## Archive for the ‘Mathematics’ Category

### Math Blog and how to write math equations using LaTeX $latex…$

Posted on: May 16, 2018

Math Blog and how to write math equations using LaTeX $latex…$

WordPress.com supports LaTeX, a document markup language for the TeX typesetting system, which is used widely in academia as a way to format mathematical formulas and equations.

LaTeX makes it easier for math and computer science bloggers and other academics in our community to publish their work and write about topics they care about.

If you’re a math blogger and expressing equations you’ve worked on, you’ve probably used LaTeX before. If you’re just starting out (or simply curious to see how it all works), we’ve gathered a few examples of great math and computing blogs on WordPress.com that will inspire you.

In general, to display formulas and equations, you place LaTeX code in between $latex and$, like this:

$latex YOUR LATEX CODE HERE$

So for example, inserting this when you’re creating a post . . .

$latex i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

. . . will display this on your site:

$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

Nifty, huh? Learning LaTeX is like learning a new language, and the bloggers below show just how much you can do. And if you’re not a math whiz, don’t worry! You’re not expected to understand the snippets below, but we hope they show what’s possible.

### Gödel’s Lost Letter and P=NP

Suppose Alice gives Bob two boxes labelled respectively ${X}$ and ${Y}$. Box ${X}$contains some positive integer ${x}$, and as you might guess, box ${Y}$ contains some positive integer ${y}$. Bob cannot open either box to see what integer it holds. Bob can shake the boxes, or hold them up to a bright light, but there is no way he can discover what they contain.

This blog, on $P=NP$ and other questions in the theory of computing, presents the work of Dick Lipton at Georgia Tech and Ken Regan at the University at Buffalo. One of their main goals is to pull back the curtain so readers can understand how research works and who is behind it.

From the recent post “Move the Cheese” to an older piece on “Navigating Cities and Understanding Proofs,” they present problems and sketch solutions, and publish thorough and thoughtful discussions that not only talk about interesting open problems, but offer context and history.

You can see LaTex in action in the example above, from the recent post “Euclid Strikes Back.”

### Math ∩ Programming

Note that we will have another method to determine the necessary coefficients later, so we can effectively ignore how these coefficients change. Next, we note the following elementary identities from complex analysis:

$\displaystyle \cos(2 \pi k t) = \frac{e^{2 \pi i k t} + e^{-2 \pi i k t}}{2}$
$\displaystyle \sin(2 \pi k t) = \frac{e^{2 \pi i k t} - e^{-2 \pi i k t}}{2i}$

Jeremy Kun, a mathematics PhD student at the University of Illinois in Chicago, explores deeper mathematical ideas and interesting solutions to programming problems. Math ∩ Programming is both a blog and portfolio, and well-organized: you can use the left-side menu to navigate Jeremy’s sections, from Primers to the Proof Gallery. The site is also clean and well-presented — can you believe he uses the Confit theme, which was originally created for restaurant sites?

The snippet above illustrates more you can do with LaTeX, taken from “The Fourier Series — A Primer.”

### Terence Tao

Definition 1 (Multiple dense divisibility) Let ${y \geq 1}$. For each natural number ${k \geq 0}$, we define a notion of ${k}$-tuply ${y}$-dense divisibility recursively as follows:

• Every natural number ${n}$ is ${0}$-tuply ${y}$-densely divisible.
• If ${k \geq 1}$ and ${n}$ is a natural number, we say that ${n}$ is ${k}$-tuply ${y}$-densely divisible if, whenever ${i,j \geq 0}$ are natural numbers with ${i+j=k-1}$, and ${1 \leq R \leq n}$, one can find a factorisation ${n = qr}$ with ${y^{-1} R \leq r \leq R}$ such that ${q}$ is ${i}$-tuply ${y}$-densely divisible and ${r}$ is ${j}$-tuply ${y}$-densely divisible.

We let ${{\mathcal D}^{(k)}_y}$ denote the set of ${k}$-tuply ${y}$-densely divisible numbers. We abbreviate “${1}$-tuply densely divisible” as “densely divisible”, “${2}$-tuply densely divisible” as “doubly densely divisible”, and so forth; we also abbreviate ${{\mathcal D}^{(1)}_y}$as ${{\mathcal D}_y}$.

Mathematician, UCLA faculty member, and Fields Medal recipient Terence Tao uses his WordPress.com site to present research updates and lecture notes, discuss open problems, and talk about math-related topics.

He uses the Tarski theme with a modified CSS (to do things such as boxed theorems).

