Tag: mathematics

3,700-Year-Old Babylonian Tablet Rewrites The History Of Math

A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals.




However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today.

Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles,” said Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

It is a fascinating mathematical work that demonstrates undoubted genius. The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.”

This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

The Greek astronomer Hipparchus, who lived around 120BC, has long been regarded as the father of trigonometry, with his ‘table of chords’ on a circle considered the oldest trigonometric table.

A trigonometric table allows a user to determine two unknown ratios of a right-angled triangle using just one known ratio.

But the tablet is far older than Hipparchus, demonstrating that the Babylonians were already well advanced in complex mathematics far earlier.

The tablet, which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822 and 1762 BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.

Plimpton 322 predates Hipparchus by more than 1000 years,” says Dr Wildberger.

It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.”

A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”

The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination.

The left-hand edge of the tablet is broken but the researchers believe t there were originally six columns and that the tablet was meant to be completed with 38 rows.

Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids,” added Dr Mansfield.

The new study is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

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Science Tackles The Hard Questions At Last: How To Create A Perfect Bubble

Blowing soap bubbles is child’s play, but surprisingly, physicists haven’t worked out the details of the phenomenon.

Now researchers have performed experiments and developed a complete theory of the process of soap bubble formation.

The team aimed a jet of gas at a soap film and observed that bubbles appear only above a threshold gas speed.

By measuring this threshold under varying conditions, the team showed that bubbles result from a competition between the pressure of the gas jet and the surface tension of the soap film.

Understanding the physics of bubbles is important for a variety of industrial processes and scientific fields, from cosmology to foam science, and the new experiments may also be useful in the classroom.

Researchers have studied related processes, such as the popping of bubbles, and examined soap films being pierced by pellets or liquid droplets.

But bubble blowing has mostly been overlooked, say Laurent Courbin and Pascal Panizza, both of the French National Centre for Scientific Research (CNRS) and the University of Rennes 1.

While watching children blowing bubbles in a local park, they realized that the phenomenon hadn’t been studied before and hurried back to the lab to tinker with soapsuds.

Following the example of previous soap film research on fluid flows and turbulence, Courbin, Panizza, and their colleagues built a large apparatus capable of creating a meter-tall, long-lived, vertical sheet of soap solution.

In this system, the soap film continually flows downward—unlike the stationary film in a standard bubble wand—and the liquid is collected at the bottom and pumped back to the top.




This laboratory setup allows the film to remain stable indefinitely, and its thickness can be adjusted, as can the speed with which it falls.

The team placed a gas nozzle at the surface of the soap film and used a high-speed camera to capture the results. At low gas jet speeds, only a small dimple appeared in the soap film.

The dimple became deeper as the team increased the jet’s speed, until bubbles finally formed.

The phenomenon, the researchers found, can be explained as a contest between the pressure the gas jet exerts on the film and the surface tension of the film, which resists any increase in curvature.

Bubbles form when the jet’s pressure is large enough to deform the film into a hemispheric dimple of the same width as the jet.

At that point, the film has reached its maximum curvature, and the bubble can fill with gas and float away.

The researchers found that wider jets, which produce larger bubbles, create them at lower gas speeds than narrower jets. These larger bubbles have less curvature, making it easier to overcome surface tension’s pull.

Repeating the study with a simple bubble wand gave similar results, suggesting that the laboratory setup is a passable proxy for real-world bubble blowing.

The thickness of the soap film had no effect on the gas speed at which bubbles formed.

Understanding how bubbles form is important for certain industrial processes, like those involving foam production, and avoiding bubble formation is necessary in glassmaking and coating solids with liquids, says Courbin.

But “this paper is really about explaining an everyday-life experiment,” rather than real-world applications, he says.

Still, says Hamid Kellay of the University of Bordeaux in France, “it’s the first time that these types of ideas can be tested correctly, because of the well-controlled experiments.

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Scientists Discover A New Shape That Might Be Inside You Right Now

The cells in our bodies are put together in all kinds of weird ways. Neuron cells have long, branching connections to other cells, bones form porous structures, and blood vessels float freely around the body.

