Category Archives: Mathematics

Why some people can’t count past “1”: Mathematical thinking is shaped by language and culture

A Tsimane’ woman is tasked with lining up beads to replicate the exact number of white buttons on the table. Credit: Benjamin Pitt, UC Berkeley.

Humans seem to have an innate system for thinking about and organizing numbers, and some scholars have proposed in the past that our brains probably have a built-in mechanism for counting. Such a mechanism would be distinct from language, so humans and other animals would be able to count without having to require language and words such as “one”, “two”, and so on. But that may be only partially true.

A new study suggests that language plays an integral part in shaping our mathematical thinking, as evidenced by members of Bolivia’s Indigenous Tsimane’ community. In this culture, people couldn’t count beyond the “number words” they knew.

“Our finding provides the clearest evidence to date that number words play an active role in people’s ability to represent exact quantities and supports the broader claim that language can enable new conceptual abilities,” said study lead author Benjamin Pitt, a postdoctoral fellow in UC Berkeley’s Computation and Language Lab.

Language drives mathematical reasoning, and not the other way around

Pitt traveled deep into the Amazon basin of Bolivia, using a Jeep, canoe, and finally some hiking to reach the remote villages of the Tsimane’ people. With the help of Tsimane’-Spanish interpreters, the researcher recruited 30 community members with little formal schooling for an experiment.

Each participant was shown a group of objects, such as four buttons, and asked to replicate what they saw using different objects like glass beads. Quite surprisingly, the participants could only match the exact number of objects only in instances when they knew the corresponding words for the numbers.

During a previous trip to the Amazon, Pitt and colleagues studied the organization of numerical information among the Tsimane’ people. Unlike children and adults in industrialized countries, who organize time and numbers or measure things from left to right, the Tsimane’ people organize them freely in either direction.

In this experiment, indigenous members received a set of five cards, where each card displayed a different number of dots. A card containing five dots was placed in the middle of a strip of Velcro, and the participants had to arrange their cards on either side of the middle card, according to their numerical value. The Tsimane’ participants were just as likely to organize the cards from left to right as they were from left to right. The same happened with cards representing fruit ripening over time, a measure of organizing size and time.

“Abstract concepts are things we cannot see or hear or touch, like time, for example. You can’t see time. You can’t touch it. The same goes for numbers. We think and talk about time and numbers constantly. But they’re abstract. So how do we make sense of them? One answer is that we use space to make them tangible — thinking of them along a line from left to right or from top to bottom. My research looks at the root of these types of concepts by studying how they vary across cultures, age groups and even across individuals within a group,” Pitt said.

Although it might sound impossible to Western people, there are to this day some cultures across the world which do not have words for numbers. These languages which do not contain words for numbers in their lexicon are known as anumeric. A prime example constitutes the Pirahã people of Brazil, which have no words for any exact number — not even the number “1”. Their language contains just three imprecise words for quantities: Hòi means “small size or amount,” hoì, means “somewhat larger amount,” and baàgiso indicates to “cause to come together, or many.” 

As a result, Pirahã people have great difficulties consistently performing simple mathematical tasks. For example, one test involved 14 adults in one village that were presented with lines of spools of thread and were asked to create a matching line of empty rubber balloons. The people were not able to do the one-to-one correspondence when the numbers were greater than two or three.

Are Pirahã adults less resourceful and intelligent than a four-year-old American toddler, for whom such a task is trivial? Of course not. In another experiment, when researchers at the University of Miami introduced numerical words, the indigenous people’s performance for mathematical tasks dramatically improved. These findings showed that language is key to mathematical reasoning.

The Tsimane’ study further strengthens this notion, and completes the jigsaw puzzle with new pieces. Unlike American or Pirahã adults, Tsimane’ adults vary greatly from one another in their ability to count. Some can count indefinitely, while others aren’t sure what follows after, say, the number 6 and can only approximate.

“We used a novel data-analysis model to quantify the point at which participants switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants’ verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language,” Pitt and colleagues wrote in their study.

If the atmosphere is chaotic, how can we trust climate models?

Before they can understand how our planet’s climate is changing, scientists first need to understand the basic principles of this complicated system — the gears that keep the Earth’s climate turning. You can make simple models with simple interactions, and this is what happened in the first part of the 20th century. But starting from the 1950s and 1960s, researchers started increasingly incorporating more complex components into their models, using the ever-increasing computing power.

But the more researchers looked at climate (and the atmosphere, in particular), the more they understood that not everything is neat and ordered. Many things are predictable — if you know the state of the system today, you can calculate what it will be like tomorrow with perfect precision. But some components are seemingly chaotic.

Chaos theory studies these well-determined systems and attempts to describe their inner workings and patterns. Chaos theory states that behind the apparent randomness of such systems, there are interconnected mechanisms and self-organization that can be studied. So-called chaotic systems are very sensitive to their initial conditions. In mathematics (and especially in dynamic systems), the initial conditions are the “seed” values that describe a system. Even very small variations in the conditions today can have major consequences in the future.

It’s a lot to get your head around, but if you want to truly study the planet’s climate, this is what you have to get into.

The Butterfly Effect

Edward Lorenz and Ellen Fetter are two of the pioneers of chaos theory. These “heroes of chaos” used a big noisy computer called LGP-30 to develop what we know as chaos theory today.

Lorenz used the computer to run a weather simulation. After a while, he wanted to run the results again, but he just wanted half of the results, so he started the calculations using the results from the previous run as an initial condition. The computer was running everything with six digits, but the results printed were rounded to 3 digits. When the calculations were complete, the result was completely different from the previous one.  

That incident resulted in huge changes for the fields of meteorology, social sciences and even pandemic strategies. A famous phrase often used to describe this type of situation is “the butterfly effect”. You may be familiar with the idea behind it: “The flap of a butterfly’s wings in Brazil can set off a Tornado in Texas”. This summarizes the whole idea behind the small change in the initial conditions, and how small shifts in seemingly chaotic systems can lead to big changes. 

Simulation of Lorenz attractor of a chaotic system. Wikimedia Commons.

To get the idea, Lorenz went on to construct a diagram that depicts this chaos. It is called the Lorenz Attractor, and basically, it displays the trajectory of a particle described by a simple set of equations. The particle starts from a point and spirals around a critical point — a chaotic system is not cyclical so it never returns to the original point. After a while, it exceeds some distance and starts spiraling around another critical point, forming the shape of a butterfly. 

Why is it chaotic?

If the atmosphere is chaotic, how can we make predictions about it? First, let’s clarify two things. Predicting the weather is totally different from predicting the climate. Climate is a long period of atmospheric events, on the scale of decades, centuries, or even more. The weather is what we experience within hours, days, or weeks. 

Weather forecasting is based on forecast models which focus on predicting conditions for a few days. To make a forecast for tomorrow, the models need today’s observations as the initial condition. The observations aren’t perfect due to small deviations from reality but have improved substantially due to increases in computation power and satellites.

However, fluctuations contribute to making things harder to predict because of chaos. There is a limit to when the predictions are accurate — typically, no more than a few days. Anything longer than that makes the predictions not trustworthy. 

Thankfully, our knowledge about the atmosphere and technological advances made predictions better compared to 30 years ago. Unfortunately, there are still uncertainties due to the chaotic atmospheric behavior. This is illustrated in the image below, the model’s efficiency is compared between the day’s ranges. The 3-day forecast is always more accurate, compared to predictions from 5 to 10 days. 

The evolution of weather predictability. Credits: Shapiro et al. (AMS).

This image also shows an interesting societal issue. The Northern Hemisphere has always been better at predicting the weather than the South.

This happens because this region contains a larger number of richer countries that developed advanced science and technology earlier than the Global South, and have more monitoring stations in operation. Consequently, they used to have many more resources for observing the weather than poorer countries. Without these observations, you don’t have initial conditions to use for comparison and modeling. This started to change around the late ’90s and early 2000s when space agencies launched weather satellites that observe a larger area of the planet.

Predicting the climate

Predicting the climate is a different challenge, and in some ways, is surprisingly easier than predicting the weather. A longer period of time means more statistical predictability added to the problem. Take a game of chance, for instance. If you throw dice once and try to guess what you’ll get, the odds are stacked against you. But throw a dice a million times, and you have a pretty good idea what you’ll get. Similarly, when it comes to climate, a bunch of events are connected on average to long-term conditions and taken together, may be easier to predict.

In terms of models, there are many different aspects of weather and climate models. Weather models can predict where and when an atmospheric event happens. Climate models don’t focus on where exactly something will happen, but they care how many events happen on average in a specific period.

When it comes to climate, the Lorenz Attractor is the average of the underlying system conditions — the wings of the butterfly as a whole. Scientists use an ensemble of smaller models to ‘fill the butterfly’ with possibilities that on average represent a possible outcome, and figure out how the system as a whole is likely to evolve. That’s why climate models predictions and projections like those from the IPCC are extremely reliable, even when dealing with a complex, seemingly chaotic system.

Comparing models

Today, climate scientists have the computer power to average a bunch of models trying to predict the same climate pattern, further finessing the results. They can also carry out simulations with the same model, changing the initial conditions slightly and averaging the results. This provides a good indicator of what could happen for each outcome. Even further than that, there is a comparative workforce between the scientific community to show that independent models from independent science groups are agreeing about the effects of the climate crisis.

Organized in 1995, the Coupled Model Intercomparison Project (CMIP) is a way of analysing different models. This workforce makes sure scientists are comparing the same scenario but with different details in the calculations. With many results pointing to a similar outcome, the simulations are even more reliable.

Changes in global surface temperature over the past 170 years (black line) relative to 1850–1900 and annually-averaged, compared to CMIP6 climate model simulations of the temperature response to both human and natural drivers (red), and to only natural drivers (solar and volcanic activity, green). Solid coloured lines show the multi-model average, and coloured shades show the range (“very likely”) of simulations. Source: IPCC AR6 WGI>

Ultimately, predicting the climate is not like we are going to predict if it will be rainy on January 27 2122. Climate predictions focus on the average conditions that a particular season of an oscillatory event will be like. Despite the chaotic nature of the atmosphere, thanks to climate’s time length and statistical predictability, long-term climate predictions can be reliably made.

New statistical approach aims to predict when low-probability, high-impact events happen

A team of researchers from the U.S. and Hong Kong is working to develop new methods of statistical analysis that may let us predict the risks of very rare but dramatic events such as pandemics, earthquakes, or meteorite strikes happening in the future.

Image via Pixabay.

Our lives over these last two years have been profoundly marked by the pandemic — and although researchers warned us about the risk of a pandemic, society was very much surprised. But what if we could statistically predict the risk of such an event happening in advance?

An international team of researchers is working towards that exact goal by developing a whole new way to perform statistical analyses. Typically, events of such rarity are very hard to study through the prism of statistical methods, as they simply happen too rarely to yield reliable conclusions.

The method is in its early stages and, as such, hasn’t proven itself. But the team is confident that their work can help policymakers better prepare for world-spanning, dramatic events in the future.

Black swans

“Though they are by definition rare, such events do occur and they matter; we hope this is a useful set of tools to understand and calculate these risks better,” said mathematical biologist Joel Cohen, a professor at the Rockefeller University and at the Earth Institute of Columbia University, and a co-author of the study describing the findings.

The team hopes that their work will give statisticians an effective tool with which to analyze sets of data when it contains very sparse points of data, as is the case for very dramatic (positive or negative) events. This, they argue, would give government officials and other decision-makers a way to make informed decisions when planning for such events in the future.

Statistics by now is a tried and true field of mathematics. It’s one of our best tools when trying to make sense of the world around us and, generally, serves us well. However, the quality of the conclusions statistics can draw from a dataset relies directly on how rich those datasets are, and the quality of the information they contain. As such, statistics has a very hard time dealing with events that are exceedingly rare.

That hasn’t stopped statisticians from trying to apply their methods to rare-but-extreme events, however, over the last century or so. It’s still a relatively new field of research in the grand scheme of things, so we’re still learning what works here and what doesn’t. Where a worker would need to use the appropriate tool for the job at hand, statisticians need to apply the right calculation method on their dataset; which method they employ has a direct impact on which conclusions they draw, and how reliably these reflect reality.

Two important parameters when processing a dataset are the average value and the variance. You’re already familiar with what an average value is. The variance, however, shows how far apart the values that make up that average are. For example, both 0 and 100, as well as 49 and 51, average out to 50; the first set, however, has a much larger variance than the latter.

