theirishtimes--2019-04-08--Go figure Why maths doesnt add up for so many of us

Go figure: Why maths doesn’t add up for so many of us

For many of us it’s a badge of honour to declare: “I’m terrible at maths. I just don’t have the brain for it.”  Yet, says Dr Pádraig Kirwan, head of Waterford Institute of Technology’s department of computing and mathematics, this isn’t the case in many other countries.  “It’s just not an issue in places like China. But in western society there seems to be a mental block that you don’t get in many Asian countries. Over there, maths is treated just like any other subject.”  So how come so many of us are happy to declare our inability to do maths? Is it true that some people just don’t have the brains to do maths? Or can anyone who puts in enough hours get by?  Dr Kirwan suggests there may be cultural or teaching issues that lie behind some of the wider anxiety around maths.  “A lot of it comes down to confidence and giving it to students, teachers and parents. If one part of that chain is weak then it is in danger in bringing it all down. If you have a teacher that conveys a lack of confidence with maths, the students will feed off that.  “Also, if the parent is not confident in helping their child with maths that can be problematic too,” says Dr Kirwan.  Psychologist Dr Kinga Morsanyi of Queen’s University Belfast says the subject has an image problem.  “Starting from parents who tell their children that they’ve always been bad at maths - and who don’t feel confident enough to help with their child’s homework - to primary school teachers who have to teach maths as well as other subjects, and they feel less confident, interested or enthusiastic about maths than other subjects,” she says.  Dr Morsanyi says in many countries, maths is taught by specialist teachers starting from early in the primary school years, but in the UK and Ireland this is often not the case.  “This means that from the start, instead of role models, children might see teachers who are not really confident in their maths knowledge and not very enthusiastic about the subject,” she says.  Maths is a cumulative subject unlike history, she adds: if a child misses some basic concepts, this will cause problems later.  “Maths might also be seen as difficult by pupils, because maths questions always have a right or wrong answer.  “Even when they go just one step wrong, and they have the wrong answer, some children might feel that they failed and give up, when they might have actually been very close to producing a correct answer,” she says.  There is also a persistent notion that girls are more likely to struggle with maths and that boys are more suited to it.  Dr Morsanyi led a team of researchers last year to see if this is borne out in reality  “Girls are less confident and feel more anxious about maths in test situations and in the classroom. These can affect measured performance both directly and indirectly,” says Dr Morsanyi.  She says anxious thoughts during exams can distract students and low self-confidence may result in an underestimation of performance. As a result, even girls with high maths ability can underestimate their chances of success. However, girls can be better than boys at some areas of maths.  “Some research shows that boys are better at spatial reasoning or that girls are better at mental arithmetic.  “However, in a recent study, we have also found no gender differences in mathematics performance in primary school children in Northern Ireland.  “This included no gender differences in average performance, and no differences in the proportion of high and low-achieving girls and boys either,” she says.  Many experts say they key to flourishing at maths is our mindset. Believing that you will be able to reach your destination is crucial.  Stanford University psychologist Carol Dweck says people tend to approach learning with one of two mindsets: growth or ‘fixed.  Those with a growth mindset believe their intelligence can be developed, while a fixed mindset reflects the belief that intelligence is unchanging.  In a number of research papers, Dweck has shown that students with a growth mindset do better in the long term academically, while also developing the sort of virtues – like perseverance and humility – that are a great help in work and life.  Dr Kirwan says our focus also needs to move away from what we can change about the curriculum to how we can best support the two main drivers in a child’s education – parents and teachers.  “The English educationalist Sir Ken Robinson’s famous phrase is ‘no education system is better than its teachers.’  “What is needed is a significant investment in teachers. As a society we invest in our teachers to raise them to the status of doctors and lawyers so that it’s a career they want to go into after college” he says.  Brendan O’Sullivan, chair of the Irish Maths Teachers Association and a teacher at Davis College, Mallow, Co Cork, adds that in the meantime there are simple changes we can make to maths more approachable during the exams.  He says the lack of choice in the Leaving Cert maths exam is a cause of stress, while the timing - on Friday afternoon - is hardly ideal.  The exam itself, he argues, is very text-heavy which can be challenging for those with weak language skills.  Overall, he is struck by how many students seem to think the key to success in maths is reams of notes.  Rather, he says students should take questions, cover up solutions and attempt them from scratch - over and over again. “Studying maths is a skill that requires practice,” he says, “like playing a musical instrument or perfecting taking shots at goal.  Project Sums is a pilot programme which is offering free maths grinds to school students who are priced out of private grind schools.  Independent Senator Lynn Ruane is part of the team behind the programme which runs in Tallaght and Trinity College with plans to open a third location.  The programme provides free maths support to students who are priced out of private grind schools, which can cost €30- €50 per hour.  An Economic and Social Research Institute (ESRI) study in 2016 found that almost half of 17- and 18-year-olds in their Leaving Cert year were taking grinds.  Independent Senator Lynn Ruane, who is part of the team behind the SUMS programme said the response to her online callout for more volunteers has been “phenomenal.”  More than 100 young people turned up for a drop-in session at the County Library in Tallaght recently.  Volunteer Rachel O’Connor says young people can work in groups or if someone has a specific question, they can drop-in, get their question answered and then head home.  “We got a great response from the call out on social media and our volunteers are mainly student maths teachers, retired maths teachers and people working in the maths industry,” she says.  “A lot of students bring in their homework or bring in their exam papers to get prepared for the leaving cert.  “We run through the main parts of the curriculum such as algebra as that is the mainly senior cycle students who come to us but if they need help with any part of the curriculum, we help in any way we can.  “We hope to start collecting data on what the main problems students are facing with the maths curriculum,” she says.  To get in touch about the programme, email project.sums.ireland@gmail.com


theguardianuk--2019-03-16--Humble Pi by Matt Parker review a comedy of maths errors

Humble Pi by Matt Parker review – a comedy of maths errors

You might think you have a phone number, but you don’t really. It’s not a number: you’re not going to perform any mathematical operations on it, and if it starts with a zero then things will go wrong if you do what you would normally do with a number that starts with zero, ie omit it. For this reason, as the “standup mathematician” Matt Parker explains with amusing pedantry, he would really rather we call them “phone digits”.  This is an innocent example of our general fuzziness about maths, where intuitions can go drastically awry. “As humans,” Parker notes for example, “we are not good at judging the size of large numbers.” A million seconds, he points out, is less than two weeks, but a billion seconds is 31 years. And even the mathematics of professionals can fail in critical situations, if our models of how things behave are incomplete. Before the Tacoma Narrows bridge in Washington State collapsed after twisting like a ribbon in the wind, no one had foreseen that kind of “flutter” feedback loop. No one imagined, either, that a single exercise class on one floor could make a whole skyscraper shake, as one did in South Korea in 2011. (The song playing, Snap’s “The Power”, encouraged people to jump up and down at a tempo that matched a resonant frequency of the building.) There may well remain other principles yet to be discovered as we make everything bigger and longer.  In the meantime, engineers continue to make mistakes as elementary as confusing units of measurement. Parker tells the alarming story, for instance, of a passenger jet on which both engines failed midflight because the fuel had been weighed in pounds rather than kilograms. (Luckily, the pilot was able to fly the plane down like a glider and land safely.) Famously, too, a Martian probe burned up in the atmosphere because one piece of software was using imperial units while the rest was expecting metric.  While such examples come with serious lessons about ways to make systems more tolerant of user failure (because users will always fail), Parker is consistently very funny. His chapter on geometrical errors reports with pride that he started a petition for the UK government to replace all football signs – which show an impossible football made entirely out of hexagons – with the mathematically correct figure, comprising 20 hexagons and 12 pentagons. He will teach you to look carefully at any PR image that involves a set of interlocking gears, because chances are the whole mechanism would not be able to move at all. Of one such picture featuring Lego people placed inside such a fatally stationary arrangement of cogs, he writes: “The longer I think about it, the more I’m convinced that this does actually make a great analogy for workplace teamwork.” There are, too, highly entertaining discussions about probability, rounding errors, randomness, correlation, and other concepts we commonly get wrong unless we think really hard.  Fair warning is here given, too, of another Y2K-style bug that is due to hit in 2038. To keep time, computer clocks are all silently counting the seconds since 1970: Parker is the sort of person, he happily tells us, who went out on a boozy night with his mates in 2009 “to celebrate 1,234,567,890 seconds having passed”. Unfortunately, the computers were only given a 32-digit binary address to keep the total number in, and it will run out of space in 19 years, shutting the computers down, unless the problem is fixed. Parker is, of course, the kind of person who knows that the original Y2K bug really was a huge danger: “Through a massive effort, almost everything was updated. But a disaster averted does not mean it was never a threat in the first place.”  Computers, indeed, are a rich source of examples of when maths goes wrong. Databases, Parker points out, are only as good as the data entered into them, and bad data can be worse than none at all. Most pragmatically, he points to a multitude of real-world threats created by the widespread habit of using Microsoft’s Excel software as an ersatz database, rather than as a simple spreadsheet manager. A lot of cell biologists use Excel, he reports, which tends to cause problems because there are genes called MARCH5 and SEP15. Type those into Excel and it will helpfully translate them into dates.  It would be easy to blame poor Excel for such garbling – except that, as Parker insists, people really shouldn’t be using it that way in the first place. Here, as so very often in all walks of modern life, the most appropriate response is the old IT support engineer’s sarcastic acronym, PEBCAK: problem exists between chair and keyboard.  • Humble Pi: A Comedy of Maths Errors by Matt Parker is published by Allen Lane (£20). To order a copy go to guardianbookshop.com. Free UK p&p on all online orders over £15.


thesun--2019-01-08--Bride accused of shaming guests by making them take maths test during dinner

