## Sunday, December 23, 2012

### Your basic 'naive set theory' exercises.

Lately, I have been seeing much of Mathematics as Philosophy, and this new attitude, although hard to describe, has been a major factor in yet another totally new level of understanding what I have studied in the past, am studying, and will be studying.

If you wish to see the philosophical roots of each and every set theory exercise you have ever been given, to the point that, while reading the philosophical paper, which contains not one formula, you keep going 'oh, this was in fact turned into an exercise' and 'this one' and 'that one' all while realizing the very painful thinness to which the whole idea was reduced before it was given to you, if you wish to read a whole chapter explaining the philosophy behind similarity, yes, the same one that is thinned down and taught in you linear algebra book, I suggest the not at all easy, but very rewarding read 'Introduction to Mathematical Philosophy by Bertrand Russell'.

I have read most of it, and now finally I see where the ideas of transfinite of Cantor that are the basis of standard analysis come from, and I see where I disagree with them, (with the help of Wittgenstein), and I can finally zoom out and see both perspectives, (me being on the non-standard, and finitist side), and I can now 'go along with' standard analysis some more steps, maybe even very large strides (which despite my previous many reading about the 'foundational problem' I was not yet ripe enough to do), knowing what I am doing, what is being fed to me, how to translate into the finitist perspective, and in general where the whole differences are leading. Finally, after days of sleepless reading, and dozens months of doubt, a bit of peace of mind.

## Wednesday, December 19, 2012

### "Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook." (http://mathoverflow.net/questions/102237/what-is-the-status-of-irrational-numbers-within-finitism-ultrafinitism by Steve L. Cowan)

"Even while considering the universe to be finite, one can do mathematics symbolically as a game with a system of rules. If the game doesn't have enough pieces we just add new pieces, with new properties or allowed moves as required. All that matters is that the enlarged system is compatible with the old system; that the smaller game is a subgame of the large one; that the smaller system can be embedded in the larger one.

When does something exist ? Well if there isn't something from amongst the objects under consideration that has the properties we want then we just create new symbols and define how they relate to the old ones.

If we were a pythagorean and the only numbers that exist are rational numbers, then we wouldn't call 2√ a number, but if we were also finitist symbolicists then we could embed any collection of numbers into a collection that contains not only numbers but also "splodges" which is what we're going to call 2√. It's important to always keep in mind that 2√ isn't a number - it's a splodge. In this new system of arithmetic we've invented we can add numbers to splodges to get new splodges like 1+2√. What a fun game. Let's add some more splodges. We're bored with algebraic splodges so let's add some non-algebraic splodges like the one in your question. Of course that expression is a bit cumbersome so we'll give it the shorthand symbol π instead.

Given a splodge x, it would make calculus easier if there were a splodge x+o that was nearer to x than any other splodge, however that isn't possible so we embed the splodges in a larger system called the hypersplodges that contains not only splodges but also vapors, and contains not only the concept nearer but also the concept "nearer". Vapors like x+o are "nearer" to x than any splodge could ever be, and when you're finished using them they evaporate leaving just a splodge.

We want a splodge that satisfies x2+1=0 however there isn't one, so we embed the splodges in a larger system called weirdums in which we've added a piece called i with the rule that i2=−1, and under the new system we can "add" splodges to weirdums to get new weirdums like 1+i.

In solving differential equations we'd like a function which is zero everywhere except at a single point but which has a non-zero area under the curve. There is no function that behaves like this so we'll go to a larger system that contains not only functions but also spikes which do have the desired property because the larger system contains a rule about spikes which says they do. Conveniently certain calculations involving spikes cancel out leaving just functions.

