Concepts of Number Line and Time May Not Be As Intuitive As Previously Thought

Mathematics, often romanticized as human intuition, may not be as natural to the human race after all. In a new study conducted by the a team of scientists, led by Rafael Nunez, director of Embodied Cognition Lab, it appears that abstract mathematical concepts like the number line, which involves the mapping of numbers onto space, need to be taught and aren’t ‘hard-wired’.

Really so intuitive? Think again!

Nunez et al studied an indigenous tribal group from Papua New Guinea called the Yupno. The group lives in a remote part of the upper Yupno Valley in Papua New Guinea. The place has no roads. The team used a small plane and then hiked, carrying heavy equipment like solar panels, since the valley has no electricity.

The study

The main study was conducted with three groups – one comprised 14 illiterate adults, another comprising 6 adults who had received very basic schooling from within the tribal community and another control group in California, comprising adults with formal schooling. All the three groups were given several objects and a long line. Then they were asked to arrange these objects on the number line.
The Yupno were given oranges. The first group (of unschooled adults) arranged the oranges, but stacked them up at the two endpoints and a put a few in the middle, totally ignoring the extension of the line in between, which is one of the most important properties of the number line. The second group did a little better, using the extension a bit more, but not quite as evenly as it should be. The control group in California treated the number line as it should be. All this suggests only one thing – that the concept of numbers, especially their mapping onto space, is a concept that has to be taught and is not ingrained in the human brain. The ability to build this intuition might be evolutionary, but not the intuition itself!

What about time?

The team also analysed the crucial concept of time. We tend to associate the flow of time with spatial position, associating ‘forward’ for future and ‘backward’ for past. Interestingly, the Aymara of the Andes, a previously studied group of Nunez et al., does the reverse. The Yupno uses ‘uphill’ and ‘downhill’, says Nunez. They even use the three-dimensional topography of the valley to describe time, obviously conflicting with just the forward and backward 2-D notions of time.

Cooperrider, a co-author in the study, says:

When confronted with radically different ways of construing experience, we can no longer take for granted our own. Ultimately, no way is more or less ‘natural’ than the Yupno way.

There you go – Mathematics, or even time, is not as universal as we thought. If some definition or notion of mathematics seems obvious, it may be because we lack imagination.

The study appears here:http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0035662

Happy Pi Day, Everyone!

At the very outset, let’s get one thing straight – ‘Pi’ is not the same as ‘Pie’. Pi is an irrational number; ‘Pie’ is a food item, horribly less important than Pi and much less faithful to a circle as well. Pi –the constant ratio between the circumference and the diameter of a circle – is always 3.14… and a few more digits. Actually, a lot more digits!

Geeky, geekier…

Now, glance at today’s date – 14th March. Write it as the Americans do – month first and then day – and you get 3.14! Ah ha, Pi!

Some people choose to take this one step further. Let’s look at a few more digits of Pi – 3.14159… Some people celebrate 1:59 AM and 1:59 PM on 14th March as “Pi Minutes”. How about continuing the trend? Sure, 3.1415926, occurring at 1:59:26 AM and 1:59:26 PM, are celebrated as “Pi Seconds”.

These are delicacies on this one day. Pi pies.

If you think all of this is extremely loony, you’d be right. However, it’s more than just fun and games. Pi day is also meant to promote math and show that it is a lot of fun. Pi day founder, Larry Shaw, a physicist at the Exploratorium, participates actively in the Pi Day celebrations by hosting a number of activities in the Exploratorium. Here’s this year’s list: http://www.exploratorium.edu/pi/pi_schedule.html

Plus, check out the homepage for pi: http://www.exploratorium.edu/pi/

… And a very happy birthday

In addition to all this fun, let us not forget that the great Albert Einstein was born on 14th March. So, it’s a double whammy and all Pi day celebrations spend some time wishing the Grand Old Man of Physics a Happy Birthday.

Wishing you a very happy Pi day!

What Is The Loneliest Number?

So which is the Loneliest Number in the world? According to Google it is the number 1. This is proven when you search Google for "the loneliest number". You can try it out here for yourself.