Bride accused of ‘shaming’ guests by making them take maths test during dinner

IT can be hard for a bride and groom to know where to seat their wedding guests – but one couple came up with a very creative way to solve the issue.  The bride decided to get her guests to complete a maths puzzle in order to find their seats.  Her Facebook post was posted on Reddit’s bridezilla forum by user u/Lattesandthelaw, who slammed the couple for “maths-shaming” their guests.  Some Reddit users said the unusual seating chart would cause them anxiety, but others said it was a sweet way to celebrate the couple’s love of maths.  The bride’s post said: “Since both XXX and I are mathematicians, our wedding (which is coming up soon!) will be perfused with mathematical titbits.  “For example, at the dinner, guests will be required to answer a mathematical question in order to find out where they sit. Every guest/couple will be presented with a unique, bespoke question: its difficulty and subject matter drawn directly from what we know their mathematical background to be.  “In fact a very large number of our guests have research-level mathematical background, so for most of them their question has been drawn directly from research papers or thesis.  “It's been a rather consuming but entertaining process to look at all kinds of paper that I wouldn't ever normally look at. Like, for example, set theory!”  Some Reddit users who saw the post said they would RSVP no to the wedding after seeing this requirement.  One person said: “I am laughing because some of the questions have been "drawn directly from their research papers or thesis.  “If I was invited to this, I would have been stuck in the lobby drinking with the caterers. That is too smart for me.”  Another added: “It does sound perfect for THEM but I would decline due to a very likely mathematically-induced anxiety attack.”  However, some defended the bride and groom and said it's up to them as to how they spend their special day.  One Reddit user reasoned: “So now math-shaming is a thing? When is everyone going to stop being offended about everything? If a couple want to celebrate their love for maths that's for them to decide, not anyone else.”  So what do you think? Would you want to go to a wedding with this type of seating chart?  Bride, 28, has dream £10,000 wedding for just £750 after winning more than 20 online competitions.  Cheeky groomsman caught checking out best mate’s new bride in wedding snap.


thesun--2019-02-06--Tiny maths test proves honeybees can do complex addition and subtraction

Tiny maths test proves honeybees can do ‘complex addition and subtraction’

BEES have a grasp of basic maths, including addition and subtraction.  The find reveals that bees have incredibly powerful brains for their size, and suggests that more animals than we thought can do maths.  Boffins trained individual honeybees to recognise colours as plus or minus symbols.  Armed with this knowledge, they went on to solve basic mathematical problems set by the scientists.  The bees completed the tasks with a success rate of up to 75%.  Understanding how a tiny bee brain can do arithmetic could lead to better artificial intelligence (AI) systems, according to the Australian and French team.  Study author Professor Adrian Dyer, from RMIT University in Australia, said: "Our findings suggest that advanced numerical cognition may be found much more widely in nature among non-human animals than previously suspected.  "If maths doesn't require a massive brain, there might also be new ways for us to incorporate interactions of both long-term rules and working memory into designs to improve rapid AI learning of new problems."  The 14 bees taking part in the study were trained to enter a Y-shaped maze consisting of a tunnel with two exits.  At the entrance the bees were shown various shapes - a square, diamond, circle or triangle - coloured either yellow or blue.  The number of elements - from one to five - was randomly altered throughout the trials.  Each colour represented a different mathematical operation - blue for "add one" and yellow for "subtract one".  Once inside the maze the bees entered a "decision chamber" where they had to choose whether to take the left or right fork of the "Y".  More shapes presented at the mouth of each fork represented the correct and incorrect answer to the problem.  If two blue shapes were displayed at the entrance, for instance, the "correct" arm of the "Y" was the one marked by three symbols.  In this case, the bees had to work out that two plus one equals three.  If the bees were initially greeted by two yellow shapes, it meant they had to fly into the arm marked by one symbol (the correct answer to the sum two minus one).  Right answers were rewarded with a tasty drink of sugar water, while making a mistake was "punished" with bitter quinine solution.  Training took place over 100 trials, during which the bees made random choices until they learned how to crack the problem.  It took up to seven hours for them to learn that blue meant "plus one" and yellow meant "minus one".  Each bee was then given four tests involving two addition and two subtraction operations.  The bees chose the correct option between 60% and 75% of the time, said the researchers writing in the journal Science Advances.  They pointed out that solving even basic maths problems required the mental ability to understand abstract rules, and an efficient short-term working memory.  "You need to be able to hold the rules around adding and subtracting in your long-term memory, while mentally manipulating a set of given numbers in your short-term memory," said Prof Dyer.  We pay for your stories! Do you have a story for The Sun Online news team? Email us at tips@the-sun.co.uk or call 0207 782 4368 . We pay for videos too. Click here to upload yours.


thetelegraph--2019-02-06--One two bee honeybees grasp maths and can do basic mental arithmetic exercises study finds

One, two, bee: honeybees grasp maths and can do basic mental arithmetic exercises, study finds

Bees understand mathematics and can navigate a series of puzzles using basic mental arithmetic, experiments have shown.  In a study which has stunned the scientific world, 14 honeybees developed a conceptual grasp of addition and subtraction, then proved they could put it to practical use.  The finding that such small-brained creatures can handle numerical cognition is significant because it suggests the ability may be shared far more widely among non-human animals than previously thought.  It also promises to enable the development of smarter artificial intelligence systems based on insights into the bees’ reasoning processes, researchers said.


thetelegraph--2019-12-03--Britain jumps up international maths rankings following Chinese-style teaching

Britain has jumped up international rankings for maths following Government efforts to import Chinese-style teaching into the classroom.  The UK came 18th, up from 27th three years ago, in the Programme for International Student Assessment (Pisa) league table.  The Pisa tests, which are administered by the Organisation for Economic Co-operation and Development, were taken by 79 countries last year.  The assessment is carried out every three years and involves more than half a million 15-year-olds across the globe taking two-hour tests.  Last year the UK scored an average of 502 points in mathematics, up from 492 in 2015.  Within the UK, the average score for maths among English pupils (504) was “significantly higher” than scores for Wales, Scotland and Northern Ireland, the Department for Education said.  Experts have said that ministers’ attempts to emulate Chinese teaching methods are likely to have driven up the UK’s test results. Countries in the Far East - including China, Singapore and Hong Kong – have long come at the top of the league tables for maths.  Prof Valsa Koshy, an expert in maths education at Brunel University, said: “Importing maths teaching from Far Eastern countries means there has been an emphasis on children needing to master the basic facts and traditional skills such as times tables. This used to be laughed at as too old fashioned.”


freedombunker--2019-08-30--Why Are There so Few Female Mathematicians

Why Are There so Few Female Mathematicians?

Hannah Fry, an associate professor in the mathematics of cities at University College London, has been chosen to present the 2019 Royal Institution’s Christmas lectures. Fry is undoubtedly a good choice to present such an abstruse subject to a lay audience. She’s one of the few mathematicians who is known outside the mathematics tower, having appeared on many BBC programs and YouTube videos. She has written three popular books in addition to the ten professional papers on her résumé. The Institution’s annual science lectures stretch back to 1825, and Fry’s talk will be only the fourth that deals with mathematical issues.  What’s rather odd is that Fry’s lectures seem geared more to warn people about mathematics rather than celebrate it, or at least her subject is being pitched that way. According to the Institution, she will be examining “how our unwavering faith in figures can lead to disaster when we get the sums wrong” and asking “big ethical questions” such as “Are there any problems maths can’t or shouldn’t solve?” In an interview with The Guardian, she argues that mathematicians should be made to take some equivalent of doctors’ Hippocratic Oath because  Now, Fry is the furthest thing from an ideologue. Indeed, her book, The Mathematics of Love, has none of the standard flaws that define many third-wave feminist tracts; the book is dispassionate, witty, and statistically accurate. So why would Fry throw out this sexist and racial language without even a beat for thought? Perhaps her perspective on privilege has been skewed by her being a minority female member of a very exclusive male group.  Even top women mathematicians, despite being highly successful within their chosen field, tend to attribute the gender disparity in mathematics to masculine bias. Karen Uhlenbeck, who in March became the first woman to be awarded the Abel Prize, the second most prestigious honor in mathematics after the Fields Medal, said,  Does the research back this up? Historically, while the first female mathematician is generally held to be Hypatia of Alexandria, who lived 1600 years ago, it was the 18th and 19th centuries that saw the emergence of several eminent women mathematicians in the supposedly inflexible patriarchy of Europe.  These included Maria Agnesi (1718-1799), an Italian philosopher and mathematician who was the first female to be appointed as a mathematics professor; Sophie Germain (1776-1831), whose paper on elasticity theory made her the first woman to be awarded by the Paris Academy of Sciences in 1816; and Emmy Noether (1882-1935), a German mathematician who worked on non-commutative algebras, hyper-complex numbers, and commutative rings and was awarded the Ackermann-Teuber Memorial Award in 1932. Considering high mathematical ability is always rare, the fact that these women’s work was recognized centuries ago suggests that the bias narrative is at least somewhat exaggerated.  Nonetheless, it remains a fact that even after decades of special programs to get women more involved in STEM fields, females are still under-represented in science, technology, engineering, and especially high-level mathematics. Lists of the world’s top mathematicians typically include just one woman, if any at all, even when those lists are compiled by female mathematicians or left-leaning newspapers. The best evidence suggests that top mathematicians are born rather than made and that more boys than girls are born that way.Indeed, feminist websites often pad their lists of female mathematicians by including celebrities like Big Bang actress Mayim Bialik, whose PhD is in neuroscience, not mathematics; Channel 4 presenter Rachel Riley, who has only an undergraduate degree in the subject; and even Nobel Peace Prize winner Malala Yousafzai, who went no further than A-Level Maths.  While social factors do explain some of the gender differences, the best evidence suggests that top mathematicians are born rather than made and that more boys than girls are born that way. In a 2008 essay in Scientific American, Diane F. Halpern, probably the world’s second-leading researcher in cognitive sex differences (after Simon Baron-Cohen), and her colleagues note that in America’s Scholastic Aptitude Tests (SATs), twice as many boys as girls score  500 or above on the math portion, and there are 13 times as many boys with scores of at least 700.  Thus, we find that in the UK, fewer than 20 percent of female students are pursuing engineering or computer science, and fewer than 40 percent are doing mathematics. The gender disparity is three to four times larger for post-graduate students. And because both sexes prefer to work in fields that they’re better at, women and men make different career choices. As a 2014 US study that tracked a top 1 percent cohort of mathematically gifted 13-year-old boys and girls to adulthood found:  So, in a field dominated by white (and Asian) males, The Guardian reports that Ms. Fry wants mathematicians to take an ethical pledge that will  Had Albert Einstein taken heed of this advice, would he have suppressed E=mc2?


