interpretations based on their own experiences and beliefs.

The largest prime number [1]—discovered by a group of scientists led by Patrick Laroche in 2018—drew considerable interest [2]. To better describe the greatness of this number with 24,862,048 decimal digits, we can give the following example: if we print this number on both sides of A4 papers with standard font sizes, the height of these papers placed on top of each other would be 83 cm. Why did these scientists go to such lengths to make this discovery? Why did such a discovery receive so many awards? These questions will lead us on a voyage into the mysterious world of mathematics.

History of prime numbers

A prime number is a positive number which is evenly divisible only by itself and 1. They start with 2, are followed by 3, 5, and 7, and seem to follow no discernible pattern after that. They have always compelled attention as many scientists have tried to come up with a formula to produce them for centuries. According to the available resources, the oldest study on the matter started with Greek polymath Eratosthenes (BC 276-195) and his famous sieve [3]. The sieve of Eratosthenes systematically eliminates composite numbers—the numbers that are not prime—and defines the remaining numbers as prime, but it is very hard to use for large numbers. In comparison, the Mersenne Theorem [4], developed by French scientist Marin Mersenne (1588-1648) and formulated as  (n is a natural number), helps us to calculate many prime numbers. Although it was later found that 2047 () is composite, all prime numbers calculated by this formula came to be referred to as Mersenne primes. The largest prime number discovered in 2018, too, is a Mersenne prime that satisfies this formula.

Another French mathematician, Pierre de Fermat (1607-1665), adopted a similar approach to Mersenne’s and stated that the formula (m is a natural number) can give us prime numbers—but exceptions to this formula have demonstrated that the formula may not always produce primes [5]. These numbers were called Fermat numbers and worked better than Mersenne numbers [6].

Today, it is generally accepted that there is no formula that can give you all prime numbers. Therefore, mathematicians focus on methods to check whether a number is prime. Researchers have come up with various primality tests including the Fermat, Miller-Rabin, AKS, and Lucas-Lehmer to determine if very large numbers are prime.

In addition to prime numbers, mathematicians also refer to the numbers that are relatively prime, i.e., coprime numbers. Take 21 and 10. Although both of these numbers are not prime, they are relatively prime as the two numbers have only one common divisor, 1. Many composite numbers can be coprime with any other number.

Prime numbers and encryption

We use prime numbers every day without even realizing it. It is possible to give examples of how they are used in many fields including physics, engineering, coding, and diverse technological applications. Cryptography or the practice of cryptography is one of the areas where primes are frequently employed. Deriving from ancient Greek words of “kryptós” meaning “hidden or secret” and “graphein” meaning “to write,” cryptography refers to the study of techniques for secure communication. One of the early applications of cryptography involves the use of the Caesar cipher [7].

Caesar cipher.

The Caesar cipher employs systematic encryption; letters are assigned to numbers:

A=0, B=1, C=2 … Z=29

Then, the message is encoded:

S E L A M

21 5 14 0 15

Using a “key” (for instance, adding 5 to each number), the message is encrypted before it is sent:

26 10 19 5 20

The recipient of the message decrypts it using the rules of modular arithmetic (in this case, by subtracting 5 from the numbers). What is crucial here is to be able to communicate the key to the recipient in a secure manner. During the Second World War, German scientists aware of this problem used the Enigma system, where the key would be changed every day during the war [8]. The British used machines developed by mathematician and logician Alan Turing (1912–1954) to decode the encrypted messages despite the fact that keys were changed on a daily basis, thereby changing the course of the war.

Enigma cipher machine used by Germans.

Prime numbers come to our rescue in solving the security problem in systematic encryption. The communication between the sender and the receiver is established using the method called “asymmetric encryption” without using any key. In 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman developed the RSA encryption method—the name deriving from their surnames—in which communication is secured between the sender and the receiver by using two primes and a third number obtained by multiplying the primes [9]. The first of these primes is called “open key” which is publicly available. The second prime is the “special key” known to each user. The third number, which may not be prime, is a prime obtained by multiplying the first two numbers. Through special operations, the system matches these prime numbers and the key is encrypted. For example, when you insert your bank card to an ATM machine, your card gives the bank’s publicly available key (prime number) to the machine. Then, the machine asks you to enter your PIN, which does not have to be a prime number, for secure communication with your bank. When you enter your PIN correctly, the system has access to the private key through some special operations in the background and allows you to perform transactions related to your account. This process relies on complex algorithms for ensuring security. Whichever algorithm is used, all encryption and decryption operations are performed using prime numbers. The larger the prime number, the harder it will be to break the code.

As is seen, we frequently use prime numbers in our daily life—often without even being aware of them. There may be undiscovered properties and novel areas of the use of prime numbers.

References

1. “Great Internet Mersenne Prime Search”, www.mersenne.org

2. “Largest known prime number”, en.wikipedia.org/wiki/Largest_known_prime_number

3. “Sieve of Eratosthenes (Eratosthenes’in Eleği)”, tolpp.com/sieve-of-eratosthenes-eratosthenesin-elegi/

4. “Marin Mersenne”, www.mathematik.ch/mathematiker/mersenne.html

5. “Pierre de Fermat”, www.spektrum.de/wissen/pierre-fermat-1607-1608/962953

6. →prime
→prime
→prime
→prime
→composite
→prime
The sequence of Mersenne primes in the binary number system.
→ prime
→ prime
→prime
→prime
→prime
→composite
The sequence of Fermat primes in the binary number system.

