I expected to be done with basic trig by now, but here I am, still memorizing formulas and doing trigonometric identity problems, for the third month in a row. I can now incorporate angle addition and subtraction, double-angle, and half-angle formulations into these workings. I assume that these will reappear someday in my studies, and I will say, while contemplating some mysterious oscillation or repeating phenomenon, wow! that must be the result of sin(A+B) or sin2A or something of that sort that I learned back in 2004.

I feel, pedantic and dull as it sounds, that I must go through all of these identities and learn them and learn how to work with them, lest I be caught short in the middle of some calculation somewhere way in the future. Will there be a TEST? Who will test me? One of my wry and imaginative friends has suggested an eschatological test. That is, at the end of time (or the end of MY time), I will face Jesus at the last judgement (or the judgement of my individual soul). And instead of the Biblical injunctions to do well by one's fellow human beings (see Matthew 25:31-46) asked of people at the Last Judgement, Jesus will approach me and say… what is the formula for the addition of the tangents of angle A and B? And what is the relationship of tangent and secant? And if I say, "Uh, I don't remember," I will fall into the lake of fire.

Well, maybe not so dramatic. And yet, Jesus according to Christian theology is a Person of a Trinity, which has three entities, thus a triangle. So Jesus Himself has a sine and a cosine, in heavenly trigonometry. In some artworks you can actually see the triangle behind Jesus' halo. If this is an equilateral triangle, which it usually is when the Trinity is symbolized in art, then Jesus is 60 degrees, thus making his sine the square root of three over 2 and his cosine one half (0.5). That means that Jesus' sine is irrational! We behold the sine of the divine, and it is irrational! Fortunately his cosine is not. He does, after all, have two natures in one divine Person.

My apologies to my Atheist or non-Christian readers for these after-midnight riffs. I am sometimes driven to distracted fantasy when facing yet another round of these trigonometric identity problems. I'm not done yet. I need to go through the Product of Sines and Cosines and the Sum and Difference of Sines and Cosines, which is the next short section in the red-backed Schaum's Outlines. After that, things might start moving, or at least oscillating, as I will be studying graphs of trigonometric functions, trigonometric equations and their graphs, and waves.

The identity problems are teaching me a way to look at mathematics which I could describe either as the "Russian nesting doll" model, or the "Chinese box" model, or perhaps the "Japanese Transformer Robot" model. These Orientalizing (OK, Russia is "Eastern" Europe) concepts have to do with nested and transforming entities. I followed an example problem in the Schaum's text which went on for a whole page in my notes, as one expression was substituted with another, which was then re-arranged with its neighbors to create yet another expression which could be substituted. And then that new entity was itself part of yet another trigonometric statement which could be re-stated to match its neighbors, thus working another transformation which eventually unfolded or infolded into the desired configuration. Entities within entities, hidden formulas I have to look for, substitutions I must not fail to make; if it's not an interlocking puzzle box, it's a labyrinth to get lost in. I must know my landmarks, hence all this going back and forth in which I have spent my winter.

For those mathematical experts who read this Weblog (I hope you're out there), this is the Orientalizing labyrinth which I was fortunate enough to be guided through: "When A + B + C = 180 degrees, show that sin2A + sin 2B + sin 2C = 4 sinAsinBsinC." Easy for you, perhaps… I hope the Last Judgement doesn't come soon.

Posted at 1:34 am | link

In trigonometry, I'm now working on angle addition and subtraction formulas, double-angle formulas, and half-angle formulas, along with their various identities and problems involving their interrelations. This is classic trig, and if I were a True Mathematician I'd probably find it entertaining just for the endless recombination of pattern and re-statement. I'm not true enough, then, because I find it truly tedious. Not only that, I keep wondering what all this stuff is for.

OK, I know, I'm not supposed to ask "what things are for" in mathematics, that's gauche. As the math mantra goes, "It will be revealed to me as my studies progress." But I can't help asking this simpleminded question: what's the point of finding the sine/cosine of the half-angle using elaborate half-angle equivalency formulas, when you already know the whole angle and could just look up the results in the table at the back of the book, let alone using your pocket calculator? There must be some importance to these calculations and formulas, otherwise the book wouldn't spend so much time and space on them. I just don't know what they are yet.

Many years ago, a certain group of people (science fiction fans, I admit it), enjoyed watching a short film called "Closet Cases of the Nerd Kind," made in 1980 as a parody of Spielberg's CLOSE ENCOUNTERS OF THE THIRD KIND (1977). I have seen it more than once, with much hilarity; in fact it is one of the funniest short films I've ever seen. (You can find a ridiculously expensive "official" video of the film and a short description at this site.)

In the film, the character parodying Richard Dreyfuss repeatedly sculpts mysterious forms with gushy substances like whipped cream, mashed potatoes, and shaving cream. He is receiving mystical visions of alien arrivals, but he doesn't know what they are yet. As he contemplates yet another skwushy disc-shaped sculpture in his hand, he almost reaches a moment of enlightenment, and he says, "This MEANS something." And then, before he says more, his hand automatically slaps the skwushy pie into his face.

