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 It is no longer being updated.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.
Wed, 30 Jun, 2004
This week, there is a big String Theory conference in Paris, called Strings 2004. Hundreds (there are that many!) of string theorists have descended on the City of Light to give presentations and practice their craft. Among them is a Canadian physicist currently at the University of Texas named Jacques Distler. He runs a Weblog, simply titled Musings, which I visit regularly. I am in awe of this guy's intellectual complexity and mastery not only of incredibly difficult mathematics and physics, but also computer theory and coding. This week he is posting updates from the Paris string theory conference. Here's a sampling of Distler's reportage:
Sen gave a very beautiful talk on 1+1 dimensional noncritical string theory. He worked out the relation between the infinite number of conserved charges of the Matrix model and the infinite number of conserved charges of the continuum theory. This relation was known at ? =0 a decade ago. Sen's contribution was to generalize the formula to nonzero.…he then went on to discuss the long-standing puzzle of the subject: where is the 2D blackhole in the Matrix model? The last talk of the day was by Ofer Aharony about the deconfinement transition in large-N gauge theories at finite volume. For small enough volume, the transition can (interestingly enough) be studied in perturbation theory…For the free theory, the path integral reduces to a unitary matrix model (for the Wilson line around the thermal circle), and one finds a Hagedorn transition (for free N=4 SYM on S 3 ) at T H =-1/(log(7-43 )R.
Gosh, I love it when he talks like that. I don't understand a single word of it, at least in its context there, but it sounds serious to me. It sounds arcane, and enticing.
One of the reasons I am studying math and physics has nothing to do with wanting to know the workings of reality. I want to know math and physics because they are difficult, arcane, esoteric, and massively complex systems. I have a "thing" for that kind of knowledge. If you read my most recent entry, from June 29, you remember my mention of Theosophy. This is an equally arcane and complex body of knowledge, but as any scientist will tell you, there is no comparison between the unreal occult phantasms of Theosophy, and the attempts of string theory to uncover the ultimate nature of the physical world. Or….is there? There are some scientists who point out that there has been no experimental confirmation of anything that string theory presents. At least, not yet. Is string theory science, or is it a kind of theosophy, except without the gods and souls and cycles of incarnations?
That last question is, from a person at my level, extremely impertinent. You might say that my entire science/math quest is an exercise in impertinence. Perhaps it would be more appropriate for me to take up spinning, knitting, and weaving, as many of my friends have. As I advance, I continually remind myself that there is no practical purpose for my doing this study. I cannot expect any career to come of it, at my age. Yet I am not doing it, as one Friendly Scientist insists on telling me, as an "exercise for my brain" so that I won't grow senile as I grow older. I am doing it, as I mentioned here last time, to be raised to a higher power.
Will I ever be raised to the intellectual power of Jacques Distler, or the other participants in Strings 2004? Will I ever understand anything they publish? Should I even bother trying to advance that far? It is almost a certainty (well, at least 99 percent, anyway) that I will never work with string theory, or any other physics/mathematics work, at the level of one of the conference participants. And yet I am willing to keep going, anyway. Impertinence indeed! As the British might say, I am getting above myself here. But I don't care. If there's even a tiny chance that I might eventually attend a conference like Strings 2004 and understand and enjoy it and even ask the right questions, I'll keep going. See you at Strings 2021, Jacques.
Posted at 2:49 am | link
Tue, 29 Jun, 2004
Involution and Evolution
I'm decoding logarithm expressions in my 1958 book. The problem sets have sneaked in e without telling me that they were going to do this; e hasn't been mentioned in the text yet. As usual with my math learning, all this is puzzle-solving with no context whatsoever. But I give things context even when they don't have any, because I'm too imaginative for my own good. Logarithms, you see, are about the very basic patterns of reality. The logarithm world is also rich in evocative words.
Much earlier in the book, when the text is explaining exponents, I find these lines:
The process of extracting roots is known as evolution. However, special methods for extracting square and cube roots were developed independently of any general theory of evolution.
