Latest news with #TheAmericanMathematicalMonthly
Yahoo
03-05-2025
- Science
- Yahoo
Mathematicians devise new way to solve devilishly difficult algebra equations
When you buy through links on our articles, Future and its syndication partners may earn a commission. Polynomial equations are a cornerstone of modern science, providing a mathematical basis for celestial mechanics, computer graphics, market growth predictions and much more. But although most high schoolers know how to solve simple polynomial equations, the solutions to higher-order polynomials have eluded even seasoned mathematicians. Now, University of New South Wales mathematician Norman Wildberger and independent computer scientist Dean Rubine have found the first general method for solving these devilishly difficult equations. They detailed their approach April 8 in the journal The American Mathematical Monthly. A polynomial is a type of algebraic equation that involves variables raised to a non-negative power — for example, x² + 5x + 6 = 0. It is among the oldest mathematical concepts, tracing its roots back to ancient Egypt and Babylon. Mathematicians have long known how to solve simple polynomials. However, higher-order polynomials, where x is raised to a power greater than four, have proved trickier. The approach most often used to solve two-, three- and four-degree polynomials relies on using the roots of exponential numbers, called radicals. The problem is that radicals often represent irrational numbers — decimals that keep going to infinity, like pi. Related: Mathematicians just solved a 125-year-old problem, uniting 3 theories in physics Although mathematicians can use radicals to find approximate solutions to individual higher-order polynomials, they have struggled to find a general formula that works for all of them. That's because irrational numbers can never fully resolve. "You would need an infinite amount of work and a hard drive larger than the universe," Wildberger said in a statement. In their new method, Wildberger and his colleagues avoided radicals and irrational numbers entirely. Instead, they employed polynomial extensions known as power series. These are hypothetically infinite strings of terms with the powers of x, commonly used to solve geometric problems. They belong to a sub branch of mathematics known as combinatorics. RELATED STORIES —Mathematicians solve vexing 'crowd problem' that explains why public spaces devolve into chaos —14-year-old known as 'the human calculator' breaks 6 math world records in 1 day —High school students who came up with 'impossible' proof of Pythagorean theorem discover 9 more solutions to the problem The mathematicians based their approach on the Catalan numbers, a sequence that can be used to describe the number of ways to break down a polygon into triangles. This sequence was first delineated by Mongolian mathematician Mingantu around 1730 and was independently discovered by Leonhard Euler in 1751. Wildberger and Rubine realized that they could look to higher analogues of the Catalan numbers to solve higher-order polynomial equations. They called this extension "the Geode." The Geode has numerous potential applications for future research, especially in computer science and graphics. "This is a dramatic revision of a basic chapter in algebra," Wildberger said.
Yahoo
01-05-2025
- Science
- Yahoo
Mathematician solves algebra's oldest problem
Most people's experiences with polynomial equations don't extend much further than high school algebra and the quadratic formula. Still, these numeric puzzles remain a foundational component of everything from calculating planetary orbits to computer programming. Although solving lower order polynomials—where the x in an equation is raised up to the fourth power—is often a simple task, things get complicated once you start seeing powers of five or greater. For centuries, mathematicians accepted this as simply an inherent challenge to their work, but not Norman Wildberger. According to his new approach detailed in The American Mathematical Monthly, there's a much more elegant approach to high order polynomials—all you need to do is get rid of pesky notions like irrational numbers. Babylonians first conceived of two-degree polynomials around 1800 BCE, but it took until the 16th century for mathematicians to evolve the concept to incorporate three- and four-degree variables using root numbers, also known as radicals. Polynomials remained there for another two centuries, with larger examples stumping experts until in 1832. That year, French mathematician Évariste Galois finally illustrated why this was such a problem—the underlying mathematical symmetry in the established methods for lower-order polynomials simply became too complicated for degree five or higher. For Galois, this meant there just wasn't a general formula available for them. Mathematicians have since developed approximate solutions, but they require integrating concepts like irrational numbers into the classical formula. To calculate