This is how I feel with these trigonometric formulas. These malleable, squeezable sets of numbers squish around on my math paper, Pythagorizing themselves and inverting themselves and sometimes melting away to a single expression. They have long since ceased to resemble hard-edged triangles or configurations of ships and lighthouses. Somewhere out there in math and physicsland, these things are important. But right now, I'm there with the Nerd Kind, contemplating the whipped-cream form before me, trying to solve problem after problem ("Prove that cos2x = cos⁴x — sin⁴x.") The ever-helpful Schaum's suggests using the "difference of squares" to deconstruct this, which melts it right away. But then there's another problem, and then another. I stare at it. This means something. The mathematical pie in the face is coming, and I know that it is calculus.

Posted at 2:47 am | link

I finished reading Joao Magueijo's steamy piece of science-porn, FASTER THAN THE SPEED OF LIGHT. The first half is fairly straight, describing in a colloquial and sometimes cutesy manner how relativity works. That's part 1. Then you get to part 2, which is his highly personal (TOO personal!) account of how he and some of his colleagues in England and elsewhere put together what could be (or could have been?) a revolutionary theory about the origin of the Universe. What if, in the very earliest mini-moments of our Universe, light had gone far faster than it does now? The reason they came up with this possibility is that according to some cosmologists, the pattern formed in the nascent universe, which eventually developed into the patterns of galaxies we now see through our telescopes and mapping satellites, could not have been imprinted into the whole thing by light going at "ordinary" speed. Information, or patterns, can travel at most at the speed of light, but in this model there just was too much universe for it to cover, even in the earliest subseconds. Therefore light must have been faster at the moment of initiation, in order to make things look the way they look now; it would have only settled down to the "conventional" speed once matter had condensed and things "froze" into what would evolve into our universe as we perceive it.

From there Magueijo and his collaborators figure out lots of different ramifications of this theory, and attempt to get it published, or at least heard by the mainstream scientific community. Naturally, the mainstream doesn't want to hear about it, or brings up objections which enrage Magueijo and send him off into splenetic and profane rants about the prejudice and incompetence of older scientists and the scientific "establishment." This is great stuff to read, as long as I'm not involved in the process; it's like reading the stories of composers and music critics and conductors, all at odds with each other. Sometimes it's funny, but more often, there's a lot of bitterness to it.

Magueijo, in telling the story of how the "variable speed of light" theory came about, is refreshingly honest about the struggles, setbacks, confusion, and blundering about that happens when theoretical physics is done. He doesn't portray it with the smooth triumphalism that you find in some of the other books. His experience of science is pervaded with high emotion: frustration and despair when things are not working out right, and elation and euphoria when they do. Many of us science consumers still have that old media-fed notion that scientists take everything in life with pure dispassionate unemotional analysis. Instead, Magueijo recounts (on page 217) Saint Einstein taking a violent tantrum when one of his papers was rejected by an important physics journal.

Magueijo is at his best when talking about these personal aspects of science. His portraits of people he works with are vivid, especially that of John Moffat, who actually had the variable speed of light idea before Magueijo did. Moffat is particularly interesting to me because according to Magueijo, John Moffat started out as a professional artist, and switched to physics in mid-career! So there is at least one physicist on this planet who was originally an artist! Other fascinating folk include physicist John Barrow, author of the excellent and fact-packed book THE ARTFUL UNIVERSE (Oxford, 1995). Magueijo even devotes some pages to Lee Smolin, the quantum gravity expert whom you have met before in this journal, at the Perimeter Institute near Ottawa, Canada.

But there are some unpleasant aspects to this book, as well. The main problem is Magueijo's attitude, which pervades the book, sometimes to the point of obnoxiousness. Perhaps this is a standard attitude for young male physicists full of the juice to buck the establishment, and only M. has the openness to show it. But this also results in cranky comments and plenty of profanity and vulgar language which will offend more sensitive readers. With this writing and attitude, I wonder whether anyone in the field will take him seriously in the future.

The most disheartening thing for me about this book is that it still portrays the world of physics as an almost exclusively male world. This is not the 1930s, it's the late 20th and early 21st century, and even now Magueijo moves in a world as male as a Masonic lodge. True, he spends some time talking about his girlfriend Kim, who was also a physicist, but after experiences of harassment and prejudice she drops out of high-level research altogether and settles for a job teaching physics in a girls' high school. Her photo appears in the book; she's wearing a low-cut tight shirt showing lots of cleavage. The male physicists pictured in the book are just modest faces. Even when M. is supposedly lamenting the problems of women in physics, he's totally immersed in the "laddie" atmosphere that continues to pervade that world, where men meet to talk physics, in rough language, late into the night in bars and clubs and gentlemen's hangouts. At one point (page 236), describing the ideological battles between different schools of thought, he calls a (presumably) female string theorist a "stringy little bitch" for making a snarky comment. What does that make snarky Joao?

Ultimately, this whole variable speed of light theory is vaporware unless it can be proven by some form of experiment or observation, and Magueijo is again honest enough to admit this. So far, neither string theory, quantum gravity, variable speed of light, multiple universes, or any other of these trendy wild ideas in physics has yet attained any definite experimental proof. So Magueijo and his boys are still playing with ideas which sound real cool but are at least for now, architecture in the clouds.