7. “Caesar cipher”, en.wikipedia.org/wiki/Caesar_cipher; “Caesar Cipher Exploration”, en.khanacademy.org/computing/computer-science/cryptography/crypt/pi/caesar-cipher-exploration

8. “Enigma machine”, en.wikipedia.org/wiki/Enigma_machine

9. “What is RSA encryption and how does it work?”, www.comparitech.com/blog/information-security/rsa-encryption/

No science teaches the harmonies of nature more clearly than mathematics. In arithmetic we explore a universe of figures by counting; in geometry we discover another universe by drawing lines and laying down definite directions in the conceptual field of the imagination; in algebra we produce magnitudes of a still more abstract nature that are expressed by letters. In all of these cases the first step is to establish the general conditions that lay down the rules which all further steps are subject to. Every one of these “universes” is dominated by a consistency that produces an excellent symmetry.

Certainly the world, the universe, and nature can be reliably understood using mathematics. In other words, we can say that nature is mathematics. For instance, the arrangement of seeds in a sunflower can be understood using Fibonacci numbers. Every number in the Fibonacci sequence equals the sum of the two previous except for the first two numbers. Sunflower heads, like other flowers, contain two families of interlaced spirals—one winding clockwise, the other counterclockwise. The shape assumed by a delicate spider web suspended from fixed points, or the cross section of sails bellying in the wind, is a catenary—a simple curve defined by a simple formula. Seashells, animal horns, and the cochlea of the ear are logarithmic spirals that can be generated using a mathematical constant known as the golden ratio. Mountains and the branching patterns of blood vessels and plants are fractals, a class of shapes that exhibit similar structures at different magnifications. Einstein’s E = mc² defines the fundamental relationship between energy, matter, and the speed of light. Additionally, a few simple constants—the gravitational constant, Planck’s constant, and the speed of light—are profound examples for better understanding our universe as a whole.

Magic squares are conspicuous instances of the intrinsic harmony of numbers. They are also noticeable instances of the inherent harmony of numbers that some consider as an interpreter of the cosmic order that dominates all existence. Can they? Maybe. Though they are a mere intellectual play, they not only illustrate the nature of mathematics but also, incidentally, the nature of reality dominated by mathematical regularity. This is incidental, because its discovery has a very extraordinary story. Magic Squares are square grids with a unique arrangement of numbers in them. These numbers are exceptional because every row, column, and diagonal adds up to the same number. They are understood from every part in their overall square shape, the sum of which gives the same number – the sum of rows, columns, and diagonals. Magic Squares, one of the Sudoku-style question types, may vary in the numbers to be placed in the square according to its size. For example, let’s say there is a 3×3 square and ask: how can you put the numbers 1 to 9 each in their own square so that all of the the rows, columns, and cross sums are always the same? I recommend you stop here for a while and work on the question. You can see that it is not as easy as it sounds. Otherwise, the answer is in Figure 1.

The historical development of magic squares is also very fascinating. The earliest record of magic squares is from China in about 2200 BC and is called Lo-Shu. It tells that the first magic square was incidentally discovered by the Chinese emperor Yu 4,200 years ago. Emperor Yu discovered a tortoise while sauntering along the Yellow River (Lo River). The tortoise’s shell had a magic square shape, and this square divided into grids contained dots instead of numbers. Imagine having the same quantity as spots instead of numbers (Figure 2).

According to the myth, Emperor Yu took the tortoise and kept caring for it as a valuable guest in his palace. Its fame spread all over the world and extends beyond time to today’s modern world. Perhaps, it deserves the title of the world’s most famous mathematical tortoise with an outstanding mark on him. After the tortoise, the magic squares have become a very significant mystical symbol in China and in the Karma philosophy. In Karma, it is said that the world is surrounded by four essential elements and this is symbolized with magic squares. If we consider the number 5 in the center as the world, the followers of this philosophy surrounded the world with numbers that were supposed to be the four elements of Yin Yang. 4 and 9 symbolize metal, 2 and 7 symbolize fire, 1 and 6 symbolize water, and 3 and 8 signify wood, as shown in Figure 3.

We see that magic squares have mystical meanings not only in China but also in many places. For example, in pre-Islamic Arabia, people hung magic squares in their homes as amulets that they believe protected their houses from evil spirits. One explanation to this superstition could be that Arab astrologers used magic squares to cast horoscopes.

Magic squares were introduced to the Western world in the work of Theon of Smyrna. Theon’s most important work is “Expositio rerum mathematicarum ad legendum Platonem utilium.” This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music, and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, which sounds rather fanciful. In the section on numbers, Theon adopts a Pythagorean approach and writes about all kinds of numbers: odd, even, prime, composite, square, oblong, triangular, polygonal, circular, spherical, solid numbers with three factors, pyramidal, perfect, deficient, and abundant numbers. The work of the Greek mathematician Moschopoulos in 1300 AD helped to spread knowledge about magic squares. So here we are now, more than 700 years later, and teachers are using them in class for problem-solving and practicing addition.

