"In exercises 9 through 20, use the method of Example C to find a general expression for the slope of a tangent line to each of the indicated curves. Then find the slope for the given values of X…" OK, let's try it. I have worked out the method of Example C a couple of times on paper, so I know what I have to do. It's painstaking. I have to state all the components in ways that can lead to a clear solution for that general expression. Not only are there X's and Y's, there are the subscripts, that is the X₁ and X₂ and the Y₁ and Y₂. And Y₂ has to be stated in its relation to Y₁.

That's where the Delta comes in, the sign for "amount of change." I'm working with intervals here rather than single numbers. Eventually, in these problems, the interval dwindles down towards zero, which is how you solve it, but Delta is always there to remind me that this is not yet familiar territory for me.

So far I have not gotten any of the problems right. It's not that I don't understand the process. It's that I continue to make mistakes working out the algebra in the problems, where numbers and letters have to be multiplied and added and subtracted and canceled out. I haven't done much of this since my algebra revival days of 2001 and 2002. The negative and positive signs are especially exasperating, as I remember from those days. And if a bunch of things are inside parentheses, if you subtract them from a bunch of other things, all the signs flip over from negative to positive and vice versa.

Sometimes I copy the problem wrongly out of the book, so I fail from the start. It probably would help if I were not doing this at 3 AM after a rather pressured and frantic time at my day job at the gourmet store. We are preparing a seasonal feature with ads all over the store. I am not using calculus to find the tangents of tangerines or the polynomials of pomegranate juice. But in spare moments, I mentally recite to myself: "Y₂ = Y₁ + Delta Y. Y₁ = f(X₁)." Add and subtract and simplify. Then divide by Delta X. As Delta X approaches zero… remember the negative and positive signs! Or are they organic tomato or pumpkin butter signs? An instantaneous rate of change for point P. Amazing fall values.

Posted at 3:25 am | link

I have always loved communications towers. Whether they are radio transmitters or cell phone relays or airport control towers, these structures inspire my respect and admiration, for their appearance, their engineering complexity, and their importance to our networked society. When I was very young, I was fascinated by radio towers, some of which could be seen from our house. My parents, steeped in Freudian ideas, laughed at my childish devotion to what could only be phallic symbols. But that never crossed my mind, though a Freudian would say that didn't matter, since it was all in the subconscious. Well, whether Freud was right or not, that isn't what enthralled me about radio towers. This old RKO film logo is more like it. It combines the things I loved best as a child and continue to love now: the Earth in space, though backed by atmospheric cumulus clouds, the Eiffel-like radio tower, and the stylized lightning bolts flashing from the top of the tower, as well as the wonderful jagged-lightning typeface.

Communications towers resemble each other in their rigorous latticework structure of squares, rectangles, triangles, and trapezoidal braces, something which my reborn mathematical self appreciates even more than my young fantasy artist self. The only curved lines in the structure are at the bottom, as the lower supports flare out to gain more stability. In the RKO and the Eiffel tower, the flare is highly conspicuous, where in other, more recent towers, the parabolic curve is just noticeable, rather like the Yamasaki building at Harvard which I described fondly in a previous Electron entry (Damn! I can't find it! How can I make a pretentious self-referential link?). The TV and radio broadcasting towers were the tallest and the most graceful: narrow needles pointing into the sky, held stable by the long rigging of guy wires. At night, they blinked with friendly red lights, to warn aviators.

As technology advanced, telephone relay towers and cell phone towers and wireless communication towers sprouted in the landscape. These were sturdier and shorter than the broadcasting towers, but just as geometrically intricate. And they were ornamented, like Christmas trees, with a variety of instruments and antennas which did the real work while the tower was just the vehicle to raise them into the heavens. It is hard for some younger people to conceive of a time when there were no cell phones, but I still remember seeing my first one, in the hands of a businessman at a Metro station, sometime in the early nineties of the twentieth century. Cell phone towers are plentiful because they have limited range and the cell phones only work along "line of sight," so that if the tower is out of sight, you will have a hard time picking up the signal. As a result, the relays are everywhere and the higher they are, the more range they command.