While the process of raising any number to any integral power, known as involution, leads to a single result, evolution does not necessarily lead to a single result… if b is positive, it has two square roots, numerically equal but opposite in sign. (Page 76)
I have not found these words used this way in any other of my math textbooks. The "process of extracting roots" sounds painful, but I must remember that numbers feel no pain, and they never protest no matter what you do to them. But what other word would you use to describe getting the square, cube, etc. root of a number? If you said "rooting," it conjures up images of trained pigs and dogs finding truffle fungus under trees. Or perhaps a gardener taking cuttings of some plant and sticking the stems in potting soil, hoping that they will sprout. But that's propagation, another branch (or offshoot) of mathematics.
And what about squaring or cubing a number, or "raising to a power?" This sounds much more high and mighty than "rooting." I myself would love to be raised to a power; it sounds rather Nietzschean. Here it involves multiplying a number by itself, any number of times. The word multiplication itself comes from the Latin for "many folds" and so you are folding something again and again when you are multiplying it. This especially seems true for squaring and cubing and so forth, where it gets folded on itself over and over. The word exponentiation may cover it, but I haven't found that in a math text either. Somehow (and this is a big "somehow," for you philosophers of science out there) the process of multiplying something by itself, or folding it, is a basic part of the world, and shows up in all sorts of monumental formulas revealed by Pythagoras and Newton and Einstein and other heroes, including Hero (of Alexandria).
The book's definition of "involution" and "evolution," to me, seems counter-intuitive. "Involution" should be the rooting, not the raising, and "evolution" should be the raising. We associate "evolution" with old-fashioned upward-striving and reaching toward "higher" forms of life, whatever they might be. But according to the 1958 book, "evolution" or getting the square/cube root makes you smaller (unless you are less than or equal to 1) and "involution" makes you larger (unless you are less than or equal to 1). Throw i into the picture and you get all sorts of mirror involutions and evolutions. Go ask Alice! It is no accident that the creator of "Alice in Wonderland" was a mathematician.
So when you take a square root, are you "evolving?" If you exponentiate, are you "involving?" Fractional exponents are even more convoluted (convolving?) because they do involution and evolution at the same time! A fractional exponent will have you square the cube root of a number, or cube the square root of a number, or any sort of root-raise combination that doesn't even out to 1 or an identity. Evidently, if you do this enough (and even exponents have exponents, and their exponents have other exponents, just like big fleas having little fleas upon their backs to bite'em) you get a logarithm. I eagerly await further clarification on this, as well as….e.
To make things even more complicated, "involution" and "evolution" were taken up and used as metaphysical terms by Madame H.P. Blavatsky, the founder of the nineteenth-century esoteric philosophy known as Theosophy. For Theosophy, "involution" describes the motion of "spirit descending into the material world" and "evolution" describes its opposite, the evolution of matter back into spirit. If you take the time to look into Theosophy, you will find that it is a vast swirling stew of pseudo-science and jumbled religions and philosophies, which continues assimilating anything it can get, even up to the present time. The pure-minded science and math devotee should turn away immediately; the purest-minded of them don't even know about it. But since I come from the netherworld of art, I am familiar with Blavatsky and her proliferating multiverse. When I see all these words and concepts, which mean so many different things depending on where you wander or wonder, it's difficult to separate myth from math.
Posted at 3:32 am | link
Fri, 25 Jun, 2004
Fighting the Commies with Mathematics
The year was 1958. Elvis was the King, "At the Hop" by Danny and the Juniors was the year's number 1 song, colorful geometric forms dominated graphic design as well as pop architecture, and "Gigi" won the Oscar for Best Picture. I was five years old, and was not a mathematical child. I was fascinated by volcanoes, lived in a fantasy world of my own, and was obsessed with superheroes. Nothing much has changed with me, except the math.
It was the height of the Cold War. The "Eastern Bloc" Communists, and the Soviets, were threatening. In the world of science and space, momentous things were happening. In the fall of 1957, the Soviets launched the first artificial Earth satellite, Sputnik I. It set off a panic in the United States as the Americans realized that they were in danger of losing the Space Race. In January of 1958, a year which was designated as the "International Geophysical Year," America launched its first satellite into space, "Explorer I." 1958 was the year that NASA was established, as America's answer to the Russian space challenge. Now the race was truly on.
America needed scientists, all types of them, to combat the Communist threat and win the Cold War not only on the earth but in space. America especially needed space scientists, who could design rockets and missiles and satellites and spy gear and all the things we would need in this new world. Astronauts and manned spaceflight were still a few years in the future. The heroes would be the men who could fight the Commies with science.