In the 16^th century the Catholic physician, astrologer, and theologian Cornelius Agrippa (1486–1535) constructed squares of orders from 3, 4, 5, 6, 7, 8, and 9 which he associated with the seven known astrological planets: Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon (the Sun and Moon were considered to be planets at that time.) Agrippa had a colorful life that included various dangerous run-ins with the Church and jobs as an occult scholar, lawyer, and military strategist. Agrippa’s De Occulta Philosophia stimulated Renaissance study of magic and got his name into early Faust legends. Agrippa believed that a magic square containing the digit 1—which exhibits the magic constant of 1 in all directions—represented God’s eternal perfection.

I do not think that anyone goes beyond the Indian people in terms of expecting help from magic squares for healing diseases and even finding a spouse. They also took first place in producing magic squares known as the Jaina inscriptions. The first 4×4 square was discovered on a door in Khajuraho, India, around 1100 AD (Figure 4). Jainism is an Indian philosophy that was established in the 6th century BC, which may imply how important magic squares are for Indians who follow this tradition.

India brought the usage of these squares to daily life. One superstition still practiced today is to write a 3×3 magic square to find someone missing and hang it everywhere like a notice. Strangely, the name of the missing person is not in the notice! The magic square’s merits are not limited to this. It is believed that a couple can save their marriage if they can get a copy of a square that is drawn on a Wednesday or Friday.

For mathematicians, it makes sense if the sums of squares across each row, column, or diagonal are equal. But Indians have different expectations from the squares. For example, the square used by young girls to get married is shown in Figure 5. This square is painted on a plate with pastels. The plate is washed in the Ganges River. The girl who is seeking her desired husband drinks the pastel-colored water from the river. This tradition is still being practiced today.

There are also squares used by Muslims in the Islamic period. One of the most beautiful examples of these squares is one designed to express God’s name as shown in Figure 6. The square with 66 letters in total on each side evokes the sum of the Arabic word Allah in Abjad calculation (In Arabic, every letter is assigned a numerical value, and this calculation is called Abjad).

Throughout history, humanity has been influenced by mathematics and numbers and has tried to reveal the unknowns by investigating the mysteries of numbers. Humanity’s long-time fascination with mathematics has arisen because the universe is constructed from a mathematical fabric. In 1623, Galileo Galilei reinforced this belief by stating his credo: “Nature’s great book is written in mathematical symbols.” Plato’s doctrine was that God is a geometer, and Sir James Jeans believed God experimented with arithmetic. Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits. Likewise, magic squares have been one such mysterious dimension of mathematics that have influenced human life in various ways.

Figure 1 – 3×3 Magic Square

Figure 2 – Spots on the tortoise

Figure 3 – Five Elements in Magic Square

Figure 4. Magic square at the Parshvanatha temple, Khajuraho

Figure 5 — Magic Square Drawn on a Plate with Pastel Paint

Figure 6- The sum of Allah’s words with Abjad calculation is 66. The grid is formed by the letters in the word “Allah,” whose numerical value is also 66 (nineteenth century, Damascus).

Sometimes I want to be a bird and fly.
Sometimes I want to be a fish and swim.
I don’t know how to fly or swim.
Right now?
I’m counting again, and I’m picking up my years and splitting them up with my disappointments.
I’m starting again today.
Anonymous

As humans, the trait from the animal world we most admire is probably the ability to fly. People have dreamed of flying like birds for thousands of years. Birds can take off and fly with movements that seem simple to our eyes. All they do is flap their wings. The size and shape of the wings affect the way a bird flies.

But it’s not just wings that allow birds to fly; they have many physical features that work together to make flight possible. They need lightweight features, a streamlined body, and a rigid skeleton to provide firm attachments for powerful muscles.

Small-winged birds usually fly by flapping their wings at least 40 times per second. On the other hand, the wide wingspan of a larger bird allows it to glide through the air without flapping its wings. A swift’s pointed wings help it dart about at great speed, while the very long wings of an albatross enable it to glide with little effort.

The rapid wing beats of smaller birds use more energy. Lowering the wings from top to bottom means lifting and consuming power; flapping from bottom to top maintains flight and balance. For example, a bat flaps its wings eight times per second while flying at cruising speed—that is, every 0.125 seconds.

The zebra finch is the bird with the highest flapping frequency. It can achieve 26.9 wing beats per second—or, 26.9 hertz (Hz), which is 3 to 3.5 times more than the bat in the same time frame. The flapping of a bird’s wings is assumed to be symmetrical and equal. However, in real life, many birds perform asymmetrical flapping and positioning. A bird can’t always fly flat, in a straight line. Slow flapping is expected in a downward stroke, and rapid flapping is upward. It should be noted that small birds use medium-sized flights to perform alternate flapping and compound movements.

The size of the wing is another crucial factor. Bigger wings produce greater lift than smaller wings. So, smaller winged birds need to fly faster to maintain the same lift as those with larger wings. When the wings are open, a bird’s wingspan is at right angles to the direction it is flying. It should also be noted that different species can fly and attack at different angles.