The Loneliest Number

While you are at it. Don’t forget to try out several of our other Math articles and jobs. Including finding your age without telling anyone, appropriate Pi Day and Mathematical combinations so interesting.

By the way, if you are mathematical enthusiastic or good at Quantum Physics and are looking to contribute, send us an email at admin [at] techie-buzz.com.

Climbing Mt. Zeta With The Man Who Wants To Prove The Riemann Hypothesis – An Exclusive Interview

It’s about being ambitious and daring to be different. I interviewed the man who intends to solve the Riemann hypothesis, using quasi-crystals. I already wrote about it here and I strongly suggest that you go through that article before reading the interview.

In the interview, I ask him about his method of attack and his methodology of recruiting people for the expedition. He openly says that the journey is just starting. We end it with a light-hearted chat about his Twitter moniker and his email ID.

The Earlier Article That You Should Read Before Proceeding:  http://techie-buzz.com/science/indian-riemann-hypothesis-quasicrystals.html

 

Here’s the full interview:

Me: Kindly say a bit about yourself. I know that you conducted a neutrino workshop a few days ago. Are you a mathematician interested in physics or a physicist interested in mathematics?  

Rohit Gupta: Yeah, the workshop I conducted on neutrinos was about  2 months prior to the OPERA results going online.  I was pretty sure that  neutrinos were the ‘ghost’ in the standard model, and well – now everyone  is talking about it.

About me, let’s just say that I am a cosmologist who is interested in connections  between physics and number theory.  My educational background  is Chemical Engineering ( IIT Kharagpur, class of 99) and after that I  have been mostly self-taught and self-employed.

 

Me: Cool! Yeah, I can’t resist the temptation. Just one more off-the-track question before the real questions. What’s your take on the faster-than-light neutrino results?  

RG: I’m no experimental physicist, but even if you read the literature on the original neutrino  experiments   like Homestake or the SN1987 at Kamiokande, we are dealing with a jaw-dropping piece of work purely in statistical terms. OPERA is no more controversial than solar neutrinos were for about  30 years.

It is too early to take sides, but personally I will try to study the work of Ettore Majorana.  That sounds promising. I don’t know it too well, though.

 

Me: Okay. First of the serious questions. You mentioned Erbium in your blog. You said that the energy levels would correspond to the pattern of zeros of the Riemann zeta function. Given that Erbium is large atom and cannot be solved exactly, how can you precisely know the energy levels? The zeros of Riemann Zeta are definitely well-defined and precise.  

RG: I am not going by the experimental data Dyson is referring to, I am going after the idea itself.  That has little to do with Erbium,  that has to do with quasicrystals and the Riemann-Zeta. And Dyson is not my only inspiration, there are other lesser known papers, there’s one by Akio Sugamoto which models “factorization of integers into prime numbers viewed as decay of a particle into  elementary particles conserving energy”. I think that is what excites me, that there is something corporeal behind numbers, a physical connection.

 

Me: And that precisely brings me to the next question.  

I am curious about the quasi-crystal strategy you intend to employ. You mention that ” if the Riemann Hypothesis is true, then the zeroes of the zeta function form a quasicrystal”. Do you intend to ‘make’ a quasicrystal in 1-D and match it with the zeros of the R-Z function? Is a 1-D quasicrystal unique?  

Further, can you be sure of the generality of the result? I mean, you cannot possibly compare all the zeros, right? How do you that the R-Z function follows the quasi-crystal to Re(s)->infinity?  

Also, I’m quite surprised to think that this hasn’t been tried by someone else, especially when someone like Freeman Dyson had suggested it.  

RG: There are gaps in the theory because quasicrystal theory is very new. What Dyson suggests as a first target  is to classify quasicrystals in 1-D, a well-known example being the Fibonacci sequence. There are many  such sequences  and a classification of these requires a new metric. Once we have the metric, we find the one that  corresponds to the Riemann Zeta. So yeah, I intend to make such a quasicrystal in 1D.  Because the quasicrystal  has a precise artificial structure, we can know for sure it corresponds to infinity.