I must admit, that while reading some of this book I was filled with intense emotion. Granted that I did my reading at around 4 AM when even a night creature like me is not at my best, but even so I was overcome with yearning and jealousy, longing to somehow be part of that world (I wanna be a SUPERMODEL!) talking physics with them on their level, sharing their macho-intellectual collaborations and combat (I wanna be a FIGHTER PILOT!) and their science and travel adventures, and filling blackboards and bar napkins with arcane mathematical graffiti (I wanna be a SECRET AGENT!). Here I am, reading about cosmological commandos when I am still dragging my way through high school trigonometry. Twenty years to go, eh? And by that time, the universe might well have ended! I think it's time for me to read something else than science porn. Maybe Harry Potter.

Posted at 2:22 am | link

In my last entry here, I left myself lying exhausted on a snowbank, trying to walk home from school over sidewalks covered with heaped-up snow rather than disobey orders by walking in the plowed street. Well, here's the rest of this story.

The residents of the house where I was stranded were home, and saw with some alarm a young girl collapsed by their driveway. They came out to investigate and found me there. They asked me who I was and where I lived, and I told the story of how I must get home, and where my family lived. Home was, by the way, only a few hundred feet from where I had stopped, if I were to take my usual route walking through brushy backyards. But the snow was too deep to take that route that day.

The kind residents called my mother, who was by that time worried that I had not come home from school yet. She immediately came there by car, picked me up, and brought me home.

I hesitate to make any analogies concerning this episode. But my mother, who is still living in the very same house as 43 years ago, cannot rescue me from the long path through the snows of trigonometry. Mother doesn't know trigonometry. I grew up among artists and musicians; math has never been our thing.

The incident does say a few things about me, though, then and now. I cite this as a counter-example to the various "portraits of the scientist as a young man" which I read in popular science writing. The authors often recount their adventurous, risky, and even anti-social behavior in their younger years, such as blowing things up in their back yards.

I, on the other hand, recount my hyperobedient, timid ways, in which I was willing to exhaust myself trying to keep to the admonition and the rule rather than break its boundaries and take the risk of walking in the street. I will walk painstakingly and patiently through a mile of snowbanks, because I am dutiful, conscientious, proper, unchallenging, and averse to any risk. Perhaps it is not a good personality for a would-be scientist. And yet the call of the particle beam has reached me, and I have set forth against all odds.

Now I must follow dutifully my book's example on how I may find the sine, cosine, and tangent of 5Pi/12 radians…

Posted at 2:39 am | link

Long ago, when I was just a little young thing in elementary school, I walked home from school one sunny winter day. There had been a snowstorm a day or two ago, and the plows had heaped up the snow over the ends of the driveways and the sidewalks. It's a typical New England winter scene, as anyone from there will attest.

My parents, in a well-meaning admonition, had told me, NEVER walk in the streets, because a car will hit you. So, being a good obedient girl, I wouldn't walk in the smoothly plowed streets, only on the sidewalks.

But the sidewalks on this day were covered with craggy piles of snow. From school to home was more than a mile. Block after block, I toiled on through the heavy deposits, climbing over some, sinking into others, crunch crunch crunch through the snowbanks, trying to get home, obeying an abstract principle I was too conscientious to break. Finally I remember collapsing onto a snow hummock, just trying to retrieve the energy to go on.

I feel this way now, more than forty years later, trying to get through trigonometry. I have just filled 16 pages with solved trigonometric identity problems. I can solve them now, at least most of them. At this point they seem as meaningless as snowbanks. They don't build anything permanent, and they melt when you apply the heat of algebra to them. Yet I have to know how they work. They are essential to what comes later in calculus and physics. And after this will come double-angle formulas, and after that, sum, difference, and product formulas. And after that, trigonometric equations and a return to graphs.

This is taking much, much longer than I thought it would. This is the type of person I am, and how I learn: not by the brilliant flashes, enthusiasm, and risk-loving energy of youthful genius (who would walk in the street regardless of cars), but by the step-by-step, stolid plodding of someone who will work by the rules of completeness and propriety even when it takes the energy right out of me.

My hope is that I will get through this, and that spring weather will come soon. The mathematical snowbanks will melt and I'll be able to walk on unobstructed sidewalks till I get where I'm going, which is not home, but a place that recedes forever into the cosmos.

Posted at 1:10 am | link

I'm currently reading a highly entertaining and informative book called FASTER THAN THE SPEED OF LIGHT by Portuguese cosmologist and physicist Joao Magueijo. I'm only about halfway through, so I can't comment on it yet, but so far, Magueijo is sailing along fine as he attempts to explain relativity and modern cosmology to us consumers, without putting in those reader-repelling equations.

This book is only the latest addition to my growing collection of popular books about modern physics and cosmology. I've got shelves of them, some read, some still unread. At this point, it's my favorite type of reading. These books are written by Real Scientists describing Real Science, how it came about, what discoveries were made, and what questions still remain. Along the way, these Real Scientists recount descriptions and anecdotes about Famous Scientists and their lives and work, some of them funny, all of them fascinating. And these authors also give me tantalizing glimpses of how scientists work together, discussing cosmic matters in the academic hallways and over beers and wherever else gentlemen meet.

If you are reading this, you probably know many of the books I'm talking about, such as THE GOD PARTICLE by Leon Lederman, or the various books by Stephen Hawking such as A BRIEF HISTORY OF TIME, or books by the Star Trek fan and prolific writer Lawrence Krauss. One of the most notable (though still unread on my shelf) of these, is THE ELEGANT UNIVERSE by Brian Greene. This was recently made into a glitzy PBS three-parter, broadcast last fall, and I was proverbially glued to the proverbial tube while it was on. I just can't get enough of this stuff. In fact, it's so attractive to me that I might as well call it as I see it: it's science porn.

