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 It is no longer being updated.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.
Thu, 27 May, 2004
I won't be updating this for the next few days, and I haven't had time to write anything Thoughtful lately anyway. I am preparing for my art show at Balticon, the yearly science fiction convention in Baltimore. I will have an art booth in a showroom with about fifty other artists. The artists in the show will range from high school aspirants, to amateurs good and bad, and some professionals including myself. My own selection is not so spectacular, since my art priorities are changing and I have not committed myself to anything "important" while I am still thinking about what to do. I don't expect to make much money. But I will have two new pieces done this month, though they are small. You can see one of them here, titled "Cometary Nucleus and Particle Shower." Acrylic on illustration board, 9 inches x 11 inches.
This middle of 2004 finds me in transition in both art and mathematics. I continue to move away from "conventional" fantasy and science fiction art, the kind that is familiar to anyone who looks at fantasy book covers, movie illustrations, comic books, or many advertisements. I am not expecting any new artistic revelations at the Balticon art show, only the same tired old motifs of babes and dragons and barbarians. Gone are the days when I could go even to a regional s.f. convention art show and marvel at spacescapes or alien worlds or mystic visions of romantic faerie. I don't know whether it is the decline of the art field itself, or whether it is my own boredom that has caused me to lose my sense of fantasy art wonder.
But then from a "fine-arts" point of view, my work is still on the facile and commercial side, and even my Deco-geometric stuff is hardly original. At least I am borrowing from the right people, art heroes of mine like Paul Klee and Wassily Kandinsky, as well as Charles Sheeler and the American Precisionists. So I am now moving in between the kitsch of fantasy art and the Serious Stuff of "fine art." It's a place of transition, but I am making progress. Four months ago I wasn't making any art at all.
As for math, I am just about done with basic trigonometry, and I am currently visiting some of the areas of "pre-calculus" which I didn't know or didn't have enough of. Right now I am solving problems about inverse functions, that is, turning them backwards, and finding their domains and ranges. After the convention action is over, I hope to review an introduction to imaginary and complex numbers, and then move on into Logarithms. At the convention I expect to meet and talk math with at least one of my Friendly Mathematicians. They are impatient with me because I don't want to start Calculus yet, but I want to make sure that I have fully covered precalculus before I do anything derivative. I will leave the "derivative" aspect to my art.
Posted at 1:41 am | link
Sun, 23 May, 2004
As most of you now know, whether you are in the affected area or not, this is the year of the seventeen-year cicadas, and we in the MidAtlantic and Midwest are now at the height of their season. I have vague memories of 1987, when they last emerged, but the old memories don't match the wildness and fascination of this cicada immersion of 2004.
You can read more than you would ever want to know about these creatures at this excellent cicada site. This site not only has detailed closeup (ugh) pictures of them, but has soundclips of their different songs.
When they first emerged, I thought they were disgusting, but now I think they're kind of cute, in a disgusting sort of way. It was fascinating to watch them heave themselves out of their holes, leave their shed husks behind, and make their way slowly up the tree trunks in the urban neighborhoods. Who could have imagined that a foot or so beneath the landscaping in your "garden apartment complex" lurked countless cicada "nymphs," growing patiently for seventeen years? When they first came out, the pin oak tree trunks were covered with them, as if the trunks had suddenly been decorated with jewels or some sort of holiday ornaments. But then you noticed that these jewels were crawling.
Many of them didn't make it. That first day, when I drove in the parking lots, my tires popped dozens of them, and the pavements were full of flattened cicadas. Others perished on the grass, either eaten by birds or attacked by other insects. But there were plenty more of the successful ones, who made it up into the trees, and within a week they were ready to rock 'n' roll.
When I first heard the "chorus" in the trees, I thought it was some mechanical engine that had been left on. But then I realized that this was the sound of cicadas by the thousands, by the ten thousands, by the hundred thousands and the powers of ten. As they warmed up, the tree chorus picked up speed and volume and sounded like ten thousand little cell phones calling (the old-fashioned ones, not the "music box" newer ones), or perhaps a thousand little industrial engines, revving in the leaves.