The image of the scientist, still powerful even now, comes from the fifties: a man (always a man) in a white short-sleeved shirt (or even that white lab coat) with the archetypal pens and pencils and plastic pocket protector, with ill-fitting black pants, white socks and black shoes, hair cut short and thick glasses. Despite decades of hip scientists trying to defy the image, Nerdman is still around. In 1958 he had a slide rule; now he has all sorts of computer, wireless, and virtual-reality gadgets to add to his arsenal. America still needs nerds!
My College Algebra book is dated 1958. When it was time for me to go seriously into algebra, back in 2001 (2001! THE FUTURE!) I went to my friendly local used bookstore and rooted in the dusty stack of math books. This one was small, compact, and best of all, it cost $5. A modern, current college algebra text, complete with CD and animated website, runs almost $100. I did the math, took "College Algebra" home, and gave it a dusting.
COLLEGE ALGEBRA, Fourth Edition, was published by Ginn and Company, and was authored by Joseph Rosenbach and Edwin A. Whitman, as well as Bruce E. Meserve and Philip M. Whitman (a relative of Edwin, perhaps?), all professors at Mid-Atlantic area universities. Meserve and Rosenbach were already known as mathematics textbook writers; my superficial research reveals nothing about the Whitmans.
The book is no bigger than a current trade hardcover, smaller than the big modern math textbooks and much easier to carry around in a bookbag. Unlike the slick graphics and bright colors of modern books, COLLEGE ALGEBRA's cover design is a dull tan, with a teal-blue spine and the title in white printed on a red rectangle. Whatever dust jacket it may have had is long gone. Also printed on its cover, in red, (as part of the original graphics) is a mysterious code number, "E 425," which has intrigued me ever since I got the book. What does "E 425" mean? Was it a course number? A library code? A signal to whomever used the book that he was at a certain level of study? I may never know.
The interior printing is small, only in black with no other colors, and occasionally poorly registered so that some numbers or letters are incomplete. This sometimes confused me, as I didn't quite realize that for instance, this "6" might really be part of an "8." There are very few typos in the book, which suggests a high level of proofreading, or perhaps just better math literacy in that antediluvian world.
The book begins with "Fundamental Operations" and goes through factoring, exponents, functions and graphs, equations linear and quadratic, systems of equations, inequalities, complex numbers, determinants, progressions, and more. This book was with me all through 2001 and 2002 as I struggled my way through college algebra without a college.
Working with this book is like going into a time warp. You might think that math is timeless, and it is, but the way it's presented is not, and it's influenced by its era. While pop culture was frivolous and flamboyant, college algebra (at least for the dedicated ones) was the realm of the slide rule men who didn't have a flamboyant thought in their crew-cut heads. The word problems reveal the world of the college algebra student of 1958: baseball statistics, exam grades, basketball players' heights and scores, airplanes and aircraft carriers, bullets and targets, and farmers plowing areas of land. The time warp really kicks in for questions like these: "Paul Jones won $64,000 in a TV contest, put aside $28,000 for taxes on this income, and split the balance with his consultant…." And then: "An office boy went to the post office and bought 425 (is that "E425?) stamps, some 2-cent, and some 3-cent…" Poor office boy, now he is in his 60s and he is buying stamps for 37 cents, if he's buying them at all! To give even more historical math perspective, the Ginn text cites some problems taken from ancient Greek and Egyptian math texts. Now that's really a time warp.
This book, in its sober determination to make engineers out of college boys, matches my own retro, second-youth determination. It also works the way I do, step by step, methodically, setting out rules and proofs and offering scads of problems, each one increasing in difficulty and complexity. Like many math books even today, it only offers answers for the odd-numbered problems, leaving me to do the even-numbered ones in mystery and suspense. An introduction at the beginning of the book states that a separate pamphlet with all of the answers was available for the teacher, but any copies of that have disappeared into the vast Cartesian plane of history. I would be fascinated to see whether any of them still exist.
Chapter 15, the chapter on Logarithms, begins on page 369. (Not page 425, which contains problems in the "Probability and Combinations" section.) The introduction reads:
"Historically, the importance of logarithms has lain primarily in their usefulness in computation. With the growth of computing machines, the value of logarithms for computation has decreased, though it still remains substantial because of the cost and complexity of machines.…"
It will be quite a while before I get to the more complex logarithmic material and the transcendental number e. This is all new to me, even though I vaguely remember something about logarithms in high school. The more problems I solve, the better. I am proceeding with interest (perhaps even compound interest) through these pages and am currently doing problems in the second subsection, about multiplying and dividing logarithms.