With each flapping of its wings, a bird’s body naturally goes up and down almost like a spring. The body’s motion also follows a fluctuating structure depending on wing movements. When the wing flaps at the initial right angle and relative to the flight path, one can observe a wave movement in the wings. The width here is simply the height of the flight waveform. While this height is 26 cm in bats, it is 12 cm in finches. Although different species have different flapping movements and flight patterns, a comparison of birds’ wavelengths can be made with that of sea animals, like dolphins and sharks, whose swimming speeds are most efficient. Optimum cruise speed in dolphins, sharks, and bony fish depends on tail swing frequency and tail width. The dimensionless number value called Strouhal constant, which is used to describe oscillating flow mechanisms, usually gives a range between 0.2 and 0.4. The Strouhal constant is also found in flying birds and bats.

Three factors are required to obtain this constant: frequency, wingspan, and speed. As an example, bats fly at a speed of 6 meters per second. When calculating Strouhal constant, the frequency value is multiplied by the wing width and divided by the bird’s speed.

Let’s calculate the Strouhal constant of the kestrel. If the frequency for the kestrel is 5.61Hz, the width is 0.339 m, and the speed is 8.1 m/s, the Strouhal constant is calculated as follows:

St =f.A ⁄ U = (5.61).(0.339) ⁄ 8.1= 0.235.

The projection was made among 42 different bird species. We observe that bats approach the upper limit of Strouhal constant, whereas large birds approach the lower limit.

Swallows are some of the most common birds. Of the 74 swallow species in the world, 47 live in Africa. Two of them—the West African swallow (Hirundo domicile) and South African swallow (Hirundo Spilodesa)—are named for the continent. Despite its name, the European swallow is also common in Africa. After 54 years of capturing and releasing 26,285 birds, the Cape Town University-supported Avian Demographics unit has a lot of data on these birds. Studies have shown that the average wing length of an adult European swallow is 12.2 cm, and its body mass is 20.3 grams.

The known frequency (f) and width (A) measurements for the European Swallow make it possible to estimate the speeds of similar bird species. It is possible to estimate the air velocity (U) with these two values. The European swallow, with a wingspan of 18 cm, can flap its wings 18 times per second.

By ignoring the mass value and looking at the actual wing lengths, not the flight wing, we see that the European swallow flaps its wings 14 times per second, and its flight wingspan is 23 cm.

In this case, we can say that the European swallow flaps its wings an average of 15 times per second, and the flight wing width is 22 cm.

The velocity formula is U = (f.A)/St.

If St=0.2 is taken for the highest speed Uy = (15.(0.22))/0.2= 16.50 m/s

If St=0.4 is taken for the lowest speed, Ud = (15.(0.22))/0.4= 8.25 m/s.

By averaging the lowest and highest speeds, the European swallow’s speed is found to be approximately 12.4 meters per second. In this case, the average speed of the swallow will be 45 km per hour. If we calculate the fuel consumption in a car traveling at the same speed, we find that you can go about 10 km with one liter of gasoline—which means it would take 4.5 liters of gasoline to cover 45 km. In this case, the cost (considering that gasoline prices change daily) is around $10. This money may not even be the amount of feed the swallow eats per year! We should marvel at the European swallow’s great design.

Sometimes unexpected things happen to us in life. We encounter situations that have almost no place in probability calculations. It’s like when the evil Star Wars character Darth Vader declares to the heroic Luke Skywalker, “I am your father.” The revelation is unexpected, and audiences are surprised.

Sometimes, life can feel that way.

This scenario can have different connotations for everyone, and it reminded me of a classical physics question. Suppose you pour a glass of tap water into the ocean. If you think about the size of the ocean, you will also imagine that a glass of water will disappear in it immediately. Five years later, fill a glass of water from another ocean shore on the other side of the world. The question is this: how many water molecules in the second glass came from the first glass? Assuming that there are around a billion cubic kilometers of water in all the world’s oceans, a glass of water is nothing compared to this total volume. Or, to put it mathematically, the probability of a molecule from the first glass being in the second glass is two chances in 1 followed by twenty-two zeros.

But then, there are an incredible number of molecules in a glass of water: they can have molecules totaling 1 followed by twenty-five zeros. Assume that a thousand molecules from the first glass are in the second. The probability decreases to only two out of 10¹⁹. That is still practically impossible. Going back to Star Wars, what is the probability that the enemy you encounter in the whole galaxy is your father?

We’d be surprised at the possibility that Darth Vader is Luke’s father, but we wouldn’t be surprised at the birth of a baby. The fact that four babies are born every second all over the world is important here. However, the probability of a baby being born is much lower than that Darth Vader is Luke’s father.

Let’s think of an ordinary person; his name is Joe, and Joe has 46 chromosomes like everyone else. Otherwise, there would be an anomaly in Joe, and then Joe would not be an ordinary person. As everyone remembers from basic biology classes, half of Joe’s chromosomes miraculously come from his mother (Mary) and half from his father (Jacob). Let’s not forget that his parents, like Joe, have 46 chromosomes from 23 pairs each. Thus, one of the chromosome pairs of each of his parents was chosen for Joe. Twenty-three chromosomes from both sides form the building blocks of Joe.

