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The only difference between me and the math-phobes so common in today's world is 1) patience, 2) stubbornness, and 3) instead of saying "Wow, that !@#$ makes absolutely no sense to me; it sounds so complicated; I'll never understand it," I say "WOW NEATO! Even though that !@#$ makes absolutely no sense to me now, it sounds so totally awesome, I really gotta learn that someday!"

[ It is extracted from my "unfiltered" or "hardcore" math entries, that detail the nitty-gritty of my progress. In effort to reduce the "irrelevant roadkill" factor in publically viewable entries, I've taken advantage of putting them under a long-forgotten LJ-friends group. Let me know if you want in... or out. Anyway generally, if I discover something that's so awesome that it would be criminal for me not to share it with the world --- and believe me, I feel that a lot --- I try to polish it up and make it a public entry. It is my (admittedly excessively optimistic) hope that, in conveying my extreme enthusiasm about a particular example, I may succeed in getting others excited about math. ]

The torus once again pays a visit. It is said that it is not a true lecture in geometry until one gets drawn on the board. Visualizing 4D space, or 3D space that curves, can be a little, shall we say, mind-bending. One of the venerable ways to visualize the unit 3-sphere (x2 + y2 + z2 + w2 = 1) is by stereographic projection to plain ol' 3-space, and its decomposition into a bunch of nested tori (in another exploration of inflating inner tubes, we further broke these tori down into a bunch of linked circles, but we won't need them this time around). The fancy-schmancy term is to say that the 3-sphere is foliated by Hopf tori. The Hopf tori are cartesian products of circles of radii a and b, such that a2 + b2 = 1, so by definition, any point on such a torus actually lives on the 3-sphere. When stereographically projected to 3-space, the Hopf tori look like inner tubes.


In the movie above, we start with an inner tube representing a Hopf torus both of whose circle-factors have radius 1/sqrt(2). The red and blue circles represent each one of these circle-factors. Now, the amazing thing is that there is a "continuous rotation" (rigid motion) of the 3-sphere (which can always be realized as a rigid motion of the ambient 4-space containing this 3-sphere) which will move the torus within in a way that the beginning and ending configurations look the same, except that the red and blue circles have been swapped. If we restrict ourselves to 3-space, such a rigid motion is impossible, but if we allow ourselves to let the torus pass through itself, then it can be done. However, visualizing the 3-sphere version in stereographic projection, with the funky 4-space rotation, we effectively allow ourselves to distort distances (actually the 4-space distance is not distorted; the distortion we see is an artifact of the stereographic projection), and add a "point at infinity." What happens is we inflate our inner tube, so a part of it gets puffed up to infinity, and wraps back around, turning the torus inside-out. In fact, after wrapping back around, we're "inflating" the outside of the torus. Or equivalently, getting back to donuts with frosting, the dough gets bigger and bigger, and when wrapping back around, almost all of space (plus a point at infinity) is dough, and the frosting bounds an inner-tube-shaped pocket of air (sort of like how complex inversion in Bubi's nose makes everything outside a circle nothing but nose).

Anyway, the full turning inside-out (which also swaps the red and blue circles, as promised) occurs exactly halfway through the movie (the rotation continues to restore the torus to its original state in the second half). Notice how the stripes on the torus which started out horizontal now are vertical, and what used to be the "apple core" shape which surrounds the donut hole now has become a "donut segment." Or, more fittingly, the donut hole has become part of the dough! (assuming, for the moment, that the dough instead stays in place and only the frosting moves through space in the weird manner shown in the movie. If we move the dough along with the frosting, then this is just our situation where, at the end of the process, we have the "nothing but dough outside a donut-shaped air pocket" situation described earlier).

