"Now in midsummer come and all fools slaughtered
And Spring's infuriations over and a long way
To the first autumnal inhalations, young broods
Are in the grass, the roses are heavy with a weight
Of fragrance and the mind lays by its trouble."

These lines were written by Wallace Stevens (1879-1955), an American poet whose work is not much read these days. This is from "Credences of Summer," written in 1946. Stevens' writing is notoriously difficult to interpret, but this first stanza is sometimes thought of as being both about the season and the middle age of a human life. The "fools slaughtered" in the first line could be anything from April fools, April flowers gone with the spring, or the foolishness of one's earlier life. The mind lays by its trouble, that is, it puts aside its trouble (our modern English often mistakes "lay" for "lie," so this needs clarification) in a summery backyard sort of leisure. And this poem of Wallace Stevens is about just now, that midsummer moment of slowness before the decline into autumn, the balmy apex of the year's parabola.

Stevens, despite the lavish proliferation of imagery in his poetry, seemed to yearn for a way of writing and a way of perception that was devoid of any encumbering metaphor or frivolity of language. Here he is later in "Credences," expressing his fervent wish for purity:

"Postpone the anatomy of summer, as
The physical pine, the metaphysical pine.
Let's see the very thing and nothing else.
Let's see it with the hottest fire of sight.
Burn everything not part of it to ash.

"Trace the gold sun about the whitened sky
Without evasion by a single metaphor.
Look at it in its essential barrenness
And say this, this is the centre that I seek.…"

As a mathematics and physics learner, I wonder whether Stevens should have been a mathematician or physicist instead of a poet. Wouldn't those disciplines, so abstract, exact, and non-metaphorical have satisfied his desire for "the very thing and nothing else?" He could have had his fill of equations and measurements and data and theoretical calculations, without any pretty poetry or colorful metaphors or sometimes, even a subject at all. Isn't that "essential barrenness" the realm of mathematics, where poetry is irrelevant? The "beauty" would then be only in the neat, pure equations or the theory that explains the phenomena. I have often envied those lucky souls (usually males) whose imagination does not clutter their minds with stories, colors, images, wordplay and shadowplay. They calculate and theorize in a world of whitened abstraction, without evasion by a single metaphor. Or do they?

Wallace Stevens' poetry goes well with calculus. Here's a quote from another verse of "Credences of Summer:"

"One of the limits of reality
Presents itself in Oley when the hay,
Baked through long days, is piled in mows. It is
A land too ripe for enigmas, too serene.…

Things stop in that direction and since they stop
The direction stops and we accept what is
As good. The utmost must be good and is…"

Oley is a place in Pennsylvania, where Stevens sometimes traveled. He sees the geometry of farmland. But this is not about mathematical limits, let alone seasonal or geographic limits, but the limits of thought, word, description. He's trying real hard, but his poetry betrays him, because the only way he can write it is with words, thoughts, specific language, description. But he at least knows where things stop.

I am still working with limits in calculus and because I am not that pure mathematical imageless type, this sounds like poetry to me. In problem 6 on page 93, my answers read like this:

"Let y = f(x). As X approaches 4 from the negative side, going right, its limit is 1. As X approaches 4 from the positive side, going left, its limit is 1. As both sides approach 4, the limit is also 1. And f(4) = 1.
As X goes off into infinity towards the negative side, the limit of the function is negative infinity. As X goes off into infinity towards the positive side, its limit is positive infinity."

I am coasting along the line of the function in the graph, in the perfect absolute grid universe, with the landmark of X = 4 in my sight. I will never reach it, yet I have already reached it and passed it. I can see my prospects towards a negative infinity of endless winter, or a positive infinity of endless summer. The direction stops, but it never really stops, for this is calculus, not solid algebra. The utmost must be good, and is…mathematical.

Posted at 10:40 pm | link

There was lightning flashing in the distance tonight, but no storm came my way. The last few days have been a time of apocalyptic deluge and dangerous floods, but nothing close to me was damaged or drenched. The weather has returned to the usual balmy soft breath of summer which I love so much. I do my mathematics in the breeze of the air conditioner and desktop fan.