such an irrational number, 'you would need an infinite amount of work and a hard drive larger than the universe,' explained Wildberger, a mathematician at the University of New South Wales Sydney in Australia. This infinite number of possibilities is the fundamental issue, according to Wildberger. The solution? Toss out the entire concept. '[I don't] believe in irrational numbers,' he said. Instead, his approach relies on mathematical functions like adding, multiplying, and squaring. Wildberger recently approached this challenge by turning to specific polynomial variants called 'power series,' which possess infinite terms within the powers of x. To test it out, he and computer scientist Dean Rubine used 'a famous cubic equation used by Wallis in the 17th century to demonstrate Newton's method.' You don't need to try wrapping your head around all that, however. Just trust Wildberger when he said the solution 'worked beautifully.' The same goes for Catalan numbers, a famous sequence of numbers that describes the number of ways to dissect any given polygon. These also appear in the natural world in areas like biology, where they are employed to analyze possible folding patterns of RNA molecules. 'The Catalan numbers are understood to be intimately connected with the quadratic equation,' explained Wildberger. 'Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.' Outside of headspinning concepts on paper, Wildberger believes the new approach to higher power polynomials could soon result in computer programs capable of solving equations without the need for radicals. It may also help improve algorithms across a variety of fields. 'This is a dramatic revision of a basic chapter in algebra,' argued Wildberger. Luckily, none of this will be your next pop quiz.
Newsweek
01-05-2025
- Science
- Newsweek
Mathematician Solves Algebra's Oldest Problem
Based on facts, either observed and verified firsthand by the reporter, or reported and verified from knowledgeable sources. Newsweek AI is in beta. Translations may contain inaccuracies—please refer to the original content. A mathematician has uncovered a way of answering some of algebra's oldest problems. University of New South Wales Honorary Professor Norman Wildberger, has revealed a potentially game-changing approach to solving higher polynomial equations. Polynomial equations involve a variable being raised to powers, such as the degree two polynomial: 1+ 4x - 3x2 = 0. Until now, a method for solving "higher order" polynomial equations, where x is raised to the power of five or higher, had proven elusive. Wildberger has developed a fresh approach to the problem though using novel number sequences. His method was detailed in an article published in The American Mathematical Monthly co-authored with computer scientist Dr. Dean Rubine. Wildberger's findings could have significant implications. Higher order polynomial equations play a fundamental role in both math and science, assisting in everything from writing computer programs to describing the movement of planets. "Our solution reopens a previously closed book in mathematics history," Wildberger said in a statement. A Brief History of Polynomials Solutions to degree-two polynomials have been around since as far back as 1800 BC, when the Babylonians pioneered their "method of completing the square", which would evolve into the quadratic formula that became familiar to many high school math students. Then in the 16th century, the approach of using roots of numbers dubbed "radicals" was extended to solve three- and four-degree polynomials. In 1832, French mathematician Évariste Galois demonstrated how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree five and higher polynomials. He concluded that no general formula could solve them. Though some approximate solutions for higher-degree polynomials have been developed in the years since, Wildberger contends that these don't belong to pure algebra. Wildberger's 'Radical' Approach He points to the issue of the classical formula's use of third or fourth roots, which are "radicals." These "radicals" tend to represent irrational numbers, decimals that extend to infinity without repeating and can't be written as simple fractions. Wildberger said this makes it impossible to calculate the real answer as "you would need an infinite amount of work and a hard drive larger than the universe." For example, the answer to the cubed root of seven, 3√7 = 1.9129118... extends forever. So although 3√7 "exists" in this formula, this infinite, never-ending decimal is being mischaracterized as a complete object. Wildberger "doesn't believe in irrational numbers" as they rely on an imprecise concept of infinity and lead to logical problems in mathematics. This rejection of radicals inspired Wildberger's best-known contributions to mathematics, rational trigonometry and universal hyperbolic geometry. These approaches rely on mathematical functions like squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine. Wildberger's