Author: IMC

There is more to where maths came from than the ancient Greeks. From calculus to the theorem we credit to Pythagoras, so much of our knowledge comes from other places, including ancient China, India and the Arabian peninsula, says Kate Kitagawa.

The history of mathematics has an image problem. It is often presented as a meeting of minds among ancient Greeks who became masters of logic. Pythagoras, Euclid and their pals honed the tools for proving theorems and that led them to the biggest results of ancient times. Eventually, other European greats like Leonhard Euler and Isaac Newton came along and made maths modern, which is how we got to where we are today.

But, of course, this telling is greatly distorted. The history of maths is far richer, more chaotic and more diverse than it is given credit for. So much of what is now incorporated into our global knowledge comes from other places, including ancient China, India and the Arabian peninsula.

Take “Pythagoras’s” theorem. This is the one that says that in right-angled triangles, the square of the longest side is the sum of the square of the other two sides. The ancient Greeks certainly knew about this theorem, but so too did mathematicians in ancient Babylonia, Egypt, India and China.

In fact, in the 3rd century AD, Chinese mathematician Liu Hui added a proof of the theorem to the already old and influential book The Nine Chapters on the Mathematical Art. His version includes the earliest written statement of the theorem that we know of. So perhaps we should really call it Liu’s theorem or the gougu theorem as it was known in China.

The history of maths is filled with tales like this. Ideas have sprung up in multiple places at multiple times, leaving room for interpretation as to who should get the credit. As if credit is something that can’t be split.

As a researcher on the history of maths, I had come across examples of distorted views, but it was only when working on a new book, The Secret Lives of Numbers, that I found out just how pervasive they are. Along with my co-author, New Scientist‘s Timothy Revell, we found that the further we dug, the more of the true history of maths there was to uncover.

Another example is the origins of calculus. This is often presented as a battle between Newton and Gottfried Wilhelm Leibniz, two great 17th-century European mathematicians. They both independently developed extensive theories of calculus, but missing from the story is how an incredible school in Kerala, India, led by the mathematician Mādhava, hit upon some of the same ideas 300 years before.

The idea that the European way of doing things is superior didn’t originate in maths – it came from centuries of Western imperialism – but it has infiltrated it. Maths outside ancient Greece has often been put to one side as “ethnomathematics”, as if it were a side story to the real history.

In some cases, history has also distorted legacies. Sophie Kowalevski, who was born in Moscow in 1850, is now a relatively well-known figure. She was a fantastic mathematician, known for tackling a problem she dubbed a “mathematical mermaid” for its allure. The challenge was to describe mathematically how a spinning top moves, and she made breakthroughs where others had faltered.

During her life, she was constantly discouraged from pursuing maths and often had to work for free, collecting tuition money from her students in order to survive. After her death, biographers then tainted her life, painting her as a femme fatale who relied on her looks, implying she effectively passed off others’ work as her own. There is next to no evidence this is true.

Thankfully, historians of mathematics are re-examining and correcting the biases and stereotypes that have plagued the field. This is an ongoing process, but by embracing its diverse and chaotic roots, the next chapters for maths could be the best yet.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Kate Kitagawa *

With mass shootings happening randomly every year in the United States, it may seem that there is no way to predict where the next horrific event is most likely to occur. In a new study published by the journal Risk Analysis, scientists at Iowa State University calculate the annual probability of a mass shooting in every state and at public places such as shopping malls and schools.

Their new method for quantifying the risk of a mass shooting in specific places could help security officials make informed decisions when planning for emergency events.

For their analysis, Iowa State associate professor Cameron MacKenzie and his doctoral student Xue Lei applied statistical methods and computer simulations to a database of mass shootings recorded from 1966 to 2020 by the Violence Project. The Violence Project defines a mass shooting as an incident with four or more victims killed by a firearm in a public place.

According to the Violence Project, the U.S. has experienced 173 public mass shootings from 1966 to 2020—with at least one mass shooting every year since 1966.

After they generated a probability distribution of annual mass shootings in the U.S., the scientists used two different models to simulate the annual number of mass shootings in each state. The results were used to calculate the expected number of mass shootings and the probability that at least one mass shooting would occur in each state in one year.