But a lot of cells are simply squished together as tightly as possible, and scientists have determined that these cells in particular come in a unique and previously unknown shape, called a “scutoid.”

Much of our bodies are covered in epithelial cells, which are cells designed to stick very closely together in order to form some type of barrier or wall.

Our skin cells are epithelial cells, as are the cells that form the walls of many of our organs.

One of the most important functions these cells have is keeping things either inside or outside of the areas they surround, so forming a tight wall is of paramount importance.

So what shape do these cells take? Most scientists previously believed that these cells were shaped like simple cylinders, but new research suggests they take a more complicated shape.




Researchers at the University of Seville ran a computer simulation to determine what the most efficient shape would be, and their simulation settled on a strange prism-like shape.

The shape has six sides at the top and five sides at the bottom, and one of the sides had a triangular protrusion.

Crucially, this shape—which the scientists named the “scutoid” after the similarly-named and -shaped scutellum of a beetle—does indeed stack much better than a simple cylinder.

But just because a computer says it’s the best shape doesn’t mean that anything in nature actually uses it, so the researchers examined cells from fruit flies and zebrafish to see if the scutoid shows up in those animals’ epithelial tissue.

To their delight, it did. They’re not certain whether these scutoid-shaped cells exist in humans as well, but there’s a good chance.

In addition to discovering what epithelial cells look like, these researchers also discovered a brand new shape new to mathematics.

Mathematical discoveries are often very abstract but can frequently have impact in other fields in science or engineering.

So scutoids are only in your body right now, but eventually we might start seeing them show up all over the place.

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Flash Recovery Of Ammonoids After Most Massive Extinction Of All Time

The study, conducted by a Franco-Swiss collaboration involving the laboratories Biogéosciences (Université de Bourgogne / CNRS), Paléoenvironnements & Paléobiosphère (Université Claude Bernard / CNRS) and the Universities of Zurich and Lausanne (Switzerland), appears in the August 28 issue of Science.

The history of life on Earth has been punctuated by a number of mass extinctions, brief periods of extreme loss of biodiversity. These extinctions are followed by phases during which surviving species recover and diversify.

The End-Permian extinction, which took place between the Permian (299 – 252.6 MY) and Triassic (252.6 – 201.6 MY), is the greatest mass extinction on record, resulting in the loss of 90% of existing species.

It is associated with intensive volcanic activity in China and Siberia. It marks the boundary between the Paleozoic and Mesozoic Eras.




Until now, studies had shown that the biosphere took between 10 and 30 million years to recover the levels of biodiversity seen before the extinction.

Ammonoids are cephalopod swimmers related the nautilus and squid. They had a shell, and disappeared from the oceans at the same time as the dinosaurs, 65 million years ago, after being a major part of marine fauna for 400 MY.

The Franco-Swiss team of paleontologists has shown that ammonoids needed only one million years after the End-Permian extinction to diversify to the same levels as before.

The cephalopods, which were abundant during the Permian, narrowly missed being eradicated during the extinction: only two or three species survived and a single species seems to have been the basis for the extraordinary diversification of the group after the extinction.

It took researchers seven years to gather new fossils and analyze databases in order to determine the rate of diversification of the ammonoids.

In all, 860 genera from 77 regions around the world were recorded at 25 successive time intervals from the Late Carboniferous to the Late Triassic, a period of over 100 million years.

The discovery of this explosive growth over a million years takes a heated debate in a new direction.

Indeed, it suggests that earlier estimates for the End-Permian extinction were based on truncated data and imprecise or incorrect dating.

Furthermore, the duration for estimated recovery after other lesser extinctions all vary between 5 and 15 million years.

The result obtained here suggests that these estimates should probably be revised downwards.

The biosphere is most likely headed towards a sixth mass extinction, and this discovery reminds us that the recovery of existing species after an extinction is a very long process, taking several tens of thousands of human generations at the very least.

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Honey Bees Can Understand Nothing

Zero, zilch, nothing, is a pretty hard concept to understand. Children generally can’t grasp it until kindergarten. And it’s a concept that may not be innate but rather learned through culture and education.

Throughout human history, civilizations have had varying representations for it. Yet our closest animal relative, the chimpanzee, can understand it.