new approach to solving polynomials avoids radicals and irrational numbers, relying on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x. Wildberger found that by truncating the power series they were able to extract approximate numerical answers to check that the method worked. He said: "One of the equations we tested was a famous cubic equation used by Wallis in the 17th century to demonstrate Newton's method. Our solution worked beautifully." University of New South Wales Honorary Professor Norman Wildberger University of New South Wales Honorary Professor Norman Wildberger University of New South Wales Wildberger's Mathematical Logic Wildberger uses novel sequences of numbers that represent complex geometric relationships in his approach. These sequences belong to combinatorics, a facet of mathematics dealing with number patterns in sets of elements. The Catalan numbers are the most famous combinatorics sequence, used to describe the number of ways you can dissect a polygon into triangles. The Catalan numbers have a number of important applications in practical life, whether it be computer algorithms, game theory or structure designs. The Catalan numbers also play a role in biology, helping count the possible folding patterns of RNA molecules. Wildberger said: "The Catalan numbers are understood to be intimately connected with the quadratic equation. Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers." This approach sees the Catalan numbers enhanced from a one-dimensional to multi-dimensional array based on the number of ways a polygon can be divided using non-intersecting lines. "We've found these extensions, and shown how, logically, they lead to a general solution to polynomial equations," Wildberger said. "This is a dramatic revision of a basic chapter in algebra." Degree five polynomials, or quintics, also have a solution under Wildberger's approach. Practical Applications of Wildberger's Method Theoretical mathematics aside, Wildberger believes this new method could have significant promise when it comes to creating computer programs capable of solving equations using the algebraic series rather than radicals. "This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas," he said. The novel array of numbers, dubbed the "Geode" by Wildberger and co-author Rubine, offers significant potential for further research. "We introduce this fundamentally new array of numbers, the Geode, which extends the classical Catalan numbers and seem to underlie them," Wildberger said. "We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years. Really, there are so many other possibilities. This is only the start." Newsweek has reached out to the University of New South Wales for comment. Do you have a tip on a science story that Newsweek should be covering? Do you have a question about algebra? Let us know via science@ Reference Wildberger, N. J., & and Rubine, D. (2025). A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode. The American Mathematical Monthly.
BBC News
20-03-2025
- Science
- BBC News
Spaghetti science: What pasta reveals about the universe
When you see pasta, your brain probably doesn't jump to the secrets of the universe. But for almost a century, world-leading physicists have puzzled over spaghetti's counterintuitive properties. You might think physicists only ask the big questions. We mostly hear about the physics of the cosmic and the miniscule, the shape of our universe and the nature of the particles that fill it. But physicists, of course, have ordinary lives outside of the laboratory, and sometimes their way of questioning the universe spills over to their daily habits. There's one everyday item that seems to especially obsess them: spaghetti. Going back at least a century, spaghetti has been the subject of rigorous studies. Through this research, physicists continue to learn new things about the solid state of matter, the chemistry of food and even draw connections to the origin of life. The steady torrent of spaghetti science helps to demonstrate that deep questions lurk in our ordinary routines, and that there are plenty of hungry physicists who can't stop asking them. For example: how thin can spaghetti get? The typical spaghetto – the word for an individual strand of spaghetti – is between one and two mm thick (0.04-0.08in). But other long noodles vary widely in diameter, from udon at 4mm (0.16in) to angel hair at 0.8mm (0.03in). The thinnest handmade strands are called su filindeu, coming in at 0.4mm (0.02in), so slender that only a few women in Nuoro, Italy know how to make them. But recently, a team of researchers at the University College London wondered if 21st Century lab equipment could do better. They used a technique called "electro-spinning". First, they dissolved flour into a special, electrically charged solution in a syringe. Then they held the syringe over a special, negatively-charged plate. "This pulls the solution through the dispenser needle down towards the collector plate in a very stringy noodle-type