I’m pretty surprised too that this hasn’t been given a serious attempt, and probably very lucky in that aspect.

 

Me: I have an objection there:

You say: Because the quasicrystal  has a precise artificial structure, we can know for sure it corresponds to infinity.How do you know that the two follow each other till infinity? A trillion zeros have been found and all follow the Riemann hypothesis, but the jury’s still out because a generic proof has not been found. How are you so sure of the exact correspondence?  

RG:  I’d be able to answer that only once I have the structure of such a sequence. Only then can we say  that it is a logical equivalent to (and implies) the Riemann Hypothesis. We haven’t even begun the  actual expedition, we merely have a conjecture and a chance to fail heroically.

 

Me: Great!   Let me now move from the method to the methodology.  

You’re trying to recruit people for this expedition. I don’t understand how recruiting people not specializing in this will help. Do you really think they can contribute? You mention about teaching them the definition of ‘primes’. If that is the case, then what hope is there for the actual solution to the problem?

RG:  My friend Edmund Harriss is a mathematician who has extensive experience with quasicrystals. You can see his opinion on the project here:  http://maxwelldemon.com/tag/riemann-hypothesis/. There are more artists, designers and of course mathematicians who are slowly joining this. I  welcome everyone.

I am asking people to share the journey with each other and learn a lot in the process by playing with it once I have explained it in a layman terms, the first three months is for that. Then there is a live GoogleDoc which the public can view ( but only the members can edit ) This public document will   start from the basics and lead up to our attempt at solving the Riemann Hypothesis.

There are a few outcomes possible 1) The hypothesis is not provable 2) it is provable 3) it is  not possible to know whether it is provable or not and 4) the proof exists but is too long by  human standards. What then? That is an open question for future generations and only collaborative  activity can resolve this.

 

Me:  What do you think the consequences of the solution might be? Except for you getting immensely famous and obscenely rich, that is.

RG: HAHAHA…I hope so my friend, I’m tired of having to charge a fee for my workshops. But that aside, the longest existing proof is currently the “classification of finite simple groups” and  that is tens of thousand of pages in hundreds of journals. I imagine that if we realize that a  proof of the Riemann Hypothesis exists, but no single human being or even several generations can never read the one million, trillion or even more pages needed to verify it, we will have to create an entirely new system  of collaboration in human society.

 

Me: Yes, the Enormous group classification proof is really enormous.  Let’s end with something light-hearted.  

I’m quite curious about your Twitter moniker, your Gmail ID and your blog name. What’s this fascination with the word ‘Fade’?

RG: I once wrote a comic book called the Doppler Effect, where the protagonist is a creature called  the Fadereu, who is only visible when he moves, or else he fades away in the air. It was my  nickname for a long time, but when I got bored I kept changing it with my age. It’s here:

 http://www.flickr.com/photos/59414774@N00/sets/72157623391956017/

 

Me:  That is cool!  

Let me end with a bit of self-appreciation, if you allow. I have to thank you for making the link from our website your GTalk chat status. Any comments?

RG: Well, it helps the project to showcase what people are writing about it. I assure you this is  all entirely in my own interest!

 

Me:  Thanks a million for the interview.  

RG: Thank you so much for your time and interest, and do let me know if you’re joining us.

 

Indian Mathematician Attempts To Solve Riemann Hypothesis Using Quasicrystals – And He Wants You In Too!

Rohit Gupta, an Indian Mathematician, is trying to solve a difficult math problem and he intends to get you involved too! The problem the Riemann Hypothesis – is one of the toughest problems ever and no one knows a solution to it yet. It regards the distribution of prime numbers on the real number line, but more of that in just a bit. More interesting is the person himself and his goal. He intends to use Quasi-crystals (this year’s Nobel recipient in Chemistry, by the way) to solve this so-far intractable problem. The call is out for anyone who wants to join him anyone really, whether a mathematician or not!

He not only plans to attack the problem, he wants to do so in public. To do that he’ll conduct workshops dubbed KNK103, with KNK standing for Kali and Kaleidoscope’. But, he points out, this isn’t just a workshop it’s much more, it’s a mathematical expedition’.