I have two other algebra books with logarithm sections at my disposal. One is a college text, unpretentious and in good condition, from the comparatively recent future of 1994. The other is another one of those Barron's study aid texts, and features the same Ruritanian fantasy setting (oh no!) that my "Trigonometry Made Difficult" text had. I don't know whether I can stand a return to Ruritania, but perhaps the exponential excitement will carry me through. Right now I prefer to work in 1958.
Three cicada cycles old
Today, June 25, is my birthday. My life so far has encompassed exactly three cycles of the periodical cicada Magicicada septendecim, or four cicada emergences. A little math will tell you how old I am. I had no experience of the first two, since I grew up in a place which did not have these creatures, and the first would have been just before I was born anyway. I just missed experiencing the next to last, in 1987, since I visited a cicada area a week or two before the grand emergence. I have been privileged, if that is the word, to live through a full manifestation of the phenomenon this year. Now the cicadas are all gone, and the trees are silent. I miss my noisy little friends, but maybe not too much. Soon the trees will again be full of noise, not only with summer katydids and crickets but with the "ordinary" annual cicadas, who are the classic soundsters of summer to me. The eggs of the periodic cicadas will hatch soon and their progeny will burrow into the earth, to grow underground for the next 17 years. They will contemplate prime numbers in their hidden lairs, during the long years ahead. Time, for mathematicians and physicists and cicadas, moves in cycles described by my other new friends, sines and cosines and their kin. If I am lucky, I will be around in 2021 to meet the myriad again. 2004 will seem, by then, as quaint as 1958 seems now. But it will still be a long prime time before I see this again.
Posted at 1:16 pm | link
Mon, 21 Jun, 2004
Back from the north woods
The Electron picks up momentum again as I have returned from New England, where I was visiting my parents in Massachusetts.
The neighborhood I grew up in has a fascinating recent history which is quite unusual for modern America. In the early fifties, developers clear-cut a whole forest and leveled the top off a hill in order to build a large development of one-story ranch houses. This is not the unusual part; after World War II, housing developments sprouted up all over the country. My family's house was built in 1954-55, on land that was reduced to an unnatural prairie. It was so open that the rare Horned Lark which nests on the ground in arctic meadows, nested there. Near the houses were spindly saplings of hardy trees like maple and less hardy trees like weeping willow, Chinese elm and yellow poplar. There were also rows of tiny conifers, and as a child I planted little pine trees which were no bigger than I was.
As I grew up, that yard was a softball sandlot and a Frisbee court; it was even open enough to fly kites there. The cherished Eastern Bluebird was persuaded to nest there, in a box in our back yard, as late as the 1960s, when meadow conditions still prevailed.
But as time went on, the trees grew large. The hardy maples and evergreens survived, while the elms and willows eventually perished. Fifty years later, the little spindly trees and the cute Christmas-tree evergreens are big forest trees, and the northern jungle of New England has returned to create a deep shady forest environment around a ranch house whose architectural ancestors were designed for open prairies. Wild underbrush has taken over my old playgrounds, and deer wander through the land, munching gardens. My old home is also home to a cartoon wonderland of little woodland creatures, such as grey and red squirrels, big-eyed chipmunks, saucy skunks, big burly bunnies, Pogo possums, and lots of mousies, as well as the occasional clever fox or wiley coyote. The meadow birds are long gone, replaced by noisy bluejays, nuthatches, cardinals, chattering wrens and woodsy woodpeckers, as well as a swooping hawk now and then who comes to do some un-cartoonlike hunting and grabbing at the bird feeders.
The unusual thing about this is that my old neighborhood is one of the few places in America which seems to become more rural with time, rather than less. Perhaps this is an illusion, since an endless garish strip of malls and shopping centers and American commercial sprawl is only a mile or two away. But at my old home, the woods that were so insensitively destroyed in the 50s have returned, part of the reforestation of America.
My parents moved into their home when it was freshly built in 1955. They are still there, and still married to each other since 1945. This in itself is an example of social stability which will probably never happen again in our society, except for a few rare cases.