And now researchers in Australia writing in the journal Science say the humble honey bee can be taught to understand that zero is less than one.

The result is kind of astounding, considering how tiny bee brains are. Humans have around 100 billion neurons. The bee brain? Fewer than 1 million.

The findings suggest that the ability to fathom zero may be more widespread than previously thought in the animal kingdom — something that evolved long ago and in more branches of life.




It’s also possible that in deconstructing how the bees compute numbers, we could make better, more efficient computers one day.

Our computers are electricity-guzzling machines. The bee, however, “is doing fairly high-level cognitive tasks with a tiny drop of nectar,” says Adrian Dyer, a Royal Melbourne Institute of Technology researcher and co-author on the study.

Their brains are probably processing information in a very clever [i.e., efficient] way.”

But before we can deconstruct the bee brain, we need to know that it can do the complex math in the first place.

How to teach a bee the concept of zero

Bees are fantastic learners. They spend hours foraging for nectar in among flowers, can remember where the juiciest flowers are, and even have a form of communication to inform their hive mates of where food is to be found.

Researchers train bees like they train many animals: with food. “You have a drop of sucrose associated with a color or a shape, and they will learn to reliably go back to” that color or shape, Dyer explains.

With this simple process, you can start teaching bees rules. In this case, the researchers wanted to teach 10 bees the basic rules of arithmetic.

So they put out a series of sheets of paper that had differing numbers of objects printed on them. Using sugar as a reward, the researchers taught the bees to always fly to the sheet that had the fewest objects printed on it.

Once the bees learned this rule, they could reliably figure out that two shapes are less than four shapes, that one shape is smaller than three. And they’d keep doing this even when a sugary reward was not waiting for them.

And then came the challenge: What happens when a sheet with no objects at all was presented to the bees? Would they understand that a blank sheet — which represented the concept of zero in this experiment — was less than three, less than one?

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The Formula For The Perfect Free-Throw

Improving your free-throw percentage is a simple matter of mathematics, according to researchers Drs. Chau Tran and Larry Silverberg of North Carolina State University.

Using three-dimensional computer simulations of hundreds of thousands of basketball trajectories, the two engineers determined the ideal characteristics of a free-throw shot.

They based their data on the assumption of a 6’6” player who would release the ball (assumed to be a men’s basketball) at a height of 7 feet.




The first variable Tran and Silverberg examined was spin. According to them, you should release the ball with about three hertz of backspin – or, so that the ball makes roughly three full backwards rotations before reaching the hoop.

This slows the ball upon contact with the backboard or rim, making it more likely that the shot will go in.

The ball should also be released at 52 degrees to the horizontal, making the peak of its arc only a few inches higher than the top of the backboard.

For aiming, they found the most successful methods put the ball towards the back of the rim, either two inches to the left or two inches to the right of the place where the rim meets the backboard.

How a mathematician sees a free throw.

Their simulation data showed that aiming straight for the center of the backboard decreases the success rate by almost three percent.

Tran and Silverberg also recommend free-throw shooters should release the ball as high above the ground as possible with a smooth, consistent release speed for best results.

Our recommendations might make even the worst free-throw shooters – you know who you are, Shaquille O’Neal and Ben Wallace – break 60 percent from the free-throw line,” Silverberg joked.

Their work is just another example of how mathematical questions can crop up in the most unexpected areas.

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Why Math Might Be Complete BS

Set up a free Brilliant account at http://www.brilliant.org/answerswithjoe/ And the first 295 to sign up for a premium account get 20% off every month!

Mathematics is the backbone of all sciences, no theories or hypotheses are proven unless there is math to back it up. But there are many who believe that math isn’t real. In today’s video, we’ll break down the arguments.

From the Mathematical Physicalists to the Platonists to the Mathematical Fictionalists, we look at all the theories behind whether numbers actually exist, and what they mean.

The Largest Prime Number Was Discovered By A FedEx Employee

A FedEx employee in Tennessee has discovered the largest known prime number.

Germantown, Tenn., resident John Pace found the number through his volunteer work with the Great Internet Mersenne Prime Search (GIMPS), a project that crowd sources computing power to search for a subset of prime numbers called Mersenne primes.