Then they put the buzz in the mix. At first only a few, then more and more got the buzz and spent the day making noise. A couple of colder, rainy days kept them quiet, but now that we have summer-like weather, they are going strong, and the racket is high in a neighborhood that already has road noise, helicopters and airplanes overhead, lawnmowers, air conditioners, and car stereos and alarms.
I thought that perhaps they would shut up at night, when it was cooler, but it is a warm night tonight, and they are howling their eerie love calls into the darkness. Every so often, the chorus starts up and there is a vast murmuring of cicadas, reminding me that there is an immense winged multitude out there, going about their business. Their business, of course, is reproduction, making more cicadas. Each day there is an orgy of unbelievable numbers in the trees above me, as they find favor in each other's bulbous orange eyes and embrace in their insectoid romantic clinch.
The noise and the orgy will go on for about a month, and by this time next month, our cicada neighbors will have finished. Later their eggs will hatch and the younglings will drop into the earth. The adults, their work finished, will die, uh, like flies and there will be an unpleasant rain of dead cicadas. Seventeen years from now, in 2021, it will happen all over again.
There are actually mathematically interesting things about cicadas (trying to stay on topic here). Much has been said about their seventeen-year cycle. Why seventeen? Seems a peculiar number for a periodic natural phenomenon. Biologists explain that seventeen is a prime number which is impossible to divide into even numbered sub-cycles which would be easier for predators to follow. (There are also thirteen-year periodic cicadas.) Seventeen years is also too long a period for predators to adapt to; most birds or small mammals don't live that long, so it's not to their evolutionary advantage to wait for 17-year prey to emerge. Nevertheless, other creatures will eat the cicadas if they can, and they do, but there are so many of the cicadas that predators can't make a big enough dent in the population. And even if fewer survived to crawl up the trees and reproduce, their own prolific eggs would re-populate the billions needed for a successful brood. Cicadas are made to multiply, and multiply, and multiply.
So much for cicada math. Now for some cicada philosophy. There is, at least in my observation, a tremendous wastage of cicadas. Millions of them die without getting up into the trees to reproduce. Some of them don't even make it out of their shells. As I said above, predators and human vehicles destroy them too. There seems to be little value for the individual cicada; their purpose, if it can be called that, is a collective one. The brood survives, not individual cicadas.
In the conventional monotheistic viewpoint, God cares for all things, especially living things. "Not one (sparrow) falls to the ground without your Father knowing," as Jesus is quoted in the Gospel of Matthew, 10:29. Is God's eye on the cicada as well as the sparrow? Is there a cicada soul, which is collective rather than individual? Jesus is quoted in the next lines: "…So there is no need to be afraid; you are worth more than hundreds of sparrows." If hundreds of sparrows, how many cicadas? Thousands, or tens of thousands?
The monotheistic religions see us sentients as having an individual, unwaste-able soul, and humankind in its billions, for a religious person, is not like cicadakind in its thousands of billions. We are not supposed to be a collective species which has so many numbers that it can waste mass quantities of its individuals while others reproduce. And yet over history, there has been quite a lot of mass human wastage, either due to natural phenomena like earthquakes and plagues or by the violence of other human beings. Does this mean that God, or evolution, did not intend us to be individuals, but to live more like cicadas and expect that for every billion born, millions would be wasted?
For this one month every seventeen years, we humans are sharing our world with a life form that seems frighteningly alien. Not only don't they think like us, they don't think at all. Their round orange eyes express no depth of feeling, at least to us, and their voices are unpoetic and harsh. It is hard to anthropomorphize them, as we do with fellow mammals, and we don't feel sad when cicadas die. Yet when it comes to numbers and scale, we humans may share more with them than we want to think.