While I was there in that dark green world I picked up my College Algebra book, which was written and published when the horned lark could still nest in my neighborhood, and began my study of Logarithms. I'll have more to say about the book, and logarithms, in my next Electron entry.
This Electron Blue nostalgic moment is dedicated to the memory of Father Robert Bullock, my beloved mentor who passed away on June 20 at the age of 75.
Posted at 2:49 am | link
Wed, 09 Jun, 2004
You are now leaving Trig City. Please visit again soon!
After a few days of half-heartedly solving problems with complex numbers expressed in rectangular or trigonometric unit-circle forms, I have finally reached a self-determined terminus for basic trigonometry. This is somewhat overdue, according to my Friendly Mathematicians, but I wanted to make sure that I didn't miss anything. I wanted to know that I could at least recognize one form or another so that when I encountered them in my next mathematical section, I could go back to my references and find it. Well then, that's enough for now.
I can't say I found trigonometry thrilling, the way algebra sometimes was. There's nothing like a polynomial pileup to make a mathematical aspirant feel as if I was riding an amusement park ride. Geometry was less thrilling, because at least the first-year work I did wasn't intensely challenging. It was endless mental origami, folding and matching and folding again and matching and proving. Trigonometry, emerging from geometry, added more numbers to it, and plotted a three-dimensional landscape filled with lighthouses, ladders, cast shadows, and seaside panoramas. But once I progressed into trigonometric identities and equations, I bogged down in a seemingly endless proliferation of interlocking relationships that gave me no information about anything except themselves. No doubt I will be glad someday that I did this (or maybe not!) and I will probably encounter something like it again soon enough.
So for now I bid farewell to the terse Brits with their story problems about coastal landmarks, fishing boats, observation towers, and memories of warfare. I also leave, for now, Schaum's red-backed book and its sparkling prose. (Example from Schaum's: "Whenever sin X is between 0 and 1, it is possible to find angles in quadrants I and II that satisfy the value of sin X and could be angles in a triangle. The first-quadrant angle is always a solution but the second-quadrant angle is a solution only when its sum with the given angle is less than 180 degrees.") And I bid a grateful good riddance to the often baffling Barron's text with its Ruritanian fantasy characters, who I hope find better employment in fantasyland than teaching algebra and trigonometry.
I will be going onward to Logarithms, which I will learn from a variety of sources, including my vintage 1958 college algebra text, which merits an entire Electron essay all by itself. Meanwhile, this Electron Blog will be taking about a week off while the Electron writer makes a trip to the "home place" in New England.
Cicadas: the party's almost over
Electron readers may be sick of hearing about the 17-year cicadas here, but I find them so fascinating that I can't help paying attention to them and writing about them. The weather has heated up, so the cicadas who are still in action are making noise again. The howler species seems to have finished, so we are left with the sustained buzzers and the clickers, who go pz-pz-pz-pz-pz and sound, in the words of one cicada text, like a lawn sprinkler. As I drove through the leafy suburban neighborhoods today, I passed through areas of high noise and less noise and no noise, as I passed by populated trees and less populated areas. The cicadas have pruned many of the ends of the branches off deciduous trees, as they pierce the new stems to lay their eggs. So everywhere in this Mid-Atlantic area you can see the drooping, dried ends of cicada-pruned branches. Stopping by a noisy tree, I was fascinated to hear that the buzzing chorus rose and fell rhythmically in volume, cycling in about a few seconds from louder to softer. Were they all co-ordinating their amplitude? Were the cicadas following a sine wave? If so, how did they know to do this? No doubt there is an explanation, though it probably doesn't involve a tuxedoed Jiminy Cricket conducting the Cicada Philharmonic.
Alas, all the adult Cicadas here will be dead by the end of the month, leaving their eggs to hatch a few weeks later. The areas under trees are littered with winged carcasses, which are soon eaten by birds and other insects. I hope that enough of their young survive to bury themselves in the earth and emerge en masse again in 2021, by which time the Electron, and the Internet, will probably be beamed directly into your brain through bio-electromagnetic implants. (What a horrifying prospect.)
I'll be spending the next week or so in cicada-free New England, and will return in due time after the Summer Solstice.