shape," says Beatrice Britton, lead author of the study. When the solution dried, the researchers were left with a crisscrossing thread of incredibly thin spaghetti. "To the naked eye, all you see is a sort of lasagna sheet," Britton says, but a powerful microscope reveals a mat made of strands as thin as 0.1mm (0.004in). These noodles are also much stiffer than regular spaghetti. Britton and her colleagues hope their research can be a step towards biodegradable alternatives to plastic "nanofibres", which are now used to filter liquids and treat wounds. A messy science The world's thinnest spaghetti is just one recent example of how physicists can't seem to stop plying their tools on everybody's favourite carb. But physicists using their noodle on their noodles is no new thing. In 1949, Brown University physicist George F Carrier posed "the spaghetti problem" in The American Mathematical Monthly, which he deemed to be "of considerable popular and academic interest". Essentially, the problem amounts to: "Why can't I slurp up a strand of spaghetti without getting sauce on my face"? His equations showed how the exposed strand swings about more wildly as it gets shorter and shorter, guaranteeing an eventual slap of the noodle against the slurper's lip – and the fateful sauce eruption Carrier so deplored. Sadly, his mathematical formulas offered no way around the face-slap. It's as deeply etched in the laws of the universe as the Big Bang. Later, two scientists inverted Carrier's pioneering study, exploring what happens when a stringy object slips out of a hole instead of being sucked in. They called their version the "reverse spaghetti problem", familiar to any impatient eater who's had to spit out burning pasta because they hadn't waited for it to cool. For now, no theoretical physicist has attempted the more complicated problem of two dogs slurping from either end of the same spaghetti strand. The great mid-century American physicist Richard Feynman helped unlock the riddles of quantum mechanics, explaining how the elementary particles that make up atoms interact with one another. But Feynman's enormous contribution to spaghetti physics is less widely known. One night, Feynman wondered why it's almost impossible to break a stick of spaghetti into two pieces instead of three. He and a colleague spent the rest of the evening snapping spaghetti sticks until they covered the kitchen floor. Feynman's interrogation into the counterintuitive physics of dry spaghetti sparked a quarter century of attempts to explain it. This finally happened in 2005, when two French researchers showed that spaghetti always breaks into two pieces – at first. But after the fracture, as the two bent pieces snap straight again, all their pent-up strain gets released in a shockwave, causing further splintering. In 2018, a team of MIT scientists figured out how to stifle the shockwave – delicately twist the spaghetti strand before snapping it. Their method required lab equipment, but it reliably produced a perfect pair of fragments. Their work provided a new and deeper understanding of brittle rods that goes beyond spaghetti; the phenomenon of three-way fracturing is well-known to pole vaulters, for example. A mechanical wonder My (Italian-American) mother taught me to break a bundle of dry spaghetti in half before putting it in boiling water, so it fits horizontally in the pot. I guess Feynman did the same, but it's an outrage to many of the world's spaghetti-eaters. If you're in the latter camp, then you place your dried spaghetti bundle upright in the pot of boiling water, then watch it slowly soften, buckle and submerge itself. This familiar spaghetti behaviour may not seem like a puzzle, but try removing a recently-curled piece of spaghetti from the pot and letting it dry. It will stay curved rather than returning to its original straight length – something in those first few minutes irreversibly changes the composition of the spaghetti. In 2020, two physicists finally explained this spaghetti transmutation. It's due to a feature called "viscoelasticity" – a name for the unique way materials like spaghetti respond to physical stress. This special property allows water to flow through the strand's outer layers. The strange mechanics of cooked spaghetti go even further. In one study, scientists dropped strands on the ground and measured how they coiled to learn about other elastic materials, from rope to DNA strands. In another, physicists tied spaghetti into knots and studied what types of strain would cause them to tear. Spaghetti physics even goes beyond the pasta itself – sauce is loaded with its own scientific mysteries. When eight Italian physicists met while doing research abroad in Germany, they found a shared frustration in the classic Roman dish cacio e pepe. The sauce requires very few ingredients – it's basically a mixture of reserved pasta water and grated pecorino cheese – but they'd all experienced its mystifying fickleness. Often the cheese irreversibly clumps