I finished the introduction to tangents and am now working on limits. This is really the first time I have seriously studied them and I am taking great care to familiarize myself with their basic concept and notation. As I said in an earlier posting, I am writing the notation over and over again to make myself used to it. Long ago, back in my college days, I did the same thing with the Greek alphabet and pretty soon those Greek letters will reappear in my math studies.

The idea of limit was known in some way to the ancient Greeks, who knew a version of it as Zeno's paradox where swift-footed Achilles races against the wise tortoise. The paradox involves a sequence of diminishing distances as one progresses, where the distance towards the goal gets smaller and smaller but never reaches the final point. And yet things actually do reach the final point. Hence the need for mathematical limits, to clear up the paradox.

The mathematics site I just mentioned is an exciting find, but I am very annoyed by their misspellings. I find this over and over again: misspellings and typos by brilliant people including science bloggers. I never misspell anythng. It probably means I am not briliant.

The first set of limit problems in the Anton book involve interpreting graphs of functions and naming where the limits are, given the notation for the different sections of the function and their limits. At first I thought that this meant putting a mark on the graph, but that's the artist being misled again. Limits are numbers, so that's what they're looking for. Once I figured that out, I was able to nail down the numbers for the limits, the goals for all those paradoxical journeys, among them the one I am making.

Posted at 2:58 am | link

There hasn't been a lot of Electron blogging lately because I've been working hard to finish a new painting. It's done now, so here you go. More information about the artwork below this image. Title: POSTCARDS FROM THE MULTIVERSE. Acrylic and watercolor on watercolor board. 20" x 12".

"Postcards" is inspired by the far-out but much-publicized modern physics idea that our universe may be only one of an infinite number of universes, each of them with different basic quantities and even basic physical laws. Some of these universes may be composed only of formless energy, others of completely solid matter, and still others might have hardly any matter at all. Many of these would disappear or consume themselves in a short while, but others would be born in energy only to run down quickly into cold inactivity. Among all of these is ours, with a life-filled planet existing at least in one place in our universe. There may be others, which we will never know about. Or there may be none. The rectangular "postcards" are each from a different universe, and the whitish one at the top is either at the end of the universe or the beginning of another one.

This picture took much longer than I thought it was going to take. There were technical problems with using acrylic as watercolor but I wanted to use it because it is bright. I decided to use purples and violets for the main color scheme because it is unusual to see these colors in a painting, and also because lightning and other energies in air make that color.
Now on to another universe. I'll write if I get work.

Posted at 2:20 am | link

One of the standard things that people tell me about Calculus is that they did fine in algebra and geometry and other high school math, but once they got to Calculus, they hit a proverbial wall and found it impossible to continue. I have been wondering where that wall is, and perhaps I have found at least one possibility in the concept and notation of limits. Having finished the section on tangents, I am now learning about limits. I have encountered the concept before, back in 1958, that is, not 48 years ago but in the pages of my 1958-dated algebra textbook. These limits were those of progressions, which were to be summed up. The limits I am learning about now are of functions and their output.

I am only beginning with this, but I can see where this would stop someone in their tracks. Instead of getting a simple yes/no or number answer, this limit process offers a moving target which keeps getting closer to that limit number L without ever really giving a complete solid result you can write into the box. The notation is also new and somewhat confusing, because an X approaching zero from the "right side" is actually positive and gets a plus sign, though it is moving leftward or in what would be a "negative" direction. And vice versa for a limit approaching the positive from the negative. Everything is in motion here, rather than sitting still in nice orderly ranks. I am making the mental adjustment so that I can live with this. I will not hit the wall, though I may find it closer and closer as if I were approaching, well, a calculus limit.

And there's also new notation to learn. I still find the "sigma" notation for sums of progressions, surrounded by the parameters for starting, doing, and ending the progression, confusing and even scary. There's something about that big Greek letter S which says "foreign difficult math." I am not taking any chances with the new notation I'm learning for limits. I have been reverently copying it over and over again onto my note pages, along with Anton's explanation: "Expression (3) is read: "The limit of f(x) as x approaches x₀ from the right equals L₁." But the little arrow that you write below the "lim" abbreviation is pointing rightward, not leftward! The whole matter of those sub-scripts continues to perplex me; I am definitely not used to them yet. Some texts use x₀ as the "original" quantity and x as the "worked-on" quantity, while others use x₁ as the "earlier" quantity and x₂ as the "next" or "worked-on" quantity. This is like the V₀ and V₁ in my physics problems, which I always read as letters rather than numbers, so that I kept thinking of Vo and Vi as peculiar hair-care products (some of you may remember Vo5 shampoo, which is still available) that somehow got mixed up in physics problems. I have got to get used to all these sub- and super-scripts. The only way for me to do that is to write them over and over again, like a scribe learning the sacred letters, so that they will no longer appear strange and embarrassing to me. They must appear on my papers as if I really knew what they meant.