If the probability of 23 chromosomes coming from his mother is calculated, the number (0.5)²³ is obtained. How so? Well, the probability of a chromosome being selected is one in two. There are 23 pairs of chromosomes; Since the probability of being selected from the first couple is ½, the probability of being selected from the second couple is ½, the probability of being selected from the third couple is ½, and the probability of being selected from the fourth couple is ½, the probability of 23 independent events is calculated: as 1/2 x 1/2 x 1/2…1/2= (1 /2)²³. This calculation means that the probability of being Joe with the chromosomes from his mother is about one in ten million. This probability is equal to a big jackpot for a single ticket purchased in the national lottery.

The same probability applies to the father’s side, too. Then the probability of being Joe by chance drops to one in a hundred trillion this time. 10⁷ x 10⁷ = 10¹⁴ or 1 in 100,000,000,000,000. As such, the probability of Joe being born as the Joe we know is 10⁷ times more difficult than winning the jackpot in a single ticket from the national lottery. In other words, the probability of Joe is ten million times less than winning the national lottery jackpot.

Let’s imagine that Mary and Jacob met in a medium-sized settlement where there is a population of at least 10,000 men and 10,000 women. Then the odds of meeting these two people would be one in ten thousand. The probability calculations for Joe’s birth then fall to one in one quintillion. The fertilization of the mother’s egg can only be achieved through an average of 15 million sperm. This situation gradually decreases the probability of Joe being born: 15 x 10⁶ x 10¹⁸ = 15 x 10²⁴—or, one chance in fifteen septillion.

Joe’s mother did not come down from out of space. The probability of Joe’s mother, Mary, is also one in 15 Septillion. Identically, if we consider that the probability of his father is the same, the probability of being Joe is 225 x 10²⁴ x 10²⁴. One in 225 Kendecillion.

Here I must add: all assumptions take place in ideal circumstances. Before knowing each other, the possibilities of the mother or father’s death, the infertility of one or both, and the like are not included. The probability is very, very low, but must not be ignored.

Mary’s parents or Jacob’s parents and their families—and their families—and all other previous families must have existed since the beginning of human life. When you consider all the possibilities, the probability becomes sooooo infinitesimal. To describe such a possibility, we can only call it a probability in infinity. This possibility is valid for every ordinary Joe. Considering that there are 7 billion people in line, it is not difficult to say that life is a miracle of creation.

The last estimated human population in the world is about 7.7 billion. Considering that four babies are born every second and two people die (according to the data in 2019), it can be predicted that the human population will increase day by day. Thinking of this complicated probability calculations, we can only conclude how precious each human being is.

People have better understood the value of life during these COVID times. Since spring 2020, nearly a million Americans have died from COVID, leaving many millions grieving. A March 2021 poll from the Associated Press–NORC (AP-NORC) Center for Public Affairs Research found that about 20 percent of people surveyed in the U.S. had lost a relative or close friend to COVID. That means a potential bereaved population of about 65 million. That number has certainly increased. Death also reminds us that life is precious, as each birth is a unique gift. Again, considering all the probabilities, the chance of a baby coming into the world healthy is much lower than the chance of coming into the world unhealthy.

The large human population is not evidence of the simplicity and mediocrity of life but rather the perfection in the creation of the universe. When considering this, it is undeniable that there is a hand of power in this magnificent functioning. Is it a coincidence, or are we living and breathing in a superb and controlled system?

And should you attempt to count God’s blessings, you could not compute them. God is indeed All-Forgiving, All-Compassionate (16:18)

Source: https://www.medindia.net/patients/calculators/worldpopulation.asp

Have you ever thought about what your world would be like without numbers? Would we be able to build our civilization without numbers? We encounter them everywhere and we need them to survive. They are not only to use them to count, measure, and do calculations; numbers rule our lives. We are identified by Social Security Numbers, bank account numbers, and telephone numbers. Science, economy, and business are all about numbers. As a society we are organized with the help of numbers.

Some define mathematics as the science of patterns – and there are observable patterns in nature. With numbers can we make sense of these designs. With pi, for instance, we can see the ratio between a circle’s circumference and its diameter, and this is possible only if we can mathematically identify it. The numbers allow us to understand countless fascinating and useful patterns of nature that we would never be able to discover otherwise.

In everyday life many occasions contribute to the growth of the mental work of knowledge and meaning associated with a number. Although numbers are always a part of our lives there are still situations that have little to do with mathematics. The numbers that help us understand mathematics are sometimes used in contexts that have nothing to do with mathematics. When you think of the number 7 for instance, you can ask why there are so many things related with it: Why are there seven days a week? Why is James Bond encoded with 007? Why are there seven dwarfs in Snow White fairy tale? Why are there seven sleepers in the Biblical story? Why is the number of circumambulations seven in the Ka’ba? Why does the rainbow have seven colors? Why are the wonders of the world are called the Seven Wonders of the World? Why is the number of notes seven in music? Why are there seven stars in each of the Big-Little Bear constellations? Why are there seven verses in Surah al-Fatiha? Why is the number seven so unique?