Posted at 2:08 am | link
Sun, 06 Jun, 2004
Urban Shamanic Music
The cicadas are silent, due to cold rainy weather. I don't know if they are all gone; it's possible that when the weather heats up, the remaining horde will be activated again. But their season is mostly over. Even so, I still seem to hear the howling call of those male cicadas, which is so loud that it is said to carry as much as a mile's distance. But when I think I'm hearing a cicada, it turns out that I am actually hearing a resonance from some mechanical item like a dishwasher, or even a police siren in the distance.
The similarities of the cicadas to other sounds which carry over distance got me thinking about evolutionary adaptations. The entire idea of an insect swarm appearing only once every 17 years is amazing enough. But along the millions of years of cycles that these creatures have evolved, the most successful stud cicadas must have had those far-reaching calls, while their softer-voiced brothers did not fare so well in the reproductive market. I hear the siren sound of the calling in my memory even now, which will reach across seventeen years if evolution and fortune are on my side.
Speaking of exotic sounds, here is a review of the latest album by Steve Roach, the American master of ambient electronic music. The album, FEVER DREAMS, is available from Steve Roach's Website, which has soundclip samples.
By Steve Roach, with Patrick O'Hearn and Byron Metcalf
Projekt Records, 2004
After 2003's monumental Mystic Chords and Sacred Spaces, Steve Roach returns to the world of consciousness-altering trance rhythms with Fever Dreams. In this collaborative album, Roach is on synthesizers and weird guitar, with the pop and jazz veteran Patrick O'Hearn on bass and shamanic drummer Byron Metcalf on percussion. Byron Metcalf is familiar to Roach fans, since he worked with Steve on two exciting drum-fests, 2000's The Serpent's Lair and 2001's Not Without Risk.
Even though Roach and company have been doing shamanic trancemusic for more than 20 years, they still have something new to say. This album moves away from the evocation of Native and aboriginal cultures, into the spooky and ominous world of our modern spiritual life, where ancient religions clash in cyberspace. The first piece, "Wicked Dream," sets a mood of foreboding which continues to remind me, whenever I hear it, of voodoo-haunted New Orleans. Once again we hear the sound of Roach's exotic stone percussion and rattles, underscored by O'Hearn's bass, which now seem to imitate not the sounds of the desert but the scratching creepiness and alarms of a city deep in the wet night.
The second piece, "Fever Pulse," features some of Roach's more familiar special effects, as well as his strange guitar bending, which has been part of his repertoire for at least ten years since his work with Suso Saiz and Jorge Reyes on the "Suspended Memories" recordings. You can also hear some of Roach's newer "fractalized" electronic rhythms, echoes of his 2001 album Core. This piece is faster and brighter in mood than the first, with nice swirls of deep reverb.
The third piece, "Tantra Mantra," is the longest on the album at just under 30 minutes. It is pure trancemusic, looping its loops and plugging along at a pace just slow enough to send the listener into a hypnotic state. Don't listen to this while driving or working! This is for long evenings of astral travel, led by the drumming of Metcalf and the odd, twisting chords of Roach's guitar which pass in and out of tonality. Over the half-hour the volume slowly builds and the sounds become heavier, with interjections of electronic twertles and glorps by Roach. It fades out as it started, leaving piece number 4 to wake you up.
This fourth and last piece, "Moved Beyond," is in my opinion the best on the album. The dark vision of the first piece returns, with eerie electronic sighs and guitar wails from Roach. Soon they are joined by Metcalf's thunderous drums, whose ancient resonance backs up the sirens and dissonance of the twenty-first century. There's no pretty sun-bleached desert nostalgia here; this is music imbued with the spirit of our Age of Mechanized Terror, lit with fluorescent lights and vectored through a virtual landscape where urban shamans battle.
Posted at 2:15 am | link
Thu, 03 Jun, 2004
Negative Space: i explained to me
As I work through the problem sets about imaginary and complex numbers, a generous commenter to Electron Blue has explained i to me in a way which disperses the mist of mysticism with which I surrounded it in my previous posting. What a relief! And there I was waxing lyrical when I didn't have to. The commenter (as usual, all commenters to the Electron are "name withheld") showed me that i and other "imaginary" numbers actually correspond to quantitative things which almost all of us can experience in ordinary life.
The square root of a negative number will occur in the calculations of removing an area from another area, for instance. The commenter quoted this aphorism: "Imaginary numbers are not mystical. Your doorway is 2i meters tall." Similarly, as I read a little about the history of i, the reference text explained that i was implied even as early as ancient Egypt, when geometers wished to find the volume of a truncated (incomplete) pyramid.