His own description:  http://fadereu.posterous.com/knk103-the-crystals-of-mt-zeta

 

An Interview with Rohit Gupta:  http://techie-buzz.com/science/riemann-quasicrystal-interview-rohit-gupta.html

The Riemann Hypothesis: The Beast and the Legend around it

The Beast

The Riemann Hypothesis is probably the toughest Math problem known. We all know that prime numbers are scattered all over the number line, but we don’t know whether there is any pattern in this or not! The prime number theorem states that all prime numbers upto some integer n’ lie below a maximum line, given by different formulae.

The statement of the Riemann Hypothesis is easy enough to state all non-trivial zeroes of the Riemann Zeta function lie on the critical line whose real part is ½.What this cryptic statement means is difficult to explain. The Riemann zeta function is shown below, valid for only Real(s)>1. (Remember, s can be a complex number.)

The Zeta Function in the Complex Plane. It has a unique analytic continuation from the Real Number line to the entire Complex Plane.

The function has Analytic continuations, which are too technical to consider here. The analytic continuation has trivial zerosfor s=-2,-4 etc. There are also non-trivial zeros (i.e. where the value of the function vanishes) of the Riemann Zeta Functions (and its analytic continuation), apart from these zeros. The Riemann Hypothesis is that these zeros should lie on a line, whose real part is ½. Don’t worry if you’ve not understood much of this section, just keep reading.

Here’s a Mathematica demonstration of the Riemann Hypothesis
Riemann Hypothesis

The Legend

The Riemann Hypothesis is listed at number 8 on Hilbert’s list of 23 greatest unsolved problems, and has on its head a prize of US$ 1 million, announced by the Clay Mathematical Institute. The hypothesis has been tested for 1 trillion zeros, but no general proof exists, i.e. how do you know there isn’t a zero that doesn’t lie on the Re(s)= ½ line beyond the trillionth zero?

The solution to the Riemann Hypothesis is thought to be a key step in many problems in topology, number theory and even cryptography. It is rumored that internet security will be revolutionized if the problem is solved.

Role of Quasicrystals!!

Here comes the most interesting part: Quasicrystals. Rohit Gupta wants to use quasi-crystals to attack the problem. He plans to use the energy levels in these crystals and look into the pattern to see if it fits the pattern of the zeros of the Riemann zeta function on the critical Re(s)= ½ line.

Quasicrystals - one of them

The idea is not new and he admits to being inspired by a conversation between Freeman Dyson and Hugh Montgomery. They noticed that the energy levels of heavy atoms, like Erbium, have energy levels which follow the pattern of the zeros of the Riemann Zeta function. Here’s Marcus du Sautoy’s account:

They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar. It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.

Yes!! Quantum mechanical solutions might already have solved the Riemann Hypothesis, loosely speaking. Gupta mentions If the Riemann Hypothesis is true, then the zeros of the Riemann Zeta function form a quasi crystal.Freeman Dyson said of this approach:

Suppose that we find one of the quasi-crystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.

Who can participate in the grand project? How about you?

Gupta wants everyone to participate in this. He wants everyone from designers and copywriters to paranoid politicians and Polar bears with Polaroid Cameras (His own words)!

The introductory fee is Rs. 4900 (about $100) for lifetime. The workshops will be held online. There is no last date to register as the project is not time bound. He can be found on Twitter as @fadesingh and his gmail id is fadebox at gmail . com. You can register here.

I’ll use Freeman Dyson’s quote taken from Gupta’s page to finish this article: the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible.

Credits: Most of the material is taken from  http://fadereu.posterous.com/knk103-the-crystals-of-mt-zeta
An Interview with Rohit Gupta about his method and methodology:  http://techie-buzz.com/science/riemann-quasicrystal-interview-rohit-gupta.html

 

Breathtaking Mandelbrot Videos – Now in Brilliant 3D

Yesterday, I saw the most amazing videos. These are called Mandelbulb videos, and half the fun of watching them is understanding the science and mathematics behind them. If you don’t want to learn how these are made, you can skip to the end of this article.

What is a Mandelbulb?