Interestingly, the number 7 shows up in a survey in which researchers asked 30 thousand participants the same question: “What is your lucky number?” Almost 10% of the participants answered it was “7,” the most preferred number in the survey. So the question comes up to the mind: why did most people choose the number 7? Does it have anything related to the “why questions” mentioned above? Or is the juxtaposition of three sevens in jackpot machines, called a lucky 7, influential in this case? Regardless of the reason, almost three thousand out of thirty thousand people feel this number unique to themselves. Researchers also asked why they thought the number was lucky and participants gave different answers: waking up at 7:00 every day, the door number of their birthplace, songs that lasted around 7:07 minutes, etc. As understood from the given responses most of them were related to people’s personal preferences. Therefore these responses seem not helpful to understanding the questions above, such as why a week is seven days or why the earth and the skies are considered to have seven levels.

When the 30 most preferred numbers were listed, the number 3 comes second after 7, then numbers 8 and 4 follow them. What are the special features of these numbers that make them to be the most preferred? I have pointed out remarkable properties of some of these numbers:

Almost 7% of participants have chosen number 8 in the survey. It is plausible in many far eastern cultures that the number 8 is considered a lucky number to the point of obsession. Number 8 means good chance in Chinese culture because the Chinese word for “eight” sounds like “wealth.” Also, turned on its side, it represents the infinity symbol – which means “forever.” The Beijing Olympics opened on 8/8/08 at 8:08 pm – not a coincidence!

It seems the number 4 is in fourth place, but surprisingly, unlike number 8, it calls for bad luck in the Far East. The pronunciation of the number four in Japanese is very similar to the word “death.” Because of this, four have been considered bad luck in Japan, Korea, and China. It is often deemed to be terrible luck to give a gift made up of four pieces to someone. Many buildings with dominantly Asian populations do not have the fourth floor, much like the number 13 in parts of North America. You might find a hotel without a 4th floor or a product line without a series 4. For example, Nokia mobiles went from series 3 to series 5 product range. In Western culture, four isn’t necessarily considered lucky or unlucky; however, there are a few unlucky fours. The Four Horsemen of the Apocalypse from the Christian Bible and most swear words are called “four-letter words.” As far as these are disadvantages for number 4, still 5.6% of thirty thousand participants in the survey selected it as their lucky number.

The next number is 5, which is in fifth place. Another weird coincidence! 5.1% of participants in the survey have chosen it as their lucky number. The number 5 is impressive because it occurs a lot in nature. Humans have five senses (sight, smell, taste, touch, and hearing); and five fingers on each hand and foot. Some fascinating creatures, such as the starfish, have five-fold symmetry which means they can be rotated five times and will still look the same.

In the sixth place we have a thrilling number: 13. Out of all participants 5% consider it to be their lucky number, but it is accepted as an inauspicious number in most western countries. Even in some buildings, there is no floor or flat with this number. The fear related with number 13 even has a name: Triskaidekaphobia. Historically, this belief is based on two events: First, according to Christian faith, there were 13 people in total at Jesus’ last supper before his crucifixion on a Friday; Jesus and the 12 apostles. The second incident related to number 13 is about the templar knights in the fourteenth century who were arrested by the orders of the French King Philip and approval of Pope Clemens, on Friday, October 13, 1307. The bad luck of the number 13 and the source of “Bloody Friday” is also attributed to this inquisition cruelty. Another event why bad luck is associated with number 13 is the conquest of Constantinople in 1453, a year whose numbers adds up to 13.

The number 42, coming after the top ten, is exceptional. To understand the story of why this number, which is considered the myth of modern times, was preferred, we need to analyze a novel which is the origin of this myth: The Hitchhiker’s Guide to the Galaxy by Douglas Adams (d. 2001). The book mentions an imaginary race of hyper-intelligent beings who lived millions of years ago and dominated all dimensions. To put an end to the futile debates on life’s meaning, two of the most intelligent and brilliant people are assigned to build an enormous supercomputer – perhaps a quantum computer – that can calculate Life, the Universe, and Everything. After the building of the computer is done, they call it “Deep Thought,” and asks it to give the answer for the meaning of life. Deep Thought’s response is, “I’ll have to think a little bit. Come back in seven and a half million years.” After 7.5 million years, everyone fills the square with enthusiasm when the big day comes. The supercomputer says, ‘You really will not like it at all,’ and says 42. Yes, the number 42. Although this answer confuses everyone, it has become very favored over time and has inspired many. So much so that an asteroid which was discovered in February 2001 was named 2001DA₄₂. In the 42nd episode of the “Doctor Who” series, the band Coldplay referred to this book with their song of the same name. “Lost,” one of the two most mysterious series in TV history, featured 42 in the enigmatic issue that will bring the world’s end: 4, 8, 15, 16, 23, and 42. By the way, all these numbers except number 15 are among the top thirty most preferred in the survey. The apartment number where the lead character, “X Files,” another mysterious TV series, was also 42. The first building of Googleplex was “Building 42.” According to some, this name was given to the building in reference to the novel The Hitchhiker’s Guide to the Galaxy. CERN Laboratories, which also includes the Large Particle Accelerator, has an office building named 42 for the same reason. How many antennas are at the “Allen Telescope Array” of SETI, which investigates extraterrestrial life? Yes, you guessed it right: 42! Also searching “answer to life the universe and everything” on Google shows number 42 with a calculator!