As stated on his About page, he uses Luca Trevisan’s LaTeX to WordPress converter to write his more mathematically intensive posts. Above, you’ll see an example of how he uses LaTeX on his blog, excerpted from the post “An improved Type I estimate.”

Terence also has a blog category for non-technical posts, aimed at a more general audience, and offers helpful advice on mathematical careers.

### Using LaTeX

From  “Euclid Strikes Back,” Gödel’s Lost Letter.

You can read a brief primer on using LaTeX on our Support site and search related forum discussions to see if a WordPress.com user has asked your question.

If you’re dipping in for the first time, we encourage you to check out these resources for help and detailed documentation:

### Tidbits and notes posted on FB and Twitter. Part 193

Posted on: May 5, 2018

Tidbits and notes posted on FB and Twitter. Part 193

Note: I take notes of books I read and comment on events and edit sentences that fit my style. I pa attention to researched documentaries and serious links I receive. The page is long and growing like crazy, and the sections I post contains a month-old events that are worth refreshing your memory.

The ancient Greeks came up with a system called the Sieve of Eratosthenes for easily determining which numbers are prime. It works by simply eliminating the multiples of each prime number. Any numbers left over will be prime. (The ancient Greeks couldn’t do this in gifs, though.)

## Robin Sloan wrote the bestselling mystery Mr. Penumbra’s 24-Hour Bookstore, in which every number was a prime (except 24 of course).

Usage of Prime Numbers:

1) Getting into Gear

Before primes were used to encrypt information, their only true practical use was at the auto-body shop. The gears in a car—and every other machine—work most reliably when the teeth are arranged by prime numbers. When gears have 13 or 17 or 23 teeth, it ensures that every gear combination is used, which helps to evenly distribute dirt, oil, and overall wear and tear.

2) Talking with Aliens

In his sci-fi novel Contact, Carl Sagan suggested that humans could communicate with aliens through prime numbers. This wasn’t a new idea. In the summer of 1960, the National Radio Astronomy Observatory searched for intelligent extraterrestrial messages by searching for prime numbers. Years later, the astronomer Frank Drake proposed that humans could communicate with aliens by transmitting “semiprimes”—that is, multiples of two prime numbers—into space.

3) Making Nature’s Music

The French modernist composer Olivier Messiaen wrote music containing transcribed birdsong and prime numbers, which helped create unusual and unpredictable rhythms, note duration, and time signatures. Messiaen, a Roman Catholic, said that musical prime numbers represented the indivisibility of God. His Liturgie de Cristal is a grand example. Listen to Messien put prime numbers into practice here.

A “prime-numbered life cycle had the most successful survival strategy” in nature, since cycles of boom and drop of resources are consistent and predictable

An emirp is a prime number that, when its decimal digits are reversed, results in a different prime. Think 13, 17, 31, 37, 71, 73, 79 …. According to Wikipedia, the largest known emirp is 10^10006+941992101×10^4999+1.

Mersenne prime numbers, named after a 17th-century French monk, are a special breed: They’re prime numbers that are one less than a power of two

Jonathan Pace is one of the volunteers participating in the Great Internet Mersenne Prime Search GIMPS. The prime he discovered (notated as 2^77,232,917-1) contains 23,249,425 digits—nearly a million digits longer than the previous record holder.

Since 1996, GIMPS volunteers have discovered 16 new numbers. “There are tens of thousands of computers involved in the search. On average, they are finding less than one a year.” (He was awarded \$3,000 for 14 years of work)

Mathematician G H Hardy wrote that he avoided “practical” mathematics: it was dull and too often exploited for military gain. His discoveries in Prime Numbers were useful: They’ve aided the fields of genetics research, quantum physics, and thermodynamics. Today, his research on the distribution of prime numbers is the bedrock for our current understanding of how prime numbers operate.

Le Chuiche (al shemmace?)

Halla2 saar fi mou3aradat: Al Moustakbal lan yet 7alaf ma3 Hezbollah bil intikhabaat. Haaza lan ya3ni 3adam al ta7alof ma3 al mouta7alifeen ma3 al Moukawamat. Kelna moukawamat.

### Babylonian tablet (3,700 year-old) has developed trigonometry

Posted on: December 5, 2017

# 3,700-year-old Babylonian tablet rewrites the history of maths – and shows the Greeks did not develop trigonometry

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

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

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

The tablet is broken and probably had more rows, experts believe  CREDIT: UNSW

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today. Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

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

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

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

“This is a rare example of the ancient world teaching us something new.”