The Violence Project also provides the percentage of mass shootings in different types of locations. MacKenzie and Lei used that data to calculate the probability of a mass shooting in nine different types of public locations (including a restaurant, school, workplace, or house of worship) in the states of California and Iowa and also at the two largest high schools in each of those states.

Their findings include the following:

• The states with the greatest risk of a mass shooting are the most populous states: California, Texas, Florida, New York, and Pennsylvania. Together they account for almost 50% of all mass shootings.
• Some states, such as Iowa and Delaware, have never experienced a mass shooting.
• The annual risk of a mass shooting at the largest California high school is about 10 times greater than the risk at the largest Iowa high school.
• The number of mass shootings in the U.S. has increased by about one shooting every 10 years since the 1970s.

Importantly, MacKenzie points out that the probability of a mass shooting at a specific location depends on the definition of a mass shooting. In contrast to the Violence Project, the Gun Violence Archive defines a mass shooting as four or more individuals shot, injured or killed, in any location, not necessarily a public location. As a result, The Gun Violence Archive has collected data on shootings that occur in both public and private locations as well as targeted shootings (i.e., a gang shooting).

When the researchers applied data from The Gun Violence Archive to their models, the predicted number of annual mass shootings was nearly 100 times greater than the forecast based on The Violence Project’s data. The models predicted 639 mass shootings in 2022 with a 95% chance that the U.S. would experience between 567 and 722 mass shootings in that same year.

MacKenzie points out that “most media appear to use this broader definition of mass shootings.” Because of this, he urges that journalists explain how they are defining a mass shooting when reporting the statistical data.

With regard to the danger posed to children at school, MacKenzie explains, that “our results show that it is very, very unlikely that a specific student will attend a K-12 school and experience a mass shooting. But to parents of a child at a school that has experienced a mass shooting, explaining that the school was extremely unlucky provides no comfort.”

While it is important to take precautions, he adds that “we should not live in fear that our children will experience such a horrific event. Mass shootings are very low probability but very high consequence events.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Society for Risk Analysis

Researchers are using mathematical models to better understand the effects of disruptions like daylight savings time, working night shifts, jet lag or even late-night phone scrolling on the body’s circadian rhythms.

The University of Waterloo and the University of Oxford researchers have developed a new model to help scientists better understand the resilience of the brain’s master clock: the cluster of neurons in the brain that coordinates the body’s other internal rhythms. They also hope to suggest ways to help improve this resilience in individuals with weak or impaired circadian rhythms. The study, “Can the Clocks Tick Together Despite the Noise? Stochastic Simulations and Analysis,” appears in the SIAM Journal on Applied Dynamical Systems.

Sustained disruptions to circadian rhythm have been linked to diabetes, memory loss, and many other disorders.

“Current society is experiencing a rapid increase in demand for work outside of traditional daylight hours,” said Stéphanie Abo, a Ph.D. student in applied mathematics and the study’s lead author. “This greatly disrupts how we are exposed to light, as well as other habits such as eating and sleeping patterns.”

Humans’ circadian rhythms, or internal clocks, are the roughly 24-hour cycles many body systems follow, usually alternating between wakefulness and rest. Scientists are still working to understand the cluster of neurons known as suprachiasmatic nucleus (SCN) or master clock.

Using mathematical modeling techniques and differential equations, the team of applied mathematics researchers modeled the SCN as a macroscopic, or big-picture, system comprised of a seemingly infinite number of neurons. They were especially interested in understanding the system’s couplings—the connections between neurons in the SCN that allow it to achieve a shared rhythm.

Frequent and sustained disturbances to the body’s circadian rhythms eliminated the shared rhythm, implying a weakening of the signals transmitted between SCN neurons.

Abo said they were surprised to find that “a small enough disruption can actually make the connections between neurons stronger.”

“Mathematical models allow you to manipulate body systems with specificity that cannot be easily or ethically achieved in the body or a petri dish,” Abo said. “This allows us to do research and develop good hypotheses at a lower cost.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of Waterloo

Graphic novels can help make math and physics more accessible for students, parents or teachers in training. Metamorworks/iStock via Getty Images

Post-pandemic, some educators are trying to reengage students with technology – like videos, computer gaming or artificial intelligence, just to name a few. But integrating these approaches in the classroom can be an uphill battle. Teachers using these tools often struggle to retain students’ attention, competing with the latest social media phenomenon, and can feel limited by using short video clips to get concepts across.

