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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.”

Patsy Z shared this link

“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
From “Royal Altess” by Thomas Mann
“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.

The Shortest-Known Paper Published in a Serious Math Journal:

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.

shortest math paper

 

Origins of Mathematics in the Kitara and Kalahari regions of sub Saharan Africa nearly 40,000 years ago

THE DISCOVERY OF A MATHEMATICAL DOCUMENT AT ISHANGO VILLAGE  

AROUND LAKE EDWARD NEAR SEMULIKI RIVER.

THE ISHAGO BONE: was discovered by Belgian archeologist in 1957 in a place called Ishango at a shore of Lake Edward near the border of D.R Congo and Uganda, part of the source of the river Nile.

The bone is 10 cm long dark brown object with a tally notches made on it and tabulated in three rows (columns). A second Ishango bone was also discovered by the same person.

A discussion about Ishango bone is still going on.

One group of researchers in Beligium thought that the bone has shown that ancient East and Central African people around Bunyoro during pre-historic time had  knowledge of mathematics .

Other researchers have argued that the bone is just a historical document. It could be some sort of a calendar according to the second group of researchers

It proves that  people around Eastern Zaire and western and Central Uganda, perhaps during or before the era of Batembuzi rule,used tools (instruments) for different mathematical and scientific purposes like counting, measuring and registering facts.

A study by international scientific researchers is apparently continuing to investigate how ancient people around this area of Africa have used tools for mathematical calculations and registrations.

We will concentrate mainly on the Ishango bone (tool / document) which was used at 22,000 BC (a date that is corrected many times by researchers).

History of this document is 22,000 BC. The historical position of this period is  the one commonly known as the Stone Age. It is roughly coinciding with the period supposedly before the discovery of agriculture.

But how in that time there was sharp increase in the diversity of artifacts is a mystery. Also evidence from elsewhere in Africa like at Blombos cave in South Africa at that time bone artifacts and a kind first art appeared in the fossils in Africa.

Recordings

On the Bone,the  first row shows a system of addition based on the number ten. The numbers in this row are:

10+1, (10 * 2) + 1, (10 * 2) – 1, 10-1 

This shows understanding of addition, subtraction and multiplication based on ten.

The second row starts with 3 notches. It is then doubled to become 6.

The same is done with the next number (4), which is doubled to become 8. The number 8 followed by 10 which halved to form 5. This is followed by 5 and 7. The operations give us evidence that people who have used this document have some understanding of multiplication and division by 2.

The last row is a set of prime numbers between 10 and 20. These numbers are 11, 13, 17 and 19. The numbers in this row as well as in the first row are odd number.

The sum of the numbers in the first and the last rows is 60. This can not be a coincidence.

The sum of the numbers in the last row in the middle is 48. This implies that all sums are multiple of 12.

From all these remarks some people suggest that this bone may have been used as an ancient counting tool or an ancient calculator or an accounting ledger document.

It may have been used to record data regarding live stock, lost number of solders, assets, etc.The bone reveals that African people at that time had an understanding of mathematics.

The technique of dividing and multiplying by 2 was used by old Egyptians in later times. This can lead to suggestions that Egyptians were largely influenced by their old  African ancestors from around the source of the Nile  (ancient Bunyoro region).

The second assumption is that Isahango bone is a calendar. But the fact that numbers in two rows have a sum of 60 is not a proof of this assumption.

Calendars were used in ancient times as widely known.

The Egyptians used a solar calendar. They knew that a year has 365 day and 12 months. Every month has 30 days. They reserved extra 5 days to complete year days to be 365 days.

SECOND ISHANGO BONE^

ISHANGO VILLAGE

Is found near the confluence of river Semliki and lake Edward in what is now Eastern DRC ( formerly Zaire) Ishango in local surounding dialect which means swampy place or in kinyoro and kinyankole in western Uganda, the word means originally found place but also the term can be derived from ekishanga which means a swampy place.

Semliki River (sometimes Semuliki) is a major river in Central Africa. It flows northwards from Lake Edward in theDemocratic Republic of the Congo, across the Uganda border, through western Uganda in Bundibugyo District, near theSemuliki National Park.