Some selected numbers may have come to the front due to their numerical properties. For instance, “Pi” and “e” (Euler’s number) come in the 18th and 28th place respectively in the list of the most preferable numbers. It is not surprising why the Pi number made it into the list considering its reputation. This number, whose number of digits after the decimal point has not yet been determined and have thought to extend to infinity, is considered sacred by many people. Assuming that it has a purely random property, if pi is long enough, you can find every string of digits in your life in pi. So your birthday, phone numbers, or any number you randomly type in are somewhere in the pi. Let’s go further. With a code that can convert letters and numbers, you can theoretically find the name of any person or institution, a word, a sentence, or even a book, in the number pi. Those who think that all life is represented inside this number are not few. Some even memorize it as worship. According to Guinnes Book of records, the record on memorizing pi belongs to a Chinese named Lu Chao with 67,890 digits and it took him 24 hours and 4 minutes to recite all the digits. In 2006, a Japanese named Akira Haraguchi said that he memorized 100,000 digits after the decimal point in almost 18 hours, but this was not officially validated by Guinness.

4 is an interesting number, too, which interestingly comes fourth among the “lucky” numbers. Some Muslims consider number 4 to have a special significance because of several reasons: The number of the Rightly Guided Caliphs is four; there are four Archangels, four holy scriptures, and four major sects in Islam, which may indicate a divine connection among all these.

“The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word,” said Galileo Galilei. This early adage is never more relevant than today, when mathematics is the key to so much of our contemporary era.

One of my most beautiful childhood memories was stone skipping with my dad. It always seemed mysterious to me when the rocks bounced smoothly on a calm sea on a warm summer day.

Skipping stones is a worldwide phenomenon. Eskimos skip rocks on ice and Bedouins on smooth sand. In England, stone skipping is known as “ducks and drakes”; in France, as “ricochet”; in Ireland, as “stone skiffing”; and in Denmark as “smutting.” Stone skipping is mentioned in the works of Shakespeare and even Homer. The British King’s bouncing gold coins on the Thames River in 1583 is known to be the first public event on skipping stones. From Hindi to Russian to Chinese, every language has a unique word or term for skipping stones. The reason for this may be that everyone wonders how a stone can glide across the water after throwing one. Of course, the secret to a successful skipping is to choose a good stone with a smooth surface, not too heavy nor too light, and the water should be as calm as possible, for wavy water prevents stones from bouncing further.

Stone skipping is normally a good application area of mathematics and physical sciences; yet Lazzaro Spallanzani (1729-1799), an Italian Catholic priest, is known to be the first person to make a scientific explanation on how stones bounce on water. After him, there seems to be a long gap in research until a French physics professor, Lyderic Bocquet, brought this subject to the scientific field in an article published in 2002 and then in another article in 2004. Dr. Bocquet attributed his interest to this topic because of his child. Similar to the father-child memoir mentioned at the beginning of our article, one day Dr. Bocquet was skipping stones on the seashore with his six-year-old son, when his son asked: “Dad, how is it possible? How does the stone skip on water?” Simple questions do not have simple answers. Bocquet reflected on this question, and his scientific interest grew on this topic. Thus, the first thing the professor did was to design small catapults that skipped stones on the water. Later, using these catapults, he bounced the rocks and continued his scientific work.

At the end of his research, he found many factors affecting whether a stone sinks or skips. Commonly known factors are the weight of the stone, its surface, and water stillness. Although everyone is on the same page for the importance of the surface structure of stone, its form is a controversial issue among scientists. Some claim the conservative-known opinion that stone surface needs to be as smooth as possible to get better results. Others, specifically Bocquet, aver its surface should be like that of a golf ball with lots of small pits to reduce the water drag of stone. The water also must be as calm as possible. Inland lakes are better choices when you think of rivers, streams, or seas. Probably almost everyone is disappointed when watching their perfect pitch hit a slight wave and bounce straight up in the air before plunging vertically down into the water.

Professor Bocquet revealed that the throw angle and initial linear velocity of the stone are also important factors. It was he who first invented the term “Magic Angle.” To have an optimum stone bounce in the experiments, the first contact of the stone with water should be at an angle of 20 degrees. Professor Bocquet and his fellow researcher Professor Christophe Clanet from the Institute of Research on Non-Equilibrium Phenomenon in Marseille have sought to deconstruct the hard physics underlying a simple game. Using a specially designed catapult, the scientists shot aluminum disks across a pool of water and then filmed the splashes with a high-speed video camera. After analyzing the results, they discovered the optimum speed and angle a stone-thrower would have to achieve the perfect skip. According to the professors, it’s all down to the conservation of momentum principle, which dictates that as the stone enters the water and pushes some of it downwards, it is, in turn, forced upwards. However, the stone needs to be moving at a certain speed – at least one kilometer an hour – otherwise, the stone “surfs” on the water for a short distance and then sinks. Spinning the stone is also important, because the “gyroscopic effect” generated with spinning ensures stability. So, the stone must rotate with a certain minimum rotational velocity – at least once during its “collision time.”