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

A trigonometric table allows a user to determine two unknown ratios of a right-angled triangle using just one known ratio. But the tablet is far older than Hipparchus, demonstrating that the Babylonians were already well advanced in complex mathematics far earlier.

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

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

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

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

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

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

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

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

### How can we make statistics lovely? Interactive graphics?

Posted on: May 17, 2017

How can we make love statistics? Interactive graphs?

Think you’re good at guessing stats? Guess again. Whether we consider ourselves math people or not, our ability to understand and work with numbers is terribly limited, says data visualization expert Alan Smith. Smith explores the mismatch between what we know and what we think we know.

Alan Smith. Data visualisation editor
Alan Smith uses interactive graphics and statistics to breathe new life into how data is presented. Full bio
Filmed April 2016

Back in 2003, the UK government carried out a survey. And it was a survey that measured levels of numeracy in the population.

And they were shocked to find out that for every 100 working age adults in the country, 47 of them lacked Level 1 numeracy skills. Now, Level 1 numeracy skills — that’s low-end GCSE score. It’s the ability to deal with fractions, percentages and decimals.

this figure prompted a lot of hand-wringing in Whitehall. Policies were changed, investments were made, and then they ran the survey again in 2011. So can you guess what happened to this number? It went up to 49.

0:57 And in fact, when I reported this figure in the FT, one of our readers joked and said, This figure is only shocking to 51 percent of the population.”

But I preferred the reaction of a schoolchild when I presented at a school this information, who raised their hand and said, “How do we know that the person who made that number isn’t one of the 49 percent either?”

1:20 (Laughter)

So clearly, there’s a numeracy issue, because these are important skills for life, and a lot of the changes that we want to introduce in this century involve us becoming more comfortable with numbers. (Can’t learn numeracy without using a pen and pencil?)

it’s not just an English problem. OECD this year released some figures looking at numeracy in young people, and leading the way, the USA — nearly 40 percent of young people in the US have low numeracy. Now, England is there too, but there are seven OECD countries with figures above 20 percent. That is a problem, because it doesn’t have to be that way. If you look at the far end of this graph, you can see the Netherlands and Korea are in single figures. So there’s definitely a numeracy problem that we want to address. (It is the method used to learning numeracy)

as useful as studies like these are, I think we risk herding people inadvertently into one of two categories; that there are two kinds of people: those people that are comfortable with numbers, that can do numbers, and the people who can’t.

And what I’m trying to talk about here today is to say that I believe that is a false dichotomy. It’s not an immutable pairing. I think you don’t have to have tremendously high levels of numeracy to be inspired by numbers, and that should be the starting point to the journey ahead.

one of the ways in which we can begin that journey, for me, is looking at statistics. Now, I am the first to acknowledge that statistics has got somewhat of an image problem.

2:52 (Laughter)

It’s the part of mathematics that even mathematicians don’t particularly like, because whereas the rest of maths is all about precision and certainty, statistics is almost the reverse of that.

But actually, I was a late convert to the world of statistics myself. If you’d asked my undergraduate professors what two subjects would I be least likely to excel in after university, they’d have told you statistics and computer programming, and yet here I am, about to show you some statistical graphics that I programmed. (You think you comprehended probability and statistics, but you forget them if Not practiced)

what inspired that change in me? What made me think that statistics was actually an interesting thing? It’s really because statistics are about us.

If you look at the etymology of the word statistics, it’s the science of dealing with data about the state or the community that we live in. So statistics are about us as a group, not us as individuals. And I think as social animals, we share this fascination about how we, as individuals, relate to our groups, to our peers. And statistics in this way are at their most powerful when they surprise us.

there’s been some really wonderful surveys carried out recently by Ipsos MORI in the last few years. They did a survey of over 1,000 adults in the UK, and said, for every 100 people in England and Wales, how many of them are Muslim? Now the average answer from this survey, which was supposed to be representative of the total population, was 24. That’s what people thought. British people think 24 out of every 100 people in the country are Muslim. Now, official figures reveal that figure to be about five. So there’s this big variation between what we think, our perception, and the reality as given by statistics. And I think that’s interesting. What could possibly be causing that misperception?

I was so thrilled with this study, I started to take questions out in presentations. I was referring to it. Now, I did a presentation at St. Paul’s School for Girls in Hammersmith, and I had an audience rather like this, except it was comprised entirely of sixth-form girls.