By 2021, when their descendants re-emerge, any number of things may have happened to me. I may be dead, which is not an exciting prospect, though a cicada doesn't think about it at all. If I'm alive, I'll be almost 68. I might not be living in an area that has these 17-year cicadas. For all I know, civilization might have fallen apart by then. I sincerely hope it doesn't, if only for the sake of my mathematics and science quest. If I'm alive, I hope that I will have at least reached quantum mechanics, or even string theory if it's still going. Perhaps by then they will have cicada theory, which proposes a 17-dimensional universe populated by vast numbers of large, winged high-energy particles.
Posted at 3:43 am | link
Thu, 20 May, 2004
Math in Bad Faith
I sometimes encounter a section of mathematics in which I cannot do anything but blindly learn a formula and then solve problems by plugging variables into it. The equations mean nothing to me, but I can figure out the pattern that is suggested to me by the worked-out problems in the book, and then solve the problem sets according to what the book tells me. This means that I am solving puzzles that have no meaning and are completely out of context. I am annoyed by this and think of it as a kind of "bad faith," in that I am doing things which have no meaning for me and in which I do not "believe."
This is what my current work in trigonometry is all about. A chapter in the BritBook introduced me to a form of equation characterized by y = a sin X + b cosX. The book shows me how to fiddle with this trigonometrically so that it becomes an equation in the form y = R sin (x + a). The book then not-so-helpfully says, "There are two main advantages in writing something like (this)…it enables you to solve equations easily, and to find the maximum and the minimum values of the function without further work."
Now this is where I get a bit uppity and out of line for a beginning math student. I think it would be rude to ask a professor this, but my question is, "What is all this for? And why do I need to find the maximum and minimum values of the function without further work? I'm already doing further work!" I am not sure this is the right attitude for a student like me to have. Since I took on this work for myself, with no hope of actually using it for anything practical like becoming an engineer or a real physicist, I should just, as another scientific blogmeister quoted in an unrelated context, "Shut up and calculate."
A similar situation arose some years ago when I was working my way through college algebra, using a book from 1958 which gave me the feeling of being in a time warp. One set of problems the time warp offered me involved deconstructing, completing the square, and finding a squared polynomial, all inside a square root sign. It was named as the "square root of a quadratic polynomial," and the book explained, "(it) will frequently be necessary in the integral calculus."
All of this goes under the heading of, "Do it now without understanding it or knowing its purpose, you will need it later on." Well, sometimes this gets a little old. Perhaps I should enjoy mathematics for the pure abstract patterns and the orderly cranking of its gears, but I am prosaic enough to want to know what it is for. What natural (or man-made) thing or process does this math describe? Is this too much to ask?
Finally one of my Friendly Experts came to my aid, when I sent a math rant e-mail to him about how I refused to go any further with this until I knew what it was for. (This is kind of funny, since no one is forcing me to make any progress in mathematics so I am rebelling against my own inner drive.) Fortunately this Expert is not an overburdened professor teaching me along with a class of 25 (or 200), so he was able to answer in some detail. It turns out that the sine + cosine formula I cited above has to do with describing waves and periodic phenomena, especially things like sound. With sound, the variables describe the amplitude (loudness), frequency (pitch) and phase shift (interaction with other sounds).
Well then, with sound I am on at least some familiar territory, since as I recounted some time ago, I spent plenty of time working with electronic sound generation and seeing the sine waves and traces on the screen of the oscilloscope. In fact, "amplitude" and "frequency" were part of our vocabulary in the electronic music studio of years past, long before those things were taken care of by computers and modern automated electronic sound manipulation. It's that old sine wave song again.
As for completing the square and achieving the square root of a quadratic polynomial, I'll look it up again when I need it. I take comfort in the fact that I can always go back and look up whatever I need in books or, if necessary, on websites. If and when I get to pondering the mathematics of periodic phenomena, I know where to go back to. But for now, I think it is finally time to move on.
