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David Smith, a retired print technician from the north of England, was pursuing his hobby of looking for interesting shapes when he stumbled onto one unlike any other in November.

When Smith shared his shape with the world in March, excited fans printed it onto T-shirts, sewed it into quilts, crafted cookie cutters or used it to replace the hexagons on a soccer ball—some even made plans for tattoos.

The 13-sided polygon, which 64-year-old Smith called “the hat”, is the first single shape ever found that can completely cover an infinitely large flat surface without ever repeating the same pattern.

That makes it the first “einstein”—named after the German for “one stone” (ein stein), not the famed physicist—and solves a problem posed 60 years ago that some mathematicianshad thought impossible.

After stunning the mathematics world, Smith—a hobbyist with no training who told AFP that he wasn’t great at math at school—then did it again.

While all agreed “the hat” was the first einstein, its mirror image was required one in seven times to ensure that a pattern never repeated.

But in a preprint study published online late last month, Smith and the three mathematicians who helped him confirm the discovery revealed a new shape—”the specter.”

It requires no mirror image, making it an even purer einstein.

‘It can be that easy’

Craig Kaplan, a computer scientist at Canada’s Waterloo University, told AFP that it was “an amusing and almost ridiculous story—but wonderful”.

He said that Smith, a retired print technician who lives in Yorkshire’s East Riding, emailed him “out of the blue” in November.

Smith had found something “which did not play by his normal expectations for how shapes behave”, Kaplan said.

If you slotted a bunch of these cardboard shapes together on a table, you could keep building outwards without them ever settling into a regular pattern.

Using computer programs, Kaplan and two other mathematicians showed that the shape continued to do this across an infinite plane, making it the first einstein, or “aperiodic monotile”.

When they published their first preprint in March, among those inspired was Yoshiaki Araki. The Japanese tiling enthusiast made art using the hat and another aperiodic shape created by the team called “the turtle”, sometimes using flipped versions.

Smith was inspired back, and started playing around with ways to avoid needing to flip his hat.

Less than a week after their first paper came out, Smith emailed Kaplan a new shape.

Kaplan refused to believe it at first. “There’s no way it can be that easy,” he said.

But analysis confirmed that Tile (1,1) was a “non-reflective einstein”, Kaplan said.

Something still bugged them—while this tile could go on forever without repeating a pattern, this required an “artificial prohibition” against using a flipped shape, he said.

So they added little notches or curves to the edges, ensuring that only the non-flipped version could be used, creating “the specter”.

‘Hatfest’

Kaplan said both their papers had been submitted to peer-reviewed journals. But the world of mathematics did not wait to express its astonishment.

Marjorie Senechal, a mathematician at Smith College in the United States, told AFP the discoveries were “exciting, surprising and amazing”.

She said she expects the specter and its relatives “will lead to a deeper understanding of order in nature and the nature of order.”

Doris Schattschneider, a mathematician at Moravian College in the US, said both shapes were “stunning”.

Even Nobel-winning mathematician Roger Penrose, whose previous best effort had narrowed the number of aperiodic tiles down to two in the 1970s, had not been sure such a thing was possible, Schattschneider said.

Penrose, 91, will be among those celebrating the new shapes during the two-day “Hatfest” event at Oxford University next month.

All involved expressed amazement that the breakthrough was achieved by someone without training in math.

“The answer fell out of the sky and into the hands of an amateur—and I mean that in the best possible way, a lover of the subject who explores it outside of professional practice,” Kaplan said.

“This is the kind of thing that ought not to happen, but very happily for the history of science does happen occasionally, where a flash brings us the answer all at once.”

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Credit of the article given to Daniel Lawler

After some exposure to group theory, you quickly learn that when trying to prove a group GG is abelian, checking if xy=yxxy=yx for arbitrary x,yx,y in GG is not always the most efficient – or helpful! – tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:

Show the commutator [x,y]=xyx−1y−1[x,y]=xyx−1y−1of two arbitary elements x,y∈Gx,y∈G must be the identity

• Show the group is isomorphic to a direct product of two abelian (sub)groups
• Check if the group has order p2p2 for any prime pp OR if the order is pqpq for primes p≤qp≤q with p∤q−1p∤q−1.
• Show the group is cyclic.
• Show |Z(G)|=|G|.|Z(G)|=|G|.
• Prove G/Z(G)G/Z(G) is cyclic. (e.g. does G/Z(G)G/Z(G) have prime order?)
• Show that GG has a trivial commutator subgroup, i.e. is [G,G]={e}[G,G]={e}.