Porn?? But this is reputable literature about science, full of good information, written by reputable scientists who are spending their precious time writing these books for us, often motivated by the best reasons: love of their subject and desire to share it with a wider audience. (If they make a few bucks too, so much the better.) How can I have the impudence to call this literature porn?

Like pornography, this literature presents, to the science-aspirant such as me, seductive words and stylized, unattainable people and breathless scenes of discovery-consummation. It's not just that young male scientists such as Brian Greene and the movie-star-handsome Joao Magueijo are marketed in magazines and on TV as intellectual matinee idols. It's that the whole enterprise (sorry, Dr. Krauss) of science is made to sound sexy and exciting. Perhaps they want young people to enter the field by making it sound romantic and fun. Or perhaps that's how the authors still see it, even after years and years of research drudgery, grant applications, publish-or-perish, and academic politics?

Invariably, the author apologizes for not putting mathematics in his book. But if he did, it would cause the enthusiasm of the potential reader to droop. It is part of the ritual of pornography that the encounter fantasy is made artificially easy. The climax of discovery gets a lot more pages than the long years of struggle that preceded it. The gossipy glimpses of life in PhysicsWorld only add to the enticement of the scene. Imagine, talking about the origin of the Universe rather than my usual boring inane conversations about the weather or what I might have for dinner.

Magueijo is quite aware of this metaphor, much to his credit. In the book, during his recounting (on page 74) of the first pictures of galaxies made by Edwin Hubble, he comments thusly:

"(Hubble) installed a telescope inside a building that rotated as a perfect clock, moving exactly to counter the Earth's rotation. He could thus automatically point his telescope in the same direction for extended periods and make observations without "attaching" the naked eye to the end of his telescope, using instead photographic plates that could be exposed for very long periods.
What came out of these unusual observations was truly pornographic…"

The "pornography," in this case, was the wildly exciting, and to that era shocking discovery that the universe was full of galaxies made of what a previous generation's science-porn-celebrity would say was BILLIONS AND BILLIONS of stars. Magueijo, though, has got the notion right. To someone excited by science and natural phenomena, the tales of galaxies, black holes, particle explosions, string theory, and esoteric cosmology are intellectual porn. You read it for the pleasure and the excitement, you want to be with it, and you keep wanting more.

But there is also an underlying element of futility. Any science aspirant has as much chance of becoming a professional cosmologist or string theorist, as a street basketball player has of playing in the NBA, or a garage guitarist of being a smash hit rock star. Between Brian Greene and his audience, a great gulf is fixed. The science-titillated can gaze and read and look and fantasize, but they cannot have it, cannot be part of the scientific work or make discoveries themselves, because for the non-mathematical public, popular science writing will always be a carefully presented illusion, which both teaches and conceals.

Posted at 1:59 am | link

I am finally making headway in this arcane section of trigonometry. I am far from the ocean breezes, distant misty landmarks, and lengthening shadows of beginning trigonometry, and into a world of rigorous abstractions in which there are only two dimensions. I began to make sense out of these trig identities and the problems when I realized that these are built just like those polynomial problems of simplification and manipulation which concerned me back in 2001.

I well remember my polynomial passage. I spent endless hours with a² and b² and a³ and b³ and ab and ab² and all those other letters and exponents and adding and multiplication and division and factoring and canceling out and then multiplying them back together. Here in Trigonometry the "a" and the "b" are replaced by "sinA" and "cosA" and other such things. And there are some different rules in this business, such as sin²A + cos²A = 1. But the process of simplification and mooshing the expression around is basically the same as polynomial-pushing, or so it seems to me.

I did polynomials in October and November of 2001, when a large part of New York City was a smoking hole in the ground, and a large chunk of the Pentagon (7 miles from my home) was a burnt-out ruin. The news in the Washington Post isn't much better now; it just isn't our turn this time. Is math an escape from the horrors of the world? It is a cold but satisfying comfort, knowing that these trigonometric relations would be true (at least in two dimensions) even if humanity disappeared from the universe.

All right, I admit it. I look at the answers in the book. Schaum's Outlines (to be precise, the authors of Schaum's) are only too happy to show me how a problem was done. This is how I'm learning what to do. In the absence of a real live teacher, the book suffices. No human being could be as patient or as helpful as the good books; they are there morning, noon, and deep into the night. They rarely make mistakes, and they never complain. Until a "virtual professor," an artificial intelligence with a realistic and responsive "human" interface is created, the books and the websites will do the job. They are unmoved by disaster, tragedy, misery, or sadness — like mathematics itself.

Posted at 2:02 am | link

My mission since 2000 is to study and learn mathematics and physics, independently working through high school and college studies and proceeding if possible to the equivalent of a graduate student level. I would eventually like to encounter and work with classical and modern physics, and even current physics such as string theory, on the same mathematical basis as scientists do, rather than simply reading non-mathematical books on the subjects directed at laypeople. I would like to be able to understand scientific papers and discourse in the field of math and/or physics, both written and in lectures and conversation.

Time frame for this project: Ten to twenty years.

Time spent so far on the project: About three years, from February 2001 to March 2004.

Level reached so far on the project: High school/early college trigonometry.

Why am I doing this? Here is a list of Reasons. Wrong reasons first.