They are the fortress and cathedral towers of our age, the heroic guardians of the hills and guarantors of civilization, for what could be more civilized than a peacefully maintained, intricate, instantaneous communications system? Wherever you go, if you see that friendly blinking light on top of a cell phone tower, you know that (if you are a cell phone user, and who isn't?) you are still joined to society, and to those who matter to you. And even as I write this, the Internet which you are reading this on is going wireless, so that those towers will bring you not only this Electron but your e-mail, news, sports, videos of all kinds, and "World of Warcraft." All brought to you through the towers which you drive by every day, hardly noticing them.

One of my most treasured books is a recent work by Brian Hayes, published 2005, called Infrastructure, just released in paperback. This wonderful book explains, chapter by chapter, with extensive photographs, the industrial features of our landscape both rural and urban. The book covers not only farms and factories and refineries, but waterworks, the electrical power grid, communications (and the towers), transportation, and waste treatment, as well as loads of other things related to keeping our civilization going. I'm using the word civilization again because these things, which I have loved since my childhood, are what make the difference between modern and pre-modern, order and disorder, even peace and war (for during war infrastructure is deliberately destroyed, or "degraded."). To paraphrase Ayn Rand, the poet of the industrial world, these are the "shapes of human achievement on earth."

And so in my art and in my writing and studies of math and science, I consciously choose to celebrate this world of engineering and function and making things work. As American artists such as Charles Sheeler knew, these structures have their own rigorous and robust beauty, even though they were designed without any consideration other than their industrial function. Here's the second in my current series of local scenes. Anyone who knows the urban environs of Northern Virginia will recognize this pair of landmarks. The tower hosts not only cellphone transmitters and microwave relays, but the faceted ball at the top houses a radar system for airplane navigation and weather reports. The round greenish shape at the tower's foot is a water tower. They are on the top of a hill which overlooks the entire area; the water tower, as Hayes explains in his book, is placed there so that the water will flow down into the urban pipes by the force of gravity. They are the modern version of the Trylon and Perisphere buildings which were the symbol of the 1939 New York World's Fair. Then, as with the RKO film tower logo, the creators looked toward a glowing, electric future. For better or worse, we are living in that future now.

Painting is ink and watercolor, with touches of acrylic, on illustration board. Dimensions are 11" x 9".

Posted at 3:09 am | link

In addition to my series of paintings depicting the Mid-Atlantic American farming countryside, I am also painting cityscapes. These depict buildings and places around where I live. This is an old practice of mine, that I used to do when I lived in Cambridge, Mass. I still have dozens of paintings of wooden architectural details, and whole houses and streets, that I did in ink and watercolor. Since I now have the prospect of showing at a gallery in my local area in Northern Virginia, I am reviving this practice and I hope to build up a small show's worth of images, at least a dozen.

I am not choosing conventionally "scenic" places or fancy houses, because these have already been done by many other artists. I choose the places that I find fascinating: commercial and industrial sites. This is where the geometry is strong and the textures are more interesting. I am very much influenced, in these ink and watercolor panels, by the watercolors of Edward Hopper and Charles Sheeler, who also painted images of the American urban-industrial world of their day. And before them was the group of American artists known as the "Ashcan School," who chose to create gritty scenes of poor neighborhoods and factories rather than slick portraits of overdressed society ladies and gilded interiors.

The city I live in, while it is historic, is neither quaint nor exceptionally beautiful. It is rapidly changing from a built-up suburb to a more concentrated area of higher-rise buildings. I have decided to highlight its commercial and industrial aspects rather than "charming" but generic architecture. This may not be conventional, but it will be different from the usual prettied-up images of leafy porches and garden gazebos. If the gallery-owner wants, I'd be happy to do a couple of that type of picture too. Remember, I am a commercial artist and have no problems with painting to a dictated style. Meanwhile, here is the first of my "ashcan school" series.

Sisler's Stoneworks is an old business in a section of town where there was once a quarry. Painting is ink and watercolor on illustration board, 9" x 11".

Posted at 3:02 am | link

Having come to mathematics late in my life, I don't have the youthful, competitive spirit of a high school math whiz. Yet in plodding through my introduction to calculus, and edging nervously away from the Rigorous Approach, I know that I am in some sort of competition anyway. I am not competing with anyone else, but with myself, and even more fundamentally, with the subject. I am currently doing review problems before moving on to the next chapter, or, perhaps, inning.