Black swan theory describes events that come as a surprise but have a major effect and then are inappropriately rationalized after the fact with the benefit of hindsight. The new research doesn’t only focus on black swans, but on all unlikely events that would have a major impact.

For typical sets, the average value and the variance can both be defined by finite numbers. In the case of the events that made the object of this study, however, the sheer rarity with which they take place can push these numbers towards ridiculous values bordering on infinity. World wars, for example, have been extremely rare events in human history, but each one has also had an incredibly large effect, shaping the world into what it is today.

“There’s a category where large events happen very rarely, but often enough to drive the average and/or the variance towards infinity,” said Cohen.

Such datasets require new tools to be properly handled, the team argues. If we can make heads and tails of it, however, we could be much better prepared for them, and see a greater return on investments into preparedness. Governments and other ruling bodies would obviously stand to benefit from having such information on hand.

Being able to accurately predict the risk of dramatic events would also benefit us as individuals, and provide important tangible benefits in society. From allowing us better plan out our lives (who here wouldn’t have liked to know that the pandemic was going to happen in advance?), to better preparing for threatening events, to giving us arguments for lower insurance premiums, such information would definitely be useful to have. If nothing bad is likely to happen during our lifetimes, you could argue, wouldn’t it make sense for my life insurance policy premiums to be lower? The insurance industry in the US alone is worth over $1 trillion and making the system more efficient could amount to major savings.

But does it work?

The authors started from mathematical models used to calculate risk and examined whether they can be adapted to analyze low-probability, very high-impact events with infinite mean and variance. The standard approach these methods use involves semi-variances: the practice of separating the dataset in ‘below-average’ and ‘above-average’ halves, then examining the risk in each. Still, this didn’t provide reliable data.

What does work, the authors explain, is to examine the log (logarithmic function) of the average to the log of the semi-variance in each half of the dataset. Logarithmic functions are the reverse of exponentials, just like division is the reverse of multiplication. They’re a very powerful tool when you’re dealing with massive, long numbers, as they simplify the picture without cutting out any meaningful data — ideal for studying the kind of numbers produced by rare events.

“Without the logs, you get less useful information,” Cohen said. “But with the logs, the limiting behavior for large samples of data gives you information about the shape of the underlying distribution, which is very useful.”

While this study isn’t the end-all-be-all of the topic, it does provide a strong foundation for other researchers to build upon. For now, although new and in their infancy, the findings do hold promise. Right now, they’re the closest we’ve gotten to a formula that can predict when something big is going to happen.

“We think there are practical applications for financial mathematics, for agricultural economics, and potentially even epidemics, but since it’s so new, we’re not even sure what the most useful areas might be,” Cohen said. “We just opened up this world. It’s just at the beginning.”

The paper Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data” has been published in the journal Proceedings of the National Academy of Sciences.

Paradoxes are contradictory.

The appeal of the paradox — mankind’s fascination with self-contradicting ideas

Just like true love, a paradox cannot be explained with logic alone. Simply put, a paradox is a self-contradicting statement. Any idea, situation, puzzle, statement, or question that challenges your ability to reason, and leads you to an unexpected and seemingly illogical conclusion, can be considered a paradox.

The classic paradox example is the so-called Grandfather Paradox. Imagine a psychotic time traveler who goes back in time and kills his grandfather before his father is conceived. This means that the traveler wouldn’t have been conceived, and if he wasn’t conceived, then who went back to kill his grandfather?

The answer to this theoretical time travel mystery is still unclear, as is the case with many other interesting paradoxes. In this information age, logic helps us understand what is known to us but a paradox serves as a reminder of what else we need to know. Let’s dive in.

Image credits: cottonbro/pexels

How do you define a paradox?

A paradox is a thought that can sound reasonable and illogical at the same time. The Cambridge dictionary defines paradox as a situation that could be true but is impossible to comprehend due to its contrary characteristics. In the Greek language, ‘para’ translates to ‘abnormal’, ‘distinct, or ‘contrary’’ and ‘dox’ means ‘idea’ or ‘opinion’. Therefore, according to some Greek philosophers, a paradox is an abnormal or self-contrary belief or idea that ultimately leads to an unsolvable contradiction.  

You don’t need time travel to create a crazy paradox. For instance, in the famous crocodile paradox (of which there are many variations), a magical crocodile steals a child and promises to return it only if the father can guess correctly what the crocodile will do. If the father says “The child will not be returned” — what can the crocodile do? If he doesn’t return the child, that means the father’s guess was true so he should have returned the child. If he does return it, then the father’s guess was false, so he shouldn’t have. It’s a paradox, nothing the crocodile does can satisfy the situation.

The face a crocodile makes when faced with an unsolvable paradox. Image credits: Pixabay/pexels.

This paradox is believed to have originated centuries ago in ancient Greece, but there are hundreds of different paradoxes that are found in literature, mathematics, philosophy, science, and various other domains as well. Though a true paradox can seem both true and false at the same time, logic tends to suggest most of the paradoxes as invalid statements. 

There are four main types of paradoxes:

  1. Falsidical paradox: A paradox that leads to a false conclusion resulting from a misconception or false belief. For example, Zeno’s Achilles and tortoise.
  1. Veridical paradox: When a situation or statement tells us about a result that sounds absurd but is actually valid by logic. Shrodinger’s cat is a famous example of a veridical paradox.
  1. Antinomy paradox: A question, puzzle, or statement that does not lead to a solution or conclusion is called an antimony paradox (also known as self-referential paradox). One of its examples is the Barber’s paradox (discussed below).
  1. Dialetheia: When the opposite of a situation and the original situation co-exist together, such a paradox is called dialetheia. No concrete examples are known but some real-life situations can be considered dialetehia (for example when you are standing at the kitchen door, and one of your family members ask you if you are in the kitchen? You are right whether you answer yes or no.    

Why paradoxes matter

Paradoxes are important because they make us think. They force us to reassess what we thought we knew and ponder things from unusual perspectives. A paradox mindset, in which we embrace contradicting (or seemingly contradicting ideas) is a key to success, some studies have shown. Leading thinkers were found to spend considerable time developing ideas and counter-ideas simultaneously, something called the Janusian process.

Studying paradoxes is also important, especially for mathematicians. Mathematicians love to break everything into small pieces and define things carefully, and they do that with paradoxes. For instance, let’s take a simple paradox called the Temperature paradox, which states:

“If the temperature is 90 and the temperature is rising, that would seem to entail that 90 is rising.”

Obviously, 90 is not rising, it’s a fixed number, it can’t be rising. We know that intuitively, but how do we prove it? American mathematician and philosopher Richard Montague dealt with this paradox (and many others), and explained that the paradox emerges from linguistic vagueness, which can be addressed through mathematical clarity. The linguistic formalization of the paradox would go something like this:

  1. The temperature is rising.
  2. The temperature is ninety.
  3. Therefore, ninety is rising. (invalid conclusion)

But the mathematical formalization implies that point 1. marks how the temperature changes over time, while point 2. makes an assertion about the temperature at a particular point in time. Therefore, we cannot draw conclusions based on this single point in time.

This type of paradox, which emerges from language issues and ambiguity is not often important, but other paradoxes, especially those that can’t be resolved through normal means, hold importance because they help us find better definitions of objects and relationships. A good example of this is Curry’s paradox.

Now that we know the types of paradoxes and why they matter, let’s look at of the most popular and insane paradoxes of all time:

Paradox examples

“This sentence is false”

This so-called liar’s paradox is the canonical example of a self-referential paradox. Other classic examples are “Is the answer to this question ‘no’?”, and “I’m lying.”

Mathematicians have tried to dissect and analyze this paradox in great detail because it can hold some importance to defining inherent limitations of mathematical axioms.  The liar’s paradox was used in 1931 by a mathematician named Kurt Gödel to define mathematical axioms, but the paradox itself dates back to at least 600 BC, when the semi-mythical seer Epimenides, a Cretan, reportedly stated that “All Cretans are liars.”

The Barber paradox

The scene of a Bucharest-based barbershop in 1842. Image credits: Charles Doussault/Wikimedia Commons

Proposed by British mathematician Bertrand Russell, this paradox states that if a barber is defined as the person who only shaves individuals who do not shave on their own, then who shaves the barber? In this case, the barber would shave himself — but then, according to the definition, he is no longer the barber as he cannot shave a person who would shave on their own. 

Now, if he is not shaving on his own, then he is among those who are supposed to be shaved by the barber. In this case, also, the barber has to shave himself. Therefore, the barber paradox suggests that no such barber can ever exist who is called a barber because he only shaves people who do not do their own shave. Well, then what the heck even is a barber?

Sorites’ paradox

If there is a heap of sand that has one million grains, and one by one, grains are being removed from the heap such that at the end of the process only one grain remains, would it still be seen as a heap? If not then when does the heap of sand become a non-heap? Sounds crazy, right? But that is the Sorites paradox given by Eubuildus of Miletus around the fourth century BCE, and till this day, no math genius has been able to give a logical solution to this problem.

Another similar type of puzzle is the so-called ship of Theseus. The mythological hero Theseus sails on to his adventures, and at some point, one of the ship parts needs replacing. It’s still the same ship, right? Just one part was replaced. But part after part, every component on the ship is replaced. Is it still the same ship? If not, when did it stop being the same ship?

Zeno’s Achilles and the tortoise

Achilles and the tortoise.

In this paradox developed by the ancient Greek philosopher Zeno, there is a race between the great Greek warrior Achilles and a tortoise. The tortoise is given a head start of 100 meters. Achilles runs faster than the tortoise so it will catch up to it. But here’s how Zeno looked at things:

  • Step #1: Achilles runs to the tortoise’s starting point while the tortoise walks forward.
  • Step #2: Achilles runs to where the tortoise was at the end of Step #1, while the tortoise goes a bit further.
  • Step #3: Achilles runs to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
  • … and so on.

The gaps get smaller and smaller every time, but there is an infinity of these steps, so how can Achilles overcome an infinite number of gaps and catch up to the turtle? How does anything catch up to anything, for that matter? Obviously, things do catch up to other things, so what’s going on here?

The ancient Greeks lacked the mathematical tools to address this paradox, but nowadays, we know better. There may be an infinite number of steps, but they are also infinitely small. It’s a bit like how 1/2 + 1/4 + 1/8 + 1/16 +… to infinite adds up to 1. It’s an infinite number of steps, but the steps become infinitely small, and in the end, they add up to something tangible.

Animalia Paradoxa – The classification of magical creatures

This is actually not a paradox but a biological classification of the beasts and magical creatures that are also mentioned in ancient storybooks. In the versions of Systema Naturae that arrived before its sixth edition, author Carl Linnaeus (father of modern taxonomy) has listed creatures like Hydra (snake with seven faces), Draco (a dragon with bat-like wings and ability to spit fire), Unicorn (beautiful single-horned horse), Lamia (half-human half-animal), etc.

From a scientific point of view, these creatures don’t exist so then why did a genius like Carl Linnaeus mention such creatures in his greatest scientific work? It seems paradoxical that the man who defined our classification of biological creatures would introduce unreal creatures; one might say it’s a bit paradoxical.

A painting of Hercules and the Lernaean Hydra by Gustave Moreau. Image credits: The Yorck Project/Wikimedia Commons

Paradoxes have a unique draw because they appeal to human curiosity and mystery. They seem to ignite the curiosity of the human mind for thousands of years and will likely continue to do so for many years to come. 

Mathematicians know how to solve traffic jams — we just need policymakers to listen

The M25 motorway circling around London, UK, can be a nightmare. Every day at rush hour, the traffic seems to stop. Drivers turn on the radio, have a sip of coffee, and prepare for a very long few miles. It’s hard to imagine going in and out of London without a traffic jam, and England’s capital is hardly an exception. Most if not all major cities are plagued by traffic jams.

Alexander Krylatov, a mathematics professor at St. Petersburg University, hates jams as much as the next guy. But unlike the next guy, Krylatov (alongside fellow mathematician Victor Zakharov) wrote a book about how traffic jams could be stopped — or at least reduced.

“Traffic flow theory began to develop 100 years ago,” the book starts off. A road network can be analyzed as a complex system composed of interdependent elements such as streets, junctions, traffic lights, traffic flows, and the like. Instead of allowing urban designers and engineers to do the job, let mathematicians take over, the book seems to say.

It makes sense. If we can realistically separate traffic into its constituent elements and understand the interplay between the different elements, we could make the whole system more efficient. To do this, we need four basic elements, says Krylatov.