Posted at 2:26 am | link
Sun, 16 May, 2004
I'm taking some time to review mathematical functions, as my Friendly Mathematicians remind me that sines, cosines, and tangents, along with their inverses, are functions. There are rather dry chapters on functions in my algebra books, but the bounteous Web is full of entertaining sites like this one which use colorful graphics and large type to explain the matter to high school youth as well as people like me. Functions are processes, rather than chunky little problems with answers. They eat domains and excrete ranges. Functions can also be stuffed into other functions, rather like those Russian dolls that fit into each other. In the center of the mathematical nesting function doll is that little variable placeholder "x."
Surfing math web sites is kind of like going to a casino; there are lots and lots of entertaining colorful animated moving things, but you don't win (or learn) much. The best math web sites I've visited have a modest amount of razzle dazzle but plenty of words, and series of well-laid-out problems with accessible solutions. One of the best of this type of site is S.O.S. Math (watch out for the intrusive ads) which is just about everything I could ask from a math help website, including sample exams to take. Looking through the "Precalculus II" exam on this site, I find that I would not be able to pass it at all. I don't know, or haven't reviewed recently, most of the material on this exam. (Sigh.) The more I attempt to go ahead, the more work is placed in my way. I get this Sisyphean feeling when I try to move ahead in mathematics or physics. Sisyphus is the mythical guy who was condemned to roll the rock up the hill forever, trying to set it on top, but it always fell back down just before he got it there. The graph of Sisyphus' rock looks kind of like… a sine curve.
Posted at 3:01 am | link
Tue, 11 May, 2004
Putting the stars in the little universes
I am back to my terse, compact BritBook for a trigonometry lesson. Unfortunately, the chapter I face presents something about which I have no clue. It is about "the form a sinX + b cosX." Evidently it has something to do with the combining of two waveforms. I am so mystified that I must have recourse to one or more of my Friendly Mathematicians who will tell me what this is all about. After eight months, I am really eager to finish basic trigonometry and get on with my math and physics studies.
I am also returning to the production of non-commercial art. I am working on a set of small pieces which are geometric abstracts painted on a black background. I have become interested in painting on black since my work at Trader Joe's often involves doing store signage on black-painted Masonite boards. Here is an image of one of my better Trader Joe signs. These signs are "painted" with opaque markers which are not permanent; they wash off with detergent and water. This sign has since been erased. But with acrylic I can do more permanent work on a black background.
I have painted spacescapes on black backgrounds with an airbrush for more than 20 years, but I never considered doing more than that. Now I will be combining spacescapes with geometric forms, and maybe adding in a spaceship, a floating planet, or futuristic buildings. If they look good, I'll post a scan of one or two of these paintings on this Weblog. These have been my stock in trade over the years, and I show them at science fiction conventions. I have a convention coming up at the end of the month, Balticon, where I will be displaying some art on a panel in the art show.
I have often wondered whether I would miss doing art if I just dropped it as a major occupation and did math and physics as my main other work (besides my day job, that is). Evidently I do miss it, because I still want to paint and draw pictures. I am not a big believer in the "divine calling" or "unsuppressable talent" theory of artists. I have always believed that I do art, and chose art as a career, because 1. I was able to, given favorable economic circumstances, and 2. I liked doing it enough to continue, and 3. I found it easy to keep doing, and 4. I didn't get bored with it.
Some things about this list have changed, especially the economic circumstances and the boredom. Whether these will continue to change, I don't know. I still like doing art, and I still find it easy to do. But it is not, at this point, an exciting challenge the way math and science are for me. However nice my current pictures come out, I have done things like them before.
