Mar 14 2016

For the Pi Day

Category: Scienceadmin @ 10:10 AM

This post may be a bit technical for general audience (as if anybody is reading this!). Although, if you do and happen to have any popular interest in Pi, the mathematical constant π=3.141592…, since it’s the day of Pi, consider to scan it through and get to the end point, if I manage to make it!

From 20 to 18 years ago I attempted to build an algebraic axiomatic system to reformulate geometry. The goal was to generalize the theorems of Euclidean geometry to a version independent from the number of its dimensions.

I didn’t know how big the world is and that my work must be redundant. So I put the effort and called it “geometry beyond dimensions”. Soon after renamed to “multidimensional Euclidean geometry” (word by word translation from Persian).

My ambitions were beyond speculating about the “flatlanders” and generalizing their problem: Oh, poor flatlanders don’t know about us three dimensional beings, so we too must learn about four dimensions and higher.

No, the point was that there is a lot more to linking geometry and algebra. Still an unacomplished mission.

Anyhow, learning two and three dimensional geometry was mandatory at school and I extended it from N=2, 3 to any number. It was a mechanical and labor intensive work using the principles of induction and a minimal set of “bridging” axioms on top of the existing literature, our school books.

Not only the concept was beyond my intuitive perception, the formulation could also get weird quickly, but it was possible after all to get familiar and use tricks to grasp the concepts and proceed.

To see how it looked like before it escalates, here is an example axiom (a bulding block for more complicated structures and proofs that came later in the book):

There exists exactly one N-dimensional space passing through any N+1 points not lying on the same straight N-1-dimensional space.

A bit weird, huh? But you could put N=1 to get the following axiom in planar geometry, more intuitive:

There exists exactly one line passing through any two distinct points.

Or N=2:

There exists exactly one plane passing through any three points not lying on the same line.

It took some 80 theorems till it covered a satisfactory area and I wrapped it up. And although I was quite obsesssed with its mechanical accuracy, I remember it still had few holes and gaps.

Now let’s get closer to the Pi:

One of the wheels I reinvented in that work was calculating the volume of n-ball, or a multidimensional hypersphere. Of course I didn’t just write an integral to solve it; I proved dozens of theorems to justify that my integral is legit and comes only from the few axioms that were introduced at the start of the book, and assumes no more.

The final result was mysterious in terms of its connection with the Gamma function and Pi. And this is where it can take us beyond a dimension-agnostic theory of geometry: discovering the nature of Pi!

Now, I refer to the pages 45 to 56 in my book (Sorry it’s all in Persian!) But I will make a simpler point here. Let’s try to formulate:

0. Consider the volume of a 0-sphere: How many dots are in a dot? 1 (or 1.R0)

1. And the volume of a 1-sphere with radius R: What is the length of a line segment with radius R: 2 R1

2. The volume of a 2-sphere: What is the surface of a circle with radius R: R2

3. How about the volume of a 3-sphere? 4/3π R3

4. And it turns out that the volume of a 4 dimensional sphere (all the points on a 4D space that are as far from one point in a 4D space) is: π2/2 R4

N. In general the volume of N-ball, an N-dimensional hypersphere with radius R turns out to be: πN/2 / Γ(N/2+1) RN

You can find the full proof in the book in Persian (pages 45 to 56), and perhaps somewhere on the net in English. Now, ignoring the trivial part of the formula (RN) we end up with a magical co-efficient as a function of N:

πN/2 / Γ(N/2+1)

Where Γ  is the Gamma function. Now the value of this function for its integer arguments is straight ahead. It ends up equal to the famous factorial function, multiplication of all integers from 1 to that number [minus one]:

Γ(n) = (n-1)! = 1*2*3*…*(n-1)
Γ(n+1) = n* Γ(n) (n) = 1*2*3*…*n

For non-integers though it will take on funny values to interpolate the factorial results between two integers. For example for the half values right in the middle of two integers, it ends up a rational number (a number that can be written in a form of an integer devided by another one) multiplied by an irrational number which is Γ(½) and happens to be the square root of π, that is not only irrational but transendental:

Γ(n+½) = (n+½)*(n-½)*…*Γ(½)

Now the strange part is that the argument of the Gamma function in our formula is N/2+1. It gets one unit higher for every second added dimension! And that for odd dimensions it will not be an integer or a rational and will include the term Γ(½)=√π.