Like a normal prime number, these can only be divided by themselves and one. What sets them apart is that they can all be expressed as the number 2 raised to a given power minus one.




The newly discovered Mersenne prime, called M77232917, can be expressed as 2 to the 77,232,917 power minus one. It’s the 50th Mersenne prime to be discovered and it’s more than 23 million digits long.

Pace might be the only person in history who went into math for the money.

He told NPR, “There was a $100,000 prize attached to finding the first prime that had a 10 million digit result, and I was like, ‘Well you know, I’ve got as much chance as anybody else.’

He has been participating in the program for 14 years and this is his first discovery.

The previous longest-known prime number was discovered in January of 2016 at the University of Central Missouri. It contains 22 million digits and is also a Mersenne prime.

Large prime numbers are important for the future of computing and cyber security, and the search is already on for larger numbers.

The Electronic Frontier Foundation is offering a prize of $150,000 for finding the first prime number with one hundred million digits and $250,000 for finding the first prime with one billion digits.

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Largest Known Prime Number Discovered With Over 23 Million Digits

A collaborative computational effort has uncovered the longest known prime number.

At over 23 million digits long, the new number has been given the name M77232917 for short.

Prime numbers are divisible only by themselves and one, and the search for ever-larger primes has long occupied maths enthusiasts.

However, the search requires complicated computer software and collaboration as the numbers get increasingly hard to find.




M77232917 was discovered on a computer belonging to Jonathan Pace, an electrical engineer from Tennessee who has been searching for big primes for 14 years.

Mr Pace discovered the new number as part of the Great Internet Mersenne Prime Search (GIMPS), a project started in 1996 to hunt for these massive numbers.

Mersenne primes – named after the 17th century French monk Marin Mersenne who studied them – are calculated by multiplying together many twos and then subtracting one.

Six days of non-stop computing in which 77,232,917 twos were multiplied together resulted in the latest discovery.

The number is the 50th Mersenne prime to be discovered, and the 16th to be discovered by the GIMPS project.

It is nearly one million digits longer than the previous record holder, which was identified as part of the same project at the beginning of 2016.

Mersenne primes are a particular focus for prime aficionados because there is a relatively straightforward way to check whether a number is one or not.

Nevertheless, the new prime has to be verified using four different computer programs on four different computers.

The process also relies on thousands of volunteers sifting through millions of non-prime candidates before the lucky individual chances upon their target.

Professor Caldwell runs an authoritative website on the largest prime numbers, with a focus on the history of Mersenne primes.

He emphasised the pure excitement that searching for prime numbers brings, describing the latest discovery as “a museum piece as opposed to something that industry would use”.

Besides the thrill of discovery, Mr Pace will receive a $3,000 (£2,211) GIMPS research discovery award.

GIMPS uses the power of thousands of ordinary computers to search for elusive primes, and the team behind it state that anybody with a reasonably powerful PC can download the necessary software and become a “big prime hunter”.

The next Mersenne prime discovery could be smaller or larger than the existing record holder, but the big target for the GIMPS team is to find a 100 million digit prime number.

The person who discovers such a number will be awarded $150,000 by the Electronic Frontier Foundation for their efforts.

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The 500-Page Proof That Only One Mathematician Can Understand

Nearly four years after Shinichi Mochizuki unveiled an imposing set of papers that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity.

Although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University’s Research Institute for Mathematical Sciences (RIMS).

Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego.

Although at first Mochizuki’s papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae.




Experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers — whole numbers that cannot be evenly divided by any smaller number except by 1.

The conjecture comes in a number of different forms, but explains how the primes that divide two numbers, a and b, are related to those that divide their sum, c.

If Mochizuki’s proof is correct, it would have repercussions across the entire field, says Dimitrov.

When you work in number theory, you cannot ignore the abc conjecture,” he says.

This is why all number theorists eagerly wanted to know about Mochizuki’s approach.”

For example, Dimitrov showed in January how, assuming the correctness of Mochizuki’s proof, one might be able to derive many other important results, including a completely independent proof of the celebrated Fermat’s last theorem.

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

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