A finitist or ultrafinitist shouldn't recognise the concept of infinite sets therefore the only sets are finite-sets and since all sets are finite the adjective finite is superfluous therefore from this point onwards we just use the term "set". Some people want to consider sets that contain things they haven't put in there themselves - which of course can't be done because a set only contains the items we've put there. So we embed the system of sets in a larger system that includes not only sets but also dafties. In this daft system the rules are that a daftie can have an "affinity" for things whether those things have been previously mentioned or not. Dafties have an affinity for things in the same way that sets contain things. A compatible embedding of a system of sets in the daft system means that a set has an affinity for the items it contains when the set is considered as part of the daft system, therefore by daft reasoning one can say things not only about dafties but also about sets. To each set one can attach a number. You can't do this with dafties so we embed the numbers in a larger system containing sinners and attach a sinner to each daftie. Sometimes there is a need for something that looks like a daftie but has no sinner - such things are called messes. A mess can have an affinity for collections of dafties that no daftie could have an affinity for. This could go on, but you need gobbledegook theory. The set system has zero gobbledegook. The daft system is level-1 gobbledegook. The messy system is level-2 gobbledegook. Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook.

There are two ways to compatibally extend a system:
1) a conservative extension adds new items but doesn't say anything new about the old items that couldn't be said before;
2) a progressive extension does say new things about the old items but only about things that were previously undecidable
P.S. We can combine the vapors, splodges and linedups in a system called the messysplodges but they haven't been studied much because they're a bit messy."

(http://mathoverflow.net/questions/102237/what-is-the-status-of-irrational-numbers-within-finitism-ultrafinitism  by Steve L. Cowan)

## Friday, December 14, 2012

### Mathematical checkpoint.

Last week, I had reached the last section of the 420 book (Linear Algebra, Hefferon) we are studying together with Tom. Also I have just now reached page 93 (Section 11, Roots, Irrational Numbers) in Zakon's 'Analysis Basics' (Our approach is very patient, complete, sequential and meticulous), along with a great deal of less meticulous reading in other books, papers and 'opinions'.
I can now say, one of the original questions that this blog carries as it's title, namely the square root of two, has been almost demystified!

As with every checkpoint, a huge huge thanks goes to my wife and love Lena.

## Sunday, December 2, 2012

### Why was 'what is the square root of two really?' the right question to ask.

"Bolzano points out that Gauss’s first proof is lacking in rigor; he then gives in 1817 a “purely analytic proof of the theorem, that between two values which produce opposite signs, there exists at least one root of the equation” (Theorem III.3.5 below). In 1821, Cauchy establishes new requirements of rigor in his fa- mous “Cours d’Analyse”. The questions are the following:
– What is a derivative really? Answer: a limit.
– What is an integral really? Answer: a limit.
– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.
– What is a limit? Answer: a number.
And, finally, the last question: – What is a number?
Weierstrass and his collaborators (Heine, Cantor), as well as Me ́ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence (see picture below), uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series." (Analysis by it's History, Hairer).

And if you like pictures, here is an indirectly related one (http://en.wikipedia.org/wiki/Algebraic_number#The_field_of_algebraic_numbers).

## Saturday, December 1, 2012

### The right comment.

Commenting 'And now in English' is never the useful thing to say, the right comment is 'And now from the beginning'.

## Friday, November 23, 2012

### Are you a 'rigor mortis' kind of guy? or maybe even a 'rigor post mortem' one?

"Certain generalities seem to have been drawn from this, namely that a concern for rigor comes at the end of a mathematical develop- ment, after the "creative ferment" has subsided, that rigor in fact means rigor mortis. Weierstrass himself provides a good counterexample to this generality, for all his work on the spectral theory of forms was motivated by a concern for rigor, a concern that was vital to his accomplishments.