Here’s a thought map which is (probably) more fun than practical. Note, pp and qq denote primes below:

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

*Credit for article given to Tai-Danae Bradley*

Ever wondered how your friends shape your music taste? In a recent study, researchers at the Complexity Science Hub (CSH) demonstrated that social networks are a powerful predictor of a song’s future popularity. By analysing friendships and listening habits, they’ve boosted machine learning prediction precision by 50%.

“Our findings suggest that the social element is as crucial in music spread as the artist’s fame or genre influence,” says Niklas Reisz from CSH. By using information about listener social networks, along with common measures used in hit song prediction, such as how well-known the artist is and how popular the genre is, the researchers improved the precision of predicting hit songs from 14% to 21%. The study, published in Scientific Reports, underscores the power of social connections in music trends.

A deep dive into data

The CSH team analysed data from the music platform last.fm, analysing 2.7 million users, 10 million songs, and 300 million plays. With users able to friend each other and share music preferences, the researchers gained anonymized insights into who listens to what and who influences whom, according to Reisz.

For their model, the researchers worked with two networks: one mapping friendships and another capturing influence dynamics—who listens to a song and who follows suit. “Here, the nodes of the network are also people, but the connections arise when one person listens to a song and shortly afterwards another person listens to the same song for the first time,” explains Stefan Thurner from CSH.

Examining the first 200 plays of a new song, they predicted its chances of becoming a hit—defined as being in the top 1% most played songs on last.fm.

User influence

The study found that a song’s spread hinges on user influence within their social network. Individuals with a strong influence and large, interconnected friend circles accelerate a song’s popularity. According to the study, information about social networks and the dynamics of social influence enable much more precise predictions as to whether a song will be a hit or not.

“Our results also show how influence flows both ways—people who influence their friends are also influenced by them” explains CSH researcher Vito Servedio. “In this way, multi-level cascades can develop within a very short time, in which a song can quickly reach many other people, starting with just a few people.”

Social power in the music industry

Predicting hit songs is crucial for the music industry, offering a competitive edge. Existing models often focus on artist fame and listening metrics, but the CSH study highlights the overlooked social aspect—musical homophily, which is the tendency for friends to listen to similar music. “It was particularly interesting for us to see that the social aspect, musical homophily, has so far received very little attention—even though music has always had a strong social aspect,” says Reisz.

The study quantifies this social influence, providing insights that extend beyond music to areas like political opinion and climate change attitudes, according to Thurner.

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Credit of the article given to Complexity Science Hub Vienna

Machines are getting better at maths – artificial intelligence has learned to solve university-level calculus problems in seconds.

François Charton and Guillaume Lample at Facebook AI Research trained an AI on tens of millions of calculus problems randomly generated by a computer. The problems were mathematical expressions that involved integration, a common technique in calculus for finding the area under a curve.

To find solutions, the AI used natural language processing (NLP), a computational tool commonly used to analyse language. This works because the mathematics in each problem can be thought of as a sentence, with variables, normally denoted x, playing the role of nouns and operations, such as finding the square root, playing the role of verbs. The AI then “translates” the problem into a solution.

When the pair tested the AI on 500 calculus problems, it found a solution with an accuracy of 98 per cent. A comparable standard program for solving maths problems had only an accuracy of 85 per cent on the same problems.

The team also gave the AI differential equations to solve, which are other equations that require integration to solve as well as other techniques. For these equations, the AI wasn’t quite as good, solving them correctly 81 per cent for one type of differential equation and 40 per cent on a harder type.

Despite this, it could still correctly answer questions that confounded other maths programs.