It’s a type of mathematical entity that’s based on ideas behind  Fractals  which were discovered in the 17th century. Fractal mathematics describe  geometric patterns that repeat at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractal patterns are often seen in nature, such as in snowflakes. Here’s an example of a Von Koch curve, which is one of the simplest forms of a fractal.

 

von koch curve

When a fractal shape is calculated, it is calculated in steps, or iterations. You can see the shape change, above, as it goes through each iteration.

Here’s another example of a complex fractal which was generated by the  GoogleLabs JuliaMap Generator.

Google Labs Juliamap web app

If you follow that link to Google Labs, your web browser will generate a fractal. One of the unique properties of fractals are that you can zoom in on the patterns and see even smaller but very similar patterns.

Over the years, many mathematicians have tried to determine the formulas for true 3D fractals. Finally, in 2007,  an amateur fractal image maker,  Daniel White, came up with some formulas that seem to generate these 3D fractal shapes. Daniel started his work by using one of the most famous fractal types, called the  Mandelbrot set. The resulting shapes are called Mandelbulbs.

A couple of months later, some software hackers and fractal enthusiasts got together to work on software to render Mandelbulbs on computers. The Mandelbulber Project has been a success, and as you’ll see below, it’s also resulted in some very fantastic images and videos.

Here’s a screenshot of what the software looks like, running on a Linux PC. The software is Open Source, completely free to download  and also runs on Windows PCs.

Mandelbulb Interface

Here are a few screen-shots of Mandelbulber’s artificial worlds.

mandelbulb 45

 

mandelbulb 14a

 

mandelbulb image

 

mandelbulb 1

 

mandelbulb 2

 

mandelbulb 3

 

mandelbulb 4

Finally, we have reached the video. As you’ll see, Mandelbulber can create entire virtual worlds that can be explored. In fact, it can create an infinity of different worlds and some of them are almost scary in their complexity.

Video:  Trip to center of hybrid fractal

Here are more Mandelbulb videos from XLACE. It’s interesting to see how his videos have evolved over the years, and how Mandelbulber has suddenly changed the world of fractals forever. Someday, I expect to see these artificial worlds appear in movies. If you need an alien place for a Sci-Fi movie, it’s pretty easy to see where you could get one.

 

Google Celebrates Fermat’s Birthday With An Awesome Doodle

In keeping up with its tradition, Google has come up with an awesome doodle today to honour the birthday of the great Pierre de Fermat (kindly pronounce as “Ferma” . The ending ‘t’ is silent.) Hailed as a genius in the world of mathematics and physics, while being virtually unknown to the world outside, Fermat’s fame rests on two basic pieces of mathematical wizardry he presented to the world Fermat’s principle and Fermat’s Last Theorem.

The doodle

The doodle looks like a board in the room of a mathematical genius. Maybe, if Fermat had a board on his wall, it would’ve looked something like this. Strangely though, out of the mathematical mess of seemingly random squiggles, emerge the letters G-O-O-G-L-Ein that order, while also maintaining complete mathematical harmony by spelling out the statement of Fermat’s Last Theorem. This is a masterstroke from the Google artist, unnamed as yet. Try a mouse-over and see the comment. Have patience – the explanation of the mouse-over comment is delicious.

The mathematician behind it

The life of Fermat is, however, way more awesome than the doodle. Starting off as a lawyer, he learned arithmetic, largely by himself. After shedding off the tag of being an amateur mathematician by discovering a method to calculate slopes of curved lines (which we regard as differential at a point), without having any knowledge of differential calculus (which came later), he moved onto things far greater. Newton would come half a century later and would develop calculus into a branch of mathematics.

Insight
A copy of Arithmetica containing Fermat's comment. (No, I don't read Latin either!)

Fermat’s great insight led him to discover the Fermat’s principle. This, in the garb of the language of modern optics, said that light always takes the path that lets it take the least time when it propagates from one point to another. Huygens, nearly two century later, would boldly propose the wave theory of light using Fermat’s principle to derive observed phenomenon of reflection and refraction. Now every branch of physics Classical mechanics, Relativity or even Quantum Mechanics uses this principle, in one form or the other.