  1. All drivers need to be on the same navigation system. Cars can only synchronize and efficiently reroute if they have access to the same information. Different systems can work, but they need to communicate together to solve traffic jams.
  2. Parking bans. Too many urban roads are too narrow and cannot be widened. Math-based traffic models can show where these ad-hoc parking spots can be turned into lanes.
  3. Green lanes. Many cities are already incentivizing electric car use. To do that, just treat them as buses and offer them special lanes.
  4. Digital twins. Build a digital copy of the real traffic. This model will be extremely useful. Traffic demands and available infrastructure can only be balanced with digital modeling that creates an entire “twin” of existing roadways. The software will be “an extremely useful thought tool in the hands of transport engineers.”

The principles aren’t necessarily new, they’re adapted from what mathematicians have been saying for half a century. In 1952, English mathematician and transport analyst Glen Wardrop first spoke of the principle of equilibrium — the idea that you can simulate traffic mathematically by assuming that each driver is selfish and focuses on their personal goals. Models that are built based on this principle are more accurate, because they incorporate a realistic component to how people drive on the roads (realistically, it’s probably safe to assume that most drivers are selfish). But driver behavior can be shifted subtly through infrastructure planning.

If drivers (or cars) can communicate with each other, they can adjust more efficiently. We’re already seeing this with the likes of mapping software like Waze or Google Maps — if the shorter road is crowded, another route is recommended, and the system flow is optimized.

Of course, measures like banning parking on some streets are somewhere between unpopular and impossible. But widening some roads and narrowing others (by shifting parking spaces from one to the other, for instance) could substantially improve traffic flow.

The first challenge, however, is to digitize everything.

‘By building digital twins of the transport system and using them to optimise flows, it will be possible to achieve a balance between the demand for using the system and the infrastructure capabilities. It is unlikely that this can be done without economic digitalisation,’ notes Victor Zakharov.

Mathematical study also reveals some counterintuitive aspects. For instance, the construction of road junctions sometimes creates more inefficiency.

All in all, the mathematical approach is better than the engineering one, researchers sum up.

‘Every year, a considerable budget is allocated for improving roads. The mathematical theory of traffic assignment suggests a set of solutions for the efficient management of these funds,’ Zakharov says ‘The mathematical approach in this case is superior to the engineering and economic one. It makes it possible to analyse the entire transport network, with respect to the complex laws of the mutual influence of its individual elements on each other. We have done a lot of work in the field of simulating traffic flows and networks. Now we want to pass on to the stage of putting our ideas into practice.’

Whether or not their approach becomes popular remains to be seen. Traffic engineers are increasingly using mathematical models, but there’s also a political aspect: sometimes, efficient change is simply unpopular, and decisionmakers are often unwilling to displease their voters.

As always, it seems, the way to improve things involves getting people to listen to math. But they rarely do.

Why is an egg shaped like an egg? Turns out, there’s some serious math behind it

Mathematicians, biologists, and even engineers have invested a surprising amount of time and effort into analyzing the shape of an egg. However, there are a few aspects that make eggs’ geometry special.

The egg is structurally sound enough to bear weight, while also being small enough to exit the body of some 10,000 species on Earth. The egg also rolls away on its side, but once it’s laid vertically, it doesn’t roll anymore.

Despite many efforts, however, eggs have resisted attempts to define their shape with an equation. Until now.

Image credits: Maria Ionova.

The new mathematical formula is based on four parameters: egg length, maximum breadth, the shift of the vertical axis, and the diameter at one-quarter of the egg length. Sounds simple enough, though the formula itself is anything but simple (as we’ll see in a moment).

Analysis of all egg shapes can be done using four geometric figures, the researchers explain — sphere, ellipsoid, ovoid, and pyriform (conical or pear-shaped). The first three are clearly defined mathematically, but the fourth is more problematic.

“Pyriform” is basically a fancy word for pear-shaped, and a mathematical formula for the pyriform profile hadn’t been developed until now. “To rectify this”, the researchers explain, “we introduce an additional function into the ovoid formula.”

Image credits: Narushin et al.

There’s a philosophical reward to mathematically understanding eggs, says Dr. Michael Romanov, Visiting Researcher at the University of Kent, and study author, explains:

“This mathematical equation underlines our understanding and appreciation of a certain philosophical harmony between mathematics and biology, and from those two a way towards further comprehension of our universe, understood neatly in the shape of an egg.”

But there’s more than just cracking one of nature’s mysteries here. There are some very useful applications to this formula. For starters, it is useful in biology research (where it could help researchers better understand egg evolution and egg incubation for desired species) and in the poultry industry, where it could help to optimize equipment that works with egg shape or volume. It could also help researchers design new bio-inspired structures.

Dr. Valeriy Narushin, a former visiting researcher at the University of Kent, added:

“We look forward to seeing the application of this formula across industries, from art to technology, architecture to agriculture. This breakthrough reveals why such collaborative research from separate disciplines is essential.”

Well, all that’s left now is to look at the formula itself. Good luck.

The formula is applicable for any egg, the researchers say.

If you’d like to take a deeper dive into what it all means, feel free to check out the full study here.

Scientists calculate Pi down to 62.8 trillion digits, set new record

Using one of the most powerful supercomputers in the world, researchers in Switzerland computed the mathematical constant pi (π) down to 62.8 trillion digits. This approximation beat the previous world record of 50 trillion decimal places, and was performed 3.5 times quicker.

Neverending digits

Pi is the ratio of a circle’s diameter to its circumference. No matter the size of a circle, its diameter will always be roughly 3.14 times shorter than its circumference — every time.

However, pi is an irrational number, meaning a fraction cannot convey its exact value. As an “infinite decimal”, after the decimal point, pi’s digits go on forever and ever.

For most applications, knowing the value of pi down to a dozen digits is precise enough to calculate uncertainty in the circumference smaller than the size of an atom. For astrophysical applications, 152 decimals of pi are probably the most you’ll ever need. At this value, you can accurately calculate a sphere whose diameter is the size of the observable universe with uncertainty in the circumference less than the Planck length — the smallest unit of distance.

However, the lack of practicality in computing pi past a dozen digits never stopped mathematicians from trying to do it. In 1873, mathematician William Shanks computer pi to 707 digits all by hand. Now, researchers affiliated with the Graubuenden University of Applied Sciences in Switzerland have calculated pi with an unprecedented level of exactitude, down to 62.8 trillion figures.

To do so, the scientists used a supercomputer that crunched pi for 108 days and nine hours without a break. Until the Guinness Book of Records officially certifies the feat, the researchers have only revealed the final ten digits they calculated: 7817924264.

But is there any practical use to this new world record? Indirectly, yes. As mentioned earlier, calculating pi to more than a dozen digits is overkill in 99.99% of cases. However, this doesn’t mean that this endeavor is just about flexing. Computing pi to these many digits involves overcoming many mathematical and computational challenges. The optimization techniques developed over the course of these computations could be applied to other “practical” fields, such as RNA analysis, fluid dynamics simulations, and textual analysis.

Oldest example of applied geometry found in 3,700-year-old Babylonian clay tablet

The Si.427 clay tablet, which was etched by an Old Babylonian scribe with a stylus sometime between 1900 and 1600 BC. Credit: UNSW Sydney.

Almost four millennia ago, two wealthy Mesopotamian landowners quarreled over a plot of land, each claiming they were the rightful owner. The dispute was solved not through sheer force and violence — this was the kingdom that laid out the very first written laws after all — but instead through rather modern-style mediation. A skilled surveyor arrived at a site and with his trusted tools, he divided the disputed lands at the border into equal plots and the two landowners were back to their happy neighborly selves.

Such Babylonian surveyors were in charge of writing up the first cadastral documents in known history, during a time when citizens were entrusted with private property which had to be delineated from common lands. These ancient surveyors, known as scribes, didn’t have total stations and GPS at their disposal, and frankly, they didn’t need them. They were very well capable of accurately measuring and dividing plots of land using a yardstick and their mathematical skill.

A 3,700-year-old clay tablet, known as Si.427, is illustrative in this regard. It shows how Babylonian surveyors must have performed geometric operations, even using Pythagorean triples to accurately make right angles, more than a thousand years before the mighty Greek philosopher was born.

In a new study published today in Foundations of Science, Dr. Daniel Mansfield, a mathematician at the University of New South Wales in Australia, explains the rich significance behind what may very well be the oldest example of applied geometry in the world.

Proto-trigonometry: the geometry for the ground

Dr. Daniel Mansfield with the Plimpton 322 Babylonian clay tablet in the Rare Book and Manuscript Library at Columbia University in New York. Image: UNSW/Andrew Kelly.

Although Mansfield is a mathematician, his research into Si.427 looked more like that of an archaeologist. The tablet was discovered in Baghdad at the end of the 19th-century but had since changed hands many times and its location remained an enigma. However, Mansfield had heard about it while studying thousands of Babylonian fragments relating to mathematical applications in the old Mesopotamian kingdom.

In 2017, Mansfield studied another similar tablet from the same period, known as Plimpton 322, revealing that its purpose was that of a trigonometric table of sorts. Babylonians did not actually use trigonometry as we know it, as in the branch of mathematics concerned with specific functions of angles and their application to calculations. In fact, these ancient scribes understood only one angle: the right angle.

While Plimpton 322 isn’t a trigonometric table in the conventional sense, it lists a table of rectangles useful in practical measurements. Specifically, it lists Pythagorean triples, right triangles whose three sides are all integers where the square of the hypotenuse equals the sum of the squares of the other two sides.

For instance, a rectangle with sides 3 and 4, and a diagonal of 5 can be divided into two equal halves at the diagonal, leaving two perfect right-angle triangles.

Plimpton 322 doesn’t list all possible Pythagorean triples but rather compiles a number of triples, both as rectangles and right triangles, that were likely commonly encountered in surveying work. It was very much practical rather than theoretical work.

Plimpton 322, a 3700-year-old Babylonian tablet. Credit: UNSW.

This limitation was owed to the sexagesimal (base 60) Babylonian number system, which means only some Pythagorean shapes can be used in practice. In this system, numbers are written by adding symbols that represent either 10 or 1, in that order. For instance, the number 5 is written as ‘blank space’ to signify no 10s and by five 1s. The number 16 is written as one 10 followed by six 1s. Then these number signs 1–59 can in turn be strung together to write numerals of any length.

The base 60 number system is actually still in use in some instances of our lives, despite the ubiquity of base 10. For instance, we still count sixty minutes in an hour and sixty seconds in a minute, and measure angles in multiples and fractions of 60. This is the legacy of Greek astronomers who adopted the Babylonian base 60 system because their own system was not as suited for astronomical calculations.

But since it is difficult to write and calculate with prime numbers bigger than 5 in base 60, only some Pythagorean triangles were used. This is why Mansfield calls the Babylonian geometry proto-trigonometry, an intermediate step towards modern trigonometry involving sin, cos, and tan.

However, it was not clear how tables such as those found on Plimpton 322 were actually used in practice. Mansfield had heard about another tablet that contained triangles and rectangles, but despite his best efforts to track it down, speaking with many officials at Turkish government ministries and museums (the last known leads for the tablet), he couldn’t find it. However, one day in mid-2018, the mathematician received a photo of Si.427 in his inbox.

“I ran out of my office and found two colleagues in the middle of a meeting. I burst into their meeting and I rambled exciting things about “Pythagoras” and “Babylon”, and my colleagues were kind enough to smile while I got all my excitement out,” he recounted.

Together, Plimpton 322 and Si.427 paint a picture of how mathematics was used in ancient Babylon. Rather than using trigonometric concepts to study the night’s sky, as the ancient Greeks had in the second century BC, the alternative proto-trigonometry employed by Babylonians seems to mostly solve problems related to the ground.

“We knew that the Babylonians were mathematically advanced. They knew all about the geometry of right triangles, but we didn’t know why. What were they doing with right triangles? What were they using them for? This question of “why” motivated me to look at Babylonian artefacts from museums, libraries and private collections around the world. What I discovered is that the Babylonians were applying their understanding of right triangles to accurately measure and subdivide land,” Mansfield told ZME Science.

“The way we understand trigonometry harks back to ancient Greek astronomers. I like to think of the Babylonian understanding of right triangles as an unexpected prequel, which really is an independent story because the Babylonians weren’t using it to measure the stars, they were using it to measure the ground. Perhaps some aspects of this knowledge were transferred to other civilizations, but I’ve not seen any evidence of this,” he added.

Although the discovery of Plimpton 322 prompted some to speculate that its purpose was linked to the construction of palaces and temples, canals, and other practical works, it was only with the discovery of Si.427 that all the jigsaw pictures came together. During the period that these tablets were etched, Babylon was undergoing social change where much land moved became private. Designating proper boundaries without affecting neighborly relationships was essential, which is where the surveyors and their right triangles came in.