Incidentally, in the derivation of the formulas needed to derive the rotation involved quaternions, which are like super complex numbers (3 imaginary units instead of just one). That is a story for another day, however. Basically I finally was able to pin down the correct sequence of rotations by using quaternionic exponentiation (taking e to the power of these guys). Of course it is also possible to do this using the log of a 4x4 rotation matrix but I didn't quite feel like fussing around with skew-symmetric matrices today.

Enjoy!

Current Mood: amusedamused

Greetings. Apple's grapher is also totally awesome. I made a movie of an example correspondence between the Möbius strip and all lines in the plane (Hollywood, here I come!).


This animation traces an example curve (the white circle) on the Möbius strip (green thing) and shows the corresponding motion through "line space." The Möbius strip is shown as a transparent "inset" in the upper right corner, overlying the plane containing the lines that it describes. The turquoise circle is the reference "core circle" of the Möbius strip. The blue dot (which may be hard to find without seeing it in motion; it starts off at the intersection of the white and turquoise circles) traces where we are on the Möbius strip "catalogue". And the line in the plane which it represents is given in blue. The red line is also there as a sort of reference and stays perpendicular to the blue line, but always goes through the origin. We'll see why we have this guy in a bit.

The details of the mapping are as follows: as you go "around" the Möbius strip (the position on the core circle closest to the dot), you correspondingly change the slope, or angle of the blue line. Its angle is always double that of the position of the dot (relative to its start point). On the other hand, as the dot slides up and down the "width" of the Möbius strip, you control how far the blue line is from the origin (measured by the distance of its closest point--which is its intersection with the red line). If the dot gets close to the "edge" of the strip, it corresponds to the line being really far away from the origin, and if the dot is on the core circle, then it passes through the origin (the closest to the origin you can get!).

Technically, in my proof, I used an infinitely wide Möbius strip, but you can always scrunch all of it down into a finite strip, provided that you leave the edges off, so the distance of the blue line to the origin varies extremely nonlinearly with the distance to the edge. If you hit the edge, the line is infinitely far away; my model comes within 1/32 of the edge, as you can see by the dramatic swings of the line the dot in the upper left part of the movie.

Enjoy!

Latest fun: I really fleshed out my knowledge about tautological bundles over Grassmannian manifolds. Now that sounds like completely abstract nonsense that has nothing to do with what possible readers could relate to. Besides, who would expect much excitement to come out of collections of linear subspaces and the like? But I discovered one gem of a theorem.

Consider all possible straight lines in the plane (by straight line I mean those that are infinitely long). My latest math musings have brought me to consider making a catalogue of all of these lines. A catalogue of all straight lines? What does that mean?

First off, surely even the most math-phobic readers remember at one time or another, and not necessarily with fondness, that almost all lines in the plane are uniquely characterized by the following famous equation, called the slope-intercept equation:

y = mx + b

The variable m if you recall is the slope of a line, and b is the y-intercept which means where the line crosses the y-axis. However few people realize that what they're doing, really, is indexing every non-vertical line in the plane with some point in another plane (called, say, the mb-plane rather than the old traditional xy-plane). That is, we have a correspondence between points in one abstract mb-plane to the lines in the xy-plane, where m is the slope, and b the y-intercept. It doesn't treat vertical lines because they have infinite slope, and you can't have a coordinate on any plane with an infinite value. That is, you have charted out the space of all possible nonvertical lines with points in a different plane.

More details of the construction of the catalogue...Collapse )

One should wonder what kind of overall "shape" our nice spiffy catalogue has, after gluing together the two possible charts we've made for it. As it turns out, its shape is the Möbius strip! That's right, the classic one-sided surface (without, as it turns out, its circle-boundary). Let's pause a moment for a bit of sober reflection:

THAT IS TOTALLY AWESOME!!!