Graphic novels – offering visual information married with text – provide a means to engage students without losing all of the rigor of textbooks. As two educators in math and physics, we have found graphic novels to be effective at teaching students of all ability levels. We’ve used graphic novels in our own classes, and we’ve also inspired and encouraged other teachers to use them. And we’re not alone: Other teachers are rejuvenating this analog medium with a high level of success.

In addition to covering a wide range of topics and audiences, graphic novels can explain tough topics without alienating student averse to STEM – science, technology, engineering and math. Even for students who already like math and physics, graphic novels provide a way to dive into topics beyond what is possible in a time-constrained class. In our book “Using Graphic Novels in the STEM Classroom,” we discuss the many reasons why graphic novels have a unique place in math and physics education. Here are three of those reasons:

Explaining complex concepts with rigor and fun

Increasingly, schools are moving away from textbooks, even though studies show that students learn better using print rather than digital formats. Graphic novels offer the best of both worlds: a hybrid between modern and traditional media.

This integration of text with images and diagrams is especially useful in STEM disciplines that require quantitative reading and data analysis skills, like math and physics.

For example, our collaborator Jason Ho, an assistant professor at Dordt University, uses “Max the Demon Vs Entropy of Doom” to teach his physics students about entropy. This topic can be particularly difficult for students because it’s one of the first times when they can’t physically touch something in physics. Instead, students have to rely on math and diagrams to fill in their knowledge.

Rather than stressing over equations, Ho’s students focus on understanding the subject more conceptually. This approach helps build their intuition before diving into the algebra. They get a feeling for the fundamentals before they have to worry about equations.

After having taken Ho’s class, more than 85% of his students agreed that they would recommend using graphic novels in STEM classes, and 90% found this particular use of “Max the Demon” helpful for their learning. When strategically used, graphic novels can create a dynamic, engaging teaching environment even with nuanced, quantitative topics.

Combating quantitative anxiety

Students learning math and physics today are surrounded by math anxiety and trauma, which often lead to their own negative associations with math. A student’s perception of math can be influenced by the attitudes of the role models around them – whether it’s a parent who is “not a math person” or a teacher with a high level of math anxiety.

Graphic novels can help make math more accessible not only for students themselves, but also for parents or students learning to be teachers.

In a geometry course one of us (Sarah) teaches, secondary education students don’t memorize formulas and fill out problem sheets. Instead, students read “Who Killed Professor X?”, a murder mystery in which all of the suspects are famous mathematicians. The suspects’ alibis are justified through problems from geometry, algebra and pre-calculus.

While trying to understand the hidden geometry of suspect relationships, students often forget that they are doing math – focusing instead on poring over secret hints and notes needed to solve the mystery.

Although this is just one experience for these students, it can help change the narrative for students experiencing mathematical anxiety. It boosts their confidence and shows them how math can be fun – a lesson they can then impart to the next generation of students.

Helping students learn and readers dream big

In addition to being viewed favourably by students, graphic novels can enhance student learning by improving written communication skills, reading comprehension and critical literacy skills. And even outside the classroom, graphic novels support long-term memory for those who have diagnoses like dyslexia.

Pause and think about your own experience – how do you learn about something new in science?

If you’re handed a textbook, it’s extremely unlikely that you’d read it cover to cover. And although the internet offers an enormous amount of math and physics content, it can be overwhelming to sift through hours and hours of videos to find the perfect one to get the “aha!” moment in learning.

Graphic novels provide a starting point for such a broad range of niche topics that it’s impossible for anyone to be experts in them all. Want to learn about programming? Try the “Secret Coders” series. Want to understand more about quantum physics? Dive into “Suspended in Language: Niels Bohr’s life, discoveries, and the century he shaped.” Searching for more female role models in science? “Astronauts: Women on the Final Frontier” could be just what you’re looking for.

With all that they offer, graphic novels provide a compelling list of topics and narratives that can capture the attention of students today. We believe that the right set of graphic novels can inspire the next generation of scientists as much as any single individual can.

For more such insights, log into our website https://international-maths-challenge.com

From avoiding the number seven to picking numbers over 31, mathematician Peter Rowlett has a few psychological strategies for improving your chances when playing the lottery.