Doing calculus on a computer isn’t especially useful in practice, but with further training AI might one day be able to tackle maths problems that are too hard for humans to crack, says Charton.

The efficiency of the AI could save humans time in other mathematical tasks, for example, when proving theorems, says Nikos Aletras at the University of Sheffield, UK.

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

*Credit for article given to Gege Li*

In the face of existential dilemmas that are shared by all of humanity, including the consequences of inequality or climate change, it is crucial to understand the conditions leading to cooperation. A new game theory model developed at the Institute of Science and Technology Austria (ISTA) based on 192 stochastic games and on some elegant algebra finds that both cases—available information and the lack thereof—can lead to cooperative outcomes.

The journal Nature Communications has published a new open-access paper on the role information plays in reaching a cooperative outcome. Working at ISTA with the Chatterjee group, research scholar Maria Kleshnina developed a framework of stochastic games, a tool to describe how people make decisions in changing environments. The new model finds that availability of information is intricately linked to cooperative outcomes.

“In this paper, we present a new model of games where a group’s environment changes, based on actions of group members who do not necessarily have all relevant information about their environment. We find that there are rich interactions between the availability of information and cooperative behaviour.

“Counter-intuitively there are instances where there is a benefit of ignorance, and we characterize when information helps in cooperation,” says Professor Krishnendu Chatterjee who leads the “Computer-Aided Verification, Game Theory” group at the Institute of Science and Technology Austria, where this work was done.

Ignorance can be beneficial for cooperation too

In 2016, Štěpán Šimsa, one of the authors of the new paper was working with the Chatterjee group, when he ran some preliminary simulations to find that ignorance about the state of the game may benefit cooperation. This is counter-intuitive since the availability of information is generally thought to be universally beneficial. Christian Hilbe, then a postdoc with the Chatterjee group, along with Kleshnina, thought this to be a worthy research direction. The group then took on the task of investigating how information or the lack thereof affects the evolution of cooperation.

“We quantified in which games it is useful to have precise information about the environmental state. And we find that in most cases, around 80 to 90% it is indeed really good if players are aware of the environment’s state and which game they are playing right now. Yet, we also find some very interesting exceptional cases, where it’s actually optimal for cooperation if everyone is ignorant about the game they are playing,” says co-author Christian Hilbe, who now leads the research group Dynamics of Social Behaviour at the Max Planck Institute for Evolutionary Biology in Germany.

The researchers’ framework represents an idealized model for cooperation in changing environments. Therefore, the results cannot be directly transferred to real-world applications like solving climate change. For this, they say, a more extensive model would be required. Although, from the basic science model that she has built, Kleshnina is able to offer a qualitative direction.

“In a changing system, a benefit of ignorance is more likely to occur in systems that naturally punish non-cooperation. This could happen, for example, if the group’s environment quickly deteriorates if players no longer cooperate mutually. In such a system, individuals have strong incentives to cooperate today, if they want to avoid playing an unprofitable game tomorrow,” she says.

To illustrate the benefit of ignorance, Kleshnina says, “For example, we found that in informed populations, individuals can use their knowledge to employ more nuanced strategies. These nuanced strategies, however, can be less effective in sustaining cooperation. In such a case, there is indeed a small benefit of ignorance towards cooperation.”

A brilliant method

Game theory is, in its essence, a study of mathematical models set up within the framework of games or exchange of logical decisions being played between rational players. Its applications in understanding social and biological evolution have been welcomed by interdisciplinary researchers given its game-changing approach.

Within the context of evolutionary game theory, many models investigate cooperation but most assume that the same game is played over and over again, and also that the players are always perfectly aware of the game that they are playing and its state at any given moment. The new study weakens these general assumptions, first by allowing the simulated players to play different games over time. And second, by accounting for the impact of information.

“The beauty of this approach is that one can combine some elegant linear algebra with extensive computer simulations,” says Kleshnina.

The new framework opens up many new research directions. For instance, what is the role of asymmetric information? One player might know the exact game being played, but another may not. This is not something that the model currently covers. “In that sense, our paper has quite [a few] future applications within theory itself,” Hilbe adds.