Posted at 3:30 am | link

A warning: This posting may be offensive to some readers. If you do not wish to view adult content, please stop reading now.

I have compiled a poem from phrases and words sent to me in spam e-mails. The scrambled word flow of unsolicited commercial e-mails provides a continuous supply of verbiage which falls into surrealistic and often hilarious combinations. The following poem is composed mostly from actual material which arrived in spams. I have added some lines to express some ideas and commentary. You may find this humorous.

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The hand on the mouse, the hand on the keyboard,
the face at the screen, the eyes on the world,
in the dim room full of books and clutter.
Girls on the screen for you, girls available and ready.
Keep your desire ALL the TIME barely legal,
Increased volume and more intense: Eternal youth.

Posted at 1:28 am | link

Even though I've been much involved in the gourmet world lately, I haven't forgotten my calculus at all. I do problems every night. My current problem work has me drawing graphs, which I love, since it's something like art. I enjoy plotting out the curves on the coordinates by hand, though I also have graphic calculators on two of my computers. As I draw these, I experience a kind of kinetic feel as they move through their curves. It's like being in motion just to look at them. Some of these, of course, actually describe motion, such as the ubiquitous parabola, or the exponential curve of the speed of falling objects. But others are less speedy, and include their negative reflections, such as hyperbolas and the up-down and suddenly across curve of f(x) = x³. Some hyperbolas have a breezy sail-like quality, gliding over a grid-patterned sea. These are only simple curves; there are plenty others which yield all sorts of wild forms. But my favorite so far is the heroic tower produced by the equation f(x) = 1/x². Here is a plot of it from one of my graph-drawing programs:

You can see how the two halves of the inverse parabola curve out at the base of the "tower." There are buildings which do this in the "real world," especially William James Hall, the building that houses Harvard University's psychology department. I lived in Cambridge for 12 years (1976-1988) and for ten of those years, I passed by this building every day. Built in 1963, its white modernism is out of place on the red-brick Harvard campus and the building is much maligned by Harvard's architectural conoisseurs. It also had the misfortune to be designed by the hapless Minoru Yamasaki, who was the architect of the doomed World Trade Center in New York. But I loved William James Hall, not only for the way the yellow-orange winter sunlight made its white surface gleam like brilliant brass, but also because of that slight parabolic curve at the bottom, which is only somewhat evident in the website's picture. My Harvard years were math-free, and yet even then, I recognized that the base of Yamasaki's building was a parabola, or at least a section of one.

The problems have me drawing secants through these curves and finding the slope of that secant, and then through the process of limits, finding the slope of a tangent line at some point on that curve. Meanwhile I am holding on to the virtual railing, or the steering wheel, as my vehicle speeds around the curve of the graph. I feel the acceleration, just as if I were actually driving my Electron Car on that trajectory. But to find my instantaneous velocity while driving, all I have to do is look at the speedometer. Really? Even that is an approximation, derived from the turn rate of the transmission shaft (or front axle, for a front-wheel drive car like mine). And even more, since my car is an Electron, I will never know just how fast it is going, as long as I know where it is (or vice versa). For now, all is approximation.

Posted at 2:02 am | link

Saturday I did a restaurant signage job, which I got through my Starbucks contacts. You've seen some of my Starbucks coffee signs on this Weblog from time to time. They're done in opaque acrylic markers on black boards. Someone from a chic restaurant in Washington DC (known to locals as "The District"), whose girlfriend's sister works at Starbucks, saw my work and thought it would be perfect for a large menu sign at his restaurant. He finally succeeded in contacting me and I brought my art kit down to the restaurant, near Dupont Circle.