theguardianuk--2019-01-28--Can you solve it Gems from a mathematical wonderland

Can you solve it? Gems from a mathematical wonderland

Today, three puzzles from Mathigon, a remarkable maths website (about which more later).  An easy one to start. Count the number of triangles in the image below.  Careful though! It’s always easy to miss a few.  Talking of (and walking on) triangles…here’s a probability question about random ants.  Finally, a very nice logic problem based on arithmetic. If 28 out of the 30 statements in the list are correct, but the two that are wrong are consecutive, which ones do they have to be? (Consecutive meaning the two statements are next to each other in the list).  Integer means whole number. I like this one since you don’t need to know the value of the large integer to work out the answer, although some of you might want to find it out.  I’ll be back at 5pm UK time with the answers. Meanwhile NO SPOILERS.  The puzzles – and the illustrations – are taken from Mathigon, a wonderful interactive maths ‘textbook’ that is free online. Rather than just presenting you with information, Mathigon challenges you with problems and allows you to interact with diagrams. It is one of the most accessible and engaging maths resources available on the web, a true mathematical wonderland. The text is well-written, the pages are beautifully designed and presented, and it covers a large array of topics.  What’s particularly impressive about Mathigon, however, is that it has been conceived, coded, written and designed by a single person: Philipp Legner, aged 26. He has a unique combination of programming skills, a gift for exposition and an eye for design.  Legner studied maths at Cambridge university. He got the idea for Mathigon when volunteering for a maths outreach project with local schools. “I always meet kids (and adults) who “hate” mathematics, so I wanted to develop a platform where every student can enjoy learning mathematics – just like I did. Rather than simply memorising equations and procedures, I want students to be able to explore, discover and be creative.”  Legner has worked as a programmer for Bloomberg and Google but recently left to work on Mathigon full-time. The site has deservedly won several awards and he hopes to secure funding to keep it free for students. If you haven’t seen it before, I recommend you check it out. The URL is mathigon.org.  I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.    I’m the author of several books of maths and puzzles, including Can You Solve My Problems. I also co-author Football School, a book series for kids that explains the world through football.


theindependent--2019-02-06--Bees are able to understand mathematics study reveals

Bees are able to understand mathematics, study reveals

Bees are able to grasp basic mathematical ideas, according to a new study, which suggests a small brain may not mean low intelligence.  After training a group of the pollinating insects scientists at the French National Centre for Scientific Research found they were able to perform addition and subtraction.  These abilities require animals to carry out intricate processes in their brains, remembering complex rules while applying them to new situations.  Maths was therefore long thought an ability unique to humans, but in recent years experiments have shown the skill is found across the animal kingdom.  Chimpanzees, parrots and pigeons are just some of the creatures that have demonstrated the capacity to add and subtract.  In the new study, a team led by Dr Scarlett Howard, first taught their bees to recognise colours as symbols for either addition or subtraction. Specifically, blue meant “more” and yellow meant “less”.  Next, their bees were trained to enter Y-shaped mazes in which they had to make a choice between two sets of shapes.  In each case, if they made the correct choice they were rewarded with a sugar water treat, whereas the wrong choice would yield a bitter-tasting quinine liquid.  At the entrance to the maze, the bees were met with between one and five shapes, coloured either blue or yellow.  Next they flew into a chamber where they could fly towards either the original number of shapes, plus or minus one, or the incorrect number of shapes.  If they first encountered blue they had to add, and if yellow they had to subtract.  Over the course of 100 trials, the researchers trained 14 bees to choose the correct option around 75 per cent of the time, the scientists wrote as they published their findings in the journal Science Advances.  Dr Howard compared the experiments to humans first learning to link mathematical symbols with concepts.  “We learn as children that a plus symbol means you need to add two or more quantities, while a minus symbol means you subtract,” she said.  Professor Adrian Dyer, who contributed to the study, added: “Our findings suggest that advanced numerical cognition may be found much more widely in nature among non-human animals than previously suspected. If maths doesn’t require a massive brain, there might also be new ways for us to incorporate interactions of both long-term rules and working memory into designs to improve rapid AI learning of new problems.”  This is not the first time bees have demonstrated a way with numbers.  Previous experiments have shown the insects are capable of counting to four, and even understand the concept of zero.  Work conducted by Dr Howard and her team showed that when bees were incentivised to choose a target with fewer shapes, they recognised that a target with no shapes had fewer than one with two or three.  This was the first time an invertebrate had been shown to understand the abstract idea of zero – a concept even humans find more difficult to appreciate than other numbers.  While the ability to undertake these kind of calculations may not be immediately useful for bees, advanced brain power may help them when foraging for flowers to remember different combinations of colours and shapes.


theindependent--2019-03-14--Pi Day Why some mathematicians refuse to celebrate 14 March and wonapost observe it

Pi Day: Why some mathematicians refuse to celebrate 14 March and won't observe it

14 March is celebrated by some as the most exciting day in mathematics – when the date lines up in the numbers of the famous constant, Pi or π. But some people would rather it isn’t celebrated at all.  The day looks like 3.14 for those that write dates in the month then day format, but that is rarely used outside of the Americas.  Many people instead opt to celebrate the potentially more interesting Pi approximation day, which takes place on 22 July or 22/7 – a way of working out an approximate value for Pi for use in rough calculations.  And still others would rather that Pi day was not celebrated at all, and say that the number should not be treated with such reverence. Instead, they say that we should use Tau – a number that serves much the same purpose and can be celebrated on 28 June, or 6/28 in America.  The complaints about the date are relatively obvious. Most places around the world use a different system to the US, where Pi Day was born.  By far the most popular date format is day/month/year, as is used in the UK as well as much of Asia, Australia, Africa and South America. It is not possible to write Pi day using that format, but it is easy to write Pi approximation day.  In China and many other countries in south-east Asia, dates are written year/month/day. It is possible to do Pi day in that format since the day (14) does follow the month (3), but the year comes first and throws that off slightly.  But the complaints about the competitor to Pi Day, Tau Day, are much more aggressive – and campaigners for the change have made a website and a full manifesto, laying out exactly why the world is wrong.  That manifesto, written by Tau fan Michael Hartl, is dedicated to the “true circle constant”, which he says should be referred to by the Greek letter Tau.  The number itself is simply Pi, but doubled – 6.28318, and so on. That makes it easier to use in many applications, campaigners claim.  But those behind the Tau manifesto admit that they are facing a difficult challenge: “a powerful conspiracy, centuries old, determined to propagate pro-Pi propaganda”.  But they hope that they can prove that Pi “is a confusing and unnatural choice for the circle constant”, they say, even if the name of the constant and its day doesn’t actually change.  To settle the argument, some people have taken to celebrating a specific “Tau time” on Pi day – 6.28pm. That is when, for instance, the Massachusetts Institute of Technology sends out its admissions decisions over the internet, which was a change made to welcome Tau into the celebrations.  We’ll tell you what’s true. You can form your own view.  At The Independent, no one tells us what to write. That’s why, in an era of political lies and Brexit bias, more readers are turning to an independent source. Subscribe from just 15p a day for extra exclusives, events and ebooks – all with no ads.


theirishtimes--2019-02-21--A mathematical formula for sharing fair and square

It is common practice in science to name important advances after the first discoverer or inventor. However, this process often goes awry. A humorous principle called Stigler’s Law holds that no scientific result is named after its original discoverer.  This law was formulated by Prof Stephen Stigler of the University of Chicago in his publication “Stigler’s law of eponymy”. He pointed out that his “law” had been proposed by others before him so it was, in a sense, self-verifying.  Occasionally, mathematical inventions are made simultaneously by more than one person. Perhaps the greatest example is the formulation of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The ensuing controversy over priority was described in a book by Jason Bardi, The Calculus Wars. The conflict between British and European mathematicians raged for decades before joint credit for independent discovery was accepted.  A remarkable numerical sequence that has been discovered several times by different mathematicians in different circumstances illustrates Stigler’s Law. It was originally called the Morse sequence when, in 1921, it was popularised by the American mathematician Marston Morse, who applied it in differential geometry. Today, it is usually called the Thue-Morse sequence: the Norwegian mathematician Axel Thue discovered the sequence while studying “combinatorics” on words; this was in 1906, well before Morse. But, more recently, the name Prouhet has been prefixed to the title of the sequence, which was first studied by Eugène Prouhet in 1851.  The Dutch mathematician and chess grandmaster Max Euwe defeated Alexander Alekhine in 1935, becoming the fifth World Chess Champion. Six years earlier, Euwe analysed the game of chess from a mathematical perspective and showed that the rules in place at that time did not exclude the possibility of infinite games. In his analysis he rediscovered the Thue-Morse sequence.  So what is the Thue-Morse sequence? It is a binary sequence, an infinite list of 0s and 1s. We start with 0. At each stage, we replace 0 by 01 and 1 by 10. At the first stage, 0 becomes 01. Then 01 becomes 01 10. Continuing, we get 01 10 10 01, then 01 10 10 01 10 01 01 10 and so on ad infinitum, the length doubling at each stage.  But what use is the Thue-Morse sequence? It arises in a remarkably wide variety of contexts. It is valuable in number theory, combinatorics, computer graphics, fractal geometry and equitable sharing. Let’s look at the last of these.  Suppose that Aodh and Bea are divorcing. They have to divide up their possessions. They agree on the value of each item. They toss for first choice and Aodh wins. The order of choice is then ABABABAB and so on. This is unfair: Aodh always gets the better of each pair of items. Better to start ABBA so that Bea gets the advantage for the second pair.  But how to continue? The answer is in the Thue-Morse sequence. Writing A for 0 and B for 1, the sequence begins ABBA BAAB, continuing according to the pattern above. If Aodh and Bea choose in this manner, a fair division of property is guaranteed.  Sequencing is important in sports: penalty shoot-outs in football, service order in tennis tie-breaks and choice of colour in chess matches. Normally, they are based on the ABABAB pattern, but this gives undue advantage to the winner of the toss. Are we ready for a fairer but mathematically more complex solution?  Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics – he blogs at thatsmaths.com


theirishtimes--2019-07-25--Limerick-based student wins silver at International Mathematical Olympiad