And I said, “Girls, how many teenage girls do you think the British public think get pregnant every year?” And the girls were apoplectic when I said the British public think that 15 out of every 100 teenage girls get pregnant in the year. And they had every right to be angry, because in fact, I’d have to have closer to 200 dots before I could color one in, in terms of what the official figures tell us.

And rather like numeracy, this is not just an English problem. Ipsos MORI expanded the survey in recent years to go across the world. And so, they asked Saudi Arabians, for every 100 adults in your country, how many of them are overweight or obese? And the average answer from the Saudis was just over a quarter. That’s what they thought. Just over a quarter of adults are overweight or obese. The official figures show, actually, it’s nearer to three-quarters.

5:56 (Laughter)

5:57 So again, a big variation.

I love this one: they asked the Japanese, for every 100 Japanese people, how many of them live in rural areas? The average was about a 50-50 split, just over halfway. They thought 56 out of every 100 Japanese people lived in rural areas. The official figure is seven.

So extraordinary variations, and surprising to some, but not surprising to people who have read the work of Daniel Kahneman, for example, the Nobel-winning economist. He and his colleague, Amos Tversky, spent years researching this disjoint between what people perceive and the reality, the fact that people are actually pretty poor intuitive statisticians. (I read many of their research papers in the late 80’s)

And there are many reasons for this. Individual experiences, certainly, can influence our perceptions, but so, too, can things like the media reporting things by exception, rather than what’s normal. Kahneman had a nice way of referring to that. He said, “We can be blind to the obvious” — so we’ve got the numbers wrong — “but we can be blind to our blindness about it.” And that has enormous repercussions for decision making.

at the statistics office while this was all going on, I thought this was really interesting. I said, this is clearly a global problem, but maybe geography is the issue here.

These were questions that were all about, how well do you know your country? So in this case, it’s how well do you know 64 million people? Not very well, it turns out. I can’t do that. So I had an idea, which was to think about this same sort of approach but to think about it in a very local sense. Is this a local? If we reframe the questions and say, how well do you know your local area, would your answers be any more accurate?

I devised a quiz: How well do you know your area? It’s a simple Web app. You put in a post code and then it will ask you questions based on census data for your local area. And I was very conscious in designing this. I wanted to make it open to the widest possible range of people, not just the 49 percent who can get the numbers.

I wanted everyone to engage with it. So for the design of the quiz, I was inspired by the isotypes of Otto Neurath from the 1920s and ’30s. Now, these are methods for representing numbers using repeating icons. And the numbers are there, but they sit in the background. So it’s a great way of representing quantity without resorting to using terms like “percentage,” “fractions” and “ratios.”

So here’s the quiz. The layout of the quiz is, you have your repeating icons on the left-hand side there, and a map showing you the area we’re asking you questions about on the right-hand side. There are 7 questions. Each question, there’s a possible answer between zero and a hundred, and at the end of the quiz, you get an overall score between zero and a hundred.

And so because this is TEDxExeter, I thought we would have a quick look at the quiz for the first few questions of Exeter. And so the first question is: For every 100 people, how many are aged under 16? Now, I don’t know Exeter very well at all, so I had a guess at this, but it gives you an idea of how this quiz works. You drag the slider to highlight your icons, and then just click “Submit” to answer, and we animate away the difference between your answer and reality. And it turns out, I was a pretty terrible guess: five.

How about the next question? This is asking about what the average age is, so the age at which half the population are younger and half the population are older. (This is the definition of the median) And I thought 35 — that sounds middle-aged to me.

9:35 (Laughter)

9:39 Actually, in Exeter, it’s incredibly young, and I had underestimated the impact of the university in this area. The questions get harder as you go through. So this one’s now asking about homeownership: For every 100 households, how many are owned with a mortgage or loan? And I hedged my bets here, because I didn’t want to be more than 50 out on the answer.

these get harder, these questions, because when you’re in an area, when you’re in a community, things like age — there are clues to whether a population is old or young. Just by looking around the area, you can see it. Something like homeownership is much more difficult to see, so we revert to our own heuristics, our own biases about how many people we think own their own homes.

the truth is, when we published this quiz, the census data that it’s based on was already a few years old. We’ve had online applications that allow you to put in a post code and get statistics back for years. So in some senses, this was all a little bit old and not necessarily new. But I was interested to see what reaction we might get by gamifying the data in the way that we have, by using animation and playing on the fact that people have their own preconceptions.

It turns out, the reaction was more than I could have hoped for. It was a long-held ambition of mine to bring down a statistics website due to public demand.