Weierstrass was dissatisfied with the kind of algebraic proofs that were com- monplace in his time. These proofs proceeded by formal manipulation of the symbols involved, and no attention was given to the singular cases that could arise when the symbols were given actual values. One operated with symbols that were regarded as having "general" values, and hence such proofs were sometimes re- ferred to as treating the "general case", although it would be more appropriate to speak of the generic case. Generic reasoning had led Lagrange and Laplace to the incorrect conclusion that, in their problems, stability of the solutions to the system of linear differential equations required not only reality but the nonexistence of multiple roots. (Hence their problem had seemed all the more formidable !) In fact, Sturm who was the first to study the eigenvalue problem (1) proved among other things the "theorem" that the eigenvalues are not only real but distinct as well. His proof was of course generic, and he himself appears to have been uneasy about it; for at the end of his paper he confessed that some of his theorems might be subject to exceptions if the matrix coefficients are given specific values. Cauchy was much more careful to avoid what he called disparagingly "the generalities of algebra," but multiple roots also proved problematic for him. As he realized, his proof of the existence of an orthogonal substitution which diagonalizes the given quadratic form depended upon the nonexistence of multiple roots. He tried to brush away the cases not covered by his proof with a vague reference to an infinitesimal argument that was anything but satisfactory.

It was to clear up the muddle surrounding multiple roots by replacing generic arguments with truly general ones that Weierstrass was led to create his theory of elementary divisors. Here is a good example in which a concern for rigor proved productive rather than sterile. Another good example is to be found in the work of Frobenius, Weierstrass' student, as I shall shortly indicate.
"
(The Theory of Matrices in the 19th Century, Hawkins)

### Oh matrix you special substitution.

"Thus by the mid-1850's the idea of treating linear substitutions as objects which can be treated algebraically much like ordinary numbers was not very novel" (The Theory of Matrices in the 19th Century, Hawkins)

## Wednesday, November 21, 2012

### And I was right about the curved light rays.

"He asked his readers to imagine some ex- periment in which a seemingly decisive result had been obtained, for example the construc- tion of a figure with light rays marking out four equal sides meeting at four equal angles for which the sum of the angles was less than 2π . This would seem to suggest that space was non-Euclidean,but,said,Poincare ́,thereis another interpretation, which was that space was Euclidean and light rays were curved. There could be no way of deciding logically between these two interpretations, and all we could do would be to settle for the geometry we found most convenient, which, indeed, he said would be the Euclidean one. His reasons were, however, unexpected, and will be con- sidered shortly." (Poincare ́ and the idea of a group, Gray)

## Thursday, November 1, 2012

### The search for maximum simplicity.

"Numerous factors compel the scientist to revise constantly his conceptual construction. Apart from general cultural predisposi- tions, conditioned by specific philosophical, theological, or politi- cal considerations, the three most important methodological fac- tors calling for such revisions seem to be: (1) the outcome of further experimentation and observation, introducing new effects hitherto unaccounted for; (2) possible inconsistencies in the logi- cal network of derived concepts and their interrelations; (3) the search for maximum simplicity and elegance of the conceptual construction. In most cases it is a combination of two of these factors, and often even the simultaneous consideration of all of them, that leads to a readjustment or basic change of the con- ceptual structure." (Max Jammer)

(3) is definitely related to the drive for abstraction when it comes to Mathematics. One could wonder why we have (3) as a goal. But the answer is simple: because it allows our poor limited intellect to make new progress while tackling increasingly complex subjects by ... making them conceptually as simple as they possibly can be.

## Wednesday, October 31, 2012

### So far yet so close. Unmeasurably happy I found this!

The legacy of Jordan's canonical form on Poincar\'e's algebraic practices. This paper proposes a transversal overview on Henri Poincar\'e's early works (1878-1885). Our investigations start with a case study of a short note published by Poincar\'e on 1884 : "Sur les nombres complexes". In the perspective of today's mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincar\'e's works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincar\'es work. It also highlights the crucial and transversal role played in Poincar\'e's works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordan's canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincar\'e's works. We first provide a micro-historical analysis of Poincar\'e's appropriation of Jordan's approach to linear groups through the prism of the legacy of Hermite's works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincar\'e between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincar\'e's approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincar\'e transformed this theorem by appealing to Hermite's.