On the other hand the gamma function in our formula is multiplied by another term of πN/2 which also introduces a √π for every added dimension. Thus, for even number of dimensions none of the terms πN/2 and Γ(N/2+1) introduce a √π and we end up with a rational number multiplied by πN/2 where N/2 is an integer. For odd numbers both of these terms introduce a √π that divides and vanishes. So, there will not be a √π in any of the integer dimensions, even or odd.

It is not a √π introduced to the formula for every added dimension, instead is it an extra π coming to multiply, for every even number of dimensions. Odd dimensions (extending from a point to a line, or from a circle to sphere) do not introduce a new π to the co-efficient, only a rational number. The even numbers (going from a line to a circle) bring in a π to the play! A strange asymmetry between the odd and even dimensions, I would say.

Ignoring the rational part of our magical co-efficient, for every second added dimension there will be just one π introduced and the co-efficient for dimensions from 0, 1, 2, 3, … will be as the following:

0 -> 1
1 -> 2
2 -> 2π
3 -> 4/3.π
4 -> 1/2.π2
5 -> 8/15.π2
6 -> 1/6.π3
7 -> 16/105.π3
8 -> 1/24.π4
9 -> 32/945.π4

Where does π come from? One intuitive way is that it comes from the comparison of the space a hypersphere takes to that of a hypercube. But one π for every second dimension. Why every second? Well, this happens in Euclidean geometry where distances are Euclidean and the ball is defined as a set of points equally far from a center, using a “two” norm distance metric. You take another distance measure and the math will change. But I would argue that Euclidean distance is the only legit metric at least when it comes to defining a ball, as it is the only metric that maintains the shape of the ball when we rotate the axes. So the key is that when you go beyond one dimension something called “shortcut” comes to existance. And there’s a straight shortcut that for some reason follows the Pithagorean theorem and that defines the perfect curvature. I couldn’t reveal how these are connected, but if I ever want to speculate about the nature of π, here would be my starting point.

p.s. I read a bit more on the topic. I opened that back door in my head and it was two decades of silence and spiders ran off quickly. My friend Sajad gave me a torch, albeit a map: Quite surprisingly the Pi day coincided this news on some weird statistical behavior of the Prime numbers. I realized that I was brought up in a typical middle class (and 3-dimensional!) family. Dimension-deprievation is the evolutionary intution of 0, 1, 2, 3 only. That is too few to realize that all dimensions do not have to be symmetric because they are all numbers. The number of dimensions, even or add, prime or divisible, affects how N-space behaves and just like number theory it doesn’t have to inherit it all from N-1-space. Do all numbers exhibit the same properties cause they are all numbers? so why should they when they count dimensions. I think this is actually what numbers are made for: counting dimensions. And the historic fact that we count 1, 2, 3 and we “…” the rest is not pure coincidence. Sounds poetic, but read it logically:

