One does not 'find' the truth. What one finds is the best spot to stand such that the truth hits one straight in the face. (J.N)

Blaakyum, Journey to Eternity (Jad's special mix)

During the production of 'Lord of Night', Bassem sent me some raw material from 'Journey to Eternity' and I had a different idea about how the song should be structured, so I mixed it up from the pieces I had, of course, my version never got anywhere, but here it is along with a few pictures from Blaakyum's tour.

Just add axiom.

"These ... considerations, however plausible, cannot be considered a rigorous proof of .... This proof also cannot be derived from the ... axioms stated thus far. On the other hand, the ... is of very great importance for the entire mathematical .... Therefore, it has to be introduced as a special axiom, which, for reasons to be explained later, is called the ,,, axiom."

From a mathematics book with some text replaced by ...

Hosting expired.

My domain name and hosting (jadnohra.net) which hosted all the downloads that can be found on this blog has expired, I am securing the files and will find a way to host them somewhere else (cheaper).

Update: The most popular files are now at:
https://docs.google.com/folder/d/0B0MEMrHTbcrzZHBwSW9acEhOUUU/edit

"Axioms are the expression of our strongest and most basic intuitions (sometimes derived from experience), we ask of them to be the only and supreme judge of all of our less strong, more complex intuitions, during a process called deduction (judged by some other axioms), with the goal of doing something useful, or just for fun."

"Those eight minutes point the road to a complete reformation of astronomy"

"If progress had been continuous and organic, all that we know, for instance, about the theory of numbers, or analytical geometry, should have been discovered within a few gen- erations after Euclid. For this development did not depend on technological advances or the taming of nature: the whole corpus of mathematics is potentially there in the ten bil- lion neurons of the computing machine inside the human skull. Yet the brain is supposed to have remained anatomically stable for something like a hundred thousand years. The jerky and basically irrational progress of knowledge is probably related to the fact that evolution had endowed homo sapiens with an organ which he was unable to put to proper use. Neurologists have estimated that even at the present stage we are only using two or three per cent of the potentialities of its built-in "circuits". The history of discovery is, from this point of view, one of random penetrations into the uncharted Arabias in the convolu- tions of the human brain."

"Thus within the foreseeable future, man will either destroy himself or take off for the stars. It is doubtful whether reasoned argument will play any significant part in the ultimate de- cision, but if it does, a clearer insight into the evolution of ideas which led to the present predicament may be of some value. The muddle of inspiration and delusion, of visionary insight and dogmatic blindness, of millennial obsessions and disciplined double-think, which this narrative has tried to retrace, may serve as a cautionary tale against the hubris of science – or rather of the philosophical outlook based on it. The dials on our laboratory panels are turning into another version of the shadows in the cave. Our hypnotic enslave- ment to the numerical aspects of reality has dulled our perception of non-quantitative moral values; the resultant end-justifies-the-means ethics may be a major factor in our un- doing. Conversely, the example of Plato's obsession with perfect spheres, of Aristotle's ar- row propelled by the surrounding air, the forty-eight epicycles of Canon Koppernigk and his moral cowardice, Tycho's mania of grandeur, Kepler's sun-spokes, Galileo's confidence tricks, and Descartes' pituitary soul, may have some sobering effect on the worshippers of the new Baal, lording it over the moral vacuum with his electronic brain."

Justification for the post multiplication of Euler angle matrices.

The basic matrix representations of rotations about the three fixed axes of an orthonormal coordinate system are easy to derive.
For right-handed rotation of an angle a, and column vectors, we get the following:

Around x,    Around y,    Around z
| 1 0 0  |   | c 0 s  |   | c -s 0 |
| 0 c -s |   | 0 1 0  |   | s c 0  |
| 0 s c  |   | -s 0 c |   | 0 0 1  |

Where c = cosine(a), s = sin(a).

Note: An easy way to derive these matrices without having to go back to pictures is use 'c' for the direction cosine of a component along it's direction, and 's' or '-s' along the remaining direction depending on the order of the components in the string x,y,z,x,y,z.
So per example, x on y gives 's' (since we have x,y in the string) while y on x gives '-s' (since we have z in between: y,z,x).

Using column vectors, matrix multiplication (as defined using linear map composition) is from right to left, so to rotate around fixed axis z (matrix Rz) and then around fixed axis x (matrix Rx), we write the multiplication to get the total rotation matrix R this way: R = Rx * Rz.

Euler angles do not operate on fixed axes, but on moving ones, as shown in the animation.  So we cannot use the fixed axes matrices, at least not directly.
The solution is to reverse the order. This means to perform an Euler angle rotation around z and then (moving axis) x we use: R = Rz * Rx.
But why does this work out? Here is a proof using basic linear algebra.

Proof:
To perform  an Euler angle rotation around z and then (moving axis) x' the matrix composition goes this way: R = Rx' * Rz, where x' is the moving axis, obtained by rotating x by Rz.

We call the basis (or frame) obtained by rotating the standard basis E3 by Rz the basis F'. Using a change of basis, we notice that in the basis F', the rotation around x' is simply a rotation around the 'fixed' axis x (of that basis).

Formally, with rx' being the linear map represented by Rx' in the standard basis, we have this change of basis:

Rep{E3,E3}(rx') = Rep{F',E3}(id) * Rep{F',F'}(rx') * Rep{E3,F'}(id)

Notice that the representation of the basis F' with respect to E3, is Actually Rz, since the column vectors of Rz are the representations in R3 of the rotated E3 vectors, so Rep{F',E3}(id) = Rz and, since rotation matrices are non-singular (invertible) Rep{E3,F'}(id) = (Rz)^-1 .
Additionally, as we explained,  Rep{F',F'}(rx) is simply equal to Rx.

In terms of matrices, this change of basis is: Rx' = Rz * Rx * (Rz)^-1.
Replacing Rx' in the expression of the total rotation, we get:

R
= Rx' * Rz
= ( Rz * Rx * (Rz)^-1 ) * Rz
= Rz * Rx * ( (Rz)^-1 * Rz )
= Rz * Rx * ( I )
= Rz * Rx.

Q.E.D

So, we can always use composition in reverse order (post multiplication) of fixed axis rotation matrices to obtain the total rotation matrix for a sequence of Euler angle rotations.

'An axiom: A trick used to catalyze the illusion of formalization of yet another trick: a proof.' (J.N)

ABAB versus BABA.

Maya is really perfecting mumbling things like MMBA, ABA, BABABA, it is very sweet.
Which brings me to this question: Let A and B be two square matrices of same size, if ABAB=0, does this imply that BABA=0?

Time to submerge.

52 'weeks' of math.

We have reached week 52  in our math project: https://sites.google.com/site/77neuronsprojectperelman/weeks/week51-matrix-inverses

To celebrate, I wish to thank Lena one billion times for tolerating this endeavor, and I invite anybody reading to prove this:

Prove that for any 2x2 matrix T, you can always find numbers c0,c1,c2,c3,c4 at least one of which not equal to zero, such that:c4.T^4+c3.T^3+c2.T^2+c1.T+c0.I=Z.

I is the identity matrix, and Z is the zero matrix.
T^n is T multiplied by itself n times.

"Logic will reliably indicate what you can't do, for the rest ..." (Jad Nohra)

A cross-product (determinant) saved my life, or did it?

Your friend has been kidnapped by evil perfectionists, they put him in the middle of a square room, with two opposite walls painted one black one blue, and the other two painted white, there is a door at each white wall, one leads to instant death by vaporization and the other to freedom. In the middle of the ceiling there is a tiny circular hole from which your friend can receive your voice and send his and nothing else.
He is facing a black wall, and if he does not escape the room within one minute he will be vaporized. Everything that can be sensed by a human is perfectly symmetrical in that room. Your vulgar friend does not know his left from his right, but you know that the door to his right leads to freedom.
What is a better use of your time? Trying to lead your friend to the door, or going for a cup of coffee?

We are in the future, 10 years later. We have established contact with a large amount of different intelligent alien species, and you even befriended an alien. To celebrate the vaporization of your last friend, evil hopeless perfectionists kidnap your alien friend, and put him in the same situation. Everything that can be sensed by this alien is perfectly symmetrical in that room.
At this point I need to tell you a bit about him: his species is Platonicus Solidus, he is a perfect sphere. He has a circular bottom on which he can bounce or rotate, and a diametrically opposed top from which he can hear, and under which his perfectly spherical brain resides. His sight is handled by 4 perfectly spherical and identical eyes, arranged symmetrically along his equator, he can never tell them apart.
However, your friend can actually feel the difference between clockwise and counter-clockwise rotation! So the kidnappers think you have better chances.
You know that the right door is to your right if you were in there facing the black wall.
Is it time for another coffee?

Releasing some old code and music.

Here is the source code for the old XNA Rendering Framework / Quake3 Renderer:
http://jadnohra.net/release/XnaFramework1-Quake3Renderer.zip

It might be useful for somebody trying to write a Quake3 level loader.

And I also stumbled upon this old strange piece of music that I made for Lena at the time where I still had time to touch the guitar:
http://jadnohra.net/release/DeadGuitars/JadNohra_ConcertoForElectricGuitar_DayOfBirth_MOV04530.MPG

Not good

0x15230600 3f800000 00000000 00000000 00000000 ..€?............
0x15230610 00000000 3f800000 00000000 00000000 ......€?........
0x15230620 00000000 00000000 3f800000 00000000 ..........€?....
0x15230630 c0499999 3fc3d70a 00000000 00000000

0x41140600 3f800000 00000000 00000000 00000000 ?€..............
0x41140610 00000000 3f800000 00000000 00000000 ....?€..........
0x41140620 00000000 00000000 3f800000 00000000 ........?€......
0x41140630 c0499998 3fc3d70a 00000000 00000000

Rotation is so much more interesting than translation, (Level up)

Randomness works as expected.

http://www.rpscontest.com/authorSearch?name=Jad+Nohra
If only I could find the right prime for each opponent hahahaha, or the right place to start in sqrt(2).

Of course genius Tom is smarter:
http://www.rpscontest.com/entry/110017

Forcing yourself to do multiple versions of the same proof is like body building.

Prediction about my future child's diary.

"Dear diary.

Monday, dad said we would go buy new tennis shoes if I find the one number that multiplied by itself, gives 9, by trial and error using multiplication, I found it very fast and I found nice shoes as well.

Tuesday, he said we would go buy a new chess board if I would do the same for 102.2121, it was more difficult but the result was the month and day of my birth, I liked that.

Today he must be going crazy, he said I can celebrate my birthday every week! if I do the same for 2, it took me all day and I am at 1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976 but I must be really really close because 76 is the year my mom was born"

Book of Proof.

Tom had the brilliant idea of not biting off too much at once with real analysis and studying pure proof techniques. He also found a very good book for it, it is even free. http://www.people.vcu.edu/~rhammack/BookOfProof/index.html.

We like it because it is geared at exactly our goal and requires little to no prerequisites.

We have been working on it in the past months quite slowly because of some personal things that needed attention, but now I am back and we are at Chapter 5.

It has not been a very difficult book for now, but it has been perfect at showing us the edge of how we currently think and how to extend it, formalized proof 'techniques' (which does not replace the need for or ), along with confidence in formal proof writing.

The book is not to be confused with the famous 'Proofs from the Book' ! we definitely do not have what it takes to tackle that one yet, but if at some point we do, we will see it as success of our project.

More good news: We have somebody (a colleague Technical Artist) who decided to join us and has been catching up starting at week 1. Unfortunately, we have not been commenting on his insightful posts, but that is the price of starting late :). We do have verbal discussions at work though...

I conclude with a Frankenstein of a proof I made, which has a much more elegant alternative found by Tom. It is a nice example of two different ways of proving the same thing, how the use of properties is often crucial to write elegant proofs, how I am using my 'debugging the truth' technique, how to be patient, how not assume that if you cannot see the path right away you will not be able to find it at all, and how you can learn from failed proofs instead of throwing the towel.

http://jadnohra.net/release/math/book_of_proof/ex_4_23.pdf

https://sites.google.com/site/77neuronsprojectperelman/weeks/week-23

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