It empties into Lake Albert in Uganda at 1.2225°N 30.5038889°E.

In places, the river has demarcated the border between Uganda and the DRC; its changing course sparked confusion in 2009 over the location of the boundary between the two countries.(wikipedia)

The structure and the techniques on  the Ishango bone closely resembles the LEBOMBO BONE which is a similar ancient African document discovered in the Lebombo mountains located between South Africa and Swaziland in the 1970s during excavations of Border Cave and dated about 35,000 B.C.,

the Lebombo bone is marked with 29 clearly defined notches. This suggests it may have been used as a lunar phase counter, in which case African women may have been the first mathematicians, because keeping track of menstrual cycles requires a lunar calendar.

Certainly, the Lebombo bone resembles calendar sticks still used by the SAN people of Botswana and  Namibia.

Educators want to pair math and music in integrated teaching method

 

As a child, before he started playing jazz, composer and musical icon Herbie Hancock was fond of taking things apart and putting them back together.

He was perpetually inquisitive and analytical, a quality that carried from his days of tinkering with clocks and watches to his playing of music, where he threw himself into jazz as a teen.

“I would always try to figure out how things work,” Hancock said. “It was that same instinct that I have that made me learn jazz more quickly. . . . It wasn’t a talent for music. It was a talent for being able to analyze things and figure out the details.”


Jazz composer Herbie Hancock addresses a group at the U.S. Department of Education on April 26, 2016, where he spoke about using music to teach math and engineering. (Paul Wood/U.S. Department of Education)

Hancock later studied electrical engineering at Grinnell College before starting his jazz career full-time. He says there is an intrinsic link between playing music and building things, one that he thinks should be exploited in classrooms across the country, where there has been a renewed emphasis on science, technology, engineering and math (STEM) education.

Hancock joined a group of educators and researchers Tuesday at the U.S. Education Department’s headquarters to discuss how music can be better integrated into lessons on math, engineering and even computer science, ahead of International Jazz Day this weekend.

Education Secretary John B. King Jr. said that an emphasis on math and reading — along with standardized testing — has had the unfortunate side effect of squeezing arts education out of the nation’s classrooms, a trend he thinks is misguided.

“English and math are necessary but not sufficient for students’ long-term success,” King said, noting that under the Every Student Succeeds Act, the new federal education law, schools have new flexibility to use federal funding for arts education.

Hancock is the chairman of the Thelonious Monk Institute of Jazz, which has developed MathScienceMusic.org, a website that offers teachers resources and apps to use music as a vehicle to teach other academic lessons.

One app, Groove Pizza, allows users to draw lines and shapes onto a circle. The circle then rotates and each shape and line generates its own distinct sound. It’s a discreet way for children to learn about rhythm and proportions. With enough shapes and lines, children can create elaborate beats on the app, all in the context of a “pizza” — another way to make learning math and music palatable to kids.

Another app — Scratch Jazz — allows children to use the basic coding platform Scratch to create their own music.

“A lot of what we focus on is lowering the barriers to creative expression,” said Alex Ruthmann, a professor of music education at New York University who helped develop the Groove Pizza app.

Other researchers discussed their experiments with music and rhythm to teach fractions and proportionality, a challenging concept for young students to grasp when it is taught in the abstract.

Susan Courey, a professor of special education at San Francisco State University, developed a fractions lesson that has students tap out a beat.

“It goes across language barriers, cultures and achievement barriers and offers the opportunity to engage a very diverse set of students,” Courey said. In a small study, students who received the music lesson scored 50%  higher on a fraction test than those who learned with the standard curriculum. “They should be taught together.”

“If a student can clap about a beat based on a time signature, well aren’t they adding and subtracting fractions based on music notation?” Courey said. “We have to think differently.”

Hancock thinks that the arts may offer a better vehicle to teach math and science to some students. But he also sees value in touching students’ hearts through music — teaching them empathy, creative expression and the value of working together and keeping an open mind.

“Learning about and adopting the ethics inherent in jazz can make positive changes in our world, a world that now more than ever needs more creativity and innovation and less anger and hostility to help solve the challenges that we have to help deal with every single day,” Hancock said.