Wrong Reasons

1. Scientists are like super-heroes to me, with intellectual and physical powers above those of ordinary people. They lead exciting lives. This is the way they are often portrayed in books and movies. I want to be like these people, be a genius and do exciting things too. Science is about power, and I want power.

2. I am trying to transcend the limitations of my gender and age. Even though there are some women in physics, for the most part it is still heavily dominated by males, and there are still many scientific studies which attempt to prove that the male mind, shaped by testosterone and by the needs of being a hunter in the prehistoric world, is naturally more gifted in math and science than the female mind. Physics is comparable to an "extreme sport" done by young, powerful, aggressive, risk-loving males. Since I have no desire and no physical ability to run marathons, snowboard or climb mountains, I am doing the intellectual equivalent of it. (It's interesting how many physicists climb mountains, as if it were a kind of testing ritual for them.) I am told that a mathematician's/physicist's best work is done when he is under 30, or at least under 40. I am just starting when I am 50. So I already have a number of limiting conditions I have to overcome. I want the challenge, even though I am an old artgirl and perhaps better suited to producing pretty, overcomplex pictures jammed full of fussy details.

3. This is something which will keep me from being bored as I trudge through my middle years. I worked on religious studies both Christian and Zoroastrian for years until I realized that since there would never be any new sacred texts for these religions, it was just one re-interpretation after another. At least with physics there will always be something new.(This is also a "right" reason.)

4. I want to know physics so that I can make the science fiction and fantasy I write or illustrate sound plausible. This includes imaginary world-building, character creation, game and plot scenarios, and images of all kinds from cosmological to technological. The trouble is, the world really doesn't need any more science fiction or fantasy. There is too much already.

5. I want to know physics so that I can talk about it to New Age people in a way they will understand but at the same time convince them that the New Age is misusing physics. At the same time I would like to connect physics legitimately with spirituality, metaphysics, and imagination. (This may actually be a "right" reason for some physicists, but many, perhaps most others wouldn't touch this with a ten-foot monopole.)

6. My world is full of silly metaphors and childish analogies to physics phenomena, such as my car as an electron or the road as a quantum path my electron takes. I put names, colors, costumes, stories, sounds, and theater to things which should remain purely abstract.

7. I want to show the world that I can do something that is more difficult and complex than art. Mathematicians and physicists often play musical instruments or do other artistic things, and often with high competence, but it is almost impossible to find an artist who does math or physics with equal competence. I want to be that person. I want to show the world that I'm not a weakling or an eccentric underachiever. Maybe I'll even get some recognition for doing it. Wanting recognition or even worse, fame, is a Wrong Thing.

Non Reasons

I only have a few of these, but they are brought up to me by other people constantly, so I feel the need to state them clearly.

1. I am NOT doing math/physics because I want to improve my mental health and efficiency by doing "exercise." I am not doing this because I want to keep my brain occupied and working hard so I don't get Alzheimer's disease when I get older. There is no guarantee on that, and if hygiene through hard mental work were my reason, I might as well have picked something more useful, like computer programming or learning Arabic.

2. I am NOT doing math/physics because I want to change careers. I realize from the start that I am better off and much more effective in the career which I chose, namely fine art, commercial art, and a bit of writing. Also, I simply don't have enough time in my life to do it. And I don't have the guts and daring and endurance it takes to undergo test after test after test under high pressure conditions.

3. I am NOT doing math/physics to improve my artistic output. I may or may not do math or physics-themed pictures, but I would not be satisfied with just using such themes in my art without understanding them. That would be like being the pretty model sitting on the fender of a powerful, shiny sports car but not being allowed to drive it or even be a passenger in it.

And now, the Right Reasons

1. I love Big Systems of knowledge, language, and law. I have always been fascinated by these systems and I like to find patterns in data and information. Systems fascinate me, and there is nothing bigger than this grand and perhaps infinite system of mathematics and physics. Not only would I learn some of it, I would work on it at the same time as I was part of it.

2. I like a challenge. I like the feeling a challenge gives me. (This is the "right" version of "wrong reason" number 2.) If I am not striving for something, if I don't have a goal, I feel lost, dim, and futile. In my life I have striven for a number of goals, and have reached them. This is another one, which is open-ended but has some recognizable milestones. Therefore I will always have something to strive for.

3. There's a lot going on in physics and cosmology right now. I want to learn enough to pay attention to it in a serious and responsible way, and to represent it to those around me correctly. In my own way, however small, I want to be part of the ongoing story of humanity learning things about our own universe.

4. Physics and mathematics hold my attention while other sciences (such as biology or psychology) don't. I love geology too, but I like math/physics more because it is more fundamental. Contemplating things like subatomic particles and galaxies makes me happy and excited. When I see a jet from an active galaxy, or the tracks of particles in an accelerator printout, I am filled with awe and joy.

5. Physics presents me with things which are aesthetically beautiful, whether they are the aforementioned particle tracks, or the spiral of a galaxy, or the glow of a cathode ray tube. As an artist I naturally love light, color, and structure. Physics provides me not only with lots of brilliant visual images but with the scientific explanation for them. There are other aesthetic attractors as well. Mathematicians and physicists often talk about "beauty" and "elegance" in regard to equations and theories. I don't know what they are talking about, yet. I want to know enough math and physics to understand, and appreciate, just what that beauty and elegance is.

6. Though most of the physicists I know are non-religious and even atheistic, they still sometimes speak of physics as a "calling." (But what is doing the calling?) Despite everything, I feel as though I have something of that "calling" too. I felt it directly after my visit to Fermilab more than three years ago and I still feel it. It is strong enough so that despite many moments of frustration in my studies, I always come back to it. I don't stop and I never get enough. I don't know whether people get legitimate "callings" in mid-life, so different from what they start out with. But I think this qualifies.

7. Seven Right Reasons to balance seven Wrong Reasons. Last but not least, I love to learn things and to find things out. The process of inquiry in itself is rewarding, and when faced with a problem, even a trivial math example, I will go at it again and again till I find a solution. Let me quote from an essay by Alan Lightman, in BEST SCIENCE WRITING OF 2001 (edited by Timothy Ferris, HarperCollins 2001).

"…When in the throes of a new problem, I was driven night and day, compelled because I knew there was a definite answer, I knew that the equations inexorably led to an answer, an answer that had never been known before, an answer waiting for me.
That certainty and power, and the intensity of effort it causes, I dearly miss. It cannot be found in most other professions.…if given a chance to start over, I would do just what I did, to be not only a young man in the shimmering of youth but a scientist. I would want again to be driven day and night by my research. I would want the beauty and power of the equations. I would want to hear that call of certain truth."

This quote by Lightman must, of course, be put in context. He left the world of science to become a creative writer, an artist, in a more realistic world where there is no certain truth. In the essay, he says that he realized that even at age 35 he was past his prime as a theoretical physicist. But as a writer, he would have a much longer creative lifespan. He wrote this essay at the age of 50, and as he writes further, he also realizes that the "purity, power, and intensity" of what he experienced in science was part of his youth, which is gone and which he misses with keen nostalgia.

I sometimes use another religious metaphor to claim that after Fermilab, when I took up mathematics beginning again with elementary school arithmetic, I was "born again." I will never know science the way Lightman knew it, from the inside where real research is done. All my "discoveries" will already be well-discovered. And yet it will all be new to me, unfolding at the forever progressing point of this electron's long journey.

Posted at 2:51 am | link

The Barron's text with the "Ruritanian" fantasy characters, titled "Trigonometry: The Easy Way," was anything but the "easy way." I have found it unhelpful on so many occasions that I have put it away, at least for now. I'm now working again with the Schaum's Outline book on trigonometry, which I referred to some time ago as the "Red Spine Book." This book's section on Trigonometric Identities contains lots of helpful, if pedantic, information on how to work these increasingly baroque statements, and also many "worked-out" problems showing the steps on how they were done. Even so, some of Schaum's identities need further working-out in order to see (for instance) just how you get from secant to tangent, and cosecant to tangent.

I am determined to learn to work with these identities, and their subsequent double-angle versions, and with trigonometric equations. Pedantic is good for me. I find that I can learn anything, as long as I take it in small enough steps. This is pre-calculus work and it's essential before I go forward. Calculus is daunting. Whenever I talk about math with someone who is otherwise well-educated, I hear a similar story: "I did well in algebra and geometry and even trigonometry, but once I tried calculus, I just gave up and dropped out. It was like hitting a wall and I couldn't go any further." I am still hoping to get to calculus sometime this year, and I wonder what I will do when I encounter this fearsome barrier.

I must admit that my difficulties with trigonometry have recently caused me some unhappy moments, and I wondered to myself and others why I was doing this whole math and science project. It's been three and a half years since Fermilab and I haven't gotten to calculus yet, and am still working on Newton's laws and other simple classical mechanics. And yet I keep going, slow as I am. Why? I went back to the section that stumped me, even though I should have just let it go for a while. I had to do it. Sometimes my drive to do this surprises even myself.

I'm working on a "mission statement" about my math and physics learning project. With it will be a series of reasons why I continue to do this. Some of them will be Wrong Reasons, some of them will be Right Reasons. There will also be Non Reasons, which are just as important.

Correction on info about Leonard Mlodinow

In my review of Mlodinow's book, I stated that M. was an "editor" for Scholastic Books. In fact, he was in charge of media research and development for Scholastic, investigating things like e-books, broadband internet, wireless transmission. He may not be doing that now, but that's what he was doing in 2001. I found an interesting, though not too recent, interview with him at this site.

Posted at 3:11 am | link

I just finished reading a highly enjoyable book by Leonard Mlodinow, called "Feynman's Rainbow" (Warner Books, 2003). Enjoyable, yes, but I admit it was a guilty pleasure. Mlodinow is a nudnik who happened also to be a new PhD in physics at Caltech. As he tells it, they gave him this spiffy fellowship to be there and study anything he wanted within the field, but he simply ran out of ideas. He had no clue what to pursue as a research interest. He had had one brilliant moment in theoretical physics, and after that nothing happened for him. But his office was right near that of the famous Richard Feynman, so he decided to ask the wise man. The book recounts Mlodinow's conversations with Feynman about creativity in physics, how scientists think, and life in general. These conversations were transcribed and edited from tapes which Mlodinow made with Feynman's permission. I find it incredible that Feynman would have allowed such a thing, but I guess he felt generous. Mlodinow bothered Richard Feynman so that we readers wouldn't have to, thus the guilty pleasure. M. has many hilarious and snarky passages describing some of his colleagues at Caltech, fortunately obscured with pseudonyms. He also bravely attempts to describe some of the new physics that was being done at the place, including trendy string theory.

This book also gave me valuable information (more guilty pleasure) about what life is like among the hallowed halls of the world of elite physicists. I've always been fascinated by these specialized slivers of society, whether it be that of priests at the Vatican, mens' secret societies, Jesuits, or Greek mystery cults. They possess privileged and incredibly complex information, understood only by their own kind, which they handle and exchange with a kind of macho nonchalance.

At the time Mlodinow was interviewing him, Feynman was already suffering from the cancer that would eventually kill him, but he kept on working. Mlodinow would appear at Feynman's door, evading the protective secretary who tried to keep unnecessary visitors away from the great man. Despite Feynman's obvious annoyance at Mlodinow's intrusion, Mlodinow managed to gather some interesting wisdom once he had gotten Feynman to talk, which evidently wasn't too hard to do. This "science wisdom literature" addresses many of the questions I've wanted to know about scientists, especially theoretical physicists, for some time. Like: how do they know what to work on? And how do they begin an inquiry? How does an experiment get conceived? What makes a person want to do physics in the first place? What keeps a person doing it when the work is often tedious and unrewarding? How does scientific creativity compare with artistic creativity?

This last one of course is a major interest of mine, and so I'll quote a passage from one of the Feynman tapes on it. Feynman is talking about science versus fiction writing:

"…I think it's much harder to do what a scientist does, to figure out or imagine what's there, than it is to imagine fiction, that is, things that aren't there…But the scientist's imagination always is different from a writer's in that it is checked. A scientist imagines something and then God says "incorrect" or "so far so good." God is experiment, of course…A writer or artist can imagine something and certainly can be dissatisfied with it artistically, or aesthetically, but that isn't the same degree of sharpness and absoluteness that the scientist deals with. For the scientist there is this God of Experiment that might say, "That's pretty, my friend, but it's not real." That's a big difference."

Feynman continues:

"Suppose there was some great God of Aesthetics. And then whenever you made a painting, no matter how much you liked it, no matter how much it satisfied you…anyway you would submit it to the great God of Aesthetics and the god would say, "This is good," or "This is bad." After a while the problem is for you to develop an aesthetic sense that fits with this thing, not just your own personal feelings about it. That is more analogous to the kinds of creativity we have in science."

Well here I must add my own two dinars' worth and say that as an artist who sells paintings and prints, and does commissions, my work is indeed judged by the great God of Aesthetics, and Her name is THE CLIENT. All right, that is a flippant comment. But seriously, I don't quite agree with Feynman. I believe there is an absolute aesthetic standard for art. Some of the criteria I would propose for something being "good" art would be seriousness, that the art addresses universal human or natural concerns especially tragic ones, difficulty, that the art is not easily appreciated by just anyone, but takes some thinking and reflection to enjoy, and technical superiority, that it's done really well. Of course these are my subjective criteria, which I learned from a particular artistic culture, and they are endlessly arguable. I'd like to think that Feynman is right when he claims that science is "sharp" and "absolute," but if so, why do scientists argue with each other so much?

This book belongs to a larger body of literature which has been called "scientific hagiography." It's the account of a poor pilgrim, Mlodinow, who visits the cell of the dying saint, Feynman. Feynman, unlike the somber, restrained saints of Christianity, is a saint for our current era and values: iconoclastic, romantic, brilliant, childlike, and sexy. There's a whole literary industry promoting Saint Feynman, including a "devil's advocate," the other brilliant physicist of his day, Murray Gell-Mann.

As for Mlodinow, after floundering about, and surviving a cancer scare of his own, he eventually gave in to his desire to be a writer, and ended up (to the horror of his physics professors) writing for TV shows including STAR TREK NEXT GENERATION. Now he is an editor for Scholastic Books, as well as a writer on mathematical and scientific subjects. Many thanks to Leonard Mlodinow for giving me an entertaining peek into the world of the scientific major leaguers.

Trigonometry grinds to a halt

I'm having a bit of a Mlodinow moment myself. After attempting twice to learn how to work with trigonometric identities, and failing both times, I'm stymied. I have no clue how to turn these formulas into other formulas, and how to solve equations with them. And yet I'm informed that this is essential for calculus. Vectors were easier. I'm wondering, what do I do now? I begin questioning my whole purpose here. Why am I trying to learn this material? Why do I want to learn physics when there is zero possibility that I will actually do work in the field? (I don't intend to change my career, I still want to do art.) I sat with some very kindly friends last night and recalled the blazing enthusiasm that filled me after my visit to Fermilab back in 2000. I am told that even those elite scientists (not just Mlodinow, but the stars) have moments where they don't know where they are in their work and what they should do next. But unlike Mlodinow, I don't have access to any scientists to ask them, nor would they have the time to tell me. I need to re-assess my motivation and my purpose, and when I do, you'll read about it here.

Posted at 2:51 am | link

Sometimes I have too damn much imagination. What? How can you have too much imagination, isn't that a precious, rare commodity? Not when you're doing mathematics, at least the low-level math that I am currently plowing through. The trigonometry that I'm doing, whether it's solving triangles or finding the components of a vector, demand that the information about any situation be reduced purely to numbers: positions on a graph, distances between the points, measures of angles, and the sines or cosines or tangents of those angles. Anything else is irrelevant. Reductionism rules.