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Credit of the article given to Institute of Science and Technology Austria

Money should be no object when it comes to the numbers game. krissyho

Mathematics has many grand challenge problems, but none that can potentially be settled by pouring in more money – unlike the case of the Large Hadron Collider, the Square Kilometre Array or other such projects.

Maths is a different beast. But, of course, you’re offering me unlimited, free dosh, so I should really think of something.

Grand Challenges in Mathematics

In his famous 1900 speech The Problems of Mathematics David Hilbert listed 23 problems that set the stage for 20th century mathematics.

It was a speech full of optimism for mathematics in the coming century and Hilbert felt open (or unsolved) problems were a sign of vitality:

“The great importance of definite problems for the progress of mathematical science in general … is undeniable … [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality … every mathematician certainly shares … the conviction that every mathematical problem is necessarily capable of strict resolution … we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought …”

Hilbert’s problems included the continuum hypothesis, the “well-ordering” of the reals, Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet’s principle and many more.

Many were solved in subsequent decades, and each time it was a major event for mathematics.

The Riemann hypothesis (which deals with the distribution of prime numbers) remains on a list of seven “third millennium” problems.

For the solution of each millennium problem, the Clay Mathematics Institute offers – in the spirit of the times – a one million dollar prize.

This prize has already been awarded and refused by Perelman for resolving the Poincaré conjecture. The solution also merited Science’s Breakthrough of the Year, the first time mathematics had been so honoured.

Certainly, given unlimited moolah, learned groups could be gathered to attack each problem and assisted in various material ways. But targeted research in mathematics has even less history of success than in the other sciences … which is saying something.

Doron Zeilberger famously said that the Riemann hypothesis is the only piece of mathematics whose proof (i.e. certainty of knowledge) merits $10 billion being spent.

As John McCarthy wrote in Science in 1997:

“In 1965 the Russian mathematician Alexander Konrod said ‘Chess is the Drosophila [a type of fruit fly] of artificial intelligence.

“But computer chess has developed as genetics might have if the geneticists had concentrated their efforts, starting in 1910, on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.”

Unfortunately, the so-called “curse of exponentiality” – whereby the more difficult a problem becomes, the challenge of solving it increases exponentially – pervades all computing, and especially mathematics.

As a result, many problems – such as Ramsey’s Theorem – will likely be impossible to solve by computer brute force, regardless of advances in technology.

Money for nothing

But, of course, I must get to the point. You’re offering me a blank cheque, so what would I do? A holiday in Greece for two? No, not this time. Here’s my manifesto:

Google has transformed mathematical life (as it has with all aspects of life) but is not very good at answering mathematical questions – even if one knows precisely the question to ask and it involves no symbols.

In February, IBM’s Watson computer walloped the best human Jeopardy players in one of the most impressive displays of natural language competence by a machine.

I would pour money into developing an enhanced Watson for mathematics and would acquire the whole corpus of maths for its database.

Maths ages very well and I am certain we would discover a treasure trove. Since it’s hard to tell where maths ends and physics, computer science and other subjects begin, I would be catholic in my acquisitions.

Since I am as rich as Croesus and can buy my way out of trouble, I will not suffer the same court challenges Google Books has faced.

I should also pay to develop a comprehensive computation and publishing system with features that allow one to manipulate mathematics while reading it and which ensures published mathematics is rich and multi-textured, allowing for reading at a variety of levels.

Since I am still in a spending mood, I would endow a mathematical research institute with great collaboration tools for roughly each ten million people on Earth.

Such institutes have greatly enhanced research in the countries that can afford and chose to fund them.

Content with my work, I would then rest.

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

*Credit for article given to Jonathan Borwein (Jon)*

A team of EPFL scientists has developed an algorithm that can identify the source of an epidemic or information circulating within a network, a method that could also be used to help with criminal investigations.

Investigators are well aware of how difficult it is to trace an unlawful act to its source. The job was arguably easier with old, Mafia-style criminal organizations, as their hierarchical structures more or less resembled predictable family trees.