Why “X” is to be the Unknown?

Question: Why is it that the letter X represents the unknown?

I have the answer to this question that we’ve all asked

Now I know we learned that in math class, but now it’s everywhere in the culture — The X prize, the X-Files, Project X, TEDx. Where’d that come from?

Terry Moore this Feb. 2012

0:34 About six years ago I decided that I would learn Arabic, which turns out to be a supremely logical language.

To write a word or a phrase or a sentence in Arabic is like crafting an equation, because every part is extremely precise and carries a lot of information.

That’s one of the reasons so much of what we’ve come to think of as Western science and mathematics and engineering was really worked out in the first few centuries of the Common Era by the Persians and the Arabs and the Turks.

This includes the little system in Arabic called al-jebra (An Arabic mathematician). And al-jebr roughly translates to the system for reconciling disparate parts.”

Al-jebr finally came into English as algebra. One example among many.

The Arabic texts containing this mathematical wisdom finally made their way to Europe — which is to say Spain — in the 11th and 12th centuries. And when they arrived there was tremendous interest in translating this wisdom into a European language.

But there were problems.

One problem is there are some sounds in Arabic that just don’t make it through a European voice box without lots of practice. Trust me on that one.

Also, those very sounds tend not to be represented by the characters that are available in European languages.

Here’s one of the culprits.

This is the letter SHeen, and it makes the sound we think of as SH — “sh.” It’s also the very first letter of the word shalan, which means “something” some undefined, unknown thing.

Now in Arabic, we can make this definite by adding the definite article “al.” So this is al-shalan — the unknown thing. And this is a word that appears throughout early mathematics, such as this 10th century derivation of proofs.

The problem for the Medieval Spanish scholars who were tasked with translating this material is that the letter SHeen and the word shalan can’t be rendered into Spanish because Spanish doesn’t have that SH, that “sh” sound.

So by convention, they created a rule in which they borrowed the CK sound, “ck” sound, from the classical Greek in the form of the letter Kai.

Later when this material was translated into a common European language, which is to say Latin, they simply replaced the Greek Kai with the Latin X.

And once that happened, once this material was in Latin, it formed the basis for mathematics textbooks for almost 600 years.

But now we have the answer to our question. Why is it that X is the unknown?

X is the unknown because you can’t say “sh” in Spanish. (Laughter) And I thought that was worth sharing.

 

Fibonacci numbers? How wonderful

Mathematics is not just solving for x, it’s also figuring out why

So why do we learn mathematics? Essentially, for 3 reasons: calculation, application, and inspiration.

0:27 Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics that we learn in school is not effectively motivated, and when our students ask, “Why are we learning this?” then they often hear that they’ll need it in an upcoming math class or on a future test.

But wouldn’t it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind?

Now, I know many people have not had the opportunity to see how this can happen, so let me give you a quick example with my favorite collection of numbers, the Fibonacci numbers.

1:12 These numbers can be appreciated in many different ways. From the standpoint of calculation, they’re as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on.

Indeed, the person we call Fibonacci was actually named Leonardo of Pisa, and these numbers appear in his book “Liber Abaci,” which taught the Western world the methods of arithmetic that we use today.

In terms of applications, Fibonacci numbers appear in nature surprisingly often. The number of petals on a flower is typically a Fibonacci number, or the number of spirals on a sunflower or a pineapple tends to be a Fibonacci number as well.

1:59 In fact, there are many more applications of Fibonacci numbers, but what I find most inspirational about them are the beautiful number patterns they display.

Let me show you one of my favorites. Suppose you like to square numbers, and frankly, who doesn’t? (Laughter)

2:15 Let’s look at the squares of the first few Fibonacci numbers.

So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on.

Now, it’s no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right? That’s how they’re created. But you wouldn’t expect anything special to happen when you add the squares together. But check this out. One plus one gives us two, and one plus four gives us five. And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues.

2:53 In fact, here’s another one.

Suppose you wanted to look at adding the squares of the first few Fibonacci numbers. Let’s see what we get there. So one plus one plus four is six. Add nine to that, we get 15. Add 25, we get 40. Add 64, we get 104. Now look at those numbers. Those are not Fibonacci numbers, but if you look at them closely, you’ll see the Fibonacci numbers buried inside of them.