Would you think I was daft if I bought a lottery ticket for the numbers 1, 2, 3, 4, 5 and 6? There is no way those are going to be drawn, right? That feeling should – and, mathematically, does – actually apply to any set of six numbers you could pick.

Lotteries are ancient. Emperor Augustus, for example, organised one to fund repairs to Rome. Early lotteries involved selling tickets and drawing lots, but the idea of people guessing which numbers would be drawn from a machine comes from Renaissance Genoa. A common format is a game that draws six balls from 49, studied by mathematician Leonhard Euler in the 18th century.

The probabilities Euler investigated are found by counting the number of possible draws. There are 49 balls that could be drawn first. For each of these, there are 48 balls that can be drawn next, so there are 49×48 ways to draw two balls. This continues, so there are 49×48×47×46×45×44 ways to draw six balls. But this number counts all the different arrangements of any six balls as a unique solution.

How many ways can we rearrange six balls? Well, we have six choices for which to put first, then for each of these, five choices for which to put second, and so on. So the number of ways of arranging six balls is 6×5×4×3×2×1, a number called 6! (six factorial). We divide 49×48×47×46×45×44 by 6! to get 13,983,816, so the odds of a win are near 1 in 14 million.

Since all combinations of numbers are equally likely, how can you maximise your winnings? Here is where maths meets psychology: you win more if fewer people share the prize, so choose numbers others don’t. Because people often use dates, numbers over 31 are chosen less often, as well as “unlucky” numbers like 13. A lot of people think of 7 as their favourite number, so perhaps avoid it. People tend to avoid patterns so are less likely to pick consecutive or regularly spaced numbers as they feel less random.

In July, David Cushing and David Stewart at the University of Manchester, UK, published a list of 27 lottery tickets that guarantee a win in the UK National Lottery, which uses 59 balls and offers a prize for matching two or more. But a win doesn’t always mean a profit – for almost 99 per cent of possible draws, their tickets match at most three balls, earning prizes that may not exceed the cost of the tickets!

So, is a lottery worth playing? Since less than half the proceeds are given out in prizes, you would probably be better off saving your weekly ticket money. But a lecturer of mine made an interesting cost-benefit argument. He was paid enough that he could lose the cost of a ticket each week without really noticing. But if he won the jackpot, his life would be changed. So, given that lottery profit is often used to support charitable causes, it might just be worth splurging.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett*

Life’s random rhythms surround us–from the hypnotic, synchronized blinking of fireflies…to the back-and-forth motion of a child’s swing… to slight variations in the otherwise steady lub-dub of the human heart.

But truly understanding those rhythms—called stochastic, or random, oscillations—has eluded scientists. While researchers and clinicians have some success in parsing brain waves and heartbeats, they’ve been unable to compare or catalog an untold number of variations and sources.

Gaining such insight into the underlying source of oscillations “could lead to advances in neural science, cardiac science and any number of different fields,” said Peter Thomas, a professor of applied mathematics at Case Western Reserve University.

Thomas is part of an international team that says it has developed a novel, universal framework for comparing and contrasting oscillations—regardless of their different underlying mechanisms—which could become a critical step toward someday fully understanding them.

Their findings were recently published in Proceedings of the National Academy of Sciences.

“We turned the problem of comparing oscillators into a linear algebra problem,” Thomas said. “What we have done is vastly more precise than what was available before. It’s a major conceptual advance.”

The researchers say others can now compare, better understand—and even manipulate—oscillators previously considered to have completely different properties.

“If your heart cells aren’t synchronized, you die of atrial fibrillation,” Thomas said. “But if your brain cells synchronize too much, you have Parkinson’s disease, or epilepsy, depending on which part of the brain the synchronization occurs in. By using our new framework, that heart or brain scientist may be able to better understand what the oscillations could mean and how the heart or brain is working or changing over time.”

Swaying skyscrapers and brain waves

Thomas said the researchers—who included collaborators from universities in France, Germany and Spain—found a new way to use complex numbers to describe the timing of oscillators and how “noisy,” or imprecisely timed, they are.

Most oscillations are irregular to some extent, Thomas said. For example, a heart rhythm is not 100% regular. A natural variation of 5-10% in the heartbeat is considered healthy.

Thomas said the problem with comparing oscillators can be illustrated by considering two markedly different examples: brain rhythms and swaying skyscrapers.