In the Internet age, however, the networks used by organized criminals have changed. Innumerable nodes and connections escalate the complexity of these networks, making it ever more difficult to root out the guilty party. EPFL researcher Pedro Pinto of the Audiovisual Communications Laboratory and his colleagues have developed an algorithm that could become a valuable ally for investigators, criminal or otherwise, as long as a network is involved.

“Using our method, we can find the source of all kinds of things circulating in a network just by ‘listening’ to a limited number of members of that network,” explains Pinto. Suppose you come across a rumor about yourself that has spread on Facebook and been sent to 500 people; your friends, or even friends of your friends. How do you find the person who started the rumor? “By looking at the messages received by just 15󈞀 of your friends, and taking into account the time factor, our algorithm can trace the path of that information back and find the source,” Pinto adds. This method can also be used to identify the origin of a spam message or a computer virus using only a limited number of sensors within the network.

Out in the real world, the algorithm can be employed to find the primary source of an infectious disease, such as cholera. “We tested our method with data on an epidemic in South Africa provided by EPFL professor Andrea Rinaldo’s Ecohydrology Laboratory,” says Pinto. “By modeling water networks, river networks, and human transport networks, we were able to find the spot where the first cases of infection appeared by monitoring only a small fraction of the villages.”

The method would also be useful in responding to terrorist attacks, such as the 1995 sarin gas attack in the Tokyo subway, in which poisonous gas released in the city’s subterranean tunnels killed 13 people and injured nearly 1,000 more. “Using this algorithm, it wouldn’t be necessary to equip every station with detectors. A sample would be sufficient to rapidly identify the origin of the attack, and action could be taken before it spreads too far,” says Pinto.

Computer simulations of the telephone conversations that could have occurred during the terrorist attacks on September 11, 2001, were used to test Pinto’s system. “By reconstructing the message exchange inside the 9/11 terrorist network extracted from publicly released news, our system spit out the names of three potential suspects; one of whom was found to be the mastermind of the attacks, according to the official enquiry.”

The validity of this method thus has been proven a posteriori. But according to Pinto, it could also be used preventatively; for example, to understand an outbreak before it gets out of control. “By carefully selecting points in the network to test, we could more rapidly detect the spread of an epidemic,” he points out. It could also be a valuable tool for advertisers who use viral marketing strategies by leveraging the Internet and social networks to reach customers. For example, this algorithm would allow them to identify the specific Internet that are the most influential for their target audience and to understand how in these articles spread throughout the online community.

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Credit of the article given to Ecole Polytechnique Federale de Lausanne

Reduce wastage and enjoy deeply satisfying neat folds by applying a little geometry to your gift-wrapping, says Katie Steckles.

Wrapping gifts in paper involves converting a 2D shape into a 3D one, which presents plenty of geometrical challenges. Mathematics can help with this, in particular by making sure that you are using just the right amount of paper, with no wastage.

When you are dealing with a box-shaped gift, you might already wrap the paper around it to make a rectangular tube, then fold in the ends. With a little measuring, though, you can figure out precisely how much paper you will need to wrap a gift using this method, keeping the ends nice and neat.

For example, if your gift is a box with a square cross-section, you will need to measure the length of the long side, L, and the thickness, T, which is the length of one side of the square. Then, you will need a piece of paper measuring 4 × T (to wrap around the four sides with a small overlap) by L + T. Once wrapped around the shape, a bit of paper half the height of the square will stick out at each end, and if you push the four sides in carefully, you can create diagonal folds to make four points that meet neatly in the middle. The square ends of the gift make this possible (and deeply satisfying).

Similarly, if you are wrapping a cylindrical gift with diameter D (such as a candle), mathematics tells us you need your paper to be just more than π × D wide, and L + D long. This means the ends can be folded in – possibly less neatly – to also meet exactly in the middle (sticky bows are your friend here).

How about if your gift is an equilateral triangular prism? Here, the length of one side of the triangle gives the thickness T, and your paper should be a little over 3 × T wide and L + (2 × T) long. The extra length is needed because it is harder to fold the excess end bits to make the points meet in the middle. Instead, you can fold the paper to cover the end triangle exactly, by pushing it in from one side at a time and creating a three-layered triangle of paper that sits exactly over the end.