A Co Limerick-based student has become only the second person to win a silver medal for Ireland at the world’s most prestigious second-level mathematics competition.  Lucas Bachmann (19), a 6th year student at Glenstal Abbey in Co Limerick, was one of 621 contestants from 112 countries who competed in the International Mathematical Olympiad (IMO) in Bath, England.  The annual competition attracts the brightest mathematical brains on the planet. It is regarded as a source of national prestige in Asia in particular, with China, South Korea, North Korea and Thailand prominent among the countries participating.  Mr Bachmann was part of the six-person Irish team selected after sitting two mathematical papers at home. He was born and brought up in Switzerland and then lived in China before his family relocated to Ireland for work purposes four years ago.  He previously won a bronze medal at the competition and hoped to win gold this time around, but was stumped by question about positive integers.  “It was one of those simple questions that had a trick to it in order to solve it. Now I know the trick,” he said. “Everything I have done over the last four years has truly cemented me in participating in this. Computer science had been my favourite subject. That was not a subject I was offered at school in Ireland. Mathematics took over for me then.”  In the three weeks leading up to the Olympiad, he did mathematics training camps with the other Irish participants.  “I would say that it is not something that any ordinary student could come in and ace. You need a lot of preparation to even understand the questions.”  Having now studied in three countries, Mr Bachmann says maths in Irish schools is too easy and involves “remembering all the information before exams and then spitting it out”. He said this was not in keeping with the analytical thinking needed to take part in the Olympiad.  The only previous time a student from Ireland gained a silver medal at the Olympiad was Fiachra Knox in 2005. Ireland has participated since 1988.  Laura Cosgrave from Midleton College in Co Cork and Tianyiwa Xie from Alexandra College in Dublin received honourable mentions for their efforts at the competition in Bath.  Three members of the Irish team were born in China, meaning four of the six participants began their mathematical education abroad.  Ireland finished 71st out of 112 countries, an improvement on past performances.  Dr Bernd Kreussler, of Mary Immaculate College in Limerick, who runs the Irish Olympiad, said the predominence of foreign-born students participating for Ireland in recent years reflects on how maths is taught in Irish schools.  “Students who have had their basic education in a school in other countries are much better in dealing with these kind of problems,” he explained.  “In standard school book situation when you are given a problem, you know exactly what to do. You have to be very creative to solve the problems in the Olympiad. This is obviously not something we are well trained at in Ireland.”


theirishtimes--2019-10-16--Irish mathematician Hamilton celebrated in Broombridge artwork

Most graffiti does not stand the test of time but one ‘Eureka moment’ 176 years ago, when an equation was scratched on a bridge in North Dublin, proves the exception. What’s more, the mathematical equation, written using a penknife by the noted Irish mathematician William Rowan Hamilton, is set to be preserved for many years to come – thanks to the latest work of tattoo artist Emma Ray.  Well-known local and Minister for Finance Paschal Donohoe unveiled the work on Wednesday, on what is Hamilton Day, at Broombridge Luas stop. It celebrates that 176-year-old act of graffiti.


thesun--2019-01-11--Alan Turing How the mathematician and computer scientist helped win WW2

Alan Turing: How the mathematician and computer scientist helped win WW2

THE work of mathematician and computer scientist Alan Turing is said to have shortened the Second World War by four years.  His work should have led him to be hailed as a national hero but instead he was hounded for his sexuality. Here’s what you need to know about him.  Alan Turing was born on June 1912 in Maida Vale, London and from an early age it was clear that he was highly intelligent.  When he was nine, his both his primary school headmistress and his secondary school maths teacher declared him to be a “genius” in reports.  He passed the time by working out complex chess problems on his own, according to the British Library.  Teachers were often annoyed with him for his high marks in exams despite him paying little attention during lessons.  Turing began studying mathematics at King's College Cambridge, in 1931 and graduated in 1934 with a first class degree  During his time at university he enjoyed an active social life, took up rowing and became an excellent long distance runner.  Later in life he finished fifth in qualifying for the 1948 London Olympics with a time of 2 hours 46.03 minutes which for the day was very respectable.  He continued to compete for Walton athletic club until forced to stop by injury.  After graduating from Cambridge he obtained a PhD in mathematics at Princeton University in 1938.  It was at Princeton that he developed the notion of a “universal computing machine” which could solve complex calculations.  Later this would become known as the Turing machine, which foreshadowed the digital computer.  At Princeton he also studied cryptology codes and cyphers which can be used to send secret messages.  Turing was asked join the Government Codes and Cypher School, a code-breaking organisation which is now known as GCHQ.  The organisation moved to Bletchley Park in Buckinghamshire, which became the top secret home of Britain’s code breakers.  He was based in the famous Hut 8 and his most notable achievement at Bletchley was cracking the German’s ‘Enigma’ code.  The Enigma was a machine used by the German armed forces to send encrypted messages securely.  Together with fellow code-breaker Gordon Welchman, developed a machine called the Bombe which from late 1940 was decoding all messages sent by the Enigma machines.  Turing’s team also cracked complex German naval signals in 1941, contributing to Allied victory in the Battle of the Atlantic.  His other work included developing a machine to encode and decode voice communications.    Turing was as awarded an OBE for his services to the country in 1945 and made deputy director of the Computing Laboratory at Manchester University.  He continued to producing ground breaking work including his famous paper “Computing Machinery and Intelligence” which was published in 1950.  Turing was gay at a time when homosexuality was illegal in Britain.  In January 1952 he began a relationship with Arnold Murray, a 19-year-old unemployed man.  Later that month, Turing reported a burglary to the police and Murray told him that the perpatrator was an acquaintance of his.  During the investigation he acknowledged a sexual relationship with Murray.  Turing was charged with gross indecency and convicted under Section 11 of the Criminal Law Amendment Act 1885.  He pleaded guilty and opted for chemical castration by a series of injections of female hormones.  The scientist was rendered impotent and his security clearance was removed barring him for continuing his code breaking work for GCHQ.  Turing died two years later from cyanide poisoning in an apparent suicide - though there have been suggestions his death was an accident.  In 2013 he was officially pardoned thanks to a campaign backed by MPs and celebrities like Professor Stephen Hawking, Stephen Fry and Benedict Cumberbatch.  Thousands of other gay and bisexual men convicted of consensual same-sex relationships were also posthumously pardoned under what became known as ‘Turing’s Law’.  We pay for your stories! Do you have a story for The Sun Online news team? Email us at tips@the-sun.co.uk or call 0207 782 4368 . We pay for videos too. Click here to upload yours.'


eveningstandard--2019-03-13--Happy Pi Day 2019 Activities facts and why we celebrate pi today

Happy Pi Day 2019! Activities, facts, and why we celebrate pi today

Maths fans rejoice, because one of the most exciting days in the number-crunching calendar is just around the corner.  March 14 is recognised annually as Pi Day, a celebration of the mathematical constant pi, which this year falls on Thursday.  Observed in some quarters by the consumption and throwing of pies, National Pi Day has also been recognised by the US House of Representatives.  Here's everything you need to know about Pi Day.  International Pi Day is a chance for maths enthusiasts to celebrate the number pi - represented by its first three digits of 3.14.  Many use the day to discuss the number, recite its infinite digits and eat plenty of Pie.  It is celebrated on the same day every year - March 14, which this week is on Thursday.  Granted, this method only works for those countries that write dates in the month then day format, but written down March 14 looks like 3.14, which are the first three digits of pi.  Since the exact value of pi can never be calculated, the numbers 3.14 are used to approximate the value of pi in other calculations.  However in a lot of countries, including the UK, the format is the other way round - meaning the day is written first, followed by the month.  Since this would make Pi Day look like 14.3, in many places outside America it doesn't make sense, which has led to calls for the celebration to be either scrapped or celebrated on Pi Approximation Day on July 22 (22/7 in the day first format) or Tau Day on June 28 (6/28 in the month first format).  Pi is a mathematical constant, which is a number that has a special meaning for calculations.  The constant Pi means the ratio of the length of a circle's circumference to its diameter - a value which is always the same for any circle.  The first 50 digits of pi are 3.14159265358979323846264338327950288419716939937510.  If you're a teacher and want to get some of the youngsters involved in Pi Day, or you just want to celebrate it with your own children, there are a number of fun activities you do on Thursday.  You could try and memorise as many digits of pi as you possibly can - a fun but nonetheless tricky activity to try.  Otherwise, you could make a paper chain with different colours representing digits and decimal points.  Or you could even bake a pie, and try to fit in as many digits on the pastry lid as you possibly can.  Subscribe to Women Tech Charge, a new podcast from the Evening Standard.  From fashion to finance, technology is revolutionising our lives. Meet the extraordinary women who are leading the charge.  Listen on Apple Podcasts, Spotify or wherever you get your podcasts.


eveningstandard--2019-03-14--Pi Day 2019 Google employee Emma Haruka Iwao breaks Pi world record