3:21 Do you see it? I’ll show it to you. Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate?

 

3:35 Now, as much fun as it is to discover these patterns, it’s even more satisfying to understand why they are true.

Let’s look at that last equation. Why should the squares of one, one, two, three, five and eight add up to eight times 13? I’ll show you by drawing a simple picture. We’ll start with a one-by-one square and next to that put another one-by-one square. Together, they form a one-by-two rectangle. Beneath that, I’ll put a two-by-two square, and next to that, a three-by-three square, beneath that, a five-by-five square, and then an eight-by-eight square, creating one giant rectangle, right?

4:17 Now let me ask you a simple question: what is the area of the rectangle? Well, on the one hand, it’s the sum of the areas of the squares inside it, right? Just as we created it.

It’s one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? That’s the area.

On the other hand, because it’s a rectangle, the area is equal to its height times its base, and the height is clearly eight, and the base is five plus eight, which is the next Fibonacci number, 13. Right?

So the area is also eight times 13. Since we’ve correctly calculated the area two different ways, they have to be the same number, and that’s why the squares of one, one, two, three, five and eight add up to eight times 13.

5:09 Now, if we continue this process, we’ll generate rectangles of the form 13 by 21, 21 by 34, and so on.

5:18 Now check this out. If you divide 13 by 8, you get 1.625. And if you divide the larger number by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians, scientists and artists for centuries.

5:41 Now, I show all this to you because, like so much of mathematics, there’s a beautiful side to it that I fear does not get enough attention in our schools. We spend lots of time learning about calculation, but let’s not forget about application, including, perhaps, the most important application of all, learning how to think.

6:02 If I could summarize this in one sentence, it would be this: Mathematics is not just solving for x, it’s also figuring out why.

The magic of Fibonacci numbers
Math is logical, functional and just … awesome.
Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)
TED · 43,696 Shares · Nov

 

Math forever? How can you start loving math?

Imagine you’re in a bar, or a club, and you start talking to a woman, and after a while this question comes up: “So, what do you do for work?”  (The most dreaded question I expect in a casual conversation)

And since you think your job is interesting, you say: “I’m a mathematician.” (Laughter)

Then 33.51% of women, in that moment, pretend to get an urgent call and leave.

And 64.69% of women desperately try to change the subject and leave.

Another 0.8%, probably your cousin, your girlfriend and your mom, know that you work in something weird but don’t remember what it is. (Laughter)

And then there’s  1% who remain engaged with the conversation.

And inevitably, during that conversation one of these two phrases come up:

A) “I was terrible at math, but it wasn’t my fault. It’s because the teacher was awful.” Or

B) “But what is math really for?”

I’ll now address Case B.

When someone asks you what math is for, they’re not asking you about applications of mathematical science.

They’re asking you, why did I have to study that bullshit I never used in my life again

That’s what they’re actually asking. So when mathematicians are asked what math is for, they tend to fall into two groups:

1. 54.51% of mathematicians will assume an attacking position, and

2. 44.77% of mathematicians will take a defensive position.

There’s a strange 0.8%, among which I include myself.

Who are the ones that attack? The attacking ones are mathematicians who would tell you this question makes no sense, because mathematics have a meaning all their own a beautiful edifice with its own logic — and that there’s no point in constantly searching for all possible applications.

What’s the use of poetry? What’s the use of love? What’s the use of life itself? What kind of question is that?

Hardy, for instance, was a model of this type of attack. And those who stand in defense tell you, Even if you don’t realize it, math is behind everything.” (Laughter)

Those guys, they always bring up bridges and computers. “If you don’t know math, your bridge will collapse.”

It’s true, computers are all about math. And now these guys have also started saying that behind information security and credit cards are prime numbers.

These are the answers your math teacher would give you if you asked him. He’s one of the defensive ones.

Okay, but who’s right then?

Those who say that math doesn’t need to have a purpose, or those who say that math is behind everything we do?

Actually, both are right.

But remember I told you I belong to that strange 0.8 percent claiming something else? So, go ahead, ask me what math is for. Audience: What is math for?