3 doesn’t get every property of 2, neither does a ball from a circle. To my previous wonder, a ball (3-sphere) did not inherit an additional π from a circle in the calculation of the volume, but 4-sphere did. Is it weird? No, 3-sphere introduces singularity too, two poles in the hairly ball theorem, that are the two ends of a segment (1-sphere), but 4-sphere doesn’t: A circle (2-Sphere) can go round on another full circle around a point and you get a 3-torus or a 4-Sphere that you can comb (no singularity) and they both happens to have π2 in the volume and surface formula. Now you try to rotate the circle, not like you just did on both dimensions of a full circle and around one point, but instead around a segment on its own disk space. And you get a ball (3-sphere) with two inevitable South and North poles (singularity) and this time it does not give you that extra π. So, 3-sphere is just a product of a circle and a line segment (thus singularity, thus no extra π). The product of two circles (3-torus or 4-sphere) gets that extra π and you can also comb it (no singularity)!
This is a short summary of the common stories that two formal proves tell. The same thing happens in both: The multiplication of a new π in the volume of n-sphere on every second dimension in my [redundant] proof (A), and the generalized hairy ball theorem for 2n-spaces (B).

Is there an established field on the intersection of algebraic topology and number theory?


Feb 24 2016

The Big Data Gold Rush

Category: Magic, Philosophy, Techadmin @ 12:21 PM

From optimizing advertisement industry to manipulating financial markets or predicting electoral results Big Data has mobilized much of its army in a highly competitive battle of exhausting opportunities. This on its good side has pushed the limits of creativity and has sparked innovations in technology and science, but the competition has also justified new evil means from exploiting the last resources and violating your privacy to forming bad habits and personal recommendation of things that you don’t need. All for the local short-term benefit of the soldiers in the forefront of this battle and their fewer commanders, and at someone else’s bigger loss.

unsustainable! Is there no tomorrow?

It is the gold rush for Big Data but much of the army at its best is redistributing the gold that others have created, looking away from the magical opportunities to create gold out of the thin air!

Let’s be on the right (and smart) side of the history and create more initiatives, for example, like this.


Jan 10 2016

Wear you later!

Category: Philosophy, Scienceadmin @ 10:24 PM

I was fantasizing and day-dreaming about exotic forms of life. This topic is very much not within my expertise, but it is fun to let your thoughts play with the idea of life somewhere else.

No, I am not going to talk about whether we are alone! There is a consensus that we are probably not. But I wanna ask who are the others. How do they look like? What do they do?

And this was not really a dream. It was rather a guided semi-concious train of thoughts with closed eyes on the way to a powernap. So it may sound trivial, or wrong, or stupid. Nevertheless I explored some fantasies and I share them with you.

Rethinking loud…

Ok, In our terrestial life on Earth we *consume* each other in different forms for our survival. We eat, we mate, we socialize…

Eating:
living organisms enjoy other creatures as nutrition to obtain energy and mass they need. So we all somehow eat for movement and for growth. Eating may have universal rules. I think creatures eat things that are not so much like them. But that can be a coincidence on our Earth and canibalism could be more widespread gallactically. And creatures don’t eat things that are so different from them, afterall they need to process the matter and rebuild they bodies, or burn it to be able to move. So some universal laws agreed on the issue of food.

Mating:
living organisms sometimes need to meet each other and do something funny in order to reproduce. Let’s be polite and without the use of the F-word remember how our fellows across the animal kingdom rape or hump or bang each other with or without consent in order to pass on their genes. And well as opposed to eating, mating (if done with another being) is probably done with something that is more alike to us, and not that different, right? Cause then a legitimate question would arise: what kind of baby would come out of that interspecious act of sex!

Mingling:
well it doesn’t have to be socializing in a bar or coexistance of ants and termites, but we sometimes need to meet each other and collaborate on overcoming the problem of survival in other creative ways.

Sure we may do other things with each other directly and indirectly and these acts have been evolved, thus formed slowly over generations and generations.

Now there are other fundamentally different actions of survival that we could do to each other cause they seem very logical to me but we don’t! Or I couldn’t find immediate examples since I don’t know biology.

And I’d like to believe these exotic acts of life are actually happening somewhere out there on another planet on other stars, albeit other galaxies, right now as you read this.

What else could we do to each other? Three guesses!