Pi Day 2019: Google employee Emma Haruka Iwao breaks Pi world record

A Google employee has calculated the value of the number pi to a new world record of 31.4 trillion digits.  Emma Haruka Iwao, from Japan, used Google's Compute engine to help smash the previous record of around 22 trillion.  It was the first time the company's cloud service had been used for a calculation of this scale - 31,415,926,535,897 digits.  The calculation took 25 virtual machines 121 days to complete, with Google celebrating the news on National Pi Day.  It reportedly required 170 terabytes of data to complete, which is roughly equivalent to the amount of data in the entire Library of Congress print collections.  Ms Iwao, who has been fascinated by the number pi since she was a child, said of the achievement: "I was very fortunate that there were Japanese world record holders that I could relate to.  "I’m really happy to be one of the few women in computer science holding the record, and I hope I can show more people who want to work in the industry what’s possible.”  Pi is a mathematical constant and is the ratio of the length of a circle's circumference to its diameter - a value which is always the same for any circle  But we can never find the true meaning of pi, because it is what is known as an "irrational number".  The first three digits - 3.14 - are well known but the number is infinite and never-ending.  Mathematicians and scientists are constantly working to calculate more and more digits of pi, which is used to test supercomputers and in engineering, physics and space exploration.  Ms Iwao, who wants to expand her work, told BBC News: "I feel very surprised. I am still trying to adjust to the reality. The world record has been really hard."  Subscribe to Women Tech Charge, a new podcast from the Evening Standard.  From fashion to finance, technology is revolutionising our lives. Meet the extraordinary women who are leading the charge.  Listen on Apple Podcasts, Spotify or wherever you get your podcasts.


fortune--2019-03-14--Google Employee Sets New World Record for Calculating Pion Pi Day

Google Employee Sets New World Record for Calculating Pi—on Pi Day

Most people know 3.14 is the beginning of the value of pi, if only because of the convenient reminder each March. Some mathematicians can take it significantly further. No one, though, has taken it as far as Emma Haruka Iwao.  The Google Japan employee has set a new world record, calculating pi to 31 trillion digits of the infinitely long number. That shatters the previous record, which took the number to 22 trillion.  Iwao had a little help in his quest, recruiting Google’s cloud computing service to assist with the math. That took 25 virtual machines 121 days to complete.  The calculation required 170 TB of data, enough to store 85,000 hours of movies or more than 105 billion photos.  To recite the number out loud, says Iwao, would take 332,064 years. So… maybe we’ll just stick with 3.14 for now and enjoy a slice of key lime while we’re at it.  Pi is more than deserts, pizzas and chicken pot pie, of course. It represents the ratio of a circle’s circumference to its diameter. And that ratio never changes.  Just 34 years ago, people had only managed to calculate pi to 1 million digits. That has, obviously, expanded as computers have gotten smarter and worldwide obsession with the number has increased.  How obsessed? There’s even a language based on Pi these days.


fortune--2019-03-14--If Youre Only Celebrating Pie on Pi Day Youre Doing It Wrong

If You’re Only Celebrating Pie on Pi Day, You’re Doing It Wrong

Pi Day, celebrated each year on March 14, marks our collective love for the mysterious mathematical constant—and also fine baked goods.  For Pi Day 2019, food establishments across the U.S. are celebrating with deals on apple pies and pizza pies, even chicken pot pies, all in the name of 3.141592653589793238462643383279. Yes, that is pi to the 30th decimal place, and yes, you should be impressed.  But there’s a lot more to Pi Day—not coincidentally celebrated on 3/14—than the food. So, while you’re enjoying the pie of your choice today, here are some pi basics to remember.  1. Pi represents the ratio of a circle’s circumference to its diameter. No matter how big or small the circle is, the ratio never changes. It will always be the same.  2. Pi is irrational, meaning its decimal representation is infinite and its exact value cannot be determined.  3. The never-ending number has been recognized for thousands of years, dating back to the Babylonians.  4. People have attempted to break records calculating and memorizing it. In 1975, pi was calculated to a million digits. Modern technology and algorithms now let us calculate pi into the trillions.  5. Rajveer Meena of India holds the Guinness world record for the most digits of pi memorized after reciting 70,000 decimal places of the ratio in 2015. It took him 10 hours and he wore a blindfold the entire time.  Think you can do better? Apply here.  6. Many Egyptians believe that the pyramids of Giza, one the seven wonders of the world, were built on the principles of pi, with the height of the pyramids having the same relationship with the perimeter of their base.  7. There is a language, called Pilish, based on pi. The numbers of letters of each word in a sentence match the digits. For example, this poem by Joseph Shipley published in 1960. The first word “but” has three letters, the second word “a” has one, the third word “time” has four and so on: 3.14…  8. Theoretical physicist Albert Einstein was born on Pi Day—March 14, 1879, in Germany.  10. Pi is the most recognized mathematical constant around the world. It is also often referenced in popular culture, including in Carl Sagan’s novel Contact.  So, there you have it—pie is great.  But so is pi.


thepoke--2019-03-14--The fabulous Brian Bilston has written the perfect poem to celebrate Pi Day

The 14th of March is Pi Day, because when the date is written in the American style – 3.14 – it matches the mathematical constant Pi (π), the ratio of a circle’s circumference to its diameter. The internet tends to celebrate by doing jokes about pies, pirates and Magnum P.I., but Twitter poet – and the world’s – Brian Bilston, has gone one better, with this beautiful new piece.  Someone noticed his nod to the vulgar fraction representation of Pi.


thesun--2019-03-14--What is Pi Day 2019 fun facts and ways to celebrate

What is Pi Day? 2019 fun facts and ways to celebrate

MARCH 14 marks the annual celebration of the mathematical constant π (pi).  Here's everything you need to know about Pi Day.  The maths event celebrates π, the term used to describe the ratio of a circle’s circumference to its diameter, commonly shortened to 3.14159.  It has been traced back to Larry Shaw, a physicist who is believed to have held the first ever Pi Day celebration in 1988.  There’s no doubt that some will be disappointed that the day has nothing to do with food.  Despite this, revellers have been known to cook up a pie or two to mark the celebration.  These are often decorated with the mathematical symbol π.  Every year, Pi Day is observed on March 14.  The event falls on this specific date because the first three significant digits of π are 3, 1 and 4.  In 2009, the US House of Representatives officially recognised the date as a national celebration.  Tech giant Google also created a special Doodle to mark the holiday in 2010.  As well as honouring pi, the celebration works to inspire further academic research and interest in maths and science.  As the exact digits of the irrational number remain unknown, the mystery of π has continued to enthral academics.  The special constant shows up across many disciplines, including chemistry, physics and maths.  Although the official premise of the day is to honour π, many believe it’s more of a celebration of academia.  For this reason, schools, universities and museums across the globe encourage festivities.  You can celebrate by baking pies, throwing pies, or if you want a more mathematical approach learning as many digits as possible.  Here's the first ten to get you started... 3.1415926535.  1. It’s the 16th letter of the Greek alphabet  2. It was proved to be an irrational number in the 18th Century  3. There are no occurrences of the sequence 123456 in the first million digits of pi  4. The World Record for memorising the digits of pi belongs to Lu Chao, from China. The maths whizz was able to recall the first 67,890 numbers  5. Using a powerful computer, the Hitachi SR 8000, a Japanese scientist found 1.24 trillion digits of Pi, breaking all the previous records  6. Pi Day falls on March 14, which is German genius Albert Einstein’s birthday  7. Pi has featured in many a Hollywood blockbuster, as it was used as the secret code in Hitchcock’s Torn Curtain and also in The Net, which stars Sandra Bullock


theirishtimes--2019-03-21--How Joseph Fourier discovered the greenhouse effect

Jean-Baptiste Joseph Fourier, French mathematician and physicist, was born in Auxerre, 251 years ago today. He is best known for the mathematical techniques that he developed in his analytical theory of heat transfer. Over the past two centuries, his methods have evolved into a major subject, harmonic analysis, with widespread applications in number theory, signal processing, quantum mechanics, weather prediction and a broad range of other fields.  In his mathematical theory of heat conduction, Fourier based his reasoning on Newton’s law of cooling, with the flow of heat between two adjacent particles being proportional to the difference in temperature. He wrote that “heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space”. He established the fundamental equation that governs the diffusion or spreading out of heat, and solved it by using the infinite series of trigonometric functions that we now call Fourier series.  Fourier published his work on heat flow in 1822 in his book Théorie analytique de la chaleur (The Analytical Theory of Heat). He claimed that any function, even one with wild variations and jumps in value, can be represented by a series of sine waves. This claim gave rise to great controversy and even Joseph-Louis Lagrange, a mathematical giant, found it incredible. Although further conditions are required to justify Fourier’s observation, it was essentially correct and was a great mathematical advance.  Fourier was the first person to study the Earth’s temperature from a mathematical perspective. He examined variations in temperature between day and night, and between summer and winter, and concluded that the planet was much warmer than a simple analysis might suggest.  Fourier calculated that it would be much colder than it is if the incoming radiation from the sun were the only warming effect. His idea that the Earth’s atmosphere acts like an insulator is the first formulation of what we now call the greenhouse effect. However, he did not use that term, which first appeared in the work of Swedish meteorologist Nils Gustaf Ekholm around 1900.  The energy from the sun peaks in the visible range and the atmosphere is transparent to this radiation, which passes through and warms the land and the oceans. They in turn radiate their energy as infra-red radiation. Some gases in the atmosphere (mainly water vapour and carbon dioxide) strongly absorb this energy and re-emit it, partly upward and partly downward to warm the surface.  The greenhouse effect is essential for sustaining life on Earth. The intensity of the warming effect depends on the concentration of the radiatively active gases, called greenhouse gases. Human activities, mainly the burning of fossil fuels, have increased their concentrations and the result is a warming of the atmosphere. There is now widespread concern that the warming will have catastrophic consequences if it is not arrested or reversed.  John Tyndall, born in Leighlinbridge, Co Carlow, conducted a series of experiments in the 1850s to measure precisely how infra-red radiation is absorbed by gases. He tested several gases and vapours, and found that while pure elements such as oxygen and nitrogen have little effect, more complex molecules are strong heat absorbers.  Tyndall showed that water vapour, carbon dioxide and ozone are strong absorbers of heat radiation. He speculated how changes in water vapour and carbon dioxide are related to climate change. This is what we now call the enhanced greenhouse effect. He realised that the strongest effect was from water vapour, without which “the Earth’s surface would be held fast in the iron grip of frost”. I leave you to ponder where the frost might come from.  Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics – He blogs at thatsmaths.com


theverge--2019-03-14--Pi Day is a lie celebrate tau the true circle constant instead