up, ruining the sauce. This is especially common when you cook it in large batches, which made the physicists hesitant to invite their German colleagues to dinner. "We can't mess up cacio e pepe in front of German people," says Ivan Di Terlizzi, who studies statistical and biological physics at the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany. Fortunately, among them were some of the world's foremost experts on the physics of "phase separation", exactly the kind of congealing phenomenon that plagued their group dinners. Arguing about the phase separation of cacio e pepe, they realised it was flummoxing from a scientific perspective as well. "This is actually a very interesting problem," says Daniel Maria Busiello, co-author on the cacio study. "So we decided to design an experimental apparatus to actually test all these things." The "apparatus" consisted of a bath of water heated to a low temperature, a kitchen thermometer, a petri dish and an iPhone camera attached to an empty box. They invited as many hungry friends as they could find to Di Terlizzi's apartment and hunkered down to cook a weekend's worth of cacio e pepe. They found that the "simple" sauce was enormously complex. Chemically, it's a water-based solution with only a few components: starch (from the pasta water), lipids (from the cheese) and two kinds of protein. Using their apparatus, they found a physical explanation for the sauce-wrecking clumps, which they termed the "mozzarella phase". Proteins, unlike most molecules, get stickier when they're hot. As the sauce is heated, the researchers found this leads to these proteins sticking to the lipids and forming mozzarella-like clumps. In a well-made cacio e pepe, what prevents this is the starch, which forms a protective coat around the lipid molecules so they can't stick to the proteins. If the sauce gets too hot, the increased stickiness of the proteins overcomes this barrier. Once they understood the science behind the sauce, it was clear how to fix it. "If you add enough starch above a certain threshold, you don't get this kind of separated state," says Di Terlizzi. Pasta water doesn't typically contain enough starch to guarantee this threshold, so they suggest adding a mixture of corn starch dissolved in water. The group decided to conclude their manuscript with a foolproof recipe for the classic dish. But in surveying the rich scientific literature, they realised they weren't the first to reach this cacio epiphany. In the name of academic integrity, they cited a YouTube video wherein the Michelin-star Roman chef Luciano Monosilio suggests the same tweak for a foolproof recipe – a dash of corn starch. "It's the only non-scientific reference in our paper," says Di Terlizzi. The physics they used connects the clumping of cacio e pepe to ideas about the origin of life on Earth. Biophysicists use phase separation to understand how droplets of liquid can congeal and divide within a solution. "A droplet dividing pretty much looks like a proto-cell," says Giacomo Bartolucci, another co-author on the study. Inside the little blobs that preceded actual cells, some believe, the building blocks of life may have come together via a process much like the Italians' mozzarella phase. The same ideas are helping biologists understand how the plaques that cause Alzheimer's coalesce in the brain. Why is spaghetti such a locus of speculation and study for physicists? For one, it's simple – flour, water and heat, says Vishal Patil, one of the discoverers of the twist-and-break method who is now a professor of mathematics at the University of California, San Diego. The fact that a combination of so few components raises so many deep questions speaks to how physics underlies everything they see and do, Patil says. More like this:• Why sarcastic teens are the smartest• The French towns giving away free chickens• YouTube statistics Google doesn't want you to know It also shows that no matter how deep physicists probe the big and the small, the answers can still fall short of explaining phenomena we see every day. When it comes to cacio e pepe, all the tools of theoretical physics can only tell us what every Italian grandmother knows: keep the stove burning low when you make it. Laboratory electrospinning can only achieve marginally thinner spaghetti than what the women of Nuoro, Italy make daily by hand. "Spaghetti is just a very accessible thing you can play with," Patil says. The low cost of flour-based noodles is what made them a democratic delicacy for so many cultures around the world – spaghetti was popularised in Naples as street food. That's why Feynman didn't hesitate to snap pounds of the stuff onto his kitchen floor. After a long day at the blackboard, plugging away at the impenetrable math of quantum mechanics or black holes, the mechanical wonders of spaghetti are the perfect fodder for scientists' mealtime probing. -- For more science, technology, environment and health stories from the BBC, follow us on Facebook, X and Instagram.