The magic angle combines with high spin rates results in an optimal toss. In 2011, I. J. Hewitt, N. J. Balmforth, and J. N. McElwaine also pointed out that, in theory, a skipping stone will bounce indefinitely across the water if it can maintain the horizontal speed. Skipping stones spend 100 times longer in the air than they do on the water. Since air is 784 times less dense than water, its impact on the rock’s flight is relatively minimal. We could say stone skipping is a miracle of fluid dynamics. But there’s more to it than math and science.

There are even Olympics around the world for skipping stones. There is even an official record for bouncing water, according to the Guinness world record book. On September 6, 2013, the record was broken by Kurt “Mountain man” Steiner (b. 1965) in Red Bridge in Allegheny National Forest, Pennsylvania, US. The number of bounces of the stone in the water was 88. His world-record skip was calculated at 43 miles per hour (69.2 km per hour). In other words, for the stone to bounce 88 times in the water, it must be thrown at a speed of 69.2 km per hour. Likewise, if you can speed the stone 100 miles per hour, you can bounce it 300-350 times on the water.

The Guinness World Record for the furthest distance skipped using natural stone stands at 121.8 m for men, established by Dougie Isaacs (Scotland), and 52.5 m for women, thrown by Nina Luginbuhl (Switzerland). These records were made on May 28, 2018, at Abernant Lake, Llanwrtyd Wells, Powys, Wales.

Up till now, we understand a functional structure for the stone to ricochet on water. The stone is unaware of this mathematical equation, nor is the water on which it bounces. But what is in this to benefit? Where will we use this knowledge in life? These are some of the questions that students often ask in math class. Let’s note that other than participating in the stone bounce Olympics or having a good time with your kids, this knowledge on skipping stones have real-world applications including for military purposes. For example, England’s Air Force used water bouncing bombs in dam raids during World War II. Operation Chastise became one of the most famous bombing campaigns of World War II. The intelligence behind the operation was simple: the Mohne, Eder, and Sorpe dams were vital to Germany’s war effort; their destruction would disrupt the water and electricity supplies to the Ruhr, Germany’s industrial heartland.

Since the Germans knew the importance of those dams, they were protecting them very well. The anti-aircraft guns that defended the dams did not bring the planes closer to them. Since the hit rate of bombs dropped from afar was low, those who hit the dams did only minor damage, and the Germans immediately fixed it. Besides, steel nets were stretched in front of the dam against torpedoes coming under the water. In this case, a bomb had to be thrown in advance, hit the dam, bounce off the water, and then explode 30 feet deep underwater. Thus, even if only one bomb hit, the dams would be destroyed by the pressure of the water.

The bombs themselves were invented specifically for the dam mission by the aircraft engineer Dr. Barnes Wallis, the designer of the Wellington bomber. They were barrel-shaped and used the principle of a “ducks and drakes” stone bouncing on the water to bypass the defenses around the dams. The bomb is dropped close to the surface of the lake. It bounces several times instead of sinking. The “bomb run” is calculated so that the bomb will reach close to the target at its final bounce, where it sinks. A hydrostatic pistol causes it to explode at the proper depth, generating destructive shockwaves. On May 16, 1943, Operation Chastise took place; 147 men flew 19 aircraft carrying “bouncing bombs.” The Mohne and Eder dams were breached, and the successful operation damaged the Sorpe dam.

Historically, it is claimed that Sultan Mehmet the Conqueror bounced the cannons of that time in the water and had the Byzantine walls beaten during his conquest of Istanbul in 1453.

Does anybody think of the math behind the stones they throw across a lake and see them bounce off water? Does the stone know how it can move on a liquid? Stones do not have brains or hearts to grasp this knowledge, but we humans do. As we are thrilled when we witness the mysteries and wonders of nature, we understand with the help of science that even a stone bounces with a functional structure as a small part of a miraculous program.e.

Verily, all things have We created in proportion and measure.

What do you need to skip stones like a pro?

1. Choosing the location

Lakes and ponds are perfect choices for any location where has a calm body of water. Though, clear and still day is a crucial factor. 

2. Picking the flawless stone

A thin and flat stone about the size of your palm, preferably oval or triangular: should be reasonably light but still heavy enough for a solid and accurate throw.

3. Adjusting your posture

Squat down slightly and adjust your throwing hand so that the angle between the stone and the water is about 20 degrees.

4. Bend and snap

Bend your wrist back to generate power (like winding up for a softball pitch) and then snap it forward, releasing your stone with your palm facing upward. As your wrist flicks on, the stone should slide right through your thumb and middle finger as the index finger sends it spinning. The bend and snap should be a swift, fluid motion. The speed of the snap determines the speed of the stone. 

5. Follow through

Don’t halt your throwing motion when the stone leaves your hand. Extend your arm across your chest to the shoulder of your other arm. Much like a proper tennis swing, you need to follow through to maintain a reasonable force and power.