Pi Day is a lie: celebrate tau, the true circle constant instead

Today is Pi Day, perhaps the most popular of geeky holidays. Hooray. But I’m here to tell you that Pi Day is wrong — or rather, the entire idea of pi as a mathematical concept is wrong.  It’s easy enough to see why people like Pi Day: the whole thing starts with a mathematical pun of sorts (The date is written as 3/14 in American notation. Pi starts with the digits 3.14. You get it.) It’s an easy, fun ritual to see how many digits you can pointlessly memorize of the famously never-ending, never-repeating number (even though 39 digits is more than sufficient for almost any calculations you’ll ever need). Plus pi sounds like pie, and who doesn’t like pie?  π as a number is bad, and therefore, so is the entire misguided day dedicated to its celebration  But here’s the thing: π as a number is bad, and therefore, so is the entire misguided day dedicated to its celebration. It’s a lot to take in, and I, too, was once like you: I was taught the virtues of pi for years, going back to Pi Day parties in middle school. But instead of pi, we should celebrate tau, an alternative circle constant referred to by the Greek letter τ that equals 2π, or approximately 6.28.  I’m not just making this up out of nowhere: the terribleness of pi as a constant was first proposed by mathematician Bob Palais in his article “π Is Wrong!” and later expounded upon in The Tau Manifesto by Michael Hartl, which serves as the basis for modern tauism. (Internet-famed mathemusician Vi Hart is also a major proponent of tau over pi, if you prefer your mathematical arguments in a more entertaining video form.)  But Palais and Hartl’s arguments both boil down to some basic math. Step back in time to when you first learned geometry and recall the simple origins: no matter what circle you’re using, if you divide the circumference of the circle by the diameter, you’ll get the same answer: an endless number, starting with the digits 3.14159265... (aka pi).  And right there is the fundamental flaw. The thing is, we don’t actually use diameter to describe circles. We use the radius, or one-half the diameter. The circle equation uses the radius, the area of a circle uses the radius, and the fundamental definition of a circle — “the set of all points in a plane that are at a given distance from a given point, the center” — is based on the radius. Plugging that into our circle constant equation gives us a new circle constant equivalent to 2π, or 6.28318530717..., colloquially referred to with the Greek letter τ (tau). Switching to τ isn’t making some arbitrary change for the sake of it. It’s bringing one of the most important constants in math in line with how we actually do math.  π isn’t really something we use in day-to-day math to start with  Now, you may be thinking that this will cause fundamental, seismic changes in math. “How on earth could you possibly replace something as important as pi!?” you could ask. But if we’re being honest, π isn’t really something we use in day-to-day math to start with. Unless you’re someone who does a lot of geometric calculations in your daily life, chances are you only ever really encounter pi when it comes time to rattle off some digits for Pi Day. Sure, it’s a good introduction to the idea of irrational numbers, but tau would work just as well for that. And if you do work with π a lot, replacing it with τ is beneficial for a whole host of reasons, mathematically speaking. Again, I’ll direct you to The Tau Manifesto for the full argument, but I’ll just point out a few here.  One big thing tau fixes is radian angles. You may remember that as “those annoying chunks of a circle represented by weird fractions of pi from high school math,” but with tau, it’s simple: everything matches up where it should fractionally. So half the circle (180 degrees) — τ/2. 1/12th? τ/12. It’s a small change, but it makes angular notation — a frustratingly obtuse part of geometry that through the use of pi demands an elitist notion of memorization of angles and conversions — a more welcoming and intuitive prospect for new students.  It also makes circle functions like sine and cosine easier, since it makes one full cycle of the function match up with one full turn of the circle (tau), instead of the seemingly arbitrary 2π that you get using π as your circle function. Like with radian angles, it makes deriving sine and cosine values a simple process from simply drawing the function, instead of demanding that students remember that 3π/2 is for some reason the three-fourths point on the wavelength.  Looking back at years of math and physics notes with the enlightened lens of tau, I weep for my former self  Similarly, it makes a bunch of other higher math — like integrals in polar coordinates, the Fourier transform, and Cauchy’s integral formula simpler, since they also already work in terms of 2π anyway. Using tau just cuts out the middleman. Looking back at years of math and physics notes with the enlightened lens of tau, I weep for my former self and the accumulated hours of needless conversions and complications introduced by pi.  It’s not just practical purposes, though. Replacing π with τ makes mathematics more elegant overall. And at the heart of it, isn’t that what we aspire to do with math? The universe is vast and almost impossible for us to ever fully comprehend, but by distilling it down into a system of logical numbers and symbols, we can make some order out of the chaos. So why not embrace a circle constant that makes our equations and formulae more beautiful?  Unfortunately, pi is probably too well-ensconced in traditional mathematics for us to ever break free from its tyrannical hold. Math textbooks still espouse the virtues of pi, and instilling such a systemic change to how we teach math is likely an uphill battle. (On the other hand, Common Core somehow seems to have managed it, despite its — to my eyes — incredibly obtuse nature, but go figure.) And that’s a shame, given how much more sense tau makes as a circle constant for even the more basic functions that we do use pi for. But the first step is to stop glorifying pi, so I won’t be celebrating Pi Day next year — and neither should you.  But all isn’t lost for those looking for a fun day to celebrate math: Tau Day (6/28, or June 28th) is just 106 days away!


theverge--2019-06-28--Tau Day is here celebrate tau not pi as the true circle constant

Tau Day is here: celebrate tau, not pi, as the true circle constant

You’re likely familiar with Pi Day, perhaps the most popular of geeky holidays. Hooray. But I’m here to tell you that Pi Day is wrong — or rather, the entire idea of pi as a mathematical concept is wrong, and why you should be celebrating Tau Day, instead (which happens to be today!)  It’s easy enough to see why people like Pi Day: the whole thing starts with a mathematical pun of sorts (The date is written as 3/14 in American notation. Pi starts with the digits 3.14. You get it.) It’s an easy, fun ritual to see how many digits you can pointlessly memorize of the famously never-ending, never-repeating number (even though 39 digits is more than sufficient for almost any calculations you’ll ever need). Plus pi sounds like pie, and who doesn’t like pie?  π as a number is bad, and therefore, so is the entire misguided day dedicated to its celebration  But here’s the thing: π as a number is bad, and therefore, so is the entire misguided day dedicated to its celebration. It’s a lot to take in, and I, too, was once like you: I was taught the virtues of pi for years, going back to Pi Day parties in middle school. But instead of pi, we should celebrate tau, an alternative circle constant referred to by the Greek letter τ that equals 2π, or approximately 6.28.  I’m not just making this up out of nowhere: the terribleness of pi as a constant was first proposed by mathematician Bob Palais in his article “π Is Wrong!” and later expounded upon in The Tau Manifesto by Michael Hartl, which serves as the basis for modern tauism. (Internet-famed mathemusician Vi Hart is also a major proponent of tau over pi, if you prefer your mathematical arguments in a more entertaining video form.)  But Palais and Hartl’s arguments both boil down to some basic math. Step back in time to when you first learned geometry and recall the simple origins: no matter what circle you’re using, if you divide the circumference of the circle by the diameter, you’ll get the same answer: an endless number, starting with the digits 3.14159265... (aka pi).  And right there is the fundamental flaw. The thing is, we don’t actually use diameter to describe circles. We use the radius, or one-half the diameter. The circle equation uses the radius, the area of a circle uses the radius, and the fundamental definition of a circle — “the set of all points in a plane that are at a given distance from a given point, the center” — is based on the radius. Plugging that into our circle constant equation gives us a new circle constant equivalent to 2π, or 6.28318530717..., colloquially referred to with the Greek letter τ (tau). Switching to τ isn’t making some arbitrary change for the sake of it. It’s bringing one of the most important constants in math in line with how we actually do math.  π isn’t really something we use in day-to-day math to start with  Now, you may be thinking that this will cause fundamental, seismic changes in math. “How on earth could you possibly replace something as important as pi!?” you could ask. But if we’re being honest, π isn’t really something we use in day-to-day math to start with. Unless you’re someone who does a lot of geometric calculations in your daily life, chances are you only ever really encounter pi when it comes time to rattle off some digits for Pi Day. Sure, it’s a good introduction to the idea of irrational numbers, but tau would work just as well for that. And if you do work with π a lot, replacing it with τ is beneficial for a whole host of reasons, mathematically speaking. Again, I’ll direct you to The Tau Manifesto for the full argument, but I’ll just point out a few here.  One big thing tau fixes is radian angles. You may remember that as “those annoying chunks of a circle represented by weird fractions of pi from high school math,” but with tau, it’s simple: everything matches up where it should fractionally. So half the circle (180 degrees) — τ/2. 1/12th? τ/12. It’s a small change, but it makes angular notation — a frustratingly obtuse part of geometry that through the use of pi demands an elitist notion of memorization of angles and conversions — a more welcoming and intuitive prospect for new students.  It also makes circle functions like sine and cosine easier, since it makes one full cycle of the function match up with one full turn of the circle (tau), instead of the seemingly arbitrary 2π that you get using π as your circle function. Like with radian angles, it makes deriving sine and cosine values a simple process from simply drawing the function, instead of demanding that students remember that 3π/2 is for some reason the three-fourths point on the wavelength.  Looking back at years of math and physics notes with the enlightened lens of tau, I weep for my former self  Similarly, it makes a bunch of other higher math — like integrals in polar coordinates, the Fourier transform, and Cauchy’s integral formula simpler, since they also already work in terms of 2π anyway. Using tau just cuts out the middleman. Looking back at years of math and physics notes with the enlightened lens of tau, I weep for my former self and the accumulated hours of needless conversions and complications introduced by pi.  It’s not just practical purposes, though. Replacing π with τ makes mathematics more elegant overall. And at the heart of it, isn’t that what we aspire to do with math? The universe is vast and almost impossible for us to ever fully comprehend, but by distilling it down into a system of logical numbers and symbols, we can make some order out of the chaos. So why not embrace a circle constant that makes our equations and formulae more beautiful?  Unfortunately, pi is probably too well-ensconced in traditional mathematics for us to ever break free from its tyrannical hold. Math textbooks still espouse the virtues of pi, and instilling such a systemic change to how we teach math is likely an uphill battle. (On the other hand, Common Core somehow seems to have managed it, despite its — to my eyes — incredibly obtuse nature, but go figure.) And that’s a shame, given how much more sense tau makes as a circle constant for even the more basic functions that we do use pi for. But the first step is to stop glorifying pi, so I won’t be celebrating Pi Day next year — and neither should you.  But all isn’t lost for those looking for a fun day to celebrate math: Tau Day (6/28, or June 28th) is today!