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.

## Monday, October 1, 2012

### Locke, On Retention

1. Contemplation. The next faculty of the mind, whereby it makes a further progress towards knowledge, is that which I call retention; or the keeping of those simple ideas which from sensation or reflection it hath received. This is done two ways.
First, by keeping the idea which is brought into it, for some time actually in view, which is called contemplation.

2. Memory. The other way of retention is, the power to revive again in our minds those ideas which, after imprinting, have disappeared, or have been as it were laid aside out of sight. And thus we do, when we conceive heat or light, yellow or sweet,- the object being removed. This is memory, which is as it were the storehouse of our ideas. For, the narrow mind of man not being capable of having many ideas under view and consideration at once, it was necessary to have a repository, to lay up those ideas which, at another time, it might have use of. But, our ideas being nothing but actual perceptions in the mind, which cease to be anything when there is no perception of them; this laying up of our ideas in the repository of the memory signifies no more but this,- that the mind has a power in many cases to revive perceptions which it has once had, with this additional perception annexed to them, that it has had them before. And in this sense it is that our ideas are said to be in our memories, when indeed they are actually nowhere;- but only there is an ability in the mind when it will to revive them again, and as it were paint them anew on itself, though some with more, some with less difficulty; some more lively, and others more obscurely. And thus it is, by the assistance of this faculty, that we are said to have all those ideas in our understandings which, though we do not actually contemplate, yet we can bring in sight, and make appear again, and be the objects of our thoughts, without the help of those sensible qualities which first imprinted them there.

3. Attention, repetition, pleasure and pain, fix ideas. Attention and repetition help much to the fixing any ideas in the memory. But those which naturally at first make the deepest and most lasting impressions, are those which are accompanied with pleasure or pain. The great business of the senses being, to make us take notice of what hurts or advantages the body, it is wisely ordered by nature, as has been shown, that pain should accompany the reception of several ideas; which, supplying the place of consideration and reasoning in children, and acting quicker than consideration in grown men, makes both the old and young avoid painful objects with that haste which is necessary for their preservation; and in both settles in the memory a caution for the future.

4. Ideas fade in the memory. Concerning the several degrees of lasting, wherewith ideas are imprinted on the memory, we may observe,- that some of them have been produced in the understanding by an object affecting the senses once only, and no more than once; others, that have more than once offered themselves to the senses, have yet been little taken notice of: the mind, either heedless, as in children, or otherwise employed, as in men intent only on one thing; not setting the stamp deep into itself. And in some, where they are set on with care and repeated impressions, either through the temper of the body, or some other fault, the memory is very weak. In all these cases, ideas in the mind quickly fade, and often vanish quite out of the understanding, leaving no more footsteps or remaining characters of themselves than shadows do flying over fields of corn, and the mind is as void of them as if they had never been there.

5. Causes of oblivion. Thus many of those ideas which were produced in the minds of children, in the beginning of their sensation, (some of which perhaps, as of some pleasures and pains, were before they were born, and others in their infancy,) if the future course of their lives they are not repeated again, are quite lost, without the least glimpse remaining of them. This may be observed in those who by some mischance have lost their sight when they were very young; in whom the ideas of colours having been but slightly taken notice of, and ceasing to be repeated, do quite wear out; so that some years after, there is no more notion nor memory of colours left in their minds, than in those of people born blind. The memory of some men, it is true, is very tenacious, even to a miracle. But yet there seems to be a constant decay of all our ideas, even of those which are struck deepest, and in minds the most retentive; so that if they be not sometimes renewed, by repeated exercise of the senses, or reflection on those kinds of objects which at first occasioned them, the print wears out, and at last there remains nothing to be seen. Thus the ideas, as well as children, of our youth, often die before us: and our minds represent to us those tombs to which we are approaching; where, though the brass and marble remain, yet the inscriptions are effaced by time, and the imagery moulders away. The pictures drawn in our minds are laid in fading colours; and if not sometimes refreshed, vanish and disappear. How much the constitution of our bodies and the make of our animal spirits are concerned in this; and whether the temper of the brain makes this difference, that in some it retains the characters drawn on it like marble, in others like freestone, and in others little better than sand, I shall not here inquire; though it may seem probable that the constitution of the body does sometimes influence the memory, since we oftentimes find a disease quite strip the mind of all its ideas, and the flames of a fever in a few days calcine all those images to dust and confusion, which seemed to be as lasting as if graved in marble.