Next, Mansfield plans on studying what other applications besides surveying the Babylonians had for their proto-trigonometric tablets. He’s also interested in whether there are any real-world applications for these simple but fast techniques in our modern era. “For example, this approach might be of benefit in computer graphics or any application where speed is more important than precision,” he said.

And as a comical illustration of the essential role surveyors had in the Old Babylon period, here’s a hilarious poem in which an older student berates a younger one for his incompetence in surveying a field. “It’s essentially a 4000-year-old diss track,” Mansfield told me.

Go to divide a plot, and you are not able to divide the plot;
go to apportion a field, and you cannot even hold the tape and rod properly.
The field pegs you are unable to place; you cannot figure out its shape,
so that when wronged men have a quarrel you are not able to bring peace,
but you allow brother to attack brother.
Among the scribes, you (alone) are unfit for the clay.

The Pythagorean cup – the vessel that spills your booze if you’re too greedy

Pythagoras, an important philosopher, mathematician, and music theorist, was born on the island of Samos, probably in 585 BC. A lesser-known fact is that he enjoyed a good prank. Credit: Wikimedia Commons.

Known as one of the most enlightened minds of antiquity, Pythagoras of Samos left an invaluable legacy to the world. As a renowned mathematician, philosopher, and mystic, Pythagoras’ ideas were tremendously influential on the ancient world. 

Whilst the 6th century BC Greek philosopher is foremost known for his mathematical innovations, such as his study of the property of numbers and his well-known geometric theorem that relates the sides of a right triangle in a simple way, not many people know he was also an accomplished inventor.

One of his many inventions is the Pythagorean cup, also known as the greedy cup – a clever and entertaining vessel designed to hold an optimal amount of wine, forcing people to imbibe only in moderation — a virtue of great regard among ancient Greeks.

If the user was too greedy and poured wine over the designed threshold, the cup would spill its entire content. Imagine the dismay and stupefaction a glutton felt when his precious brew perished on the floor.

Indeed, Pythagoras was always a professor at heart, and his contraption may have taught a couple of fellows about the virtue of moderation. But the ‘greedy cup’ also gives a lesson in ingenuity.

On the outside, the 2,500-year-old design looks like any other cup. However, when you make a cross-section, it becomes clear this is no ordinary vessel. At the center of the cup lies a mechanism consisting of a hollow pipe-like chamber that follows an opening, starting from the bottom of the liquid-holding part of the cup, up to the top of the central column that makes up the cup’s core, and back down 180 degrees out the bottom.

pythagorean_cup_cross_section

As the pipe curls over the top of the U-shaped central column, its floor marks an imaginary line. If you fill the cup over this horizontal line, the liquid will begin to siphon out the bottom and onto your lap or feet — the entire content of the cup, even the liquid below the line.

A Pythagoras cup you can find in Greece or online.

cross_section_pythagorean_cup

Cross-section of a Pythagorean cup.

The siphon is created due to the interplay between gravity and hydrostatic pressure. Water tends to flow from the area of high pressure to the area of low pressure. When the liquid level rises such that it fills the U-shaped chamber, the liquid will start to fall due to gravity. As gravity pulls the water column down the pipe of the Pythagorean cup, the lower pressure thus created on the other end causes the liquid to be overpowered, subsequently allowing itself to be “dragged” along, stopping only when the water level either falls below the intake or the outlet. Some modern toilets operate on the same principle: when the water level in the bowl rises high enough, a siphon is created, flushing the toilet.

According to one account, which may be more myth than history, Pythagoras got the idea for his fabled cup while supervising workers or students at a water supply project in Samos island. There, he was troubled by the debauchery of the workers, so he came up with this ancient prank to ensure they only drank in moderation.

Today, Pythagorean cups can be bought all over Greece at souvenir shops and can even be ordered on eBay. If you’re up for pranks, this is a great gift. Be wary of wine though since it stains.

Eco-friendly geometry: smart pasta can halve packaging waste at no extra cost

Pasta comes in a variety of shapes and sizes — from the plain and simple to all sorts of quirky spirals. But for the most part, they have one thing in common: they’re not using space very effectively. But a new study may change that.

Researchers from Carnegie Mellon University have found a way to change that, designing new types of pasta that use less packaging and are easier to transport, reducing both transportation emissions and packaging plastic.

Unconventional pasta shapes use up less space but spring to life in water. Image credits: Morphing Matter Lab. Carnegie Mellon University.

Pasta is big business. In 2019, nearly 16 million tons of pasta were produced in the world — up from 7 million tons produced 20 years ago. That adds up to billions of packets that are transported, stored, and ultimately discarded across the world.

Since pasta often has such odd shape, pasta packages often end with a lot of wasted space, which also has to be transported and stored. Using less space means less trucks driving across states and less plastic.

Carnegie Mellon University’s Morphing Matter Lab director Lining Yao had an idea on how that could be reduced — with a bit of help from an old friend: geometry.

“By tuning the grooving pattern, we can achieve both zero (e.g., helices) and nonzero (e.g., saddles) Gaussian curvature geometries,” the study reads. It then goes on to translate what this means. “This mechanism allows us to demonstrate approaches that could improve the efficiency of certain food manufacturing processes and facilitate the sustainable packaging of food, for instance, by creating morphing pasta that can be flat-packed to reduce the air space in the packaging.”

They started out with a computer simulation to see how different shapes would achieve the goal. They tried various designs, including helixes, saddles, twists, and even boxes. After they settled on a few efficient shapes, they put it to the boiler test — quite literally.

Flat-packed pasta before and after boiling. Image credits: Morphing Matter Lab. Carnegie Mellon University.

Speaking to Inverse, Yao says wasted space could be reduced by 60% by flat-packing pasta — and that’s just the start of it. The method could also be used for things like wagashi or gelatin products. The method could also be used to design more complex and fancy shapes for special occasions. In dry form, a piece of pasta could look like a disc, but when boiled, it could become a rose flower.

Credits: Carnegie Mellon University..

However, there are limitations to study. Flour dough is known to be a complex material. It can have different proportions of water, starch, gluten, fiber, and fat. Flour dough also has variable, nonlinear properties, which makes it hard to anticipate how different types of pasta would behave — this was just a proof of concept.

Researchers recommend more quantitative models of assessment to see how different materials with complicated groove shapes and patterns would behave.

The study was published in Science.

Scientists develop origami inflatable structures that are stable both inflated and deflated

This inflatable shelter is out of thick plastic sheets and can pop up or fold flat. (Image courtesy of Benjamin Gorissen/David Melancon/Harvard SEAS).

In 2016, cyclism fans watched in shock as an inflatable arch at the Tour de France deflated and fell down on a cyclist, throwing the race into disarray. Organizers later blamed the accident on a passing spectator’s wayward belt buckle, but we all know who the real culprit was: physics.

Inflatable structures, which are used for everything from temporary hospitals to weddings and parties, are monostable — they are stable in one state and one state only, when they are inflated. If they deflate, they fall down.

It makes sense: if you think about common inflatable structures, they have a completely different shape when they’re inflated versus when they’re not. But what if, through some clever geometry, you could design a structure that’s stable in both configurations? In other words, one that’s bistable.

That was exactly the reasoning of a team of researchers at Harvard. Inspired by origami, they started with the simplest geometrical shapes (triangles) and developed a library of triangular building blocks that can be used to build bistable shapes.

Their idea is all the more impressive since it doesn’t need specific materials to work.

“We are relying on the geometry of these building blocks, not the material characteristics, which means we can make these building blocks out of almost any materials, including inexpensive recyclable materials,” said Benjamin Gorissen, an associate in Materials Science and Mechanical Engineering at SEAS and co-first author of the paper. 

They put their idea to the test and developed a tent-sized shelter out of plastic sheets.

The origami approach comes in handy because structures of different shapes and sizes can be designed. Researchers built several structures based on their design, including an archway, an extendable boom, and a pagoda-style structure.

This clever invention could be put to good use. Arches and emergency shelters can be safely locked in place after deployment, without needing a stable inflation source. It’s more robust and easier to install.

“This research provides a direct pathway for a new generation of robust, large-scale inflatable systems that lock in place after deployment and don’t require continuous pressure,” said Katia Bertoldi, the William and Ami Kuan Danoff Professor of Applied Mechanics at SEAS and senior author of the paper.  

The shelters require just one or two people to set up, as opposed to about a dozen, which is the case with current military inflatable hospitals.

“You can imagine these shelters being deployed as part of the emergency response in disaster zone,” said David Melancon, a PhD student at SEAS and co-first author of the paper. “They can be stacked flat on a truck and you only need one pressure source to inflate them. Once they are inflated, you can remove the pressure source and move onto the next tent.”

The study was published in Nature.

Scientists find no difference in math ability in the brains of boys and girls

Credit: Pixabay.

There’s a pervasive folk belief that girls are less biologically equipped than boys at math — and this may explain the gender gap in STEM fields. A new study by researchers at Carnegie Mellon University puts such myths to rest, showing no gender difference in brain functionality when performing math.

“We see that children’s brains function similarly regardless of their gender so hopefully we can recalibrate expectations of what children can achieve in mathematics,” said Jessica Cantlon, professor of neuroscience at Carnegie Mellon University and lead author of the new study.

We’re more similar than we are different

The researchers used functional MRI to scan the brains of 104 children, aged 3 to 10, while they watched an educational video covering various early math topics, such as counting and addition. The scans of the boys and girls were compared to evaluate any differences in brain activity. What’s more, the scans were compared to those taken from a group of adults who watched the same videos to examine brain maturity.

Cantlon and colleagues employed a range of statistical methods and comparisons, but none rendered any differences in brain development between boys and girls. According to the researchers, boys and girls were equally engaged with the educational material and exhibited the same brain functions when processing math skills. Lastly, the children’s brain maturity was statistically equivalent to either men or women in the adult group.

“It’s not just that boys and girls are using the math network in the same ways but that similarities were evident across the entire brain,” said Alyssa Kersey, postdoctoral scholar at the Department of Psychology, University of Chicago and first author on the paper. “This is an important reminder that humans are more similar to each other than we are different.”

Besides brain activity, the researchers also examined potential gender differences in mathematical abilities, as measured by standardized tests for 3- to 8-year-old children in a study involving 97 participants. The results showed that math ability was equivalent among boys and girls.

Cantlon says that as children grow up, gender differences in science and math abilities can surface due to the way boys and girls are socialized. She mentions studies showing that most American families encourage young boys to play games that involve spatial cognition. Parents also generally have different expectations from their children in terms of math abilities.

“We need to be cognizant of these origins to ensure we aren’t the ones causing the gender inequities,” Cantlon said.

In the future, the researchers would like to extend their study using a broader array of math skills, such as spatial processing and memory. They would also like to follow children over many years, preferably into adulthood to see how their math abilities and brain functions differ by gender.

The findings appeared in the journal Science of Learning.

What is Mass-Energy Equivalence (E=mc^2): the most famous formula in science

In a series of papers beginning in 1905 Einstein’s theory of special relativity revolutionized the concepts of space and time, uniting them into a single entity–spacetime. But, the most famous element of special relativity–as famous as the man himself–was absent from the first paper.

Mass-energy equivalence, represented by E=mc2, would be introduced in a later paper published in November 1905. And just as Einstein had already unified space and time–this paper would unite energy and mass.

So what does the mass-energy equivalence tell us and what is the equation E=mc2 saying about the Universe?

The Basics

If you wanted to walk away from this article with one piece of information about the equation E=mc2 (and I hope you won’t) what would that be?

Essentially the simplified version of the equation of special relativity tells us that mass and energy are different forms of the same thing– mass is a form of energy. Probably the second most important piece of information to take away is the fact that these two aspects of the Universe are interchangeable, and the mitigating factor is the speed of light squared.

Still with us? Good!

Perhaps the most surprising thing about the equation E=mc2 is how deceptively simple it is for something so profound. Especially when considering that as the equation that describes how stars release energy and thus make all life possible. Mathematical formulae don’t get much more foundational.

Gathering Momentum: Where Does the Mass-Energy Equivalence Come From?

There are actually a few ways of considering the origin of E=mc2. One way is by considering how the relationship it describes can emerge when comparing the relativistic equation for momentum and its Newtonian counterpart. The major difference between the two, as you’ll see below, is multiplication by the Lorentz factor — you might remember in the last part of this guide to special relativity —concerning space and time— I told you it gets everywhere in special relativity!