WEARING:
So among other ways of consuming another live being, one animal could possibly wear another animal to protect against hazards, such as some poisonous matter, a colony of contagious and alive microorganism or some deadly radiation. I am aware that in our vicinity crabs move into new shells but this is not quite the same thing as shells are dead. And we wearing fur doesn’t count either. I am talking about life forms that are alive both as non-wearable and wearable. Or at least in the latter form.

CLEANING WITH:
Next time you take a shower imagine that water was a life form. And that your interaction was not that boring and static, like now that you two (you and water) are linked simply by gravity. Or by drowning. Let’s say the drops or the shower head could escape from you, or you had to trick and manipulate it somehow to wash and clean your body. If this example is not clear, try wiping your ass with a soft and fluffy rabbit next time.

SENSING THROUGH:
My inspiration here is a cool gif animation of an E.T. that put on a pair of eyes from his plate into his hands and started to see the world (I can’t find it now). And this is where there is no limit to imagination. And gamble with a risky bet that: Whatever you imagine exists somewhere out there!

So, imagine an animal that wears another compatible animal temporarily or lifetime, to sense the environment better. What if some animal takes onother poor creature like a pair of glasses to see or hear or touch better? Or to recieve electrical signals more effectively? This must be more painful than joyful if it doesn’t somehow endanger the survival of the pray. Then pray will not decide and volunteer to be worn and well it will suffer. Hunted against its will, just like food or even worse if it us an unpleasant lifetime imprisonment!

Or let’s hope that karma is not that bitch and in most of such colonies of life, creatures enjoy being “sensed through”.

Thanks for following till this point. Now wake up and get back to life. To this very form you are used to.

Wear you later!


Jan 06 2016

Smart house

Category: Personal, Techadmin @ 10:56 AM

I just woke up from a dream that we were in a smart house and lots of weird things were connected to the Internet of Things (IoT). Several interesting products in that house that all doesn’t make sense to me now. I mention two:

There were smart plates and people were playing an eating game based on the dart game 301. So we started eating random portions of a soup and the goal was to finish it up just with the right portion on the last spoon. It was not allowed to tilt the plate at the end of the game.

And this one: Upon exiting the house there were animated carpets!


Dec 19 2015

Facebook will die

Category: Culture, Techadmin @ 5:57 PM

I have used Facebook for more than 8 years now, constantly and regularly. I have used it for laid-back surfing, as a gaming platform, a political forum and for business. I have also spent a lot of money on it to promote the cafe page and particularly its events. Facebook charged me more and more at the same time that was charging my competitors more and more to reach out to the same audience. It made it really obvious lately to the extent that people who were not coming from Tech and had no familiarity with such tricks also felt something is very shady and got pissed off. This is when you wish the same opportunity that was taken advantage of a decade ago in sillicon valley, was siezed by other people so may the dominant social network of our time have been in the hands of a humbler and less greedy company.

Facebook is investing a lot in other things and so it will be very dissapointing if the future of innovation in this company is not super bright. Still, given the current state of the social network service it does not seem that they can last this way for so long. Facebook as a company may last as long as Coca Cola. And  the concept of digital social networking will also continue with the future human just like alphabet stayed with us. So say both will survive but not necessarily in link with each other. Why social networking does not have to stay locked in the hands of the Facebook company?

For many reasons.

Facebook as we know it (the service) although has grown, improved algorithmically and then monetized agressively, hasn’t provided a different user experience from day one. Now it has hit the limit and is losing the new generations, alarming that it will lose even more of the future generations. All Facebook has is the momentom, piles of cash, huge user-base and invaluable data. This is enough to stay big for a long time, but data can get old like food and a user-base can die out. And most importantly big data insights will be more widespread and be accessible to many more companies, bug and small and that will not be a competitive advantage in a long run.

What if a small service grows so fast and feels the gap, like what Snapchat only much better.