It is also possible to wrap large, flat, square-ish gifts using a diagonal method. If the diagonal of the top surface of your box is D, and the height is H, you can wrap it using a square piece of paper that measures a little over D + (√2 × H) along each side.

Place your gift in the centre of the paper, oriented diagonally, and bring the four corners to meet in the middle of your gift, securing it with one piece of tape or a sticky bow. This will cover all the faces exactly, and look pretty smart too.

For maximum mathematical satisfaction, what you want is to get the pattern on the paper to line up exactly. This is easier for a soft gift, where you can squash it to line up the pattern, but will only work with a box if the distance around it is exactly a multiple of the width of the repeat on the pattern. Otherwise, follow my example (above) and get your own custom wrapping paper printed!

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

*Credit for article given to Katie Steckles*

Algorithms have become integral to our lives. From social media apps to Netflix, algorithms learn your preferences and prioritise the content you are shown. Google Maps and artificial intelligence are nothing without algorithms.

So, we’ve all heard of them, but where does the word “algorithm” even come from?

Over 1,000 years before the internet and smartphone apps, Persian scientist and polymath Muhammad ibn Mūsā al-Khwārizmī invented the concept of algorithms.

In fact, the word itself comes from the Latinised version of his name, “algorithmi”. And, as you might suspect, it’s also related to algebra.

Largely lost to time

Al-Khwārizmī lived from 780 to 850 CE, during the Islamic Golden Age. He is considered the “father of algebra”, and for some, the “grandfather of computer science”.

Yet, few details are known about his life. Many of his original works in Arabic have been lost to time.

It is believed al-Khwārizmī was born in the Khwarazm regionsouth of the Aral Sea in present-day Uzbekistan. He lived during the Abbasid Caliphate, which was a time of remarkable scientific progress in the Islamic Empire.

Al-Khwārizmī made important contributions to mathematics, geography, astronomy and trigonometry. To help provide a more accurate world map, he corrected Alexandrian polymath Ptolemy’s classic cartography book, Geographia.

He produced calculations for tracking the movement of the Sun, Moon and planets. He also wrote about trigonometric functions and produced the first table of tangents.

Al-Khwārizmī was a scholar in the House of Wisdom (Bayt al-Hikmah) in Baghdad. At this intellectual hub, scholars were translating knowledge from around the world into Arabic, synthesising it to make meaningful progress in a range of disciplines. This included mathematics, a field deeply connected to Islam.

There are no images of what al-Khwārizmī looked like, but in 1983 the Soviet Union issued a stamp in honour of his 1,200th birthday. Wikimedia Commons

The ‘father of algebra’

Al-Khwārizmī was a polymath and a religious man. His scientific writings started with dedications to Allah and the Prophet Muhammad. And one of the major projects Islamic mathematicians undertook at the House of Wisdom was to develop algebra.

Around 830 CE, Caliph al-Ma’mun encouraged al-Khwārizmī to write a treatise on algebra, Al-Jabr (or The Compendious Book on Calculation by Completion and Balancing). This became his most important work.

A page from The Compendious Book on Calculation by Completion and Balancing. World Digital Library

At this point, “algebra” had been around for hundreds of years, but al-Khwārizmī was the first to write a definitive book on it. His work was meant to be a practical teaching tool. Its Latin translation was the basis for algebra textbooks in European universities until the 16th century.

In the first part, he introduced the concepts and rules of algebra, and methods for calculating the volumes and areas of shapes. In the second part he provided real-life problems and worked out solutions, such as inheritance cases, the partition of land and calculations for trade.

Al-Khwārizmī didn’t use modern-day mathematical notation with numbers and symbols. Instead, he wrote in simple prose and employed geometric diagrams:

Four roots are equal to twenty, then one root is equal to five, and the square to be formed of it is twenty-five.

In modern-day notation we’d write that like so:

4x = 20, x = 5, x² = 25

Grandfather of computer science

Al-Khwārizmī’s mathematical writings introduced the Hindu-Arabic numerals to Western mathematicians. These are the ten symbols we all use today: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The Hindu-Arabic numerals are important to the history of computing because they use the number zero and a base-ten decimal system. Importantly, this is the numeral system that underpins modern computing technology.