The restaurant is Nage (Warning: This is a noisy website! Better turn your speakers down) which is attached to a Marriott Courtyard in Washington, and has a sister restaurant in Rehoboth Beach, Delaware. The food is best described as "Mediterranean eclectic." According to the restaurant people, "Nage" (pronounced "naj"), is a tasty bouillon liquid in which seafood is poached.

The sign, which is about four feet by three, is painted on the wall in "blackboard paint," which is used for chalkboards. This is what we use at Trader Joe's, too. My design was to go around the borders as well as on frames for the lists of beers and cocktails. I used an "art nouveau" style to do the borders, since I find that customers like this style more than other more modern forms. This is a summer and seafood themed border, so there are shells, a fish, and a lobster in the blue waves. I am scheduled to renew the border each season, so I'll return in the fall. I did all the writing this time, but the drink writing will remain while the food items, which are written in real chalk, will be erased and re-written (by the restaurant staff) every night.

I was not only paid in money for this work, but was given a complimentary dinner there. I had a deep-fried lobster and cornmeal appetizer with jicama and mango julienne. For my main meal, I was served the "wolf-fish" that I wrote about on the sign. I had never heard of "wolf-fish" before, but it was white, tender, and tasty. It was served along with a chunk of hearty eggplant and cheese casserole. I sipped a Sauvignon Blanc wine and finished with a double espresso. No starving artist here!

Now back to calculus and ambient music. I am also planning some Electron entries on philosophy, science, and religion, lest I digress too much into gourmet indulgences.

Posted at 3:36 am | link

I feel guilty because I hardly ever cook. The cooking magazines and even this week's issue of TIME feature endless articles about how one is supposed to eat. Here I am working in a gourmet store and I still can't figure out what to eat. Certainly it's not because I don't have enough choices; it's because I have so many. I walk the aisles dumfounded by so many good things that I might eat. If I go to a major supermarket such as Safeway, it's even worse. There is too much, and I can't decide.

A good socially conscious person would always keep in mind the brute fact that a supermarket shopper in America probably has more food available to her than anyone in any other country, and that her trip through the aisle is by itself a horrific reminder that in the other half of the world, where not even Safeway can reach, people are starving. The guilt is even more pronounced because other than throwing money into an envelope for some worthy cause (and receiving thousands of address labels and pounds of junk mail in return) there is nothing you can do about it. This creates stress, which at least in my case makes me lose my appetite.

The nutritionists who are ever busy telling us what we should eat follow me like shadows as I wander down the aisles. No, you can't have that, it has too much salt. No, you can't have that, it has too much sugar. No, you can't have that either, because it is fatty and loaded with empty calories. How about this? It has too many processed food chemicals. Don't eat that. You should eat salad, salad, salad. The nutritionists all agree that I am supposed to eat leaves, vegetables and grains. Lots of them, kind of like a browsing animal. Unfortunately I am not supplied with a cow's digestion so cow food does not agree with me. And all vegetables and most fruits taste bitter to me, for reasons I have never been able to fathom. Only if I overcook it the way the English do does a vegetable taste halfway decent to me. But then, as those nagging nutrition shadows say, it has far less nutritional value. And put down that bottle of soda. Guilty again.

So because I despair of cooking, with its greasy splatter and mess in a small kitchen, I go out to eat. Unfortunately, I have enough money to afford this. But the nagging shadows say that restaurant portions are too big. So don't eat it all, take it home in a styrofoam container (which pollutes the environment with its trash, by the way.). But even then, restaurant food is too rich. And I should have prepared it myself, as a conscious eater. Paid the bill, gained more guilt.

I have often wished for some universal food, instantly edible and portable over long distances, which fulfilled all the myriad health requirements and required no preparation other than opening up its package. It has to be low-calorie and nutritious, and have vegetable elements but hardly any salt or fat. Tofu, that staple of pallid vegetarians, comes close, but it has to be refrigerated. Bananas also come close, but I don't like them. There is a disturbing amount of meat jerky in stores these days, but I'm not that desperate, and it's really salty too. And then my own gourmet store carries a wide array of protein bars, which also come close to my desired ideal, but they are filled with sweets, nuts, and chocolate flavoring, when I really want spinach and garlic flavoring.