11:06 (Laughter)

This URL contains the words “statistics,” “gov” and “UK,” which are three of people’s least favorite words in a URL. And the amazing thing about this was that the website came down at quarter to 10 at night, because people were actually engaging with this data of their own free will, using their own personal time.

I was very interested to see that we got something like a quarter of a million people playing the quiz within the space of 48 hours of launching it. And it sparked an enormous discussion online, on social media, which was largely dominated by people having fun with their misconceptions, which is something that I couldn’t have hoped for any better, in some respects. I also liked the fact that people started sending it to politicians. How well do you know the area you claim to represent? (All candidates to public office must go through such quizzes in their locality and the nation)

then just to finish, going back to the two kinds of people, I thought it would be really interesting to see how people who are good with numbers would do on this quiz. The national statistician of England and Wales, John Pullinger, you would expect he would be pretty good. He got 44 for his own area.

12:16 (Laughter)

Jeremy Paxman — admittedly, after a glass of wine — 36. Even worse. It just shows you that the numbers can inspire us all. They can surprise us all.

12:31 So very often, we talk about statistics as being the science of uncertainty. My parting thought for today is: actually, statistics is the science of us. And that’s why we should be fascinated by numbers.

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“Whether we consider ourselves math people or not, our ability to understand and work with numbers is terribly limited.”

Alan Smith explores the mismatch between what we know and what we th…
ted.com

### Are you obsessed with symmetry? This new Math language?

Posted on: May 5, 2017

Are you obsessed with symmetry?

On the 30th of May, 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris.

A peasant, who was walking to market that morning, ran towards where the gunshot had come from, and found a young man writhing in agony on the floor, clearly shot by a dueling wound. The young man’s name was Evariste Galois, a mathematician who started modern math. He was a well-known revolutionary in Paris at the time. Galois was taken to the local hospital where he died the next day in the arms of his brother. And the last words he said to his brother were, “Don’t cry for me, Alfred. I need all the courage I can muster to die at the age of 20.”

0:55 It wasn’t, in fact, revolutionary politics for which Galois was famous. But a few years earlier, while still at school, he’d actually cracked one of the big mathematical problems at the time. And he wrote to the academicians in Paris, trying to explain his theory. But the academicians couldn’t understand anything that he wrote. (Laughter) This is how he wrote most of his mathematics.

the night before that duel, he realized this possibly is his last chance to try and explain his great breakthrough. So he stayed up the whole night, writing away, trying to explain his ideas.

And as the dawn came up and he went to meet his destiny, he left this pile of papers on the table for the next generation. Maybe the fact that he stayed up all night doing mathematics was the fact that he was such a bad shot that morning and got killed.

But contained inside those documents was a new language, a language to understand one of the most fundamental concepts of science — namely symmetry. Now, symmetry is almost nature’s language. It helps us to understand so many different bits of the scientific world. For example, molecular structure. What crystals are possible, we can understand through the mathematics of symmetry.

In microbiology you really don’t want to get a symmetrical object, because they are generally rather nasty. The swine flu virus, at the moment, is a symmetrical object. And it uses the efficiency of symmetry to be able to propagate itself so well. But on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information.

I’ve taken two pictures here and I’ve made them artificially symmetrical. And if I ask you which of these you find more beautiful, you’re probably drawn to the lower two. Because it is hard to make symmetry. And if you can make yourself symmetrical, you’re sending out a sign that you’ve got good genes, you’ve got a good upbringing and therefore you’ll make a good mate. So symmetry is a language which can help to communicate genetic information.

Symmetry can also help us to explain what’s happening in the Large Hadron Collider in CERN. Or what’s not happening in the Large Hadron Collider in CERN.

To be able to make predictions about the fundamental particles we might see there, it seems that they are all facets of some strange symmetrical shape in a higher dimensional space.

I think Galileo summed up, very nicely, the power of mathematics to understand the scientific world around us. He wrote, “The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometric figures, without which means it is humanly impossible to comprehend a single word.”

it’s not just scientists who are interested in symmetry. Artists too love to play around with symmetry. They also have a slightly more ambiguous relationship with it. Here is Thomas Mann talking about symmetry in “The Magic Mountain.” He has a character describing the snowflake, and he says he “shuddered at its perfect precision, found it deathly, the very marrow of death.”

what artists like to do is to set up expectations of symmetry and then break them. And a beautiful example of this I found, actually, when I visited a colleague of mine in Japan, Professor Kurokawa. And he took me up to the temples in Nikko. And just after this photo was taken we walked up the stairs. And the gateway you see behind has eight columns, with beautiful symmetrical designs on them. Seven of them are exactly the same, and the eighth one is turned upside down.