“In San Francisco, modern skyscrapers sway in the wind, buffeted by randomly shifting air currents—they’re pushed slightly out of their vertical posture, but the mechanical properties of the structure pull them back,” he said. “This combination of flexibility and resilience helps high-rise buildings survive shaking during earthquakes. You wouldn’t think this process could be compared with brain waves, but our new formalism lets you compare them.”

How their findings might help either discipline—mechanical engineering and neuroscience—may be unknown right now, Thomas said, comparing the conceptual advance to when Galileo discovered Jupiter’s orbiting moons.

“What Galileo realized was a new point of view, and while our discovery is not as far-reaching as Galileo’s, it is similarly a change in perspective,” he said. “What we report in our paper is an entirely new point of view on stochastic oscillators.”

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Credit of the article given to Case Western Reserve University

Everyone knows that arithmetic is true: 2 + 2 = 4.

But surprisingly, we don’t know why it’s true.

By stepping outside the box of our usual way of thinking about numbers, my colleagues and I have recently shown that arithmetic has biological roots and is a natural consequence of how perception of the world around us is organized.

Our results explain why arithmetic is true and suggest that mathematics is a realization in symbols of the fundamental nature and creativity of the mind.

Thus, the miraculous correspondence between mathematics and physical reality that has been a source of wonder from the ancient Greeks to the present—as explored in astrophysicist Mario Livio’s book “Is God a Mathematician?”—suggests the mind and world are part of a common unity.

Why is arithmetic universally true?

Humans have been making symbols for numbers for more than 5,500 years. More than 100 distinct notation systems are known to have been used by different civilizations, including Babylonian, Egyptian, Etruscan, Mayan and Khmer.

The remarkable fact is that despite the great diversity of symbols and cultures, all are based on addition and multiplication. For example, in our familiar Hindu-Arabic numerals: 1,434 = (1 x 1000) + (4 x 100) + (3 x 10) + (4 x 1).

Why have humans invented the same arithmetic, over and over again? Could arithmetic be a universal truth waiting to be discovered?

To unravel the mystery, we need to ask why addition and multiplication are its fundamental operations. We recently posed this question and found that no satisfactory answer—one that met standards of scientific rigor—was available from philosophy, mathematics or the cognitive sciences.

The fact that we don’t know why arithmetic is true is a critical gap in our knowledge. Arithmetic is the foundation for higher mathematics, which is indispensable for science.

Consider a thought experiment. Physicists in the future have achieved the goal of a “theory of everything” or “God equation.” Even if such a theory could correctly predict all physical phenomena in the universe, it would not be able to explain where arithmetic itself comes from or why it is universally true.

Answering these questions is necessary for us to fully understand the role of mathematics in science.

Bees provide a clue

We proposed a new approach based on the assumption that arithmetic has a biological origin.

Many non-human species, including insects, show an ability for spatial navigation which seems to require the equivalent of algebraic computation. For example, bees can take a meandering journey to find nectar but then return by the most direct route, as if they can calculate the direction and distance home.

How their miniature brain (about 960,000 neurons) achieves this is unknown. These calculations might be the non-symbolic precursors of addition and multiplication, honed by natural selection as the optimal solution for navigation.

Arithmetic may be based on biology and special in some way because of evolution’s fine-tuning.

Stepping outside the box

To probe more deeply into arithmetic, we need to go beyond our habitual, concrete understanding and think in more general and abstract terms. Arithmetic consists of a set of elements and operations that combine two elements to give another element.

In the universe of possibilities, why are the elements represented as numbers and the operations as addition and multiplication? This is a meta-mathematical question—a question about mathematics itself that can be addressed using mathematical methods.

In our research, we proved that four assumptions—monotonicity, convexity, continuity and isomorphism—were sufficient to uniquely identify arithmetic (addition and multiplication over the real numbers) from the universe of possibilities.

• Monotonicityis the intuition of “order preserving” and helps us keep track of our place in the world, so that when we approach an object it looms larger but smaller when we move away.
• Convexityis grounded in intuitions of “betweenness.” For example, the four corners of a football pitch define the playing field even without boundary lines connecting them.
• Continuitydescribes the smoothness with which objects seem to move in space and time.
• Isomorphismis the idea of sameness or analogy. It’s what allows us to recognize that a cat is more similar to a dog than to a rock.