Somehow, those other countries where people are worse off have solved this problem by having lots of small street food stands or little stores where you can get things like meat or spinach pastry rolls, roast corn on the cob, slices of coconut, skewers of toasted chicken, or tacos. But these things depend on a way of life and urban arrangements which just don't exist in most parts of the USA. To get the snacks, you have to have a city or village concentrated enough for people to walk through on their daily business. McDonald's takeout in the car is not an equivalent.

So there I am standing paralyzed in the aisle of Safeway, or even in my own workplace, faced with both an overload of choices and an overload of cautions. Instead of being the consumer, I am consumed by indecision and overcaution. Most choices will taste good, but no choice (other than those hated vegetables) is right. Perhaps I will lose weight on this diet of guilt.

Posted at 2:27 am | link

After some updating, which is an ongoing project, I'm happy to announce that there is a new section of art visible on my Website at pyracantha.com. This features my newer work in mathematical and space abstraction. Mathematical curves such as those in conic sections and exponential graphs give me endless ideas for abstract art. And physics concepts, for an artist, turn into bright-colored fantasies which would never occur (or would be dismissed as silly, superficial, and irrelevant) to a physicist. I try to be careful not to disrespect either math and physics by making it into trivial, pretty, or cute decorative designs. All these pictures come from things I have worked out in my math and physics with real calculations, though I don't calculate anything with numbers and equations while designing the images. Design elements include conic sections, intersecting lines, inequalities, and exponents. If I were a scientist, I'd specify the colors for the artwork in nanometers or RGB number sets, but I don't go that far. What I do want to convey, by using high contrast, bright colors, and starry black backgrounds, is the bold, heroic quality of science, which dares to investigate the universe and find order in the world.

You are invited to visit my new gallery section at Abstract Geometric Spaces.

Posted at 3:00 am | link

I just ordered this course from "The Teaching Company," which makes courses available on DVD for anyone who can afford them, to view and learn from. I have been wanting to get "Change and Motion: Calculus Made Clear" for many years, but now that I am actually studying calculus, it's time for the recorded course. This course, taught by a professor with the wonderfully evocative name of "Michael Starbird," (Archangel flying with an eagle on starry wings!) is not meant to be a math-crunching ordeal with lots of problems, it is about the practical applications of calculus in everyday life. For the "real" work in mathematics, I stay with the less fabulously named Dr. Anton's book.

Currently I am carefully reading the introductory chapter on tangents and limits. I must remember that a tangent is a line touching a curve at one point, where that line is at a right angle to another line perpendicular to that curve at the same point. I'm not sure whether that is a proper definition of a tangent but that's what fixes it in my mind. I notate all sorts of redundent or personal explanations of material in the book, since this copy of Dr. Anton is well-used and I don't have to return it to anyone.

There was lightning in the sky tonight, but no heavy rain at all. The humid, hot weather I love is now here, but my enjoyment of it is somewhat reduced by the repulsive, mind-numbing geysers of heat and sweat that I must endure every forty minutes or so, namely the hot flashes which have returned as the weather is warmer. My soymilk and soy extracts worked when the weather was colder, but they don't do much now. I continue to consume soy, though, because I think it makes me feel somewhat better. There may be other remedies, though at this time hormone treatments are not an option. One of the most annoying things about hot flashes is that it makes me feel so un-hip and dorky. You do not hear about "cool" people having hot flashes.

In other news, I am about to get Photoshop and Adobe Illustrator for my new Macintosh. These fancy graphics programs are the "industry standard" for illustrators and digital artists. I have been cranking along on CorelDraw and Corel PhotoPaint for all these years, and I've done plenty of good graphics on these programs, but I'm eager to work with the more sophisticated Adobe software. Meanwhile, I'm working on new concepts for geometric/mathematical abstractions on the Corel programs. The nice thing about geometric art is that it's real easy to do a "sketch" for a painting on the computer, and you can play around with the elements without having to do many pencil or watercolor roughs. Then when it's ready, I just print it on my printer and transfer the design to canvas or board, where I paint it with old-fashioned brushes and slightly newer-fashioned acrylic. By "cool" standards, I shouldn't have to push paint around at all, and all my art should be digital, but somehow the old paintjobs still appeal to the old buyers.

Posted at 2:44 am | link