And I said to Professor Kurokawa, “Wow, the architects must have really been kicking themselves when they realized that they’d made a mistake and put this one upside down.” And he said, “No, no, no. It was a very deliberate act.” And he referred me to this lovely quote from the Japanese Essays in Idleness” from the 14th century, in which the essayist wrote, “In everything, uniformity is undesirable. Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth.” Even when building the Imperial Palace, they always leave one place unfinished.

if I had to choose one building in the world to be cast out on a desert island, to live the rest of my life, being an addict of symmetry, I would probably choose the Alhambra in Granada.

This is a palace celebrating symmetry. Recently I took my family — we do these rather kind of nerdy mathematical trips, which my family love. This is my son Tamer. You can see he’s really enjoying our mathematical trip to the Alhambra. But I wanted to try and enrich him. I think one of the problems about school mathematics is it doesn’t look at how mathematics is embedded in the world we live in. So, I wanted to open his eyes up to how much symmetry is running through the Alhambra.

You see it already. Immediately you go in, the reflective symmetry in the water. But it’s on the walls where all the exciting things are happening. The Moorish artists were denied the possibility to draw things with souls. So they explored a more geometric art. And so what is symmetry? The Alhambra somehow asks all of these questions. What is symmetry? When [there] are two of these walls, do they have the same symmetries? Can we say whether they discovered all of the symmetries in the Alhambra?

it was Galois who produced a language to be able to answer some of these questions. For Galois, symmetry — unlike for Thomas Mann, which was something still and deathly — for Galois, symmetry was all about motion. What can you do to a symmetrical object, move it in some way, so it looks the same as before you moved it? I like to describe it as the magic trick moves. What can you do to something? You close your eyes. I do something, put it back down again. It looks like it did before it started.

for example, the walls in the Alhambra — I can take all of these tiles, and fix them at the yellow place, rotate them by 90 degrees, put them all back down again and they fit perfectly down there. And if you open your eyes again, you wouldn’t know that they’d moved. But it’s the motion that really characterizes the symmetry inside the Alhambra. But it’s also about producing a language to describe this. And the power of mathematics is often to change one thing into another, to change geometry into language.

I’m going to take you through, perhaps push you a little bit mathematically — so brace yourselves — push you a little bit to understand how this language works, which enables us to capture what is symmetry.

let’s take these two symmetrical objects here. Let’s take the twisted six-pointed starfish. What can I do to the starfish which makes it look the same? Well, there I rotated it by a sixth of a turn, and still it looks like it did before I started. I could rotate it by a third of a turn, or a half a turn, or put it back down on its image, or two thirds of a turn. And a fifth symmetry, I can rotate it by five sixths of a turn. And those are things that I can do to the symmetrical object that make it look like it did before I started.

for Galois, there was actually a sixth symmetry. Can anybody think what else I could do to this which would leave it like I did before I started? I can’t flip it because I’ve put a little twist on it, haven’t I? It’s got no reflective symmetry. But what I could do is just leave it where it is, pick it up, and put it down again. And for Galois this was like the zeroth symmetry. Actually, the invention of the number zero was a very modern concept, seventh century A.D., by the Indians. It seems mad to talk about nothing. And this is the same idea. This is a symmetrical — so everything has symmetry, where you just leave it where it is.

this object has six symmetries. And what about the triangle? Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise. But now this has some reflectional symmetry. I can reflect it in the line through X, or the line through Y, or the line through Z. Five symmetries and then of course the zeroth symmetry where I just pick it up and leave it where it is. So both of these objects have six symmetries. Now, I’m a great believer that mathematics is not a spectator sport, and you have to do some mathematics in order to really understand it.

here is a little question for you. And I’m going to give a prize at the end of my talk for the person who gets closest to the answer. The Rubik’s Cube. How many symmetries does a Rubik’s Cube have? How many things can I do to this object and put it down so it still looks like a cube? Okay? So I want you to think about that problem as we go on, and count how many symmetries there are. And there will be a prize for the person who gets closest at the end.

let’s go back down to symmetries that I got for these two objects. What Galois realized: it isn’t just the individual symmetries, but how they interact with each other which really characterizes the symmetry of an object. If I do one magic trick move followed by another, the combination is a third magic trick move. And here we see Galois starting to develop a language to see the substance of the things unseen, the sort of abstract idea of the symmetry underlying this physical object. For example, what if I turn the starfish by a sixth of a turn, and then a third of a turn?