Thus, arithmetic is special because it is a consequence of these purely qualitative conditions. We argue that these conditions are principles of perceptual organization that shape how we and other animals experience the world—a kind of “deep structure” in perception with roots in evolutionary history.

In our proof, they act as constraints to eliminate all possibilities except arithmetic—a bit like how a sculptor’s work reveals a statue hidden in a block of stone.

What is mathematics?

Taken together, these four principles structure our perception of the world so that our experience is ordered and cognitively manageable. They are like colored spectacles that shape and constrain our experience in particular ways.

When we peer through these spectacles at the abstract universe of possibilities, we “see” numbers and arithmetic.

Thus, our results show that arithmetic is biologically based and a natural consequence of how our perception is structured.

Although this structure is shared with other animals, only humans have invented mathematics. It is humanity’s most intimate creation, a realization in symbols of the fundamental nature and creativity of the mind.

In this sense, mathematics is both invented (uniquely human) and discovered (biologically based). The seemingly miraculous success of mathematics in the physical sciences hints that our mind and the world are not separate, but part of a common unity.

The arc of mathematics and science points toward nondualism, a philosophical concept that describes how the mind and the universe as a whole are connected, and that any sense of separation is an illusion. This is consistent with many spiritual traditions (Taoism, Buddhism) and Indigenous knowledge systems such as mātauranga Māori.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Randolph Grace, The Conversation

A team of physicists and mathematicians at the Institute for Basic Science’s Center for Soft and Living Matter, in South Korea, working with a colleague from the University of Geneva, has developed an algorithm that can be used to find the shape of an object to cause it to roll down a ramp following a desired path.

In their paper published in the journal Nature, the group describes how they developed their algorithm, and possible uses for it. Elisabetta Matsumoto and Henry Segerman with the Georgia Institute of Technology and Oklahoma State University, respectively, have published a News & Views piece in the same journal issue outlining the work done by the team on this new effort.

In this new effort, the research team started with an interesting puzzle—one that begins by envisioning a sphere rolling down a ramp. If the sphere is imagined to be made of clay, it can be manipulated (deformed) as it rolls to make it conform to a given path.

If the sphere is then rolled down the ramp again, it will follow the previous path due to the new deformities in its shape. The researchers noted that the paths that could be taken by the sphere could be nearly limitless due to the nearly limitless possible deformations.

That realization led them to wonder if the deformations that form in such a sphere could be corelated mathematically with its path. And if so, if such math could be used to create an algorithm that could be used to 3D print a sphere with deformations that would force it to follow a predetermined path.

It turned out the answer to both questions was yes. The team used math and physics principles to create formulas that described deformations to a given object that would result in the object following a desired path down an inclined plane. They then created a computer program that could be used to create such an object in the real world, using 3D printing.

The team named the objects trajectoids. Each had a solid metal ball-bearing inside to give it weight. They also found that they could create trajectoids that traveled over a given path twice, and named them “two-period trajectoids.”

The research team suggests their formulas and algorithm could be used in robotics applications and also in physics research associated with the angular moment of an electron—or in quantum research centered around the study of evolution of a quantum bit.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Bob Yirka , Phys.org

Terry Loring, distinguished professor of mathematics and statistics, published and co-authored a new research piece involving his research on K-theory with the major advances in applications to critical problems in physics.

The study titled, “Revealing topology in metals using experimental protocols inspired by K-theory,” was published in Nature Communications. Loring used mathematical properties of K-theory to help advance the understanding of topological materials in the physics world.

The main focus of the study was to discover how electricity, sound, or light can be trapped in a portion of a material. “This experiment was done in what is called a meta-material, built from individual sound resonators coupled in a fashion that mimics how atoms can come together to form a crystal. Three-dimensional printing allows us to make customized resonators that we join in a precise way to make the physics match the mathematics,” explained Loring. The study was part of a larger project that covered many areas of physics.

According to Loring, there are different forms of K-theory that arise in many different mathematical fields, however, the form of K-theory that he used in this study was focused on being best suited for studying matrix models of physical systems.

Loring explains that matrices are simply square tables of numbers, with a peculiar rule for how two matrices are multiplied. This rule has an asymmetry in it that leads to having AB and BA sometimes being very different, meaning that the commutative law for multiplication is violated.