Al-Khwārizmī’s art of calculating mathematical problems laid the foundation for the concept of algorithms. He provided the first detailed explanations for using decimal notation to perform the four basic operations (addition, subtraction, multiplication, division) and computing fractions.

The contrast between algorithmic computations and abacus computations, as shown in Margarita Philosophica (1517). The Bavarian State Library

This was a more efficient computation method than using the abacus. To solve a mathematical equation, al-Khwārizmī systematically moved through a sequence of steps to find the answer. This is the underlying concept of an algorithm.

Algorism, a Medieval Latin term named after al-Khwārizmī, refers to the rules for performing arithmetic using the Hindu-Arabic numeral system. Translated to Latin, al-Khwārizmī’s book on Hindu numerals was titled Algorithmi de Numero Indorum.

In the early 20th century, the word algorithm came into its current definition and usage: “a procedure for solving a mathematical problem in a finite number of steps; a step-by-step procedure for solving a problem”.

Muhammad ibn Mūsā al-Khwārizmī played a central role in the development of mathematics and computer science as we know them today.

The next time you use any digital technology – from your social media feed to your online bank account to your Spotify app – remember that none of it would be possible without the pioneering work of an ancient Persian polymath.

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Credit of the article given to The Conversation

This shows a stretch of the M30. In the bottom left-hand corner, you can see the square frames under which the detectors are placed.

Technicians from Madrid City Council and a team of Pole and Spanish researchers have analysed the density and intensity of traffic on Madrid’s M30 motorway (Spain) throughout the day. By applying mathematical algorithms, they have verified that drivers should pay more attention to the road between 6pm and 8pm to avoid accidents.

Detection devices installed by the Department of Traffic Technologies of Madrid City Council on the M30 motorway and its access roads were used to conduct a scientific study. Researchers from SICE, the traffic management company in charge of this thoroughfare, used past records to develop a new device that determines the time during which more attention should be paid to the road.

This period is the same as the shortest lifetime of spatio-temporal correlations of traffic intensity. In the case of the M30, it has proven to be between 6pm and 8pm, according to the study published in the Central European Journal of Physics.

“Between 6pm and 8pm, the most ‘stop and go’ phenomena occur. In other words, some vehicles break and others set off or accelerate at different speeds,” as explained to SINC by Cristina Beltrán, SICE engineer, who goes on to say that “vehicle speeds at consecutive stretches of the motorway are less correlated during these periods.”

The researcher clarifies that traffic conditions that vary quickly in space and time means that “drivers should always pay more attention on the roads as to whether they should reduce or increase their speed or be aware of road sign recommendations.”

Reference data were taken during a ‘typical week’ on the 13 kilometre stretch of the M30 using detectors at intervals of approximately 500 metres. These devices record the passing speed of vehicles and also how busy the road is (the time that vehicles remain stationary in a given place). Then, using algorithms and models developed by AGH University of Science and Technology (Poland), correlations were analysed.

Free flow, Passing and Congested Traffic

The team focused mainly on the intensity of traffic (vehicles/hour) and density (vehicle/km) during the three phases of traffic: free flow, congested and an intermission named ‘passing’ or synchronised. The easiest to categorise is the first, where intensity and density grow exponentially with hardly any variation, but the other two also show correlations.

This information helps us to take traffic control measures during rush hours, provide speed recommendations that can alter traffic characteristics and offer alternative routes via less congested areas,” outlines Beltrán. “This is all part of Madrid City Council’s objective to actively research new systems for improving traffic flow on the M30.”

The study enjoyed the support of the European Union’s 7th Framework Programme through the SOCIONICAL Project (www.socionical.eu) and the results were cross-referenced with data from the USA’s Insurance Institute for Highway Safety. The work of this scientific and educational organisation is geared towards reducing human and material loss as a result of road accidents.

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Credit of the article given to Spanish Foundation for Science and Technology (FECYT)