Eduardo Sáenz de Cabezón: Okay, 76.34% of you asked the question, 23.41 percent didn’t say anything, and the 0.8 percent — I’m not sure what those guys are doing.

Well, to my dear 76.31% — it’s true that math doesn’t need to serve a purpose, it’s true that it’s a beautiful structure, a logical one, probably one of the greatest collective efforts ever achieved in human history.

But it’s also true that there, where scientists and technicians are looking for mathematical theories that allow them to advance, they’re within the structure of math, which permeates everything.

It’s true that we have to go somewhat deeper, to see what’s behind science.

Science operates on intuition, creativity. Math controls intuition and tames creativity.

Almost everyone who hasn’t heard this before is surprised when they hear that if you take a 0.1 millimeter thick sheet of paper, the size we normally use, and, if it were big enough, fold it 50 times, its thickness would extend almost the distance from the Earth to the sun.

Your intuition tells you it’s impossible. Do the math and you’ll see it’s right. That’s what math is for.

It’s true that science, all types of science, only makes sense because it makes us better understand this beautiful world we live in.

And in doing that, it helps us avoid the pitfalls of this painful world we live in. There are sciences that help us in this way quite directly.

Oncological science, for example. And there are others we look at from afar, with envy sometimes, but knowing that we are what supports them.

All the basic sciences support them, including math. All that makes science, science is the rigor of math. And that rigor factors in because its results are eternal.

You probably said or were told at some point that diamonds are forever, right? That depends on your definition of forever!

A theorem — that really is forever. (Laughter) The Pythagorean theorem is still true even though Pythagoras is dead, I assure you it’s true. 

Even if the world collapsed the Pythagorean theorem would still be true. Wherever any two triangle sides and a good hypotenuse get together the Pythagorean theorem goes all out. It works like crazy.

Well, we mathematicians devote ourselves to come up with theorems. Eternal truths.

But it isn’t always easy to know the difference between an eternal truth, or theorem, and a mere conjecture. You need proof.

For example, let’s say I have a big, enormous, infinite field. I want to cover it with equal pieces, without leaving any gaps. I could use squares, right? I could use triangles. Not circles, those leave little gaps. Which is the best shape to use?

One that covers the same surface, but has a smaller border.

In the year 300, Pappus of Alexandria said the best is to use hexagons, just like bees do. But he didn’t prove it. The guy said, “Hexagons, great! Let’s go with hexagons!” He didn’t prove it, it remained a conjecture.

“Hexagons!” And the world, as you know, split into Pappists and anti-Pappists, until 1700 years later when in 1999, Thomas Hales proved that Pappus and the bees were right — the best shape to use was the hexagon. And that became a theorem, the honeycomb theorem, that will be true forever and ever, for longer than any diamond you may have. 

But what happens if we go to 3 dimensions?

If I want to fill the space with equal pieces, without leaving any gaps, I can use cubes, right? Not spheres, those leave little gaps. 

What is the best shape to use? Lord Kelvin, of the famous Kelvin degrees and all, said that the best was to use a truncated octahedron which, as you all know — (Laughter) — is this thing here!

Come on. Who doesn’t have a truncated octahedron at home? (Laughter) Even a plastic one.

“Honey, get the truncated octahedron, we’re having guests.” Everybody has one!  But Kelvin didn’t prove it. It remained a conjecture — Kelvin’s conjecture.

The world, as you know, then split into Kelvinists and anti-Kelvinists (Laughter) until a hundred or so years later, someone found a better structure.

Weaire and Phelan found this little thing over here, this structure to which they gave the very clever name “the Weaire-€“Phelan structure.”

It looks like a strange object, but it isn’t so strange, it also exists in nature. It’s very interesting that this structure, because of its geometric properties, was used to build the Aquatics Center for the Beijing Olympic Games.

There, Michael Phelps won eight gold medals, and became the best swimmer of all time. Well, until someone better comes along, right?

As may happen with the Weaire-€“Phelan structure. It’s the best until something better shows up.