prepareforchange--2019-07-13--The Dimensional Structure of Consciousness

A colossal transcendence of material thinking and clear connection to higher consciousness has to be achieved if we want to attempt to contemplate a topic as abstract as how dimensions are structured within the God Source. Today, I’m addressing my understanding of how the various frequency bands of existence are set up within the energetic field that is Source. I want to note that even though my research, readings, and learnings informed and influenced my conceptualization of this idea, none of what I’m about to propose is being regurgitated from elsewhere, nor have I found this particular imagining or “theory” anywhere else on the web. The contents of this post come from my personal downloads.  The body of Source is a spiraling vortex of energy, beginning from a singular point which can be called the Godhead and which uncoils like a cone to form the energy body. The central filament is a golden cord, which serves as The Creator’s figurative spinal column, which is the essence of unconditional love.  Within itself, Source created frequency bands where it could apportion and then experience itself from various levels of awareness and from differing perspectives. 11 of the dimensions are experiential, and the 12th is the essence of Unconditional Love.  Each frequency band or strata of light spins at higher frequencies as they’re ascended, and each represents a different color, along with separate makeup and activity.  Via a process of energetic division as well as frequency reduction, Source put aspects of its own consciousness inside the various levels of its creation.  These levels I’m referring to can also be called dimensions, and they exist as strata of light or spectrums of frequency that stream from the Godhead through its body.  Dimensions are not places or locations; they are levels of consciousness vibrating at certain oscillatory speeds. Consciousness acquires greater photonic density it ascends upward, and in so doing, gains a broader perspective of reality.  As Consciousness ascends the dimensions, it accumulates greater density. This means it re-acquires increased Godsource energy — more light, love, manifestation power, wisdom, and understanding.  I am not a mathematician or physicist, but there are a number of concepts that might help you begin to form a mental image of this concept.  Energy. First and foremost, energy is everything. All is made of energy. Do not think about dimensions as places or material layers of a cake; think of them as energy bands stacked atop one another but in the same ‘space’, like radio/TV stations.  369. Tesla’s ‘369 Keys to the Universe‘ — unveiled and explained in detail by Gary Lite — reveal the how energy is expressing itself mathematically. This math has no anomalies and shows the dimensional shape and function of the universe, which is one of many situated within Source’s omniverse.  Sacred math. Vortex-based mathematics and Sacred Geometry offer insights we can take into account to understand the dimensional structure of existence.  In the same construction as a galaxy, a hurricane, or a seashell, dimensions are spirals of energy inside of Source.  In following the Fibonacci sequence, each dimension spirals and swirls in an unfurling manner.  Understand: up until now, I have NOT been talking about what happens in each dimension or what kind of entity/collective would correspond with or populate each. I am addressing the pattern/structure of the light-energy that constitutes dimensions within Source.  However, if you’re interested, here’s a couple graphics showing the manifestation/form that consciousness takes to experience each dimension:  I’ll publish a separate piece down the line explaining how consciousness would experience at each dimension as it ascends.  Imagine Observing the Godhead as if you were viewing it head on — directly in front of it. You’d see a golden dot in the very middle as a point, and which extends backward as the central filament of golden light. You’d also see a rotating spherical vortex filled with different and beautiful energy structures, all of different oscillations, speeds, and colors.  Dimensions spin along the rotation of Source as it rotates itself, catapulting itself through the un-manifest. Dimensions are in constant movement, too — their placement is not fixed. But consciousnesses works among each in relatively steady environments. Think about the solar system and the way the Earth is in a state of constant motion. Dimensions work in a similar fashion.  Dimensions are non-existent unless there’s something there to experience them. The First Dimension — rocks, minerals, and elementals — will be eliminated from the Matrix once that experience has been completely achieved by consciousness and once all 2D and 3D beings have no further learning objectives with that matter.  Once all individuated units/portions of consciousness are complete with 3D experiences, then there will no longer be any need for First, Second, or Third dimensional observation/experience/presence. Once all aspects of the Source have ascended beyond the 3D, anything less than 3D will be “closed” down, just like someone flipped a switch and the lights go off.  Dimensions exist as rotational energetic frequencies within Source and can be likened to our internal cells, arteries, and musculoskeletal system. They exist everywhere but are in constant movement. Dimensions provide different experiences and purposes, and they all work together for the totality.  I will update this article as I continue to gain new insights, downloads, and information about this complex subject.  Here’s a few recommended resources/articles to check out if you want to understand more about the dimensions of reality:  -> What is the Meaning of Life? [article]  Disclaimer: We at Prepare for Change (PFC) bring you information that is not offered by the mainstream news, and therefore may seem controversial. The opinions, views, statements, and/or information we present are not necessarily promoted, endorsed, espoused, or agreed to by Prepare for Change, its leadership Council, members, those who work with PFC, or those who read its content. However, they are hopefully provocative. Please use discernment! Use logical thinking, your own intuition and your own connection with Source, Spirit and Natural Laws to help you determine what is true and what is not. By sharing information and seeding dialogue, it is our goal to raise consciousness and awareness of higher truths to free us from enslavement of the matrix in this material realm.


prepareforchange--2019-08-31--7 SECRET WAYS SACRED GEOMETRY HELPS EVOLVE YOUR CONSCIOUSNESS