Lasting legacy

But this was for technicians in the field. Fermat left behind a delicious puzzle for future generations. He conjectured (and never proved) that three positive integers, x, y and z, cannot possibly satisfy the equation xn + yn = zn, for any n>2 (For n=2, you’d recognise it as the Pythagoras theorem). Fermat supplied a proof for it for n=4, for not a general proof. In his copy of Arithmetica, a book written by the Greek Diophantus, he scribbled on the margin something which said that he had a proof but it was too big to fit in the margin.

Mouse over the doodle, and you’ll see that it says that the discovered proof is too big to fit in the doodle.

The general proof of Fermat’s last theorem is a stuff of legends now, with Andrew Wiles’ proof and his struggles to get to it having been made into TV shows, documentaries and books.

Fermat, pot-bellied and round-nosed, left behind a legacy too big to fit into this one article.

Wish You A Very Happy Approximate Pi Day, 22/7

It’s a number an infinitely long number crucial to the lives of mathematicians, physicists and people with bulging bellies. Without it, a circle would be a square, the poetic beauty and mathematical austerity of the twain never meeting would be lost. Yes, it’s Pi that horribly omnipresent irrational number, which is deduced through pure rational logic, beloved to practitioners of math, loathed by those forced to use it and understood by none. It’s long, infinitely long, having one of the shortest names you can imagine, and just to pile the woes on, it’s irrational. The absolute worst news is this last bit: it’s everywhere!!!

Knock, Knock! Who’s there? Pi! Pi who?

Get a few chips (choose circular ones and avoid the Silicon ones) and sit down with pi. Let’s get to know the apparently harmless beast. Pi is the length of the boundary of a circle, which has unit diameter. Meaning? Draw a circle with a diameter of 1 cm (use a compass please!) and then you can sleep tight with the knowledge that the boundary will be pi cm long. Why? Because a circle is a circle is a circle. Not satisfied? Don’t worry, no one knows.

No, 22nd July is not Pi Day. It’s approximate Pi day. Written in day/month format, 22/7 represents a rational approximation to pi. It’s right to the first three digits! A more popular cousin is 14th March. Written in month.day format, 3.14 represents pi. Interestingly, 14th March is also Einstein’s birthday.

Yet, many people aren’t happy! They say that it is not pi, but twice pi that is more fundamental! Let’s called that tau’. Why? The peripheral length can simply be tau x radius, or the area of a circle, simply ½ tau x (diameter)2 ­ . So is there a Tau Day? You bet! It’s 28th June. Write it in month.date format 6.28. This is exactly twice pi correct to the first three digits. While there is no approximate tau day, and though that should’ve been a disappointment, tau-ists proudly advertise that their number needs no approximation. It’s not for greenhorns! (Guys, it’s just twice pi!)

Up close and personal with Pi

Pi is a part of the most beautiful equation ever written.

The most beautiful equation in math. (Photographed without make-up)

Look at it with glazed eyes! People familiar with complex numbers (overhyped name!) would point out that the first term (eiÏ€) equals -1. (Take my word for it, or look up a on book on Complex numbers The de Moivre’s Theorem). So in only one equation, you have two most used fundamental constants in Math (e and Ï€) along with the imaginary unit (i), combining to give a really nice real number (-1), which when added to unity (1) gives the omnipresent zero (0). Phew! Overcome with emotions? Please don’t shed tears – it happens to everyone!

However, the greatest utility of pi for the human race is the test of memory! Since pi doesn’t end or even repeat, it has got all braniacs interested and there are international competitions testing one guy’s knowledge of pi with another guy’s. If you’re interested, remember the following rule of thumb: If you remember pi as 22/7, you’re school child. If you know the first three digits(3.14), you’re normal. If you know three more digits after that (3.14159), you’re a physicist. Knowing up to twenty digits (3.1415926535897932385) might brand you as a mathematician or a math freak. If you know a million more (oh, forget it!), you’re a nerd.

Techie-buzz wishes you the very best on your quest to memorize a million digits of pi, Nerd…