But be careful, because this one really stands a chance that in a hundred or so years, or even if it’s in 1700 years, that someone proves it’s the best possible shape for the job. It will then become a theorem, a truth, forever and ever. For longer than any diamond.

So, if you want to tell someone that you will love them forever you can give them a diamond. But if you want to tell them that you’ll love them forever and ever, give them a theorem! (Laughter)

But hang on a minute! You’ll have to prove it, so your love doesn’t remain a conjecture.

Reine Azzi shared this link on FB

For the love of math (I sucked at it) in Spanish with English subtitles.

With humor and charm, mathematician Eduardo Sáenz de Cabezón answers a question that’s wracked the brains of bored students the world over: What is math for?
He shows the beauty of math as the backbone of science — and shows that…
ted.com|By Eduardo Sáenz de Cabezón

Hacking OkCupid: And Chris McKinlay finding “True Love” 

What large-scale data processing and parallel numerical methods have to do with falling in love?

OkCupid was founded by Harvard math majors in 2004, and it first caught daters’ attention because of its computational approach to matchmaking.

Members answer droves of multiple-choice survey questions on everything from politics, religion, and family to love, sex, and smartphones.

OkCupid lets users see the responses of others, but only to questions they’ve answered themselves.

KEVIN POULSEN posted this Jan. 21, 2014

How a Math Genius Hacked OkCupid to Find True Love

Chris McKinlay was folded into a cramped fifth-floor cubicle in UCLA’s math sciences building, lit by a single bulb and the glow from his monitor.

It was 3 am, the optimal time to squeeze cycles out of the supercomputer in Colorado that he was using for his PhD dissertation.

(The subject: large-scale data processing and parallel numerical methods.)

While the computer chugged, he clicked open a second window to check his OkCupid inbox.

Mathematician Chris McKinlay hacked OKCupid to find the girl of his dreams

McKinlay, a lanky 35-year-old with tousled hair, was one of about 40 million Americans looking for romance through websites like Match.com, J-Date, and e-Harmony, and he’d been searching in vain since his last breakup 9 months earlier.

He’d sent dozens of cutesy introductory messages to women touted as potential matches by OkCupid’s algorithms. Most were ignored; he’d gone on a total of 6 first dates.

On that early morning in June 2012, his compiler crunching out machine code in one window, his forlorn dating profile sitting idle in the other, it dawned on him that he was doing it wrong.

He’d been approaching online matchmaking like any other user. Instead, he realized, he should be dating like a mathematician.

On average, respondents select 350 questions from a pool of thousands—“Which of the following is most likely to draw you to a movie?” or “How important is religion/God in your life?”

For each, the user records an answer, specifies which responses they’d find acceptable in a mate, and rates how important the question is to them on a 5-point scale from “irrelevant” to “mandatory.” OkCupid’s matching engine uses that data to calculate a couple’s compatibility. The closer to 100 percent—mathematical soul mate—the better.

But mathematically, McKinlay’s compatibility with women in Los Angeles was abysmal.

OkCupid’s algorithms use only the questions that both potential matches decide to answer, and the match questions McKinlay had chosen—more or less at random—had proven unpopular.

When he scrolled through his matches, fewer than 100 women would appear above the 90 percent compatibility mark. And that was in a city containing some 2 million women (approximately 80,000 of them on OkCupid).

On a site where compatibility equals visibility, he was practically a ghost.

He realized he’d have to boost that number. If, through statistical sampling, McKinlay could ascertain which questions mattered to the kind of women he liked, he could construct a new profile that honestly answered those questions and ignored the rest.

He could match every woman in LA who might be right for him, and none that weren’t.

Chris McKinlay used Python scripts to riffle through hundreds of OkCupid survey questions. He then sorted female daters into 7 clusters, like “Diverse” and “Mindful,” each with distinct characteristics.  Maurico Alejo

Even for a mathematician, McKinlay is unusual.

Raised in a Boston suburb, he graduated from Middlebury College in 2001 with a degree in Chinese. In August of that year he took a part-time job in New York translating Chinese into English for a company on the 91st floor of the north tower of the World Trade Center.

The towers fell 5 weeks later. (McKinlay wasn’t due at the office until 2 o’clock that day. He was asleep when the first plane hit the north tower at 8:46 am.) “After that I asked myself what I really wanted to be doing,” he says.