“Physicists like Heisenberg realized that matrices are terrific at modeling uncertainty in molecular- and atomic-scale physics. K-theory can tell us when certain matrices can be connected except by a path that goes through what we call a singular matrix. This guaranteed singularity turns out to have an important meaning when the matrices come from models of physical systems,” Loring said.

The researchers were mainly looking at topological materials which include topological insulators. A topological insulator can have an index associated to it, which is a number computed using K-theory. If a device is built from two topological insulators that each have a different index, there is guaranteed to be a conducting region where the two materials come together.

“This conducting region exactly corresponds to where a certain matrix goes singular. To demonstrate this fact we use results about determinants one learns in linear algebratogether with the intermediate value theorem that people learn in their first calculus class,” said Loring.

This research is attempting to advance the theory of topological metals. Topological metals mix up conducting and insulating properties in very confusing ways. Loring and team built an acoustic crystal that had a specific pattern, they then deliberately broke the pattern in the middle thus inserting a defect in the system.

“During the experiment, and computer simulations, we were able to show how sound can get trapped at the defect. The hope is that it teaches us how to better trap light in small-scale photonic devices, and more generally start to manipulate light in a similar way to how electronic circuits manipulate electricity. There are advantages to moving information with light, as this can sometimes eliminate/reduce the energy wasted by the heat associated with electronics,” Loring stated.

Another part of the experiment which was more delicate included modifying the acoustic resonators by a formula from K-theory. The modified system removed the metallic properties in many parts of the crystal, isolating the binding metallic nature of the defect.

“Of course our acoustics system is not a metal, but shares mathematical properties with metals that harbor topology in their electronic structure. The hope is we will be able to devise experimental probes of photonic and electronic systems that bring the K-theory off the blackboard and into the lab,” explained Loring.

Mathematics was central to the design of this experiment. The project began with a discussion of formulas in K-theory that might lead to a matrix that can describe the energy in an acoustic system.

“We started with the analysis we would use to explain the system and then built a system that could be analysed this way. This backwards flow is somewhat common in the field of ‘topological physics’ where clean formulas in math suggest the search for physical systems that match that formula,” Loring stated.

In finding, Loring and his team discovered that new mathematics can classify very local patches of material as insulating or conducting. Loring points out that initially it was not clear if this classification had any meaning that a physicist would care about.

“This experiment showed that we can manufacture materials where this local classification is physically meaningful. While this material has no practical application, it is expected that materials and devices will be discovered or manufactured that have these local variations and that these local variations will give us even more control over light and electricity than we now enjoy.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Dani Rae Wascher, University of New Mexico

Even though “x” is one of the least-used letters in the English alphabet, it appears throughout American culture—from Stan Lee’s X-Men superheroes to “The X-Files” TV series. The letter x often symbolizes something unknown, with an air of mystery that can be appealing—just look at Elon Musk with SpaceX, Tesla’s Model X, and now X as a new name for Twitter.

You might be most familiar with x from math class. Many algebra problems use x as a variable, to stand in for an unknown quantity. But why is x the letter chosen for this role? When and where did this convention begin?

There are a few different explanations that math enthusiasts have put forward—some citing translation, others pointing to a more typographic origin. Each theory has some merit, but historians of mathematics, like me, know that it’s difficult to say for sure how x got its role in modern algebra.

Ancient unknowns

Algebra today is a branch of math in which abstract symbols are manipulated, using arithmetic, to solve different kinds of equations. But many ancient societies had well-developed mathematical systems and knowledge with no symbolic notation.

All ancient algebra was rhetorical. Mathematical problems and solutions were completely written out in words as part of a little story, much like the word problems you might see in elementary school.

Ancient Egyptian mathematicians, who are perhaps best known for their geometric advances, were skilled in solving simple algebraic problems. In the Rhind papyrus, the scribe Ahmes uses the hieroglyphics referred to as “aha” to denote the unknown quantity in his algebraic problems. For example, problem 24 asks for the value of aha if aha plus one-seventh of aha equals 19. “Aha” means something like “mass” or “heap.”

The ancient Babylonians of Mesopotamia used many different words for unknowns in their algebraic system—typically words meaning length, width, area or volume, even if the problem itself was not geometric in nature. One ancient problem involved two unknowns termed the “first silver thing” and the “second silver thing.”