I’ve given names. The capital letters, A, B, C, D, E, F, are the names for the rotations. B, for example, rotates the little yellow dot to the B on the starfish. And so on. So what if I do B, which is a sixth of a turn, followed by C, which is a third of a turn? Well let’s do that. A sixth of a turn, followed by a third of a turn, the combined effect is as if I had just rotated it by half a turn in one go. So the little table here records how the algebra of these symmetries work. I do one followed by another, the answer is it’s rotation D, half a turn. What I if I did it in the other order? Would it make any difference? Let’s see. Let’s do the third of the turn first, and then the sixth of a turn. Of course, it doesn’t make any difference. It still ends up at half a turn.

And there is some symmetry here in the way the symmetries interact with each other. But this is completely different to the symmetries of the triangle. Let’s see what happens if we do two symmetries with the triangle, one after the other. Let’s do a rotation by a third of a turn anticlockwise, and reflect in the line through X. Well, the combined effect is as if I had just done the reflection in the line through Z to start with. Now, let’s do it in a different order. Let’s do the reflection in X first, followed by the rotation by a third of a turn anticlockwise. The combined effect, the triangle ends up somewhere completely different. It’s as if it was reflected in the line through Y.

it matters what order you do the operations in.

And this enables us to distinguish why the symmetries of these objects — they both have six symmetries. So why shouldn’t we say they have the same symmetries? But the way the symmetries interact enable us — we’ve now got a language to distinguish why these symmetries are fundamentally different. And you can try this when you go down to the pub, later on. Take a beer mat and rotate it by a quarter of a turn, then flip it. And then do it in the other order, and the picture will be facing in the opposite direction.

Galois produced some laws for how these tables — how symmetries interact. It’s almost like little Sudoku tables.

You don’t see any symmetry twice in any row or column. And, using those rules, he was able to say that there are in fact only two objects with six symmetries. And they’ll be the same as the symmetries of the triangle, or the symmetries of the six-pointed starfish. I think this is an amazing development. It’s almost like the concept of number being developed for symmetry. In the front here, I’ve got one, two, three people sitting on one, two, three chairs. The people and the chairs are very different, but the number, the abstract idea of the number, is the same.

we can see this now: we go back to the walls in the Alhambra. Here are two very different walls, very different geometric pictures. But, using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same.

For example, let’s take this beautiful wall with the triangles with a little twist on them. You can rotate them by a sixth of a turn if you ignore the colors. We’re not matching up the colors. But the shapes match up if I rotate by a sixth of a turn around the point where all the triangles meet. What about the center of a triangle? I can rotate by a third of a turn around the center of the triangle, and everything matches up. And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees. And all the tiles match up again. So rotate along halfway along the edge, and they all match up.

let’s move to the very different-looking wall in the Alhambra. And we find the same symmetries here, and the same interaction. So, there was a sixth of a turn. A third of a turn where the Z pieces meet. And the half a turn is halfway between the six pointed stars. And although these walls look very different, Galois has produced a language to say that in fact the symmetries underlying these are exactly the same. And it’s a symmetry we call 6-3-2.

Here is another example in the Alhambra. This is a wall, a ceiling, and a floor. They all look very different. But this language allows us to say that they are representations of the same symmetrical abstract object, which we call 4-4-2. Nothing to do with football, but because of the fact that there are two places where you can rotate by a quarter of a turn, and one by half a turn.

this power of the language is even more, because Galois can say, “Did the Moorish artists discover all of the possible symmetries on the walls in the Alhambra?” And it turns out they almost did.

You can prove, using Galois’ language, there are actually only 17 different symmetries that you can do in the walls in the Alhambra. And they, if you try to produce a different wall with this 18th one, it will have to have the same symmetries as one of these 17.

these are things that we can see. And the power of Galois’ mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two-dimensional, three-dimensional, all the way through to the four- or five- or infinite-dimensional space. And that’s where I work. I create mathematical objects, symmetrical objects, using Galois’ language, in very high dimensional spaces. So I think it’s a great example of things unseen, which the power of mathematical language allows you to create.

like Galois, I stayed up all last night creating a new mathematical symmetrical object for you, and I’ve got a picture of it here. Well, unfortunately it isn’t really a picture. If I could have my board at the side here, great, excellent. Here we are. Unfortunately, I can’t show you a picture of this symmetrical object. But here is the language which describes how the symmetries interact.

this new symmetrical object does not have a name yet. Now, people like getting their names on things, on craters on the moon or new species of animals. So I’m going to give you the chance to get your name on a new symmetrical object which hasn’t been named before. And this thing — species die away, and moons kind of get hit by meteors and explode — but this mathematical object will live forever. It will make you immortal. In order to win this symmetrical object, what you have to do is to answer the question I asked you at the beginning. How many symmetries does a Rubik’s Cube have?