Whilst you could argue that the only difference between the two is that velocity (v) has been replaced with a more complex counterpart that approaches v when speeds are far less than light–everyday speeds that we see everyday objects around us move at–but some physicists find this more significant than a mere substitution.

These scientists would argue that this new factor ‘belongs’ to the mass of the system in question. This view means that mass increases as velocity increases, and this means there is a discernable difference between an object’s ‘moving mass’ and its ‘rest mass.’

So, let’s look at that equation for momentum again with the idea of rest mass included.

So, if mass is increasing as velocity increases, what is responsible for this rise?

Let’s conduct an experiment to find out. Our lab bench is the 2-mile long linear particle accelerator at SLAC National Laboratory, California. Using powerful electromagnetic forces, we take electrons and accelerate them to near the speed of light. When the electrons emerge at the other end of the accelerator we find that their relativistic mass has increased by a staggering factor of 40,000.

As the electrons slow, they lose this mass again. Thus, we can see it’s the addition of kinetic energy to the object that is increasing its mass. That gives us a good hint that energy and mass are interconnected.

But, this conclusion leads to an interesting question; if the energy of motion is associated with an object’s mass when it is moving, is there energy associated with the object’s mass when it is at rest, and what kind of energy could this be?

Locked-Up Energy

An object at rest without kinetic energy can, with the transformation of an infinitesimally small amount of mass, provide energy enough to power the stars.

As the equation E=mc2 and the fact that the speed of light squared is an extremely large number implies, in terms of energy just a little mass goes a very long way. To demonstrate this, let’s see how much energy would be released if you could completely transform the rest mass of a single grain of sugar.

That’s a lot of energy!

In fact, it is roughly equivalent to the amount of energy released by ‘little boy’– the nuclear fission bomb that devastated Hiroshima on the 6th August 1945.

That means that even when an object is at standstill it has energy associated with it. A lot of energy.

As you might have guessed by this point, as energy and mass are closely associated and there are many forms of energy there are also many ways to give an object increased mass. Heating a metal rod, for example, increases the rod’s mass, but by such a small amount that it goes unnoticed. Just as liberating a tiny bit of mass releases a tremendous amount of energy, adding a relatively small amount of heat energy results in an insignificant mass increase.

We’ve already seeen that we can accelerate a particle and increase its relativistic mass, but is there anything we can do to increase a system’s rest mass?

E=mc2: Breaking the Law (and Billiard Balls)!

Until the advent of special relativity two laws, in particular, had governed the field of physics when it comes to collisions, explosions, and all that cool violent stuff: the conservation of mass and the conservation of energy. Special relativity challenged this, suggesting instead that it is not mass or energy that is conserved, but the total relativistic energy of the system.

Let’s do another experiment to test these ideas… The first location we’ll travel to in order to do this… a billiard table at the Dog & Duck pub, London.

At the billiard table, we strike a billiard ball 0.17 kg toward a stationary billiard ball of the same mass at around 2 metres per second. We hit the ball perfectly straight on so that all of the kinetic energy of the first ball is transferred to the second ball.

If we could measure the kinetic energy of the initial ball, then measure the kinetic energy of both balls after the collision, we would find that–accounting for the small losses of energy to heat and sound–the total energy of the system after the collision is the same as the energy before the collision.

That’s the conservation of energy.

Let’s rerun that experiment again, but this time we launch the billard ball so hard that instead of knocking the target ball across the table, it shatters it. Collecting together the fragments of the shattered ball and remeasuring the mass of the system, we would find the final mass is exactly the same as the initial mass.

And that’s the conservation of mass.

We’re starting to get funny looks from the Dog & Duck regulars now, and the landlord looks angry about the destruction of one of his billiard balls. Luckily, the third part of our test requires we relocate to CERN, Geneva. So we down our drinks, grab our coats and hurry out the door.

Trying the experiment a third time, we are going to replace the billiard table with the Large Hadron Collider (LHC)–that’s some upgrade– and the billiard balls with electrons and their equal rest mass anti-particles– positrons.

Using powerful magnets to feed these fundamental particles with kinetic energy we accelerate them to near light speed, directing them towards each other and colliding them. The result is a shower of particles that previously weren’t present. But, unlike in our billiard ball example, when we measure the rest mass of the system it has not remained the same.

Just one of the particles we observe after the collision event is a neutral pion–a particle with a rest mass 264 times the rest mass of an electron and thus 132 times the initial rest mass we began with.

Clearly, the creation of this pion has taken some of the kinetic energy we poured into the electrons and converted it to rest mass. We watch as the pion decays into a muon with a rest mass 204 times that of an electron, and this decays into particles that are lighter still. Each time the decay releases energy in the form of pulses of light.

Relativistic Energy .vs Rest Energy

By now it is probably clear that in special relativity rest mass and relativistic mass are very different concepts, which means that it shouldn’t come as too much of a surprise that rest energy and relativistic energy are also separate things.

Let’s alter that initial infographic to reflect the fact that the equation E=mc2 actually describes rest energy.

This raises the questions (if I’m doing this right that is) what is the equation for relativistic energy?

It’s time for another non-surprise. The equation for relativistic energy is just the equation for rest energy with that Lorentz factor playing a role.

Ultimately, it is this relativistic energy that is conserved, thus whilst we’ve sacrificed earlier ideas of the conservation of mass and the conservation of energy, we’ve recovered a relativistic version of those laws.

Of course, the presence of that Lorentz factor tells us that when speeds are nowhere near that of light — everyday speeds like that of the billiard balls in the Dog & Duck–the laws of conservation of mass and energy are sufficeint to describe these low-energy systems.

The Consequences of E=mc2

It’s hard to talk about the energy-mass equivalence or E=mc2 without touching upon the nuclear weapons that devasted Hiroshima and Nagasaki at the close of the Second World War.

It’s an unfortunate and cruel irony that Einstein–a man who was a staunch pacifist during his lifetime–has his name eternally connected to the ultimate embodiment of the most destructive elements of human nature.

The Sun photographed at 304 angstroms by the Atmospheric Imaging Assembly (AIA 304) of NASA's Solar Dynamics Observatory (SDO). This is a false-color image of the Sun observed in the extreme ultraviolet region of the spectrum. (NASA)
The Sun photographed at 304 angstroms by the Atmospheric Imaging Assembly (AIA 304) of NASA’s Solar Dynamics Observatory (SDO). This is a false-colour image of the Sun observed in the extreme ultraviolet region of the spectrum. (NASA)

Nuclear radiation had been discovered at least a decade before Einstein unveiled special relativity, but scientists had struggled to explain exactly where that energy was coming from.

That is because as rearranging E=mc2 implies, a small release of energy would be the result of the loss of an almost infinitesimally small amount of rest mass –certainly immeasurable at the time of discovery.

Of course, as we mention above, we now understand that small conversion of rest mass into energy to be the phenomena that power the stars. Every second, our own star–the Sun– takes roughly 600 tonnes of hydrogen and converts it to 596 tonnes of helium, releasing the difference in rest mass between the two as around 4 x 1026 Joules of energy.

We’ve also harnessed the mass-energy equivalence to power our homes via nuclear power plants, as well as using it to unleash a terrifying embodiment of death and destruction into our collective imaginations.

We could probably ruminate more about special relativity and its elements, as its importance to modern physics simply cannot be overstated. But, Einstein wasn’t done.

Thinking about spacetime, energy and mass had open a door and started Einstein on an intellectual journey that would take a decade to complete.

The great physicist saw special relativity as a great theory to explain physics in an empty region of space, but what if that region is occupied by a planet or a star? In those ‘general’ circumstances, a new theory would be needed. And in 1915, this need would lead Einstein to his greatest and most inspirational theory–the geometric theory of gravity, better known as general relativity.

Sources and Further Reading

Stannard. R., ‘Relativity: A Short Introduction,’ Oxford University Press, [2008].

Lambourne. R. J., ‘Relativity, Gravitation and Cosmology,’ Cambridge University Press, [2010].

Cheng. T-P., ‘Relativity, Gravitation and Cosmology,’ Oxford University Press, [2005].

Fischer. K., ‘Relativity for Everyone,’ Springer, [2015].

Takeuchi. T., ‘An Illustrated Guide to Relativity,’ Cambridge University Press, [2010].

Computation pioneers awarded ‘Nobel of Mathematics’

László Lovász and Avi Wigderson have been awarded the Abel Prize, sometimes referred to as the ‘Nobel of Mathematics’. The two were recognized for groundbreaking contributions in theoretical computer science and discrete mathematics, as well as “their leading role in shaping them into central fields of modern mathematics.”

Mathematics can often seem like an occult field of research, thoroughly detached from mundane day-to-day life. But even though we may not realize it, mathematics affects our daily lives in more ways than one.

Take, for instance, the platform I wrote this article on, and your browser or app that allows you to read it. Both are secured by algorithms which we are blissfully unaware of, but without which, it would be impossible to communicate over the internet. As the internet itself takes a more central role in our lives, the unseen algorithms that help it run smoothly also become more important — and some of those algorithms were pioneered by Lovász and Wigderson.

“Lovász and Wigderson have been leading forces in this development over the last decades. Their work interlaces in many ways, and, in particular, they have both made fundamental contributions to understanding randomness in computation and in exploring the boundaries of efficient computation,” says Hans Munthe-Kaas, chair of the Abel committee.

It was at some point in the 1970s that a generation of mathematicians realized that the emerging field of computer science was essentially a new area of application for mathematics. Randomness, for instance, became not only a curiosity of mathematics but an area with direct applications. Randomness is essential for cryptography and plays a fundamental role in the design of many algorithms.

Avi Wigderson was one of the mathematicians who understood the importance of this field. He’s part of a keen type of mathematician able to see links between seemingly unrelated areas. He has published papers with over 100 collaborators, in fields ranging from complexity theory to quantum computation. Wigderson also made contributions to a concept called zero-knowledge proof, in which one party can prove to another party that they know a value, without conveying any information apart from the fact that they know the value30 years on, this concept is used in blockchain technology.

Every big problem in complexity theory, Wigderson has had a go at it — often with success. His contribution to this field has been instrumental, and this is owed in part to his approachable, collaborative, and curious nature.

“I consider myself unbelievably lucky to live in this age,” he says. “[Computational complexity] is a young field. It is a very democratic field. It is a very friendly field, it is a field that is very collaborative, that suits my nature. And definitely, it is bursting with intellectual problems and challenges.”

Meanwhile, László Lovász proved himself to be a stellar mathematician from his teenage years. He published his first paper when he was 17, and it wasn’t a coincidence: he published two more in the next two years. By the time he graduated from the Eötvös Loránd University, he was awarded a Candidate Degree of Mathematical Science by the Hungarian Academy of Sciences — a degree higher than a doctorate. University regulations did not allow a student to apply for a Ph.D. until he finished his undergraduate studies, but no such rules existed in the Academy of Science because it was assumed that it wasn’t necessary.

Lovász went on to publish over a dozen papers and hold several talks at prestigious conferences before being awarded any degree, and he never really stopped. His meeting with the famous “nomadic mathematician” Paul Erdös, who was known for his insatiable hunger for mathematical problem-solving, not only inspired Lovász, but set a direction for his working style, which became open and collaborative.

Lovász is interested in connections between discrete mathematics and other branches of mathematics. His work focused especially on combinatorics (the mathematic of patterns) and graph theory (the mathematic of network). For Paul Erdös, these fields were an intellectual curiosity, but Lovász saw that they could be applied practically in the field of computer science. In the 1970s, graph theory became one of the first areas of pure mathematics to impact computer science. As the years went on, Lovász’s work established many ways in which pure mathematics can address fundamental theoretical questions in computer science. He traveled widely, held positions in several countries, and became known for his generosity and openness.

“I was very lucky to experience one of those periods when mathematics was developing completely together with an application area,” he says.

The work of the two pioneers was crucial to a then-nascent and now thriving, but the still-young field of computer science. They laid out the groundwork for the theoretical framework and continued to solve long-standing problems in various fields of mathematics.

But perhaps the most striking trait of the two laureates is their approach to mathematics: collaborative, democratic, and open. There is a lesson that can benefit all of us, regardless of our mathematical ability: mathematics, like science, and like all discovery, works best when it is collaborative.

This AI invents unique math we’ve never seen before

Credit: Pixabay.