Mathematical know-how developed somewhat independently in many lands and in many languages. Limitations in communication prevented any immediate standardization of notation. However, over time some abbreviations crept in.

In a transitional syncopated phase, authors used some symbolic notation, but algebraic ideas were still presented mainly rhetorically. Diophantus of Alexandria used a syncopated algebra in his great work Arithmetica. He called the unknown “arithmos” and used an archaic Greek letter similar to s for the unknown.

Indian mathematicians made additional algebraic discoveries and developed what are essentially the modern symbols for each of the decimal digits. One especially influential Indian mathematician was Brahmagupta, whose algebraic techniques could handle any quadratic equation. Brahmagupta’s name for the unknown variable was yãvattâvat. When additional variables were required, he instead used the initial syllable of color names, like kâ from kâlaka (black), ya from yavat tava (yellow), ni from nilaka (blue), and so on.

Islamic scholars translated and preserved a great deal of both Greek and Indian scholarship that has contributed immensely to the world’s mathematical, scientific and technical knowledge. The most famous Islamic mathematician was al-Khowarizmi, whose foundational book Al-jabr wa’l muqabalah is at the root of the modern word “algebra.”

So what about x?

One theory of the genesis of x as the unknown in modern algebra points to these Islamic roots. The theory contends that the Arabic word used for the quantity being sought was al-shayun, meaning “something,” which was shortened to the symbol for its first “sh” sound. When Spanish scholars translated the Arabic mathematical treatises, they lacked a letter for the “sh” sound and instead chose the “k” sound. They represented this sound by the Greek letter χ, which later became the Latin x.

It’s not unusual for a mathematical expression to come about through convoluted translations—the trigonometric word “sine” started as a Hindu word for a half-chord but, through a series of translations, ended up coming from the Latin word “sinus,” meaning bay. However, there is some evidence that casts doubt upon the theory that using x as an unknown is an artifact of Spanish translation.

The Spanish alphabet includes the letter x, and early Catalonian involved several pronunciations of it depending on context, including a pronunciation akin to the modern sh. Although the sound changed pronunciation over time, there are still vestiges of the sh sound for x in Portuguese, as well as in Mexican Spanish and its use in native place names. By this reasoning, Spanish translators conceivably could have used x without needing to resort first to the Greek χ and then to the Latin x.

Moreover, although the letter x may have been used in mathematics during the Middle Ages sporadically, there is no consistent use of it dating back that far. Western mathematical texts over the next several centuries still used a variety of words, abbreviations and letters to represent the unknown.

For instance, a typical problem in the algebra book “Sumario Compendioso of Juan Diez,” published in Mexico in 1556, uses the word “cosa”—meaning “stuff” or “thing”—to stand in for the unknown.

I think that the most plausible explanation is to credit the influential French scholar René Descartes for the modern use of x. In an appendix to his major work “Discourse” in the 17th century, Descartes introduced a version of analytic geometry—in which algebra is used to solve geometric problems. For unspecified constants he chose the first few letters of the alphabet, and for variables he chose the last letters in reverse order.

Although scholars may never know for sure, some theorize that Descartes may have chosen the letter x to appear often since the printer had a large cache of x’s because of its scarcity in the French language. Whatever his reasons for choosing x, Descartes greatly influenced the development of mathematics, and his mathematical writings were widely circulated.

Xtending beyond algebra

Even if the origins of x in algebra are uncertain, there are some instances in which historians do know why x is used. The X in Xmas as an abbreviation for Christmas definitely does come from the Greek letter χ. The Greek word for Christ is Christos, written χριστοσ and meaning “anointed.” The χ monogram was used as a shorthand for Christ in both Roman Catholic and Eastern Orthodox writings dating back as far as the 16th century.

There are also some contexts in which x was chosen specifically to indicate something unknown or extra, such as when the German physicist Wilhelm Roentgen accidentally discovered X-rays in 1895 while experimenting with cathode rays and glass.

But there are other cases in which scholars can only guess about the origins of x’s role, such as the phrase “X marks the spot.” And there are other contexts—such as Elon Musk’s affinity for the letter—that may just be a matter of personal taste.

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Credit of the article given to Peter Schumer, The Conversation