That is to say, if you give me a point on the Möbius strip, it specifies one and only one line in the plane. One would not, initially, be able see why non-orientability enters the picture. But a little interpretation is in order. First, if we take a particular line and rotate it through 180 degrees, we get the same line back. Everything in between gives every possible (finite) slope. It so happens that as far as slopes of lines is concerned, ∞ = -∞, and if you go "past" this single "projective inifinity" as they call it, you go to negative slopes. In other words, if you start on a journey on rotating a line through 180 degrees, from vertical back to vertical, you come back to the same line, except with orientation reversed (because what started out as pointing up now points down).

If you fix an origin and declare that it correspond to a certain special line in the plane, and then select a "core circle" for the Möbius strip, then as you travel around this circle, the distance traveled represents rotation angle for this special line. Traveling from the origin along the core circle and making one full loop should correspond to rotating the special line by 180 degrees. If you instead move up or down on the core circle, you instead end up sliding the line along a perpendicular direction, without changing its angle. So moving up and down the strip corresponds to parallel sliding of lines, and moving around the strip along a circle corresponds to rotating a line.

Current Music: Mozart Sonata in C - Michael Powell

... we might actually have a shot at world peace.

... now looking for ways to anticommute... ω /\ η = -η /\ ω

BLARRR!!!!

I finally solved that !@$%ing mystery of pseudo-BS today, namely the machinery surrounding integration over non-orientable manifolds, without sacrificing tensor character (namely cheating via use of absolute values). That bothered me for years. It wasn't exactly productive use of my time (how did you guess?)... since all complex manifolds (the concept I'm supposed to be learning right now) are orientable anyway. But it's kind of ironic that the reason why I was able to solve it was because I generalized the method of complexifying the tangent bundle or any vector space in general. Except instead of tensoring with C, you tensor with the pseudoscalar algebra, which basically says, instead of having an imaginary unit i satisfying i2 = -1 and i * -i = +1, you have instead, a "pseudo-unit," call it a, satisfying a2 = +1 and a * -a = -1. Such a simple change, swapping the occurrence of a +1 and -1, profoundly changes the character of the algebra (for one thing, it isn't a field). It is like the minus sign acquiring a new friend which behaves in the same way but yet isn't minus. And -a, call it b, also satisfies exactly the same relations, except it's not a. a and b are like new participants in the old classic

Minus times minus is plus;
The reason for which we need not discuss.


The resolution to the problem of nonorientability is choosing the pseudo generators a and b to be the two possible orientations of an individual tangent space. Individual tangent spaces are always orientable; it's just that for a manifold to be orientable, the orientations must be "consistent." Every manifold has tangent spaces, each with a pair of opposing orientations, but a priori, there is nothing to distinguish between them---they're like an infinite collection of achiral socks. All you know is that they are two distinct objects, nothing more. An orientation on a manifold is a universal decree of left sock. If such a decree cannot be made (without the Axiom of Choice---sorry, Bertrand Russell), then the manifold is not orientable.

But being able to tell left from right is irrelevant to being able to define the pseudo algebra generated by the pair (both of them squaring to 1, and when multiplied by each other, giving -1), and letting this vary over all tangent spaces at all points, we get a nice new line bundle whose transition functions are, gasp, the SIGN of det(dy/dx) !!!! Instead of artificially generating that line bundle via brute force and equivalence classes and whatnot (you can define a vector bundle to have any dang transition functions you want, so long as it's smooth), there is something a little nicer. Even the (otherwise very nice) Bott&Tu just do it the equivalence class way (they make up for it by defining d and cohomology and even Poincaré Duality using these guys, I've seen that done nowhere else... except maybe in the original papers that define it!)

Like I said, it's bothered me for years. So I guess today, I can say that I no longer have orientation issues...............