6. Constantly repeated ideas can scarce be lost. But concerning the ideas themselves, it is easy to remark, that those that are oftenest refreshed (amongst which are those that are conveyed into the mind by more ways than one) by a frequent return of the objects or actions that produce them, fix themselves best in the memory, and remain clearest and longest there; and therefore those which are of the original qualities of bodies, vis. solidity, extension, figure, motion, and rest; and those that almost constantly affect our bodies, as heat and cold; and those which are the affections of all kinds of beings, as existence, duration, and number, which almost every object that affects our senses, every thought which employs our minds, bring along with them;- these, I say, and the like ideas, are seldom quite lost, whilst the mind retains any ideas at all.

7. In remembering, the mind is often active. In this secondary perception, as I may so call it, or viewing again the ideas that are lodged in the memory, the mind is oftentimes more than barely passive; the appearance of those dormant pictures depending sometimes on the will. The mind very often sets itself on work in search of some hidden idea, and turns as it were the eye of the soul upon it; though sometimes too they start up in our minds of their own accord, and offer themselves to the understanding; and very often are roused and tumbled out of their dark cells into open daylight, by turbulent and tempestuous passions; our affections bringing ideas to our memory, which had otherwise lain quiet and unregarded. This further is to be observed, concerning ideas lodged in the memory, and upon occasion revived by the mind, that they are not only (as the word revive imports) none of them new ones, but also that the mind takes notice of them as of a former impression, and renews its acquaintance with them, as with ideas it had known before. So that though ideas formerly imprinted are not all constantly in view, yet in remembrance they are constantly known to be such as have been formerly imprinted; i.e. in view, and taken notice of before, by the understanding.

8. Two defects in the memory, oblivion and slowness. Memory, in an intellectual creature, is necessary in the next degree to perception. It is of so great moment, that, where it is wanting, all the rest of our faculties are in a great measure useless. And we in our thoughts, reasonings, and knowledge, could not proceed beyond present objects, were it not for the assistance of our memories; wherein there may be two defects:-
First, That it loses the idea quite, and so far it produces perfect ignorance. For, since we can know nothing further than we have the idea of it, when that is gone, we are in perfect ignorance.
Secondly, That it moves slowly, and retrieves not the ideas that it has, and are laid up in store, quick enough to serve the mind upon occasion. This, if it be to a great degree, is stupidity; and he who, through this default in his memory, has not the ideas that are really preserved there, ready at hand when need and occasion calls for them, were almost as good be without them quite, since they serve him to little purpose. The dull man, who loses the opportunity, whilst he is seeking in his mind for those ideas that should serve his turn, is not much more happy in his knowledge than one that is perfectly ignorant. It is the business therefore of the memory to furnish to the mind those dormant ideas which it has present occasion for; in the having them ready at hand on all occasions, consists that which we call invention, fancy, and quickness of parts.

## Saturday, September 29, 2012

### 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.

## Wednesday, September 19, 2012

"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 ...

## Friday, August 24, 2012

### 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:

## Friday, August 10, 2012

### "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."

## Tuesday, June 26, 2012

### 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.

## Thursday, May 24, 2012

### 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?

## Saturday, May 12, 2012

### 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.

## Friday, March 30, 2012

### 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?

## Thursday, February 2, 2012

### Releasing some old code and music.

Here is the source code for the old XNA Rendering Framework / Quake3 Renderer: