My weblog ELECTRON BLUE, which concentrated on science and mathematics, ran from 2004-2008. It is no longer being updated. My current blog, which is more art-related, is here.

Fri, 30 Apr, 2004

String Theory Gets Me a Cup of Fancy Coffee

Yesterday (April 29) I was wandering in the mall as I sometimes do, being a good American consumer, and after my fast food meal I stopped into the mall Starbucks. This particular Starbucks runs a trivia contest every day. They put a question on a chalkboard on the counter, and if a customer gets the question right, they get a free drink from Starbucks, even one of the expensive ones.

When I saw the chalkboard I was delighted. The question was, "In superstring theory, how many dimensions do the theorists propose there are?" I had just recently seen a TV talk by Brian Greene about string theory, so it was fresh in my mind. "Eleven dimensions!" I said to the Starbucks barista. "Right!" she said. I could have any drink I wanted, for free. "You are the only person to get this question right, this evening," she said. "It helps to watch Brian Greene on TV," I replied.

I got a Double Tall Cappuccino with Almond Flavoring, a luxury coffee that would have cost a small fortune. I don't know whether Brian Greene would be thrilled, amused, or appalled to find string theory being used as a trivia question at a mall Starbucks. Should esoteric science be part of pop culture? I would love to sit down with a cup of coffee and discuss string theory with Brian Greene; thankfully for him, it won't ever happen. But thanks, Brian, for the fancy Starbucks Coffee.

Posted at 7:40 pm | link


If I am to reach the promised land of calculus, I have to know and be able to work what is called "precalculus." I have actually been in the realm of precalculus for the last couple of years, plugging my way through algebra, geometry, and trigonometry. Right now, I am working with trigonometric equations, which is more nuts 'n' bolts and playing with virtual Japanese transformer robot toys. When I am through with that, I will finally be more or less through with basic trigonometry. If I am weird enough to hanker for more, it will always be there when I want to return. I will also have my archives of trig notes to refer to when I need them.

I have been looking up various curricula of precalculus studies on the Web, helpfully provided by universities in the English-speaking world. I'm trying to figure out what I am still missing. Some of them are encouragingly familiar to me; I can work out almost all the problems in their placement tests. But they also point out what I still need to know, or what I need to review. I am still quite uncertain about working with functions, function notation, and reverse functions. And I need to learn logarithms. Once I get familiar with functions, and row my way through logarithms, then I will finally be ready for Calculus. I will no longer be "pre-calculus."

I vaguely remember logarithms from high school math, mainly because of the craggy Greek syllables of the word. It sounded like an old train going down the track: logarithm logarithm logarithm logarithm… But I dozed my way through high school math, already long-lost to the promised land back there in my teen years. I forgot any bit of information I had managed to learn about logarithms.

If I had somehow seen into my future, back when I was in high school, that I would not only re-visit the hard territory of trigonometry and logarithms, but do it enthusiastically and of my own will, I would be astonished. But my Math Journey is in many ways a return to childhood as well as a re-affirmation of adulthood. Even though high school students now routinely study Calculus, it was not so in my day, especially for girls (why would they need calculus anyway?). So in my mind, calculus, if I ever get there, will be the first "adult" mathematics that I will study. I get to grow up all over again.

Posted at 1:13 am | link

Wed, 28 Apr, 2004

Herons Flight

I've been spending the last couple of days going over proofs for "Heron's Formula," which is a way to find the area of any triangle if you know or can figure out the lengths of all three sides. If s stands for the "semi-perimeter" of the triangle, that is, half the sum of the lengths of all three sides, then Heron's formula for area is the square root of: s multiplied by s minus the length of side a, s minus the length of side b, and s minus the length of side c. That's (square root) (s(s-a)(s-b)(s-c).).

Heron of Alexandria was a mathematician, physicist, and engineer back in the late Hellenistic world of the first century CE. (That's "A.D." for the older types.) He inherited a Greek and Near Eastern mathematical tradition which stretches back not only to Archimedes and Pythagoras but to ancient Babylon where the Pythagorean theorem and algebra may already have been practiced. Reading this biography of Heron, I find it amazing how technologically and mathematically sophisticated the ancients were. And they didn't even have slide rules, let alone pocket calculators.

If I were asked to prove Heron's formula by myself, I'd be completely baffled. I can do, on my own, only the simplest algebraic and trigonometric proofs; I have yet to really get into the technique of doing more complex ones. But fortunately the helpful Schaum's text has a detailed proof of the formula, and there's another, equally good proof at Jim Wilson's math course site at the University of Georgia.

The proof flies through all sorts of mathematical twists and turns, often relying on trigonometric and algebraic identities and substitution. It multiplies things out, and re-arranges them. It uses old algebra standbys like solving for a single variable, factoring out quantities, finding the difference of squares, and canceling out common factors in a fraction. It also uses trigonometric standards like the cosine formula, which in turn depends on the majestic Pythagorean theorem, which was ancient even when Heron mathematized two thousand years ago.

I could just memorize the formula, which I have indeed done, but the proof of this formula carefully followed is a lesson in proof technique which I will find valuable in my later mathematical work. It demonstrates the use of identity and transformation, how to make one thing into another thing which is equivalent but fits with yet another thing. The proof of Heron's formula, to use another metaphor, meshes, turns, and clicks like antique gears, like one of Heron's own late Hellenistic inventions. It's the technique that I need to remember. Take the identity equivalent of something, and plug it into another expression where you find it, and multiply it out. See if what you get fits anywhere else or looks like something else that could be relevant. I wonder if this is where mathematical creativity starts.

In my usual way of learning math, I wrote the entire proof down on a sheet of notebook paper, turned horizontally. Even with all that space, I had to cram a lot onto the page, and it looks rather like an oriental carpet pattern. But I can read it, and it will go into my archives as a reference.

By the way, Heron's name has nothing to do with long-necked waterfowl. The name "Heron" is probably the English version of the ancient Greek name "Hieron," which comes from the Greek word hieros, which means "holy" or "divine."

Posted at 2:56 am | link

Sun, 25 Apr, 2004

Mockingbird Jazz

Math has been rather slow lately, with much going on in my other, commercial work that is taking my time up. I have been reviewing some material I may not have learned well the first time: tangents and secants and their segments in circles, solving non-right triangles with the sine and cosine formulae, and areas of triangles found through trigonometric formulas. Geometry shows me tangents and secants, but I have yet to figure out why tangent and secant are also names of trigonometric ratios. None of my trig books have seen fit to tell me. In fact Barron's Ruritanians were misleading, talking about "going off on a tangent" rather than explaining how the geometry led to trigonometry.

I feel the need to know this material well enough so that if I see it again next year or later on, I'll at least recognize it and be able to look it up in my reference books. Some important formulae, I must remember forever and always have available, just as I learned the quadratic formula in years past. All told, trigonometry has featured a lot of formula-crunching, with more to come.

Spring is really here now, with the leaves finally on the trees in that ever-so-brief color of brilliant chartreuse, which seems so bright as to be unnatural. But then, the blasting pink and red of azaleas also seems unnatural, and those colors are there too. There are certain colors of paint (or markers) which you can only use unaltered, at this time of year.

The urban spaces are filled with birdsong at dawn and dusk, and sometimes all through the night as well. There is an indefatigable mockingbird who sings all night long in back of my residence, when it is warm enough. He goes on for hours. As a birdwatcher trained from childhood in song identification, I can recognize almost all of the birdsongs he repeats. Some of his imitations are so good that they are indistinguishable from the real thing, such as his red-bellied woodpecker "barks" or his carolina wren calls. Other imitations, such as his "robin," are too loud and frantic to match the sweeter tones of the real bird. This mockingbird also imitates baby birds, crickets, frogs, and even a car alarm siren.

Listening to him for an hour or so, I charted his "song list" to see whether he followed any pattern. I was interested to see that he continued to vary his song imitations, in length, complexity, and dynamics (loudness or softness). For instance he would follow a loud, repetitive carolina wren chant with single soft notes of a woodpecker call, then the squeaky tweets of a titmouse, then the chattering of a robin alert call. It was as if he had some sort of variety generator which reminded him not to go on too long with the same type of call. He reminded me of a jazz trumpeter, riffing away in syncopated rhythms in a long solo. And like that jazz player, it seemed as though Mockingbird had an aesthetic sense about it all — he knew what to play at what time, when to play soft and slow or when to shout it out.

Can a bird have an aesthetic sense? Perhaps he can. Or perhaps, as is more probable, the variations of the mockingbird song are evolutionarily determined as the best strategy to win the attention of female mockingbirds and proclaim one's territory. I suppose one could say the same thing about the jazz musician.

Posted at 1:47 am | link

Wed, 21 Apr, 2004

Making charts

It's back to the studio for me, as I take a short break before proceeding to trigonometric equations. I am also looking forward to re-introducing myself to logarithms, which I briefly encountered in high school but have long since forgotten. But first before actually making any art, I need to update my color charts for markers.

I am a "marker junkie." I can go on and on about markers. I've used this medium for years, especially on commercial jobs. Most markers are made with non-permanent colors and will fade rather quickly, but for commercial work this is not an issue, since (as I explained in my last entry here) commercial signs are temporary. Also, with the availability of good computer scanning, even illustrations with fade-able (the official term for it is "fugitive") markers can be made into more or less permanent digital images. One of the good things about doing commercial work with markers is that you can get the client (your employer) to subsidize your marker habit. I now have hundreds of them.

The variety and technology of markers have improved greatly in the last twenty years. Most of them no longer use the toxic solvents they used to, which made marker work in a closed place into inhalant abuse. New generations of water-based markers have also been created which provide not only the garish colors of the older type of marker but a wide palette of neutralized colors, pastels, and many shades of cool, warm, and neutral grey. With enough of a collection, you can create something in markers which looks almost as good as a watercolor painting. The only thing you can't do is a wide area of blending (a "wash") which still requires blending paint rather than markers.

The art-minded among you can see a great selection of these at the Jerry's Artarama Marker Center. I've used almost every single one of them. I buy compartmentalized plastic boxes at The Container Store, one of the best stores in the world, and I put the markers in there sorted by color and shade and lightness. Then I make color charts which show what colors each marker writes, again sorted by color and shade and lightness, with identifying notes as to which marker it is. This is a painstaking process but it makes using the markers so much easier. When you need a specific color, you just look at the chart, and grab the marker from its compartment in the box. And since the color on the outside of the marker often doesn't match the real color of the marker, your annotated chart will show you the true color.

At this point you may be thinking that I am excessively organized and probably edging into obsessive-compulsive, and you may be right. However, I think that anyone who loves order and color, or who has any artistic impulses at all, will respond positively to a color chart. It brings back the crayons of one's youth, or the colorful toys left behind in childhood. A color chart tempts you with visions of artistic play; it has the sweet potential of an ice-cream menu.

The analogue of these color charts in mathematics may well be trigonometric tables or logarithmic tables. I am not referring to the synesthesia which makes numbers into mosaics of colors (which I have fortunately been able to put aside) but the idea of an ordered array of gradations and their associations. The "Trigonometric Functions of Some Particular Angles" table in the back of one of my math books is filled with allusions to those venerable Pythagorean triangles and their famous angles, 30, 60, 90 degrees, or 45, 45, 90 degrees. The wheel spins, and the table gives us the sine, cosine, and tangent, etc. of the other quadrants' angles too. To a mathematician, each one of these angles has its flavor and its associations. And unlike markers, these angles and functions never run dry and their art never fades.

Posted at 2:39 am | link

Sat, 17 Apr, 2004

The whisper of the calling

It's been about six months since I started my day job doing commercial art for my local Trader Joe's. In that time I've done a huge amount of work for them — everything from more than a hundred reproducible decorative price tags, to small advertising signs for wine and individual products, to large "billboards" advertising whole sections of the store. I've had a lot of fun doing this work, and I continue to enjoy it. There's always something new to do and invent. (Not to mention the free sample goodies and wine-tasting.)

But (there's always a "but") for the last four months, I haven't done a single piece of "original" work for myself. Not only have I not had the time, I haven't been able to decide what to do. Viewing the "Art Renewal" neo-retro art only brought back this feeling of indecision. I used to run my life by the deadlines of clients and the exhibits I would have at science fiction conventions. Now I have largely scaled back my science fiction convention shows, and for now I have only a couple of private clients. The money from the day job gives me more than I made doing freelance art, most of the time, and this should allow me to do what I want to do artistically. I can work around the time constraints, but it's the idea constraints that are currently troubling me.

And of course there's that other matter I've been attending to, that is, mathematics and science. Trigonometry, after these last months where I was stuck in the labyrinth of trigonometric identities, continues to be a struggle. It's not that I can't do the work once I've learned it, it's that I am faced with a diffuse cloud of information, little of it connected with material reality. I look in different books, and it doesn't fit together. It's not like the solid, progressing architecture of algebra and geometry; in trig, it seems one section doesn't follow logically into the next, and one book describes a process (for instance, the plotting of amplitude, period, and phase of a sine curve) in quite different terms from another. The trigonometric equations which are my next subject just show up without any explanation; they are there to solve, like words that make grammatical sense in a language which I don't understand. I patiently learn the formulas and solve the problems, after all, that's what I'm here for, to solve problems.

Or is that what I'm here for? When I read that some of the neo-retro artists believe that their art is a "calling from God," I'm embarrassed. No matter what my religious beliefs, I think that attributing your talent and artistic output to God is a sure way to embarrass God and possibly yourself as well. What if your art is bad? Is God well-served by bad art? And even more confusing is that some scientists refer to their work as a "calling," even though they don't believe in a God or any entity which would do the calling. They are not in it for money or fame, they're in it because they have some inner need to do the work.

I grew up among artists who were like that. Even if their work is never shown and never recognized in public, they keep doing it, because they have an inner, perhaps inborn, need to make paintings or music. I, however, am more pragmatic, and probably more vulgar as well; I need to feel that my art connects and communicates with a wider public than just a few friends. I am not a "pure" artist who creates without thinking of recognition or sales. At the moment, I am uncertain whether I should do work which is designed to be showable and sellable, whether to known patrons or a known audience. Or should I do pieces which are more esoteric but more in keeping with what I am thinking about math, geometry, and physics?

It is not enough, as some friends who think they are encouraging me say, to "just do it." And I am sick of the other ones who offer the platitudes of "one to pay the bills, one to feed the soul." I am not the kind of artist who just sits down and creates stuff on the spot. Doing a major painting for me is like waging a campaign, or perhaps running an elaborate experiment. It takes planning, assembly of materials and references, logistics, and lots of time. I am a very unspontaneous artist. Not that I don't do quick sketches, preliminary drawings, and on-site notes, but all of those are subordinate to the "big picture." "Just do it" or "feeding the soul" will tie up months of my time. It had better be a good "it" that I do.

My commercial art communicates and entertains, but is also ephemeral. Even my best signs for Trader Joe's are only up for a few months, after which they go out of date and are removed, often painted over for another sign. I photograph them all for my archives, but their purpose is gone and they are history. The art that I am beginning to miss doing is the work that I intend to last beyond my lifetime, the work that will carry on the values and qualities and information which I personally stand for. Is that a "calling?"

The same goes for mathematics and science. No matter how confused I get, I never want to give up. There is always something more that beckons to me. In this case, it is the looming wall of calculus. Is math/science a "calling" for me as well? Can one get an additional "calling" late in life, or is everything set out for you at birth and in childhood, by the genetic and environmental lottery? A physicist acquaintance of mine once said to me that he would become depressed and disoriented if he were not able to do physics. Is he "called" to do physics? Am I? What is calling to me, if anything, and how much responsibility do I have to it?

Posted at 3:00 am | link

Thu, 15 Apr, 2004

We will control the horizontal, we will control the vertical

First, a couple of Electron corrections: In my previous entry, I wrote:
"It seems that while us "serious" artists were sleeping in the comfortable darkness of abstract modernism…"
that should be "while WE "serious" artists were sleeping…"

And when I wrote, regarding the new "retro" art and its compliance with my somewhat sarcastic "rules for Seriousness,"
"All of the pieces fit part 2: they appeal to just about everyone, there's nothing baffling or difficult about them.…"
I meant to say that NONE of the art pieces fit part 2, because they are not "difficult" and anti-popular, but are meant to appeal to the public, rather than an elite.

Sines of spring

Here in MidAtlantica we've had an unusually cold, wet, and blustery Spring so far, which has dampened spirits and caused us to yearn for the balmy days that usually begin around this time. The spent flowers are flying off the cherry and plum trees in the wind, causing more Pseudo-Japanery on my part:

Cold day in spring;
A white flurry
of petals in the air.

I turned to Barron's TRIGONOMETRY THE DIFFICULT WAY again to learn about trigonometric graphing, hoping that our little fantasy characters in their Ruritanian world would have something I could work with. But once again I was bewildered by them. The Schaum's book was equally cryptic. So I returned to the very helpful Website which I mentioned in my previous post, : Sharon Walker's math site. The sine curve pages in her "beginning math" tutorial (they're somewhere in the "Math 170" course pages) were more helpful to me than either of my trig books. She explained clearly and with good mnemonic anchoring just how the coefficients in a sine curve equation determine how high the curve is (amplitude) how stretched-out or squished a sine curve is (period) and how shifted it is to left or right (phase).

As I remember from my electronic music days, each of these variables changed the sine wave sound on the oscillator. I wish I had that dusty old oscillator again! I could turn these sine curves into noise. Remember "The Outer Limits" with the sine tone on the TV screen. "We will control the horizontal, we will control the vertical."

I'll be returning to Ms. Walker's math site often. She has just what I need, including review tests. Yes, just what I wanted, a way to re-live the pressure and horror of high school math exams!

Posted at 2:52 am | link

Tue, 13 Apr, 2004

Is this art bad, or what

I often wander through the Web looking for art and have returned again and again to the Art Renewal Center site. Just recently they have held their first "Salon Competition" (just as one did in the nineteenth century in Europe) and the results are displayed here. The winning piece is by Daniel Gerhartz, a Wisconsin-based artist whose paintings recall the style of John Singer Sargent or even some of the French Impressionists. The winning Salon picture is inspired by a mystical, ultra-sentimental twentieth-century religious allegorical tale, "Hind's Feet on High Places" by Hannah Hurnard, a book I detested. However, I am quite fond of the painting, especially for its depiction of evening light and the graceful figures of the angelic ladies.

It seems that while us "serious" artists were sleeping in the comfortable darkness of abstract modernism, a whole new movement has sprung up in the arts, both in visual art and in music. The visual artists who paint work like that which is depicted in the Salon site are unabashedly realistic and retro. Some of them, like Gerhartz, actually advertise that they are religious and they dedicate their work to God. These "realistic art" sites (there are many others, for instance the gorgeous site of former Disney artist Christophe Vacher) often venture into surrealism, but all of these artists paint with a nineteenth-century precision and "classicism."

Similarly, you may hear, if you are one of the few people paying attention to contemporary "classical" music, pieces which are no longer written in the jagged, harsh, atonal style popular in the middle to late twentieth century. An example of this newer (or older?) style of music is Christopher Theophanidis' "Rainbow Body" which won the 2003 "Masterprize," a new award for modern classical work. "Rainbow Body" is outright tonal, using conventional harmonies, and has melodies based on the medieval chants composed by the visionary abbess Hildegard of Bingen whose work has been rediscovered by artists and trendy religious types in the last twenty years or so. Tonal harmonies, realistic art — what's going on here?

If you are used to the modernism that has ironically become "conventional" (!) you may think that these works of art and music are, well, kitsch. The visual pieces are tepid re-makes of Sargent or Monet or Ingres or Albert Bierstadt, who were all fine for the nineteenth century but are irretrievable as style in the twenty-first. The music sounds like movie and TV music, easy listening for a crowd weaned on the commercial mass banalities of the big and little screen. Does this art pass the "seriousness" test? To recap this test, from an earlier posting of mine on this Weblog: "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."

A few of these pieces fit part 1, universal and tragic concerns; if you look through the Salon collection you will find a 9/11/01 memorial, or an image of love and loss, done in a surrealistic or allegorical style. But most of these pieces seem unconnected with specific themes. All of the pieces fit part 2: they appeal to just about everyone, there's nothing baffling or difficult about them. And part 3: most of them are done with amazingly good technique, in front of which I grumble with envy.

So why do I, and many other artists inevitably raised with twentieth-century standards, feel so ambivalent about this art? Would I do this kind of art if I had the freedom to do so? Would I buy this art? I am amazed (and even more envious) when I see that pieces by these artists and others like them are being sold in galleries around the country for big bucks!

More grumbling. Of course they sell for big bucks, 'cause they've sold out to the tacky, commercialized, media-sloppy mainstream. Their main subject matter seems to be lovely ladies, gracefully posed in swirling garments (or no garments at all). Pretty women always sell! Damn, I've never been able to paint images of lovely women, no matter how many times I've tried. Cows sell, too. Should I paint images of cows? Moooooo, oh, so boring.

But wait a minute. Is this really bad art? Is this selling out to the media-numbed public rather than elevating the aesthetically inferior masses to the difficult abstraction and erudition of "high" art? Honestly, I'm not quite sure. The artists who paint this "middlebrow" art write manifestos (try the ARC Philosophy section on the main art site) about how they reject the sterility, harshness, hateful irony, elitism, and just plain meanness of the "contemporary art" scene. Well, there's a lot of that out there, you can't deny it. And these retro artists also stand against the current trend of "soulless" mechanization of art in digital images, "installations," and other forms of gimmickry. These artists actually learn to paint the old-fashioned way, with paint and brushes on real canvas and boards.

So I am left with an unsettling mixture of snobbish contempt and artistic envy. It's kitsch, and I'd do it if I could.

Meanwhile, I'm contemplating other visual images: sine curves in all their undulating glory. I think I'll put a sine curve into my next painting. It passes the "seriousness test" on all three counts. It's universal, it's difficult, and it contains within its mathematical purity, technical perfection.

Posted at 2:57 am | link

Sun, 11 Apr, 2004

My Lenten reading and spiritual light

Lent is over, the six weeks of spiritual work are done, and I wish all my Christian friends a Happy Easter.

The usual notion about Lent is that you give something up, but I don't usually find much improvement in my life even if I do give something up. Rather, I take something on for Lent, something good that I would not otherwise do. This Lent, I read writings of the fourth century AD Christian theologian, Gregory of Nazianzus.

Gregory of Nazianzus (c. 329-390 AD or CE) is not well known in the West, but Eastern Orthodox Christianity holds him in high honor. He was one of a group of Eastern Christian theologians and writers around the end of the fourth century whose learned culture was very much influenced by Greek philosophy. He lived in what is now southeastern Turkey. Reading Gregory's writing is a look back into an era when Christianity was still a fairly new religion and was still working out its doctrines using the intellectual tools of the time: Stoic and especially Neo-Platonic philosophy.

The text I read was a series of eight theological and philosophical poems on Christian teachings and doctrines, known as the "Poemata Arcana." They are "arcane" not in the sense of occult or weird, but in the Greek sense of "mystery," trying to make sense of what is ultimately ineffable. The poems are written in the same hexameter (six-measure) meter as Homer's Iliad and Odyssey, in a resounding and Byzantinely baroque style. My book has both Greek text and English translation, plus plenty of notes.

I first encountered this text when I was in my senior year at Brandeis and was working on Late Roman Christian Latin poetry. One reference pointed to Gregory as a source that the Latin poets copied, especially with his treatment of light and radiance-related images and themes. Returning to these poems I see again how impressive his light-metaphors are, especially since he had to mix literary poetry with heavy-duty theology. Gregory was intimately involved in the forging of the Nicene Creed and other Christian doctrinal formulas, and these poems reflect some of the bitter conflicts which characterized the doctrine forming process. This can lead to long passages of polemics and dry logic-twisting, but it can also lead to shining lines of bright metaphor:

"For, just as in former times teaching brought to light the full Godhead of the sovereign Father, while enlightening only a few wise mortals… so later, when revealing more clearly the Godhead of the Son, it manifested only half-hidden gleams of the shining Spirit's Godhead.…" Poem number 3, "On the Spirit," lines 24-27, translation by D.A. Sykes.

In Greek I think this sounds wonderful even to those who don't know the language:

"… pauroisin pinutoisi phaeinomenon meropessin
hos kai Paidos epeita phainoteren anaphainon…
(poem 3, lines 26-27)

Which means, "to a few wise mortals, so later when revealing more clearly the Godhead of the Son…"; note how the root-word phaino is repeated in three different forms in just two lines. It's where our word "phenomenon" comes from and means "bring to light" or "appear."

The most important philosophy of Gregory's day was Neo-Platonism, which believed (as did the original Platonism) that our world was not the only world, and that there was a higher world above (or in another "dimension" of) ours in which the perfect archetypes and "blueprints" of everything existed. This higher world especially contained ideas of mathematics, music, art, beauty, and science. This was a "world of Light," and to the mystical Neo-Platonists, you could access that world directly with your "Nous" or intellect, if you lived a disciplined, educated, and virtuous life.

Here's Gregory again, this time talking about the "rational natures" of beings both human and angelic:

"Even as a sunbeam, traveling through rain-heavy, calm air, encountering clouds in its refracted, revolving movements, produces the many-coloured rainbow curve, everywhere around, the upper air gleams brightly with many circles dissolving towards the edges; such is the nature of lights also, the highest light always shining brightly upon minds which are lesser beams. There is one who is the source of lights, a light inexpressible, eluding capture, fleeing the speed of a pursuing mind whenever it approaches, for ever outstripping the minds of all, that we may be drawn by desires to a height which is ever new.…" (Poem 6, lines 1-11)

The poet is using the metaphors not only of philosophy but of the science of his day to (literally) illuminate his reasoning. Gregory's description of the rainbow and the atmospheric effects of light refraction, awkwardly translated by Sykes, are almost "scientific" even by our standards. Indeed, in a previous poem in the series, Gregory takes apart astrology with a logical attack worthy of any modern Skeptic. Of course, since Gregory uses all of his philosophical and poetic resources to support a supernaturalist Christianity, he will never be acceptable to a modern scientific atheist; nevertheless, his brilliance and invention are stirring. I am filled with admiration and nostalgia for a rare Christian intellectual and philosophical life which has long since disappeared.

I can't help being reminded of the modern quest to solve cosmological and quantum problems when I read Gregory's lines about "the source of lights… eluding capture, fleeing the speed of a pursuing mind whenever it approaches." This is a nouetic passion of the mind, which some people even now will recognize. The yearning of both the theologian and the scientist is beautifully described by the quintessentially Neo-Platonic spirit of the last lines I quote here: "That we may be drawn by desires to a height which is ever new."

Again, Happy Easter, and may all your worlds be renewed!

Posted at 3:31 am | link

Thu, 08 Apr, 2004

My life as a step-by-step proof

Studying mathematics has profoundly changed my life, but I'm not sure whether it is for the better. It has brought out some personality traits I always had, and enhanced them. I have always been somewhat un-spontaneous and methodical, and fond of precision and neatness; now I am terribly methodical and determined to seek precision, not only in mathematics and ideas but in language. I am constantly proofreading everything I read, including throwaway memos and graffiti.

I have always been prone to overelaborate details; just a glance at my fantasy city panoramas will show just how far I can get carried away. This wild proliferation of details attracts people, and has sold a lot of prints for me, so I can't fault it altogether. However I have gotten quite tired of hearing the same thing for years and years: "Look at the DEEEE-TAILS!" What if I made art work which did not have all those deeee-tails? Would it be rejected by viewers and not sell? Would it suggest that I was "slacking off" and being lazy, making a less detail-work-intensive piece? I used to jokingly say to clients that the price of a picture was "a dollar for each detail." Would I have to charge less for a simpler picture?

At work, I have the task of lettering hundreds of price tag signs, in which the name of the item, the price, and the weight must all be correct. The management knows that I am the one to do it. I also proofread all the other signs in the store, and catch many mistakes. I can catch mistakes in other stores, or in books, Websites, or even in advertisements, but I can't correct them, which causes me much distress. Woe to those who use the Dread Apostrophe, "it's" as the possessive of "it!"

Doing proof after proof seems to fit my methodical, step-by-step temperament. It's not a temperament that is attractive to our pop-romantic modern society, so fond of hot-sexy impulse and dash, high intense emotion and following "the heart" and "it's" intuitive, passionate urges. I know I am a nerd, and doing math increases nerdness. Well, so there it is; I celebrate nerd pride, along with my nerd heroes who ponder quantum cosmology and string theory.

I set out the details of my ordinary life in proof-like lists. Here's an example. I have a small water fountain which sits on a desk in my studio next to my drawing table and my math space. I have to clean the fountain every month or so to keep the water clear and fresh. Last night I cleaned the fountain. Now the fountain is sitting in its place, dry and disassembled. I won't assemble, fill, and activate it yet, because the collection of little glass cat sculptures (details! details!) that I keep around the fountain is all dusty and needs to be cleaned. If I assemble and fill the fountain with the sculptures in place nearby, I might knock one of them over and break it. But I can't wash the sculptures, because the sink is full of dishes from tonight's dinner, which I have to clean first. I can't do math because the sculptures and fountain parts are all over the place where I usually do my math. And there are papers and finished but yet-unsent tax forms (arrrgh!) under the sculptures and fountain parts. So what is the proof here? Given: dirty dishes, dusty tchotchkes, math paper and books buried under essential but uncleaned stuff. Prove: you can get your studio neat and in working order before the end of the known universe, or better, before the weekend.

Posted at 1:46 am | link

Tue, 06 Apr, 2004

FIDO Trigonometry

I am delighted that ELECTRON BLUE is attracting thoughtful readers, some of whom have sent me comments. I'm even more delighted that so far the comments have all been helpful and kind. No rude remarks yet.

My Friendly Mathematicians have also been helpful in my latest tough passage. One of them explained the matter to me, and I will sum up what he said. Trigonometry is one of those kinds of math that can go on and on into years' worth of complexity, but unless I am one of those very rare people who have a passion for trigonometry itself (there may actually be some of these) I can learn the basics on my way to calculus and then go back if necessary to learn what I missed, if it's required. The Friendly Mathematicians advised me to go on into trigonometric graphing and equations, which is more "pre-calculus," after which point I can start the Big C itself.

What? You mean I don't have to do all the proofs about the half-angle re-statements of the products, sums, and differences of sines and cosines?
Well, not right now. I've had enough of this.
One of my friends once taught me an Army expression for what I must do now and the proper attitude to have. It's an acronym, spelled FIDO. It has nothing to do with someone's dog. The acronym stands for F**k It, Drive On.

So I'm getting back in my virtual shining blue Electron car and starting it up again. I've got some very winding switchback roads to travel.

Zoroastrian Connections

Those readers who know me well, or who visit my main Website, know that one of my major interests is the ancient Persian (and still-living) religion of Zoroastrianism, which may well be the first monotheistic faith. Zoroastrians come from Iran, but also from Bombay and the Gujarat region of India, where they are the famous Parsees. In the course of my religious studies I've met Zoroastrians from all over the world, and have visited Zoroastrian communities all over the USA and also in Canada.

Many of these Zoroastrians are scientists, and Zoroastrians have made major contributions to science both in India and in the countries they have emigrated to. I've met Zoroastrian physicists, chemists, doctors, medical researchers, biologists, engineers, and of course many computer professionals.

There are social reasons why so many Zoroastrians are scientists, engineers, and other professionals; the first generations of South Asian immigrants are highly motivated to enter these fields and do well in their adopted countries. But I believe there are also religious and cultural reasons. Zoroastrianism, ancient though it is, exalts rational understanding as a Divine virtue, and its scriptures, at least the earliest ones and the words of the founder, Prophet Zarathushtra, encourage inquiry and reflection, rather than blind faith. The material world shows forth the work of God's Law, and Zoroastrians are encouraged to work within the world and redeem it (from evil and chaos) rather than turn away from it.

You may have heard of Zoroastrians as "fire-worshippers," but that is an insult. Zoroastrians use fire, or a flame set in a vessel, as the central symbol of God. Their temples have eternal flames, some of which in India have been kept burning without interruption for centuries. Fire for Zoroastrians, maintained and under control, is a symbol of light, warmth, energy, and protection. They pray before it the way Christians pray before a cross or Jews before a Torah or Muslims in the direction of Mecca. It is a symbol. They don't worship it.

One of my Zoroastrian friends, who is a physical chemist, has given me a wonderful big color poster of the "Visual Elements Periodic Table." Those of you with broadband connections can enjoy this rendition of the Periodic Table of the elements, done in dazzling and colorful computer graphics with clever mythological and cultural allusions. It is a fine example of the LOGOS (the rational, abstract pure structure) of science meeting the MYTHOS (culture, color, mythology, and theater) of art.

I wish my Jewish friends a very happy Passover, and I also wish my Zoroastrian and Persian friends a happy NoRuz, or Persian New Year, whose two-week festival has just concluded.

Posted at 2:29 am | link

Sun, 04 Apr, 2004

The Trigonometric Slot Machine

But first, a bit of pseudo-Japanery:

Sky clears,
Moonlight on cherry blossoms;
I'm still doing trigonometry.

I was talking today with a good friend of mine in Starbucks (my home away from home) about my mind-numbing quest to learn and solve all the intricacies of trigonometric identities and formulas. It seems, I told her, that the more I do these things, the more complex and recondite they get. It's not enough now to just work with sines and cosines and tangents, I now have to work with addition and subtraction of angles, double angles, squares and square roots of of sines and cosines and their many equivalent re-statements, and now half-angles and their many equivalent re-statements. The problems are proliferating in an almost fractal-like pattern of complexity, where equivalents of equivalents of equivalents combine and recombine in hierarchical intricacies.

I have no idea of whether these are going to be important in my later studies or not, so I have set myself to learning all of them in the book. For all I know, someday a piece of math or physics will hinge on whether I know the deconstruction of cos4x into (cos2x)2 and then into half-angle restatements. I'll probably have forgotten it anyway by then, and have to go looking it up in one of my old textbooks. Yet I have to go through it: it's in the book, so it must be important! I have to go through it, yet I am getting slower and slower and getting less out of it day by day.

My friend said, this sounds like you're playing the slots at Atlantic City! You're putting coin after coin in the machine, and pulling the handle, and spinning the wheel, but you're not winning anything. You keep playing, though, hoping that you'll hit the jackpot and a flood of mathematical understanding will pour out of the machine. But it won't happen. It's like an addiction.

I said, You know, it's true. It's like playing the slots. Every so often, I put in the coin and spin the wheels, and a handful of coins comes out in a little win. That's like when I actually solve one of these on my own. But then, since I'm feeling lucky, I try again, and again, and again, and I find more and more complex problems, and I don't solve them, and I've lost my money again.

So what do I do? What if I just skipped some of the stuff in the book about the more complex trigonometric identities, and went on to what I need more (at least in the context of my math studies) which is trigonometric graphing and equations? Eventually, I'll have to do just that. Because I can see that there's something about this ever-multiplying, ever-complexifying quality of trigonometric identities which can drag the unwary math-student in and never let them go. I have not had this type of experience recently, and it's kind of spooky to recognize it again.

Posted at 3:01 am | link

Fri, 02 Apr, 2004

Role Models

The April 5, 2004 issue of NEWSWEEK has an article by a woman engineer who describes how she chose engineering and became "the only female engineer at (her) company." Since this article is only going to be available for another few days, I'll quote from it here. The author recounts how she had no idea of engineering as a high school student until she went to a "six-week summer program designed to interest girls in engineering." She did well and got her degree, and plenty of job offers, in engineering. Her story is so inspiring to me that I will quote it at length:

"I can't help shuddering when I hear about studies that show that women are at a disadvantage when it comes to math. They imply that I am somehow abnormal. I'm not, but I do know that if I hadn't stumbled into that summer program, I wouldn't be an engineer.
When I was growing up I was told, as many students are, to do what I am best at. But I didn't know what that was. Most people think that when you are good at something, it comes easily to you. But this is what I discovered: just because a subject is difficult to learn, it does not mean you are not good at it. You just have to grit your teeth and work harder to get good at it…"

She goes on to tell how a sympathetic teacher allowed her to re-study for and re-take math tests she had failed, and how she eventually succeeded, with determination, in college. She continues:

"But the guys in my classes had to work just as hard, and I knew that I couldn't afford to lose confidence in myself… So I reminded myself that those studies, the ones that say that math comes more naturally to men, are based on a faulty premise: that you can judge a person's abilities separate from the cultural cues that she has received since she was an infant…
Here's a secret: math and science don't come easily to most people. No one was ever born knowing calculus. A woman can learn anything a man can, but first she needs to know that she can do it, and that takes a leap of faith…"

I'm not so sure about that last paragraph though. The biographies of male physicists that I've read (including Wheeler, Feynman, Freeman Dyson, Murray Gell-Mann, and the mouthy Magueijo among others) almost always state that the young budding scientist taught himself calculus and differential equations at a very young age, (some by eleven, others at sixteen) just by himself, in a matter of just a few months! It is as if these guys WERE born knowing calculus, and only needed a little review to pick it up again.

When I ask why I can't do this and learn higher mathematics at such dazzling speed and with such ease, the usual counter-question is, "Who do you think you are? You think you're as smart and gifted as (name that physicist)? Don't compare yourself to geniuses! Get real!"

I can't help comparing my situation, though, to the woman engineer who wrote the Newsweek article. She's young and mentally strong. She isn't fifty years old. I had no idea, in high school, that there were summer programs designed to interest girls in engineering, and if I had known, I wouldn't have signed up — when I was in high school, I wanted to be a writer and artist. I have no idea what has come over me in my baggy middle years. It's not like I'm going to go back to school or have a career in any form of engineering or science.

I am still struggling over the same trigonometric identity, multi-angle and double-angle problems that I have been working with for the last three months. I can solve some of them, but not all of them. I don't know how much of this I have to do before I can go on. Do I have to successfully solve every one of them? These interlocking fences of trigonometry stand in the way of my proceeding in my math studies. I may have a great deal of determination, perhaps to the point of obsession, but I have limited energy. And yet I know I will grit my teeth and be back at it tomorrow.

Posted at 1:45 am | link

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