Sacred geometry patterns are everywhere, and we will find them in nature, in our own lives, and in the entire universe. Think about your own life; what patterns, have you found, whether in the things you pass by every day, or in technology, or anywhere else, that are either symmetrical and balanced or asymmetrical and unbalanced?  Developing a keen interest in patterns is the first step to knowing which ones are inherently good and should be maintained and which ones are bad and should be avoided or broken.  “It is the constant dance between symmetry and asymmetry that leads to the twin processes of life and death and the evolution that emerges as a result,” says Anne Durham, an alternate news writer from https://www.a-writer.com.  That is where sacred geometry comes in: it gives us the opportunity to contemplate that dance and, in the process, to grow and evolve our own consciousness and become one with the totality of existence.  That said, here are 7 useful ways that an understanding of sacred geometry will help to evolve your consciousness.  The first step is to have an intimate understanding of what sacred geometry is. That is definitely where I started when I first stumbled upon the subject while working at BestEssay.  Sacred geometry is simply the collection of all the patterns that are found prolifically in the design of everything in the universe and are therefore considered sacred. You will find them a lot in art and architecture, but that’s because the artists and architects were copying the ultimate artist and architect: Mother Nature. In fact, most of the most beautiful works of art and architecture have borrowed heavily from patterns that are originally found in nature.  The belief behind sacred geometry is that things like patterns of proportion, harmonics, and mathematical ratios are also found in phenomena as disparate as light, music, and nature itself. You will find them in our bodies, in plants, and just about everywhere. They are believed to be the sole source of energy and animation in the physical world much as the spirit is believed to be the source of energy and animation in the body.  Sacred geometric patterns are able to captivate us and seduce us in deep ways. Whether we’re looking at a sacred geometric pattern in nature or a work of art, we are so emotionally moved and awed by the geometry that we inevitably attach it to some higher identity, considering it to be of a transcendent nature.  There are many things in these patterns that can be measured, copied, and even defined, making them at once both esoteric and mathematical. It is by seeking to study this world that you will be forced to change and refine your point of view on the world.  2. Knowing about the History of Sacred Geometry  Being able to evolve your consciousness does not just stop at knowing what sacred geometry is; part of it is knowing about the history of this field of study.  A simple essay review of recorded history will show you how prolific sacred geometry has been throughout history. Sacred geometry was independently discovered, studied, and understood by many cultures from all around the world. These included the Egyptians, the Greeks, the Hindus, and even the Christians.  These cultures were able to find different patterns that were not only deep and mystifying but also rather prolific throughout nature, their repetitive natures being rather hard to miss. These cultures went on to study these patterns in depth and soon found out that there seemed to be very clear connections between the elements that could be found on Earth and the patterns in the sky. It was as if the heavens and the earth mirrored each other. This led earlier civilizations to believe that the sacred patterns they had found here on Earth existed all over the universe.  If you are a student of science then this will sound familiar. After all, science has found its own patterns that repeat themselves throughout the universe and that the most basic molecular shapes are so universal that they are responsible for all life in nature.  To be sure, there is a certain sense of spirituality and awakening that comes when anyone beholds a sacred pattern for the first time. They give you a giddy sense of awareness as if you are witnessing the universe unfold before your very eyes. However, due to the fact that different people will interpret these patterns in different ways, depending on their specific beliefs and backgrounds, I won’t assert that they have any particular meaning at the exclusion of all other possible meanings.  The point here is to try them out and find out why they are sacred for you. One thing I thoroughly enjoyed during my school years was geometry because it was all about using simple tools like a compass, a pencil, and some paper to draw beautiful shapes. It was a way to bring something as abstract as a number into the visual world where we could see it in a concrete way.  In fact, there are many examples of abstract notions from mathematics finding their way in the physical world through such things as music or the study of physics. Mathematics has proven itself time and again in its ability to dance in the space between the world of the tangible and that of the intangible; between the world of the physical and that of the non-physical; between the world of the concrete and that of the abstract.  What makes this so interesting is that that space, and that dance, are often reserved for less academic things, such as spirituality or art. That is why it is so fascinating to find mathematical formulas dancing in the same spaces. It is also the reason why this particular geometry is considered sacred. Try getting assignment help on sacred geometry on the internet to help you go deeper into it. Going deep into it and finding out why it is sacred for you with this in mind will help you expand your mind and consciousness to such great proportions that it can never shrink back to where it had been before.  The golden ration goes by many names, such as the Golden Ratio, the Golden Section, the Golden Mean, or simply Phi. This ratio was revered, almost worshipped, by the ancient Greeks as a symbol of the mathematical beauty of the universe.  The Golden Ratio itself is an irrational number, which means it can never be exactly achieved in real life. Everything can only approach it with varying degrees of accuracy. It is, therefore, the embodiment of infinity and perfection. In that sense, it is transcendent.  The Golden Ratio, through the Golden Rectangle, is responsible for one of the most common spirals in nature: the Fibonacci Spiral. The Fibonacci Spiral can be found in lots of different places in nature, including pinecones, flowers, pineapples, shells, and many others.  Humans too use the Fibonacci Spiral and the Golden Rectangle to design beautiful things, such as musical instruments, works of art and architecture, musical chords, and much more. The Golden Ratio can be found in many forms in all of these places and it has been used in things like architecture for thousands of years. Buildings as old as the Greek Parthenon and the Pyramids of Egypt incorporate Phi in numerous ways.  The Flower of Life is yet another symbol that is found within human cultures, dating as far back as 10,000 years. This symbol consists of 6 overlapping circles that are evenly spaced and contain six-fold symmetry, causing them to look like a hexagonal flower.  You will find the flower of life in the synagogues of Israel, in the Forbidden City of China, in the Temple of Osiris in Egypt, in La Mezquita in Spain, in temples all over India, in Japan, in Turkey, and a whole host of other places around the world and across different cultures and epochs in history.  Leonardo Da Vinci, the very face of the Italian Renaissance, spent a lot of time thinking about the Flower of Life and playing around with it in his designs and works of art. You can find out a lot more about these patterns by reading some of the articles on Australian Writings. Just looking at it is enough to evoke strong emotions and images in your mind, which is probably why people from all over the world have been fascinated by it since antiquity.  6. Understanding the Seed of Life and other Patterns within the Flower of Life  Within the Flower of Life is the Seed of Life, which is just one among many symbols that you will find within the flower of life. These symbols consist of polygons (a polygon is a geometric shape composed of straight sides of equal length).  In the Flower of Life, you will find the cube, tetrahedron, dodecahedron, icosahedron, and octahedron. Together these 5 solids are known as the platonic solids, named after the Greek Philosopher Plato who studied them in depth. They can be found within the Flower of Life when you connect the centers of different circles within the flower with straight lines.  The most amazing thing about all of the patterns within the Flower of Life is how they can all be found within a single pattern, revealed by nothing more than a few mathematical manipulations. Another interesting point to note is that the Flower of Life contains both curves and straight lines. Many cultures associate circles and curves with the feminine and polygons and straight lines with the masculine. Both can be found in perfect balance within the Flower of Life.  Sacred geometry isn’t just beautiful; it also transmits higher frequencies of awareness and energy. It shows us the very basic mathematical patterns that belie everything we see, and much that we don’t see, in our universe.  This can teach us a lot about our own selves and our journeys through life.  It shows that the patterns we create with our internal consciousness and beliefs are rippled outward into the universe and ultimately manifest themselves as things and experiences in the physical world. The visible is informed by the invisible, and all of life and existence is connected. Sacred geometry is a visual and mathematical metaphor for this concept. It is the interface between concrete and abstract, between physical and non-physical, and between visible and invisible. When we change the mathematical patterns, the relationships between them, and even the physical form change.  As you can see, there is much to be learned about the world and the flux between the symmetrical and the asymmetrical in sacred geometry. Together they inform life, death, and evolution. Likewise, learning about sacred geometry will inform the evolution of your own consciousness and lead you to access higher and higher planes of awareness and existence.  Scott Mathews is a writer and editor at essay paper and term paper writers. He runs an essay writing service and offers dissertation writing help. He is also the author of 6dollaressay reviews where he shares insightful information about education and technology.  Disclaimer: We at Prepare for Change (PFC) bring you information that is not offered by the mainstream news, and therefore may seem controversial. The opinions, views, statements, and/or information we present are not necessarily promoted, endorsed, espoused, or agreed to by Prepare for Change, its leadership Council, members, those who work with PFC, or those who read its content. However, they are hopefully provocative. Please use discernment! Use logical thinking, your own intuition and your own connection with Source, Spirit and Natural Laws to help you determine what is true and what is not. By sharing information and seeding dialogue, it is our goal to raise consciousness and awareness of higher truths to free us from enslavement of the matrix in this material realm.


theverge--2019-04-16--The Night Kings eerie sigils on Game of Thrones could have a basis in math

The Night King’s eerie sigils on Game of Thrones could have a basis in math

As we near the end of Game of Thrones, there are still a few huge mysteries left unsolved. The audience still knows relatively little about the White Walkers. Their origins were revealed in season 6, but the extent of their powers hasn’t been explained, nor have the mysterious symbols they leave behind in their bloodshed. The season 8 premiere, “Winterfell,” prominently features one of these symbols, made out of human body parts. But it’s only one of many times we’ve seen it, and there are various ways to interpret what it means.  The first time Game of Thrones associates a curious symbol with the Army of the Dead is in season 1, episode 1, “Winter is Coming” when body parts of wildings are left in a strange circle with a line through the middle. It looks like Φ. A member of the Night’s Watch uncovers this symbol in the snow and sees a wilding girl who has been pinned to a tree, dead, behind him. The imagery of a strange circle plus an additional sacrifice appears again in the premiere of season 8 when Ned Umber’s body is found pinned to a wall at the center of a spiral.  Φ is the Greek symbol, phi, which has various uses in science and mathematics. Most famously, it’s used to represent the golden ratio, which frequently appears in geometry, art, and architecture throughout history. As seen in season 1, it appears that the White Walkers are perverting the golden ratio and sending some sort of foreboding message to the humans who observe it.  The spiral we see at the end of “Winterfell” also appears to have roots in mathematics. It bears a resemblance to the Fibonacci spiral, a golden spiral whose growth factor is Φ, the golden ratio. It grows wider by a factor of Φ for every quarter turn it takes. We could interpret this as more humans being killed and absorbed into the Army of the Dead, which increases with every new slaughter.  Andrew Beveridge, a math professor at Macalester College, says the White Walker spiral goes in the opposite direction of a golden spiral, but it’s “pretty darn close” to a clockwise golden spiral. Beveridge measured the expansion of each quarter turn and found it to be 1.616, which is close to the golden spiral’s expansion factor of 1.618. “Not bad for the frozen undead,” he says. “So if you ask me: yes, the White Walkers have made eight clockwise golden spirals.”  Still, the fan theory isn’t 100 percent perfect. As for whether the first symbol seen in “Winter is Coming” is indeed Φ, Beveridge says, “I agree that it is pretty close. But I wish that either the circle didn’t complete at 11:00, or that the vertical line extended above the circle. That way it would match one of the two lower-case phi variants. I personally wouldn’t put my money down on that one.”  So far, the show’s inclusion of these weird symbols seems to be a departure from George R.R. Martin’s source novels. The books do mention of whorls and swirling patterns in dragon eggs and fabrics, but there’s no greater meaning ascribed to them. On the HBO show, though, we know that before the White Walkers started using these symbols, their creators, the Children of the Forest, had also used them — carved in stone rather than with flesh.  In season 6, Bran observed that the stones around a weirwood tree are positioned in a peculiar spiral pattern. After he spots the symbol, he sees one of the Children of the Forest, Leaf, stabbing one of the First Men with dragonglass, turning him into the first White Walker, aka the Night King. Later on in season 7, when Jon takes Daenerys down into a cave in Dragonstone, they study the ancient carvings of spirals on the walls, which were drawn by the Children of the Forest.  Showrunner David Benioff explained in a behind-the-scenes clip on season 7, episode 4 that he showrunner D.B. Weiss pulled these ideas from the real world. “These are patterns that have mystical significance for the Children of the Forest. We’re not sure exactly what they signify, but spiral patterns are important in a lot of different cultures in our world, and it makes sense that they would be in this world as well,” Benioff said. The symbols also appear naturally when Dothraki riders surround Daenerys in season 5 and in season 3 when she’s embraced by a crowd.  One fan theory suggests that the spirals symbolize magic and its proximity to humans. Others note that the symbols bear a resemblance to the Targaryen sigil’s multiple dragon heads. Another possibility is that the spirals simply signify the circle of life (and the undead). As Varys says in the season 8 premiere, while looking at Jon and Daenerys, “Nothing lasts.” There’s also the idea that history repeats itself and life is cyclical. In season 1, episode 1, Ned Stark tells Bran that the White Walkers haven’t appeared in thousands of years. Now, seven seasons later, they’re back. The symbol, in other words, could be a warning that the White Walkers will always return to make humans part of their undead army.