A friend at Columbia recruited him into an offshoot of MIT’s famed professional blackjack team, and he spent the next few years bouncing between New York and Las Vegas, counting cards and earning up to $60,000 a year.

The experience kindled his interest in applied math, ultimately inspiring him to earn a master’s and then a PhD in the field. “They were capable of using mathema­tics in lots of different situations,” he says. “They could see some new game—like Three Card Pai Gow Poker—then go home, write some code, and come up with a strategy to beat it.

Now he’d do the same for love. First he’d need data.

While his dissertation work continued to run on the side, he set up 12 fake OkCupid accounts and wrote a Python script to manage them. The script would search his target demographic (heterosexual and bisexual women between the ages of 25 and 45), visit their pages, and scrape their profiles for every scrap of available information: ethnicity, height, smoker or nonsmoker, astrological sign—“all that crap,” he says.

To find the survey answers, he had to do a bit of extra sleuthing.

OkCupid lets users see the responses of others, but only to questions they’ve answered themselves.

McKinlay set up his bots to simply answer each question randomly—he wasn’t using the dummy profiles to attract any of the women, so the answers didn’t mat­ter—then scooped the women’s answers into a database.

McKinlay watched with satisfaction as his bots purred along. Then, after about a thousand profiles were collected, he hit his first roadblock.

OkCupid has a system in place to prevent exactly this kind of data harvesting: It can spot rapid-fire use easily. One by one, his bots started getting banned.

He would have to train them to act human.

He turned to his friend Sam Torrisi, a neuroscientist who’d recently taught McKinlay music theory in exchange for advanced math lessons.

Torrisi was also on OkCupid, and he agreed to install spyware on his computer to monitor his use of the site. With the data in hand, McKinlay programmed his bots to simulate Torrisi’s click-rates and typing speed.

He brought in a second computer from home and plugged it into the math department’s broadband line so it could run uninterrupted 24 hours a day.

After 3 weeks he’d harvested 6 million questions and answers from 20,000 women all over the country.

McKinlay’s dissertation was relegated to a side project as he dove into the data. He was already sleeping in his cubicle most nights. Now he gave up his apartment entirely and moved into the dingy beige cell, laying a thin mattress across his desk when it was time to sleep.

For McKinlay’s plan to work, he’d have to find a pattern in the survey data—a way to roughly group the women according to their similarities.

The breakthrough came when he coded up a modified Bell Labs algorithm called K-Modes.

First used in 1998 to analyze diseased soybean crops, K-Modes takes categorical data and clumps it like the colored wax swimming in a Lava Lamp. With some fine-tuning he could adjust the viscosity of the results, thinning it into a slick or coagulating it into a single, solid glob.

He played with the dial and found a natural resting point where the 20,000 women clumped into 7 statistically distinct clusters based on their questions and answers. “I was ecstatic,” he says. “That was the high point of June.”

He retasked his bots to gather another sample: 5,000 women in Los Angeles and San Francisco who’d logged on to OkCupid in the past month.

Another pass through K-Modes confirmed that they clustered in a similar way. His statistical sampling had worked.

Now he just had to decide which cluster best suited him. He checked out some profiles from each. One cluster was too young, two were too old, another was too Christian.

But he lingered over a cluster dominated by women in their mid-twenties who looked like indie types, musicians and artists. This was the golden cluster. The haystack in which he’d find his needle. Somewhere within, he’d find true love.

Actually, a neighboring cluster looked pretty cool too—slightly older women who held professional creative jobs, like editors and designers. He decided to go for both.

He’d set up two profiles and optimize one for the A group and one for the B group.

He text-mined the two clusters to learn what interested them; teaching turned out to be a popular topic, so he wrote a bio that emphasized his work as a math professor.

The important part, though, would be the survey.

He picked out the 500 questions that were most popular with both clusters. He’d already decided he would fill out his answers honestly—he didn’t want to build his future relationship on a foundation of computer-generated lies.

But he’d let his computer figure out how much importance to assign each question, using a machine-learning algorithm called adaptive boosting to derive the best weightings.

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