Researchers have taken things to the next level by developing an AI that is basically a mathematical conjecture generator. Conjectures are mathematical statements that are suspected to be true but have not yet been rigorously proven. Any mathematician will tell you that these are their bread and butter, which they use to develop mathematical theorems. Now, we have computers that can feed mathematicians with new conjectures, which they’ll have to prove, and in the process might revolutionize the field.

The AI developed by the team at Technion-Israel Institute of Technology specifically deals with conjectures surrounding another fundamental element of mathematics: constants. In math, constants are key numbers with fixed values that emerge naturally from different mathematical computations and structures.

Take for instance pi, arguably the most important constant in mathematics. It gives the ratio between a circle’s circumference and diameter, which stays the same value for every circle, no matter how large. Other important fundamental constants include Euler’s number and the golden ratio.

Not anyone can make conjectures about such fundamental constants. In fact, this is something typically reserved for geniuses like Newton, Riemann, Gauss, or Srinivasa Ramanujan. The latter was so good at it that Ramanujan is credited for the discovery of thousands of innovative formulas in number theory — and he did so with no formal training, starting from a poor family background.

Srinivasa Ramanujan. Credit: Wikimedia Commons.

In honor of the great mathematician, the researchers named their AI the Ramanujan Machine. Like the late Indian genius, they hope that the AI becomes just as prolific at conjecturing unproven mathematical formulas.

The software has made its own conjectures that independently formulate well-known mathematical constants such as pi, Euler’s number (e),  Apéry’s constant, and the Catalan constant, as well as a couple of original universal constants.

“Our results are impressive because the computer doesn’t care if proving the formula is easy or difficult, and doesn’t base the new results on any prior mathematical knowledge, but only on the numbers in mathematical constants. To a large degree, our algorithms work in the same way as Ramanujan himself, who presented results without proof.”

“It’s important to point out that the algorithm itself is incapable of proving the conjectures it found — at this point, the task is left to be resolved by human mathematicians,” said Assistant Professor Ido Kaminer from the Faculty of Electrical Engineering at the Technion.

For thousands of years of mathematical history, conjectures were reserved for rare genius. This is why we only have a few dozen important formulas discovered in the last hundred years of research. But in a few hours, the Ramanujan Machine “re-discovered” all the formulas for pi discovered by Gauss, which took him a lifetime of work, as well as dozens of new formulas that were unknown to Gauss.

“Similar ideas can in the future lead to the development of mathematical conjectures in all areas of mathematics, and in this way provide a meaningful tool for mathematical research,” wrote the researchers in their study published in Nature.

The researchers launched a website where the public can find algorithmic tools that anyone can use for the advancement of mathematical research.

The pandemic changed the way we dream. A new study tries to make sense of it all

Using math, researchers in Brazil connected a surge in disturbing dreams with people's troubled waking lives.
Credit: Weid de souza.

Since the pandemic suddenly upended our lives, there has been a surge of reports from people recalling very vivid, bizarre dreams. In one way or the other, these dreams are connected to the coronavirus and the restrictions the virus has forced upon us. Now, a new study employed math-based computer models to analyze the pandemic dreams, finding connections between what happens in the dream world and people’s mental health during these troubled times.

The COVID nightmares

It’s quite common for a person’s mental and emotional state to become reflected in dreams, with fear, sadness, and anxiety creeping up in various guises.

These sorts of troublesome feelings have been widely reported at the start of the pandemic in early 2020 when things were a lot more frightening and uncertain than they are today. Terms such as “coronavirus dreams” and “lockdown dreams” have been widely used to describe reports on social media and mainstream outlets of important disruptions to dream patterns.

A poll conducted in March 2020 found that 29% of Americans recalled more dreams than usual and 37% had dreams related to the pandemic.

Had a dream about returning as a sub teacher in the fall, unprepared. Students were having a difficult time practicing social distancing, and teachers couldn’t stagger classes or have one-on-one meetings,” one person wrote on Twitter. “My phone had a virus and was posting so many random pictures from my camera roll to Instagram and my anxiety was at an all-time high,” tweeted another.

In a new study, researchers led by Natália Bezerra Mota, a neuroscientist and postdoctoral fellow at the Brain Institute of the Federal University of Rio Grande do Norte (UFRN), in Brazil, used natural language processing techniques to analyze 239 dream reports by 67 volunteers.

The dream reports were recorded between March and April 2020 using a smartphone app. The reports were then transcribed and analyzed using three different tools, responsible for breaking down discourse structure, word count, and connectedness, as well as interpreting the meaning of the content.

One of the natural language processing tools classed words in certain emotional categories, such as positive and negative emotions. Another tool employed a neural network to identify words that were related semantically to pandemic-related keywords, such as contamination, cleanliness, sickness, health, death, and life.

Perhaps unsurprisingly, this analysis revealed that dream narratives in this period involved a large proportion of terms related to cleanliness and contamination, as well as anger and sadness.

These results strengthen the idea that dreams reflect the challenges of waking-life experience.  The prevalence of negative emotions such as anger and sadness during the period reflects a higher emotional load to be processed, the authors wrote in the journal PLOS ONE.

“It’s the first study on the subject to look empirically at these signs of mental suffering and their association with the peculiarities of dreams during the pandemic,” Mota told Agência FAPESP.

“The significant similarity to ‘cleanness’ in dream reports points towards new social strategies (e.g. use of masks, avoidance of physical contact) and new hygiene practices (e.g. use of hand sanitizer and other cleaning products) that have become central to new social rules and behavior. Taken together, these findings seem to show that dream contents reflect the different sources of fear and frustration arising out of the current scenario,” the researchers wrote in their study.

Another striking finding was that women seem to have had more pandemic-related dreams and nightmares.

“There are studies on gender difference in the literature. Women report more negative dreams and nightmares. I think this has to do with women’s history and daily lives, with working a double or triple shift, and the heavier mental burden entailed by concerning themselves with a job plus the home and children. The pandemic has made this worse,” Mota said.

What are Cistercian numbers — the forgotten ciphers of Medieval monks

Virtually the entire world uses the Hindu-Arabic numeral system to represent digits in base-10 and perform algebra. You probably also heard of the Roman numeral system, which lacks a symbol for zero, and the placement of numerals within a number can sometimes indicate subtraction rather than addition. Throughout history, however, people have employed all sorts of systems, most of which have been forever forgotten. These include the intriguing Cistercian numbers used by monks in the Middle Ages, but also Nazis and occultists in the 20th century.

Various Cistercian numbers from 1 to 9999 can be represented by superimposing up to 36 basic forms.

Cistercian numbers are definitely very odd, but once you grasp the logic behind them, they can be quite fun to use if you’re keen on using a ‘coded’ language. These numbers are represented by nine appendages to vertical stems that each correspond to units, tens, hundreds, and thousands.

Cistercian numbers are represented on a quadrant using horizontal and vertical staves.

Each of the four different orientations (1-9, 10-99, 100-999, 1000-9999) can be represented by changing the coordinates. So, practically, changing the coordinate of each ligature — either by rotating or mirroring them — can alter the digit from being a unit to a thousand. When the appendages are combined on a single stem, you get a cipher representing any number from 1 to 9999.

Changing the coordinate of each ligature can alter the type of digit.

This peculiar number-notation was invented in the late 13th century by Cistercian monks, near the border region between France and Belgium. For two centuries, the number system was used by monks belonging to the order across all of Europe as an alternative to the well-known Roman numerals and the ‘novel’ Hindu-Arabic numerals, the latter of which were just beginning to get adopted at the time.

The monks used this system to represent year-numbers in dates, and to number staves of music and pages of manuscripts, but it also proved useful outside of the monasteries. We know the numerical notation was used in astronomy because Cistercian numerals were etched on astrolabes — a handheld medieval astronomical instrument — and used for astronomical tables compiled in Salamanca in the late 15th century.

But since Cistercian numbers are useful for representing only compact numbers up to 9999, as well as due to the fact that they were challenging to print, the system soon fell out of grace in favor of the much more flexible Hindu-Arabic numbering system. However, Cistercian ciphers would survive in use as secret codes. The Freemasons in Paris adopted it in 1780, as well as occultists and Nazis who re-discovered the Cistercian numeral system in the early 20th century.

This website has a nifty online tool that converts Arabic numbers into Cistercian Monks numbers.

The Hidden World of Mathematics in The Simpsons

The Simpsons is one of the most beloved shows in entertainment history. Its writers have masterfully blended a whole spectrum of humor with memorable characters while poking fun at society’s biggest shortcomings.

No doubt, The Simpsons shines thanks to its humor, which mixes low-level gags with many intellectual references. Many times I’ve found myself caught in deep philosophical thought after watching one the show’s weekly episodes.

For instance, mathematics often creeps into episodes of the show, from pi to Mersenne primes, from Euler’s equation to the unsolved riddle of P vs NP.

TThese clever math jokes were smuggled in by the show’s cast of talented writers, many of whom completed PhDs in mathematics, physics, or computer science at Ivy League universities.

Here are some of the most memorable math jokes from The Simpsons, as a tribute to a show that’s been running for three-decades straight with no sign of stopping anytime too soon.

Homer’s last theorem

Credit: The Simpsons.

Let’s kick it off with what’s probably the most dazzling mathematical interlude from the show. In The Wizard of Evergreen Terrace (1998), a nod to the Wizard of Menlo Park (Thomas Edison’s nickname), Homer seems apt to follow in Edison’s footsteps, inventing various gadgets, from an alarm clock that beeps every three seconds to a shotgun that shoots makeup on the face.

During his intense research and development, viewers are given a glimpse of Homer’s blackboard where several mathematical equations are scribbled.

These aren’t some random symbols and numbers pasted as filler content. In fact, they’re all extremely consequential.

The first equation, which combines the Planck constant, the gravitational constant, and the speed of light predicts the mass of the Higgs boson. If you crunch the numbers you get 775 GeV, which is much higher than the 125 GeV measurement found when the Higgs boson was finally discovered in 2012. Still, not too off.

The last equation concerns the density of the universe. If Ω(t0) is greater than 1, this means that the universe will eventually implode under its own weight. This equation is quickly changed to “less than 1” after an implosion in Homer’s basement. In this form, the equation suggests that the universe expands for eternity.

What about the doughnuts? That’s a funny quip to topology, the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting. Under a topological lens, the letters “A” and “R” are essentially the same since they’re both made of a loop with two legs (topologists call the two objects homeomorphic). If you stretch carefully one of the two, you end up with either an A or and R. In contrast, you can never make an A out of an H because Hs have no loops to speak of. Likewise, you should never be able to turn a doughnut, which has a hole, into a sphere, which has no holes.

In topology, you’re never allowed to cut — just stretch or twist. However, Homer says that nibbling is not against the rules, so a munched doughnut is turned into a banana, which can then be morphed into a sphere.

I saved the second equation for last because it is the most interesting, although it looks like some random numbers. It reads:

3,98712+4,36512= 4,47212

Those who are versed in mathematics or its history, however, will instantly recognize this as Fermat’s last theorem.

In 1637, Pierre de Fermat scribbled a few sentences in Latin on the margin of his copy of Diophantus’ Arithmetica essentially stating that while there is an “infinite number of solutions” to the following equation:

x² + y² = z²

It is impossible to find three different whole numbers that satisfy the following equation:

x³ + y³ = z³

or

x4 + y4 = z4

or any combination of the form

xn + yn = zn

Fermat also left one of the most frustrating notes in mathematical history, writing: “I have discovered a truly remarkable proof which this margin is too small to contain.” The proof has never been found and for four centuries the world’s greatest minds have tried to crack the equation with no luck. But if you grab a pocket calculator and crunch the numbers on Homer’s blackboard, you’ll see it fits. You can try it out yourself.

Wait a minute, did Homer, who literally has a pea-sized brain, solve one of the greatest mathematical mysteries in history? Not so fast.

If you have a more powerful calculator that can compute more than ten digits, the actual value for the third term in the equation is closer to 4,472.000000007057617187512. So, what Homer found is a near-miss solution to Fermat’s equation — not bad for a pea brain! And, by the way, Fermat’s Last Theorem was solved in 1995 by Andrew Wiles.

Apu’s perfect memory of delicious π

“Now, Marge, you’ve come to the right place.  By hiring me as your lawyer, you also get this smoking monkey.  Better cut down there, smokey, ha ha ha.” – Lionel Hutz
“Mr. Hutz-” – Marge Simpson
“Look, he’s taking another puff!” – Lionel Hutz

In Marge in Chains (1993), Marge goes on trial for shoplifting after she walked out of Kwik-E-Mart having forgotten to pay for a bottle of bourbon.