Okay, I’m going to sort you out. Rather than you all shouting out, I want you to count how many digits there are in that number. Okay? If you’ve got it as a factorial, you’ve got to expand the factorials. Okay, now if you want to play, I want you to stand up, okay? If you think you’ve got an estimate for how many digits, right — we’ve already got one competitor here. If you all stay down he wins it automatically. Okay. Excellent. So we’ve got four here, five, six. Great. Excellent. That should get us going. All right.

Anybody with five or less digits, you’ve got to sit down, because you’ve underestimated. Five or less digits. So, if you’re in the tens of thousands you’ve got to sit down. 60 digits or more, you’ve got to sit down. You’ve overestimated. 20 digits or less, sit down. How many digits are there in your number? Two? So you should have sat down earlier. (Laughter) Let’s have the other ones, who sat down during the 20, up again. Okay? If I told you 20 or less, stand up. Because this one. I think there were a few here. The people who just last sat down.

how many digits do you have in your number? (Laughs) 21. Okay good.

How many do you have in yours? 18. So it goes to this lady here. 21 is the closest. It actually has — the number of symmetries in the Rubik’s cube has 25 digits. So now I need to name this object. So, what is your name? I need your surname. Symmetrical objects generally — spell it for me. G-H-E-Z No, SO2 has already been used, actually, in the mathematical language. So you can’t have that one. So Ghez, there we go. That’s your new symmetrical object. You are now immortal. (Applause)

17:26 And if you’d like your own symmetrical object, I have a project raising money for a charity in Guatemala, where I will stay up all night and devise an object for you, for a donation to this charity to help kids get into education in Guatemala. And I think what drives me, as a mathematician, are those things which are not seen, the things that we haven’t discovered.

It’s all the unanswered questions which make mathematics a living subject. And I will always come back to this quote from the Japanese “Essays in Idleness”: “In everything, uniformity is undesirable. Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth.”

“One of the problems about school mathematics is it doesn’t look at how mathematics is embedded in the world we live in.”

The beauty of math is everywhere:

The world turns on symmetry — from the spin of subatomic particles to the dizzying beauty of an arabesque.
But there’s more to it than meets the eye.
t.ted.com|By Marcus du Sautoy

### “I have the taste of humanity and kindness…” Thomas Mann

Posted on: March 31, 2017

##### “You think? Your are wrong. I’am not an aristocrat, I’m all the opposite, by reason and by taste.

Proof is for mathematical theorems and alcoholic beverages. It’s not for science.”

You’ll agree with me that if I decline the vivas of the people, it’s not out of pride: I have the taste of humanity and kindness.

The greatness of mankind is a miserable thing and often, it seems to me that men should know it , and conduct themselves humanly and with kindness and not to seek to be humiliated or to cow tow to one another.

##### Littérature et Poésie‘s photo.
In order to suffer being the object of simagrees that surround this greatness, we have to thicken our skin. I’m a little fragile by nature.
I don’t feel able to confront the ridicule of my situation. The person posted on my door and who expect to see me pass without extending him any attention or respect commensurate to the chamberlain at the door, embarrasses me. It is my way of loving the people…”
From another Mann, and from post over at Peter Guest’s blog, the scientist Michael “Hockey Stick” Mann is quoted saying:

Proof is for mathematical theorems and alcoholic beverages. It’s not for science.”

Michael Mann goes on to explain that science is all about “credible theories” and “best explanations” and his critics are not offering up any of those.

Mann’s attempt to separate proof from science stems from increasing public awareness that the warming predicted by the high-sensitivity models that Mann and others have championed just have not occurred over the last 15 years.

### Euler’s Conjecture? Debunking Sums of like Powers?

Posted on: January 10, 2017

# Two Succinct Sentences

Euler’s conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years.

Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences.

Their article — which is now open access and can be downloaded here — appeared in the Bulletin of the American Mathematical Society.

If you’re wondering what the conjecture and its refutation are all about, you might want to ask Cliff Pickover, the author of 45 books on math and science. He brought this curious document to the web last week.

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