[NO, NO NO, not those kinds of orientation issues! ;-)]

Current Mood: amusedamused
Current Music: Finite Simple Group (of Order Two) - The Klein Four Group

I happened across a "popular math" book, The Poincaré Conjecture, by Donal O'Shea, and gave it a good going-over. It does a much better job of explaining the kind of math I do, in non-expert terms, than I can muster, complete with the historical perspective and development of geometry, and relating it to some really cool current questions, such as the shape of the entire universe. The phrase "in non-expert terms" should of course by no means be an insult to the reader, as I don't believe in the distressingly common hostile mantra "You wouldn't understand, so I won't even try to explain it to you!" or the belief that there is no way to condense 8+ years of hard work into a 5-minute or even a book-length summary in a sensible manner. I can't force people down the same path that I have taken. Of course, things are going to be lost in translation, but this doesn't mean we shouldn't try.

Why teaching is good...Collapse )

I can't say for certain (because I am hardly an unbiased source) whether or not the book makes good bedside reading--not in the sense that it would be so boring that it would make an excellent cure for insomnia (I can't stand it when people say something like that), but rather in the sense that it is an adventure that one can enjoy with comfort at the end of a long day. Maybe even too exciting for the bedside. However I'm guessing one will need a bit of patience to read some parts, because there are points where he presents a flurry of mathematical definitions. I don't want would-be readers to slam the book shut at that point and so I'll engage in a bit of ranting and "mathematical moralizing" about what I see as one of the major problems in learning: people don't have the patience for anything, especially math- or science-related, anymore. If they don't understand what is being said as they read it, they give up, telling themselves that they're either too stupid, or, (possibly true =) it's all pure nonsense anyway. It would be a tremendous benefit for science education (or even all education) if everyone could be confident that it is okay to stop at a line and ponder for a moment to sort things out.

Continuing the post last time... Well, my place is still chaos; it's always a fight to keep things from sliding back there. I'd left the Bay Area to go back to SD for a conference, and it was one disaster after another as I realized I'd forgotten this and that, plus not setting the alarm clock so I could leave early, driving from (the) OC where I'd flown to and stayed for the night. And I thought I was well-prepared, for once!

Anyway, Omnivore's Dilemma has induced me to try something very interesting: grass-fed beef. I just tried some hamburgers made from such beef. It seems hilarious that beef has to be qualified with grass-fed when everyone thinks of cows eating grass, anyway. But that's exactly the problem... due to our corny food economy, what we eat has become mysteriously transformed, behind the scenes. Most cattle are fed corn... and it was worse, until very recently (due to outbreaks of mad cow disease), feeding animal parts to the poor herbivores. It has been noted that this is more than reason enough to become a vegetarian (I must admit, however, that my self-discipline is too poor to commit myself to such an undertaking...). But after this experience, this may offer an alternative to "political vegetarianism."

Enough with the food politics ranting though; the burning question that you might have on your mind is.... "How did it taste?"

Interesting. It still tasted beefy, yes, although I could tell there was something different about it, kind of hard to put in words, though. I'd read others' opinions on it; though I haven't tried grass-fed steak, I hear it is a little more chewy in texture and is less fatty. The burger was a little chewier but the fat content seemed perfectly fine (one of the goals of adapting cattle to eat corn has been to make them fatter, too, so it would taste better). In any case, though, I liked it!

Ok, so I think I can say I've suitably ingrained an exercise routine these days. And it was not, believe it or not, a new year's resolution... in keeping in accordance to my resolution to have none about 6 years ago... but rather a build-up since Oct. of last year. However, Congress would not have approved of my activities since the new year, though, as I have escalated the fitness routine, bahaha. But something else occurred today... I successfully exceeded the exercise machine's numeric representation capability, such that it had to move the decimal point to continue counting calories.

Therefore, we've reached the tipping point... it is no longer at all about health, weight loss, looking good, and all that nonsense.... no, not at all!

It's about SETTING NEW WORLD RECORDS!!!!!!!!!!!!!!!!!!!!!!!!!!@~!#!#!~@#@!#!~@#@

Oh yeah, speaking of new year... HAPPY CHINESE NEW YEAR! *0ink*

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Current Mood: mischievousmischievous
Current Music: .shur - Manoochehr Sadeghi
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