The trial isn’t going too well, especially since Marge employs a sketchy lawyer by the name Lionel Hutz. The lawyer’s strategy is to discredit the memory of convenience store owner Apu Nahasapeemapetilon. Joke’s on him, though, because Apu actually has a perfect memory. “In fact, I can recite pi to forty thousand places. The last digit is 1,” Apu told the judge and jury.

This is actually all true. First of all, believe it or not, there are people who can memorize thousands of digits for pi. Indian mnemonist Sharma Suresh Kumar is the world record holder for reciting the infinite decimals of pi from memory. Kumar recited 70,030 digits in 2015.

Secondly, the 40,000th digit of pi really is 1. This is more impressive than it sounds because in 1993 when the episode was written, there was no google or ‘book of pi’ in libraries where you could just randomly look up digits for pi. The writers actually received a letter back from NASA with printed pages for the first 40,000 digits of pi, confirming that the last digit is 1.

Lisa, the sabermetrics coach

In the show, 8-year-old Lisa Simpson has a reputation for being a bookworm who is skilled in mathematics (and just about any subject). In MoneyBART (2010), Bart’s Little League baseball team loses their coach and Lisa steps up to fill the position, hoping the credentials will earn her an Ivy League scholarship once she’s of age.

Although Lisa doesn’t know the first thing about baseball, she employs deep mathematical analysis and sabermetrics, the use of complicated statistics to make managerial decisions. During one scene in the episode, Lisa immerses herself in a pile of technical books, among them The Bill James Historical Baseball Abstract. The book is considered the most important and thorough title of statistics in baseball.

Lisa’s strategy based on baseball statistics pays off, transforming Bart’s team into winners. But her use of statistics and probabilities is fumbled by Bart, who else, during the state championship game. Instead of following Lisa’s instructions, Bart acts on his gut feeling, costing the team the winning game.

Bart… the genius?

Math is so prevalent in the show that The Simpsons’ second episode, Bart the Genius (1990), opens with a scene in which Maggie, the eternal one-year-old, stacks alphabet blocks to form EMCSQU. Without any letters or equal sign, this is the closest Maggie could get to form the most famous equation in physics E = mc2.

However, this episode’s central character is Bart who decides to cheat on his school aptitude test by swapping his answer sheet with Martin Prince, the smartest kid in class. This would earn Bart an IQ of 216, which fast tracks him to the Learning Center for Gifted Children. As you might imagine, all hell breaks loose from here on.

At his new school, his first lesson is in mathematics. This is when we’re treated to the first example of an overt mathematical joke in the show. While writing an equation on the board, the teacher says: “So y equal r cubed over three, and if you determine the rate of change in this curve correctly, I think you will be pleasantly surprised.”

After a moment of silence, all the children in the class laugh out loud with glee — all but one. The teacher, looking to help Bart, turns to him and says: “Don’t you get it, Bart? Derivative dy equals three r squared dr over three, or r squared dr, or r dr r.”

Get it?

Not only is “r dr r” the correct answer, it also sounds like har-de-har-har, a sarcastic laughter in reaction to a bad joke, popularized by the 1950s TV sitcom The Honeymooners.

Leave it to the writers of the The Simpsons to make gags out of advanced calculus.

There are many other instances of math, some more or less overt, in the animated series. For more, you can order “The Simpsons and Their Mathematical Secrets” by Simon Singh.

What are prime numbers and why they matter — yes, even in your day to day life

A prime number must satisfy three conditions:

  • it must be a natural number (so numbers like 1.2, -7, or √3 are out of the question);
  • it must be greater than 1;
  • it must not be the product of any two other numbers.

So essentially, a prime number is any natural number starting from 2 that isn’t divisible with any numbers other than 1 or itself. For instance:

  • 5 is a prime number, it can only be written as a product of 1 x 5;
  • 6 is not a prime number, it can be written as 2×3.

A number that is not prime is called composite. All natural numbers are either primes or composites.

Understanding the basics of prime numbers

All natural numbers are either primes or composites — but what does that really mean? Well, it basically means that if we take any natural number, it can be either written as a product of two other numbers, or it’s a prime. There’s no middle way. If we look at the image above, 12 is written as 4 x 3. It can also be written as 6 x 2, but it doesn’t matter.

Whether there’s one way to write a number as a product or a million ways to write it as a product, it’s still a composite number, and therefore not prime.

Every prime number can only be written as ‘1’ times itself. Every composite number can be written as a product of prime numbers (something called prime factorization).

Checking if a number is a prime

The rough way to search for prime numbers is to take any number and try and see if any numbers divide it evenly. In other words, if it has any divisors other than 1 and itself, it’s not prime.

Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly.

But we can get a bit more clever about our search for prime numbers.

We can weed out half of them right from the start. How? By using the number ‘2’. Every number is either odd or even, by definition. Even numbers are divisible by 2, which means they can be written as 2 times something — and are therefore composite, not prime. The number ‘2’ is the only even prime number.

With the odd numbers, we can use a similar approach, but using other numbers instead of ‘2’. For instance, ‘3’ is a prime number, so it can be used to weed out other numbers that aren’t prime. This is actually a very old algorithm used to discover prime numbers: you start from 2, and then every prime number you encounter, you use it to weed out its multiples. You don’t need to use ‘4’ since it is an even number, and we already know that other than ‘2’, no prime number is even. So we can just move on to 5, 7, and move on. This is called a ‘sieve’, and commonly, the ‘sieve of Eratosthenes’, since the ancient Greek mathematician Eratosthenes first described it. You can see a visual representation of how it works in the image below:

You can get a bit more clever with your prime-finding sieves too. For instance, let’s say you want to check if the number 100 is prime (spoiler alert, it’s not). You don’t need to check every number from 2 to 100, you can stop with your checks at 50, which is the half of 100. Why? Look at it this way:

  • every number can be written as: number (n) = ‘half’ x 2. Every number is twice its half. If you reach a number beyond ‘half’ of n’s value, you’d need to multiply it by something smaller — but there’s nothing smaller than 2. So you don’t need to look further than the half of the number.

We can get even smarter than this: you don’t even need to check until half of the number, you only need to check to its square root (√n).

Looking for prime number is a common exercise not just in mathematics, but also in programming, where the goal is learning how to devise and optimize an algorithm.

A list of prime numbers up to one thousand:

 23571113171923
29313741434753596167
717379838997101103107109
113127131137139149151157163167
173179181191193197199211223227
229233239241251257263269271277
281283293307311313317331337347
349353359367373379383389397401
409419421431433439443449457461
463467479487491499503509521523
541547557563569571577587593599
601607613617619631641643647653
659661673677683691701709719727
733739743751757761769773787797
809811821823827829839853857859
863877881883887907911919929937
941947953967971977983991997

Some properties of prime numbers

Prime numbers may seem like just a new quirk, but mathematicians have been fascinated with them for millennia (as we’ll see in a bit). They have several properties that make them special — here are just a few of them.

  • There are infinite prime numbers

The sequence of prime numbers never ends. It can get harder and harder to find prime numbers, but there will always be an even larger one. This was first proven by the Ancient Greek mathematician Euclid. The longest prime number we know of is 22 million digits long.

  • There is no efficient formula for calculating prime numbers

Mathematicians have devised several ways of encoding or looking for prime numbers, but none of them are very efficient. This is one of the things that makes prime numbers so attracticve. We don’t know when and how prime numbers occur, we can just calculate them.

  • 1 is not technically a prime number

This sometimes starts weird debates and is not truly consequential, but 1 is not typically considered a prime number. There’s no strict reason for that, however. Ancient Greeks (and Arabic mathematicians as well) did not consider 1 to be a number in the sense that all other numbers are — it was more a unit for numbers than a number itself. In the 18th and 19th centuries, some mathematicians considered it prime, while others did not. Nowadays, we don’t really consider it a prime, if only for the fact that we would have to rewrite the definition of prime numbers in a way that’s more awkward.

  • Numbers can be prime among themselves (but not technically primes)

Two numbers that don’t have any common divisors are considered to be ‘coprime’ — and this doesn’t really mean that they are really primes (though they can be). For instance, the numbers 25 and 4 are coprimes, because 25=5×5, while 4=2×2, and they have common divisors.

History of prime numbers

The Rhind Mathematical Papyrus, from 1550 BC.

You may think that prime numbers are a modern quirk that mathematicians came up with, but they’ve actually fascinated people for centuries — nay, millennia. An Egyptian papyrus dating from 3,550 years ago references prime numbers, and Euclid’s landmark Elements work demonstrates that there are infinitely many prime numbers. Several well-known Greek mathematicians dealt with prime numbers, including Eratosthenes, whose sieve we’ve just looked at.

Coincidentally or not, the period in which people didn’t seem really interested in prime numbers is a period we now call the Dark Ages. Some Islamic mathematicians looked at prime numbers, which Fibonacci brought to Europe, translating and analyzing the work. After that, most brilliant mathematicians looked at prime numbers: Pierre de Fermat stated (but did not prove) Fermat’s little theorem, which was later proved by Leibniz and Euler.

Many mathematicians have also tried to find ways to predict the emergence of prime numbers, but this has proven a hidden Graal. Since 1951, all the largest known prime numbers have been discovered using computers.

Why prime numbers matter so much

For a very long time, prime numbers were canonical and pure — a quirk of mathematics, but with little significance outside of mathematics. Now, we know this to be untrue and prime numbers have important applications in several fields.

For instance, the number of cogs on two gears is often chosen to prime among themselves, to create an even contact between every cog of both wheels and avoid unnecessary wear and damage — but this is not technically an application of prime numbers.

The moment when primes became really important was in the 1970s when it was first announced that prime numbers could serve as the basis of public-key cryptography algorithms. In other words, every time you use the internet to send an encrypted message (like you probably do daily), it uses an algorithm based on prime numbers, specifically because they’re so mysterious and hard to assess.

Prime numbers also play a part in fields such as quantum mechanics and abstract algebra. Perhaps even more intriguing, prime numbers have been shown to also play a role for some biological species. The cicadas of the genus Magicicada, for instance, spend most of their time as grubs beneath the ground. They emerge after 7, 13, or 17 years — all prime numbers. This is not a coincidence, biologists believe: cicadas emerge in this fashion to prevent predators from synchronizing with their lifecycle. In addition to the many applications of prime numbers in mathematics, modern research seems to indicate that at least in some cases, prime numbers also play a role in other fields.

Prime number research is far from over. Undoubtedly, there are many more prime mysteries waiting to be uncovered — this is only scratching the surface.

Time Travel Without the Paradoxes

Time Travel Without the Paradoxes

It’s one of the most popular ideas in fiction — travelling back through time to alter the course of history. The idea of travelling through time — more than we do every day that is — isn’t just the remit of science fiction writers though. Many physicists have also considered the plausibility of time travel, especially since Einstein’s theory of special relativity changed our concept of what time actually is. 

Yet, as many science fiction epics warn, such a journey through time could carry with it some heavy consequences. 

Ray Bradbury’s short story ‘A Sound of Thunder’ centres around a group of time travellers who blunder into prehistory, making changes that have horrendous repercussions for their world. In an even more horrific example of a paradox, during an award-winning episode of the animated sci-fi sitcom Futurama, the series’ hapless hero Fry travels back in the past and in the ultimate grandfather paradox, kills his supposed gramps. Then, after ‘encounter’ with his grandmother, Fry realises why he hasn’t faded from reality, he is his own grandfather. 

Many theorists have also considered methods of time travel without the risk of paradox. Techniques that don’t require the rather extreme measure of getting overly friendly with one’s own grandmother Fry. These paradox-escape mechanisms range from aspects of mathematics to interpretations of quantum weirdness. 

ZME’s non-copyright infringing time machine. Any resemblance to existing time travel devices is purely coincidental *cough* (Christopher Braun CC by SA 1.0/ Robert Lea)

Before looking at those paradox escape plans it’s worth examining just how special relativity changed our thinking about time, and why it started theoretical physicists really thinking about time travel. 

Luckily at ZME Science, we have a pleasingly non-copyright infringing time machine to travel back to the past. Let’s step into this strange old phone booth, take a trip to the 80s to pick up Marty and then journey back to 1905, the year Albert Einstein published ‘Zur Elektrodynamik bewegter Körper’ or ‘On the Electrodynamics of Moving Bodies.’ The paper that gave birth to special relativity. 

Don’t worry Marty… You’ll be home before you know it… Probably.

A Trip to 1905: Einstein’s Spacetime is Born

As Marty reads the chronometer and discovers that we have arrived in 1905, he questions why this year is so important? At this point, physics is undergoing a revolution that will give rise to not just a new theory of gravity, but also will reveal the counter-intuitive and somewhat worrisome world of the very small. And a patent clerk in Bern, Switzerland , who will be at the centre of this revolution,  is about to have a very good year. 

The fifth year of the 20th Century will come to be referred to as Albert Einstein’s ‘Annus mirabilis’ — or miracle year — and for good reason. The physicist will publish four papers in 1905, the first describing the photoelectric effect, the second detailing Brownian motion. But, as impressive those achievements are–one will see him awarded the Nobel after all–it’s the third and fourth papers we are interested in. 

1905: young Albert Einstein contenplates the future, unaware he is about to change the way we think about time and space forever. (Original Author Unknown)
1905: young Albert Einstein contemplates the future, unaware he is about to change the way we think about time and space forever. (Original Author Unknown)

In these papers, Einstein will first introduce special relativity and then will describe mass-energy equivalence most famously represented by the reduced equation E=mc². It’s no exaggeration to say that these works will change how we think of reality entirely — especially from a physics standpoint. 

Special relativity takes time — and whereas it had previously been believed to be its own separate entity — unites it with the four known dimensions of space. This creates a single fabric— spacetime. But the changes to the concept of time didn’t end there. Special relativity suggests that time is different depending on how one journeys through it. The faster an object moves the more time ‘dilates’ for that object. 

This idea of time running differently in different reference frames is how relativity gets its name. The most famous example for the time difference is the ‘twin paradox.’

Meet twin sisters Stella and Terra. Stella is about to undertake a mission to a distant star in a craft that is capable of travelling at near the speed of light, leaving her sister, Terra, behind on Earth. 

A spacetime diagram of Terra’s journey through spacetime, against her twin Stella’s. Less ‘proper time’ passes for Stella than Terra meaning when she returnes to Earth Terra has aged more than she has. (Robert Lea)

After travelling away from Earth at near the speed of light, then undertaking a return journey at a similar speed, Stella touches down and exits her craft to be greeter by Terra who has aged more in relation to herself. More time has passed for the ‘static’ Earthbound twin than for her sister who underwent the journey into space.

Thus, one could consider Stella to have travelled forward in time. How else could a pair of twins come to be of considerably different ages? That’s great, but what about moving backwards through time? 

Well, if the faster a particle in a reference frame moves, the ‘slower’ time progresses in that frame, it raises the question, is there a speed at which time stands still? And if so, is there a speed beyond this at which time would move backwards? 

A visualisation of a tachyon. Because a tachyon would always travel faster than light, it would not be possible to see it approaching. After a tachyon has passed nearby, an observer would be able to see two images of it, appearing and departing in opposite directions.
(Wiki CC by SA 3.0)

Tachyons are hypothetical particles that travel faster than the speed of light — roughly considered as the speed at which time would stand still — and thus, would move backwards rather than forwards in time. The existence of tachyons would open up the possibility that our space-bound sister could receive a signal from Terra and send her back a tachyon response. Due to the nature of tachyons, this response could be received by Terra before she sent the initial signal.

Here’s where that becomes dangerous; what if Stella sends a tachyon signal back that says ‘Don’t signal me’? Then the original signal isn’t sent, leading to the question; what is Stella responding to?

Or in an even more extreme example; what if Stella sends a tachyon signal back that is intercepted by herself before she embarks on her journey, and that signal makes her decide not to embark on that journey in the first place? Then she’ll never be in space to send the tachyon signal… but, if that signal isn’t sent then she would have embarked on the journey…

And that’s the nature of the causality violating paradoxes that could arise from even the ability to send a signal back through time. Is there a way out of this paradox?

Maybe…

Interlude. From the Journal of Albert Einstein

27th September 1905

A most astounding thing happened today. A young man in extraordinary attire visited me at the patent office. Introducing himself as ‘Marty’ the youngster proceeded to question me about my paper ‘On the Electrodynamics of Moving Bodies‘– a surprise especially as it was only published yesterday.

In particular, the boy wanted to know about my theory’s implications on time travel! A pure flight of fancy of course… Unless… For another time perhaps.

If this wasn’t already unusual to the extreme, after our talk, I walked Marty to the banks of the Aare river where he told me that his transportation awaited him. I was, of course expecting a boat. I was therefore stunned when the boy stepped into a battered red box, which then simply disappeared.

I would say this was a figment of my overworked imagination, a result of tiredness arising from working the patent office during the day and writing papers at night. That is, were I the only witness!

A young man also saw the box vanish, and his shock must have been more extreme than mine for he stumbled into the river disappearing beneath its surface.

His body has not yet been recovered… I fear the worst.

Present Day: The Self Correcting Universe

As the battered old phone box rematerializes in the present day, Marty is determined to seek out an academic answer to the time travel paradox recounted to him in 1905. 

He pays a visit to the University of Queensland where Bachelor of Advanced Science student Germain Tobar has been investigating the possibility of time travel. Under the supervision of physicist Dr Fabio Costa, Tobar believes that a mathematical ‘out’ from time travel paradoxes may be possible.

“Classical dynamics says if you know the state of a system at a particular time, this can tell us the entire history of the system,” Tobar explains. “For example, if I know the current position and velocity of an object falling under the force of gravity, I can calculate where it will be at any time.

“However, Einstein’s theory of general relativity predicts the existence of time loops or time travel — where an event can be both in the past and future of itself — theoretically turning the study of dynamics on its head.”

Tobar believes that the solution to time travel paradoxes is the fact that the Universe ‘corrects itself’ to remove the causality violation. Events will occur in such a way that paradoxes will be removed.

So, take our twin dilemma. As you recall Stella has sent herself a tachyon message that has persuaded her younger self not to head into space. Tobar’s theory — which he and his supervisor Costa say they arrived at mathematically by squaring the numbers involved in time travel calculations — suggests that one of two things could happen.

Some event would force Stella to head into space, she could accidentally stumble into the capsule perhaps, or receive a better incentive to head out on her journey. Or another event could send out the tachyon signal, perhaps Stella could accidentally receive the signal from her replacement astronomer. 

“No matter what you did, the salient events would just recalibrate around you,” says Tobar. “Try as you might, to create a paradox, the events will always adjust themselves, to avoid any inconsistency.

“The range of mathematical processes we discovered show that time travel with free will is logically possible in our universe without any paradox.”

The Novikov self-consistency principle
The Novikov self-consistency principle (Brightroundircle/ Robert Lea)

Tobar’s solution is similar in many ways to he Novikov self-consistency principle — also known as Niven’s Law of the conservation of history — developed by Russian physicist Igor Dmitriyevich Novikov in the late 1970s. This theory suggested the use of geodesics similar to those used to describe the curvature of space in Einstein’s theory of general relativity to describe the curvature of time. 

These closed time-like curves (CTCs) would prevent the violation of any causally linked events that lie on the same curve. It also suggests that time-travel would only be possible in areas where these CTCs exist, such as in the presence of wormholes as speculated by Kip Thorne and colleagues in the 1988 paper “Wormholes, Time Machines, and the Weak Energy Condition”. The events would cyclical and self-consistent. 

The difference is, whereas Tobar suggests a self-correcting Universe, this idea strongly implies that time-travellers would not be able to change the past, whether this means they are physically prevented or whether they actually lack the ability to chose to do so. In our twin analogy, Stella’s replacement sends out a tachyon signal and travelling along a CTC, it knocks itself off course, meaning Stella receives rather than its intended target.

After listening to Tobar, strolling back to his time machine Marty takes a short cut through the local graveyard. Amongst the gravestones baring unfamiliar dates and names, he notices something worrying–chilling, in fact. There chiselled in ageing stone, his grandfather’s name.

The date of his death reads 27th September 1905. 

Interlude: From the Journal of Albert Einstein

29th September 1905

This morning the Emmenthaler Nachrichten reports that the body of the unfortunate young man who I witnessed fall into the Aare has been recovered. The paper even printed a picture of the young man. 

I had not realised at the time, but the boy bares the most remarkable resemblance to Marty — the unusually dressed youngster who visited with me the very day the boy fell…

Strange I such think of Marty’s attire so frequently, the young man told me his garish armless jacket, flannel shirt and ‘jeans’ were ‘all the rage in the ‘86.’ 

Yet, though I was seven in 1886 and have many vague memories from that year, I certainly do not remember such colourful clothes…

Lost in Time: How Quantum Physics provides an Escape Route From Time Travel Paradoxes

Marty folds the copy of the Emmenthaler Nachrichten up and places it on the floor of the cursed time machine that seems to have condemned him. The local paper has confirmed his worst fears; his trip to the past to visit Einstein doomed his grandfather. 

After confirming his ancestry, he knows he is caught in a paradox. He waits to be wiped from time…

After some time, Marty wonders how it could possibly be that he still lives? Quantum physics, or more specifically one interpretation of it has the answer. A way to escape the (literal) grandfather paradox. 

The double slit experiment (Robert Lea)

The ‘many worlds’ interpretation of quantum mechanics was first suggested by Hugh Everett III in the 1950s as a solution to the problem of wave-function collapse demonstrated in Young’s infamous double-slit experiment.

As the electron is travelling it can be described as a wavefunction with a finite probability of passing through either slit S1 or slit S2. When the electron appears on the screen it isn’t smeared across it as a wave would be. It’s resolved as a particle-like point. We call this the collapse of the wavefunction as wave-like behaviour has disappeared, and it’s a key factor of the so-called Copenhagen interpretation of quantum mechanics.

The question remained, why does the wavefunction collapse? Hugh Everett asked a different question; does the wavefunction collapse at all?

The Many Worlds Interpretation of Quantum Physics (Robert Lea)



Everett imagined a situation in which instead of the wavefunction collapsing it continues to grow exponentially. So much so that eventually the entire universe is encompassed as just one of two possible states. A ‘world’ in which the particle passed through S1, and a world where the particle passed through S2.

Everett also stated the same ‘splitting’ of states would occur for all quantum events, with different outcomes existing in different worlds in a superposition of states. The wavefunction simply looks like it has collapsed to us because we occupy one of these worlds. We are in a superposition of states and are forbidden from seeing the other outcome of the experiment.

Marty realises that when he arrived back in 1905, a worldline split occurred. He is no longer in the world he came from– which he labels World 1. Instead, he has created and occupies a new world. When he travels forward in time to speak to Tobar he travels along the timeline of this world–World 2.

This makes total sense. In the world Marty left, a phone box never appeared on the banks of the Aera on September 27th 1905. This world is intrinsically different than the one he left.

What happens as a result of Marty’s first journey to 1905 according to the Many World’s Interpretation (Robert Lea)

He never existed in this world and in truth he hasn’t actually killed his grandfather. His grandfather exists safe and sound back in 1905 of World 1. If the Many World’s Interpretation of quantum physics is the correct solution to the grandfather paradox, however, then Marty can never return to World 1. It’s intrinsic to this interpretation that superpositioned worlds cannot interact with each other.

With reference to the diagrams above, Marty can only move ‘left and down’ or ‘right’–up is a forbidden direction because it’s his presence at a particular moment that creates the new world. This makes total sense, he has changed history and is in a world in which he appeared in 1905. He can’t change that fact.

The non-interaction rule means no matter what measures he takes, every time he travels back into the past he creates a new state and hops ‘down’ to that state and can then only move forward in time (right) on that line.

Marty’s multiple journey’s to the past create further ‘worlds’ (Robert Lea)

So what happens when Marty travels back to the past in an attempt to rescue his world? He inadvertently creates another state–World 3. This world may resemble World 1 & 2 in almost every conceivable way, but according to the application of the interpretation, it is not the same due to one event–one extra phone box on the banks of the River Aare for each journey back.

As Marty continues to attempt to get back to World 1 — his home — he realises he now lives in a world in which one day in September 1905 on the streets of Bern, hundreds of phone box suddenly appeared on the banks of the Aare, and then simply disappeared.

The sudden appearance of hundreds of red telephone boxes around the banks of the River Aare really started to affect property prices. (Britannica)

He also realises that his predicament answers the question ‘if time travellers exist why do they never appear in our time?’ The truth being, that if a person exists in the world from which these travellers departed they can never ‘get back’ to this primary timeline. 

To someone in World 1, the advent of time travel will just result in the gradual disappearance of daring physicists. That’s the moment it dawns on Marty that as far as World 1 — his world — is concerned, he stepped in a phone box one day and vanished, never to return.

Marty escaped the time travel paradox but doomed himself to wander alternate worlds.

Hey… how do we get our time machine back?

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