What Facebook (the service) is doing can be done much better. But they don’t innovate unless it’s about making short-term cash. Very important, life changing and profitable predictions must be possible now with their valuable data and they are only showing ordinary improvements. Nothing really amazing comes out of that company, except for more advanced methods to trick and rob the addicted page-owners so they pay more cash and get less, to make them want to pay even more. This will not sustain and they will quit once a better channel appears.

I doubt if the company that owns the next generation of social networks is Facebook. The people in that future company right now lack huge capital, their active userbase, and their valuable data. But there are many other shortcuts as well as enough smart people out there to bridge that gap to grow and eat Facebook. Such process doesn’t have to show its symptoms to the naked eye before it’s too late. I can’t wait to see how and when it is going to happen but it will soon.

Of course any innovation that takes place in social networking or related technologies if not already in Facebook, is gonna be aquired immediately by them. The current way we are using Facebook will not last long, either they keep the continuum intact and manage to ride the next wave of social need and technological advancement or will eventually leave it to the next creative player.

Facebook is huge, rich and powerful. Too big to fail? The exact opposite. Big things can also change rapidly. Such as the inevitable downfall of the corporate Empire. It is in the proceess already and can’t be stopped, although how will it fall through what candidate and which pathway is not clear.


Dec 19 2015

Darvin IV

Category: Philosophy, Scienceadmin @ 1:53 PM

There are billions of galaxies out there, billions of stars in each of them. There are trillions or quadrillions of planets in our universe and some of them harvest life. What happens in the bottom of our own oceans surprises us, let alone far planets around other stars in other galaxies (and assume that’s the only recipe for life).

Other life forms are extremely far and unreal, as if they don’t exist. But they most likely do, and so many of them indeed. But how do they look like? I think although our universe is ruling them all similarly, the potential is so huge that anything we can imagine proably exists somewhere. And anything that our imaginary creatures can imagine, could as well.

What other life forms may look like has not really captured our imaginations. Alien Planet – Darwin IV is the best (realistic, still very earthly) animation I have come across. There are many documantaries out there but no fictional motion pictures that I know of. If you know of some pleases hint me. If you haven’t watched this, give it a try. Don’t think fiction but more science. Think reproduction, growth, survival, energy, memory, intelligence. Think life! It’s fun.


Dec 12 2015

Meeting

Category: Humoradmin @ 11:00 AM

I have an important meeting at 9AM tomorrow. But I don’t have anyone to meet. Anyone knows anyone important that I can meet around 9AM tomorrow?


Oct 29 2015

The long tail of terrestial life

Category: Philosophy, Scienceadmin @ 11:48 PM

If you are an organic molecule, a molecule of terrestial life, is it more likely for you to be a part of a big animal, or a microorganism?

Let’s say you break (or not), you will travel from body to body, from a plankton to a fish, then bacteria, a tree, to a pig, or to a human. You spend there short or long. But where will you spend most of your lifetime? A big or a small host?

I would say both.

Could there be a simple answer to this, that applies to every other livable planet, at any stage of their evolution?

On ours, among uniqe species krills consist most of the biomass, human are second, arguably more than pigs and cows (still farmed by us). Though if you count thousands of species of ants as one, they win over all.

Still, seems all sizes are involved at this stage of life.

Geometrically I would say big should win at the end of the game. (Feel a jar with big marbles, then smaller, then sands, etc.)

Economy of scale aside.


Oct 18 2015

Non-existentialism

Category: Philosophyadmin @ 9:19 PM

I had a nightmare that I didn’t exist. But I was there hearing an explanation justifyng why I don’t exist. It was not that I had died, but I was never even born:

“The exact combimation of the matter like you – statistically – has ended so many times before, but has never started once yet. That’s why you never existed.”


Jun 22 2015

Elevator Mel

Category: Philosophyadmin @ 10:17 PM

So there was this smart elevator in 2025 that was designed to machine learn the user behavior with some degree of freedom. Rumors has it that the code started to compose Clayderman style music, as it was easier to learn than the people’s floor preferences.


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