Here's how to create a universe on a panel of illustration board. Spray your panel with matte-surface black paint. I spray this over black board, so I don't have to do multiple coatings. Once the surface is good and dry, take a fairly big brush (big, that is, for artistic rather than interior decorating projects), the fiber part maybe about an inch long, and mix up some middle-grey acrylic paint. You can add some blue to the grey mix if you want. Take your brush and get a moderate amount of paint on it, not too thick but not too thin, about the texture of heavy cream. Hold your brush over the panel and gently tap the UNDERSIDE of the brush with another brush handle or stick. Droplets of paint will sprinkle from your brush onto the black panel in starlike "random" patterns. You can increase the magnitude of the "stars" by tapping harder, and more paint (larger droplets) will fall off. You can sprinkle paint closer or farther away, and concentrate on an area with lots of droplets for a globular or open star cluster. Or you could use that time-honored high tech art device, an old toothbrush, and put some paint on that, then scrape across the bristles while holding the brush above your panel. A galaxy of tiny particles will appear on your panel. You can then add a few stars of other colors like yellow, red, or blue, and top it off with a few pure white ones. I then go on to spray colorful nebulae onto the panels with my airbrush.
I don't use a computer to do this work, though everyone else who does space art does. I don't see the point of it; when a low-tech process like this does so well, why overdo it with expensive computer hardware and software?
I am always delighted and filled with wonder when I first sprinkle the paint onto the black panel. It's like watching the stars come out. I feel like a lower-echelon creator god, making the stars appear in a sub-universe. I suppose that once I have learned something about physics, I will want to design the physical laws of that universe as well.
Posted at 1:57 am | link
Wed, 05 May, 2004
A stylized reality: art deco equations
I'm doing trigonometric equations. LOTS of trigonometric equations. You can find a list of them at the SOS Math trigonometric equations page. I can solve at least the first six, but after that it gets into things that I haven't learned yet, and wonder whether I need to learn at all. I am also working through a list of about fifty of them in Schaum's Red-spine book. Not only can't I solve all of them, they ask for answers in many different forms. They are not satisfied with simple degree measurements, they also want their answers listed in radian measurements. And the SOS Math page lists its answers in general formulas for where you can find the answers in radian measurement, using negative-sine notation. This isn't in my books anywhere, at least I haven't found it yet. Just as in my previous experience with trigonometry, this is like biting off something that is too big for me to chew. If I want to do all of these equations correctly and learn every angle, corner, and radian, I'll be doing this for the rest of the year. Maybe for the rest of the life of the universe, from how it looks to me.
But what is interesting, and also convenient for me and whoever else toils with these books and sites, is that the answers to these trigonometric equations are often recognizable angles and sets of angles, such as the series of angles which fit sin x = 0.5 or cos x = 1. I often visualize the unit circle's indicator spinning round and round, stopping at the familiar spots of 30, 60, 90 degrees, and 120, 150, 270 degrees. It doesn't usually stop at 47.43957349 degrees, or something like that. True, if you translate these familiar angles into radian measures, you get something suitably irrational, but in sensible old degrees, they are the stuff of the Pythagorean ideal, that divine 30-60-90 form that makes up the drafting triangle on my art table.
I am very fond of the Art Deco style and these trigonometric manipulations and equations, with their stylized, repeating patterns of solutions, remind me of the geometric gestures of Deco. While other art styles use plenty of geometry, hidden in their compositions, Deco is openly geometric. It uses circles, triangles, rectangles and squares, and repeating patterns to build its constructions. I haven't seen a sine wave in any Deco design, but I'm sure that they are there somewhere. This is art which speaks of mathematical confidence and power. The answers to the trigonometric equations, whether degree, radian, or pattern, have that same kind of artificial clarity.
These equations describe something in the "real world" relevant to physics; I'm told that they have to do with oscillations, vibrations, and anything periodic in nature. Sooner or later, if I persevere, I'll be solving these equations regarding experimental data and getting solutions which are no longer in the artificial realm of Art Deco style. I am looking forward to it. One of the reasons I want to study physics is so that I can talk about "energy," "frequency," and "vibrations," and not sound like an idiot.
Posted at 1:47 am | link