But trigonometry takes place in a scene, a world, a universe. And the universe is full of sense-details, which are endlessly distracting to a would-be reductionist. In this situation, I envy those intellectual souls (mathematical physicists, perhaps) who are naturally oblivious to the world and who find reductionism and abstraction easy. Doing word problems, I find myself involved in problems in ways which aren't mathematical at all. Let's take the beginning of a problem from Schaum's Outlines, the "Red Spine" book:

From a boat sailing due north at 16.5 km/h, a wrecked ship K and an observation tower T are observed in a line due east…

The mathematics student will plot these on a graph or do some other thing which is appropriately reductionist. But when I read this, I am on the boat sailing due north on the calm blue-grey water, into a golden afternoon sky. To the east is the mysterious wreckage of that old ship. How had it gotten wrecked? What had been lost? And further east is the red and white iron openwork of the observation tower, pale in the coastal haze. I smell the sea-brine and feel the wind as our ship steams along at the 16.5 km/h pace. Where are we going? The waves are low, yet I feel the ship's motion under my feet. I take a breath of fresh salt air…

But all I need to do is find the distances the words and data are asking for. A word problem puts me into a little world, whether it's the world of a trigonometric seascape or a chemistry lab or a car accelerating along a highway. Like the synesthetic colors of numbers I talked about some time ago, these irrelevant yet distracting sense-associations accompany most word problems for me.

I thought perhaps this was only a problem for me. Maybe it is a "girl thing," that females are supposedly more concerned with "holistic" sense-data than men are. Or maybe, more likely, it is an "artist thing." I am so used to depicting scenes and creating realistic visual pictures that I can't turn off the artistic imagination when I am doing math problems.

But I am not alone. This phenomenon of getting lost in the details has been documented in an excellent book by Sheila Tobias called OVERCOMING MATH ANXIETY. This book was one of the first ones I turned to at the very beginning of my math journey, as I faced the ordeal of recapitulating painful and difficult parts of my childhood and youth. On page 125, in a chapter discussing different styles of thinking and problem-solving, Tobias cites a (probably autobiographical) passage by Philip Roth, about "Nathan, a sickly and feverish young boy, whose father tried to sharpen his mind by giving him arithmetic problems to solve." The boy, instead of seeing the problem as a simple exercise in arithmetic, turns the details of the problems' situations (a merchant discounting unsold merchandise, a lumberjack buying lengths of chain) into intriguing and poignant stories. Roth continues: "My father … was disheartened to find me intrigued by fantasies and irrelevant details of geography and personality and intention, instead of the simple beauty of the arithmetic solution.…" Author Tobias consolingly comments: "(To distinguish)…"mathematical" and "nonmathematical" minds is to miss what they represent: two beacons in a continuum of human curiosity in search of meaning."

Hmmmm, two beacons. A ship is passing at 16 kilometers/hour on a northwesterly course between two beacons… For Roth, born to be a storyteller, the math problems became stories. For me, accustomed to painting pictures, the math problems become landscapes. Yet, like that steady ship passing by the wreck and the tower, I am determined to sail the sea of mathematics and direct myself in the vector-world of physics. But what do I do when I am faced with the first line of this trigonometry problem in the Barron's book:

You are lost on an endless, flat plain.

I cannot help but feel a sense of surrealistic dread, from which mathematics and physics, two pure abstract beacons, can rescue me.

Posted at 2:11 am | link

Before I get back to math, I have sad news to report. "Spinoza," my Golden Barrel cactus, expired in February 2004. I regret to say that it is my fault. In a frenzy of re-potting, I re-potted Spinoza as well as numerous other plants. The cactus seemed to do well in its new pot but I made the mistake of overwatering it in the wintertime and soon it began to turn brown and finally it collapsed.

"Spinoza" was given to me by a dear friend from the Boston area in the mid-'90s. It was my first cactus, though I've been keeping lots of indoor plants for more than 30 years. The cactus was an Echinocactus grusonii, known as the "Golden Barrel" among other names. Though it's a popular domestic plant, it's nearly extinct in the wild. It grows very slowly, and a specimen as big as a football might be over 20 years old. Mine was probably about 10 to 12 years old at the time of its demise.

I called it "Spinoza" because it was so spiny or in Latin, "spinosus." Unfortunately, I didn't know how to take care of it. In the beginning I placed it in direct sunlight, assuming that as a cactus it would belong there, but I soon learned that this cactus is actually damaged by direct sunlight. It's also affected by cold. It survived sun and cold damage, as well as an infestation of mealy bugs. It grew to about twice the size it was when I acquired it, and I thought that despite its hard life I would grow old with it. But it won't happen, at least not with this cactus.

The carcass is not a pretty sight. I have been watching it collapse for a couple of weeks now. Eventually it will be an unsightly mass of decayed spiky vegetable matter, and I'll have to throw it into the ditch out in back of my house where I throw all my other garden waste to return it to "nature."

I have other cacti now, many of them grown from seed. I know more about taking care of them now. I don't know whether I'll get another Golden Barrel, though. I will conclude with a note about cacti and mathematics. In many cacti, including the Golden Barrel, the tufts of spines are arranged in the famous Fibonacci sequence which also shows up in the patterns of seeds in a sunflower's center as well as in the seed patterns of pine cones. Cacti die, but Fibonacci's sequence is forever.

Posted at 1:24 am | link