Basically, i is involved in the measure of a negative area, that is, an area that is taken away. To put it simply, if you have a measured flat area, and you take away a square area from that area, then the measurement of the side of that removed square will be an "imaginary" number, the square root of a negative number. The area of a doorway removed from the plane of a wall is an i number, too.
This is, then, about "negative space," not some mystical imaginal world. (Alas.) Most artists are familiar with the concept of "negative space," either instinctually or because they had to sit through lectures about it in art school. "Negative space" in the context of art composition, is the space around a figure, the shape or shapes that are formed not by the figure itself but by the remaining space on the plane it occupies. Especially with geometrical or abstract art, "negative space" becomes as important as "positive space." You don't just fill negative space with a background. The example I quoted about the area of a square removed from a plane is a very simplified version of this "negative space." Either the figure (square) or the ground (remaining space around the square) could be "negative," and described with an i -number. I'm glad to have received some background on this matter.
Posted at 2:14 am | link
Wed, 02 Jun, 2004
I is imaginary. No, I AM imaginary.
After a very full weekend, I am back in the studio and at work on math. I have reviewed the initial information about Imaginary and Complex Numbers and am encountering, with the help of Schaum's Red Book, the expression of complex numbers as vectors, and their relation to trigonometry and the unit circle. This last material is all new to me, so it will take a modest bit of time to get over the shock factor of seeing imaginary numbers spinning around the unit circle where rational and irrational once ruled.
As I reach the end of basic trigonometry (finally!) I will now award the rating of "Best of my Trigonometry Books" to the Schaum's Outlines text. It may lack character or atmosphere, and its type is tiny and hard to see, but it delivers the math, with enough explanation for me to understand, and oodles of exercises to work through. The BritBook comes in second. Of my three, the worst was Barron's with its obnoxious Ruritanian characters, whose fate I now consign to the dust of my bookshelf. I now intend to dutifully apply myself to the imaginary number problems in the Schaum's text. Yeah, I really know how to have fun.
It's a masterpiece of human creativity, to me, that anyone ever thought up such a thing. Evidently it's been around for quite a long time. The history of i and imaginary numbers is documented in the book AN IMAGINARY TALE by Paul Nahin, which I hope to read in the near future when I have enough mathematical knowledge to get through Nahin's examples.
Naturally I am tempted to apply all sorts of metaphysics and fiction to i and its imaginary and complex numbers. Is there another universe where i and its fellows are the real numbers and our real numbers are imaginary? Is i magical? Does a complex number participate in two universes? Perhaps i is the bright shadow of the imaginal world of the Neoplatonists, a world made not out of prosaic material reality but the stuff of myth, dreams, and visions. Well, I should stop right now, before I embarrass myself by saying something truly gauche. I must remember that whatever I imagine or invent about mathematics and science, whatever puns or humor or metaphors or wordplay or wry observations… whatever it is, has been said before, by someone, somewhere. If I am humble enough, my ego can become a lowercase i which even when multiplied by itself is still a negative quantity.
Balticon and cicada update
Balticon, the science fiction convention I attended over Memorial Day weekend, was more successful than I had expected. I sold five out of the seven pieces I had on display, and a number of prints as well. My "regular clients" continue to buy my art, even when it is in a style they are not used to. I did not sell the "Cometary Nucleus" picture which I referred to in my previous entry, but I did sell this one, "A Passage Through Warped Space." Acrylic on illustration board, 11 inches by 14 inches.
I also met with many of my Friendly Scientists and Friendly Mathematicians at Balticon, and though I didn't have time to sit down and work through any math with them, I gave them progress reports about what I have been doing.
The cicadas are winding down, though they still are making a lot of noise. Their orgy is ending, and their little winged bodies litter the ground beneath their trees. By the end of this month they will all have perished, leaving their eggs waiting to hatch in the twigs above. When they hatch, the larvae will drop to the earth and burrow underground again, to sip sap from roots and grow in their hidden depths for the next long cycle. What do you say to the neatly folded remains of a cicada who has fulfilled his or her earthly duty? Farewell, one of the myriad, and may some of your genetic material persist and survive to emerge again in seventeen years, the prime of your next generation's life.
Posted at 2:01 am | link