I was never the competitive sort, especially when I was young. Not only is it, as the evolutionary psychologists and neuro-scientists assert, a characteristic of my gender not to want to compete, but I was never raised that way either. Winning and losing were reserved only for tennis, where there was no real meaning to the wins or losses among friends, and for our Boston Red Sox, who were experts at losing. The worlds of math and science, though, are from the beginning as competitive as organized sports. It is related to the gender issue which I cannot speak of lest I be accused of wrong thinking and those dreadful "sweeping generalizations."

The great English mathematician G.H.Hardy, in his autobiographical and mathematical memoir, "A Mathematician's Apology," expresses this in a famous paragraph:

"…I thought of mathematics in terms of examinations and scholarships; I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively."

Hardy was also a "court tennis" and squash player, and wild about cricket, which only reinforces the image of mathematician as athlete, or perhaps athlete as mathematician. For a young mathematician or physicist nowadays, not only are there examinations and scholarships, but science fairs and math tournaments and finally, the Math Olympiad which tests high school students in the USA, which then leads to the International Mathematical Olympiad, where the best youths (and a few maidens) from all over the world compete. I looked at the problem sets that these fortunate few tackled, and it would take me ages to solve just one of them, let alone a whole test full of them. Math is a competitive sport.

And then of course after ace-ing the examinations, winning the scholarships, and scoring in the Olympiad or the science fair, there is the ever-more-brutal competition of undergraduate and graduate schools, which winnows out all but the (pun) Hardiest of people. There in the classrooms and the laboratories and the conferences, competition goes on at all levels. After the PhD, the post-doc. After the post-doc, if he keeps "winning," the academic or industrial job. And after years of competing in research, publishing, politicking, jockeying for position, the favored few reach the high point: academic tenure. But that is only the lower reaches of the science and math competition, for in the lofty heights above live the superstars: Nobel Prize winners, or in the case of mathematics, Fields Medal winners. It is amazing to contemplate the recent case of Grigory Perelman, who, against all sense of competition, refused this most-coveted prize in mathematics, even though he earned it.

But my approach to my extremely modest math effort is rather like my participation in tennis, where I could hit a ball off the backboard, bouncing it back to myself, for hours on end. I didn't do much actual playing. My experience of any kind of competitive play was very limited. There's plenty of competition in the art world, but as with math or science, I have avoided it. I don't know whether that's a good or a bad thing.

There is a lot going on in my life currently, with day job, art, and learning new graphics programs taking up much of my time. Some days, I only get to do a few math problems. I have been wondering whether I should contrive some sort of artificial pressure to make me do more math work and progress faster. There is no Olympiad for me to win. There is nothing about my math and science work that matters to anyone but myself, and there are no coaches who encourage me to stay in training, or even ask about my progress. There is no career ahead of me no matter how much I learn. And yet every time I approach those math problems, I tackle each one with some anxiety, since the review problem will test me to see whether I have really learned anything. If I get it wrong, I'm disappointed, as if I've hit the tennis ball into the net. But if I get it right, I feel as though (to change the game to one I have never played) I've sunk the ball into the basket.

Posted at 3:08 am | link

I've finished the first of my "formal" paintings adapted from the sketches and photographs I did on my tour through Pennsylvania and Maryland. I hope that this will be the start of a long series of landscape, architectural, and townscape art inspired by the American countryside. It's called "Cornfield's Edge," and it is acrylic on illustration board, 11" x 14".

Here's some information for the artists who might be looking at this Electron entry. "Cornfield's Edge" is painted in acrylic gouache, which is a recently developed form of acrylic paint that dries not only opaque but with a flat, non-shiny surface. Conventional acrylic paint is almost always somewhat transparent, and I have to build up colors with lots of layers. With acrylic gouache, a single coat will do, on which I then add other layers of color and details. The paint is rather thick, though it can be thinned with water, and does not blend easily on the painting's surface. It won't be opaque if I add too much water, though. It dries very quickly, except in humid weather. When it dries, it's waterproof. The non-reflective surface tends to be delicate, so I spray a matte fixative over it when the painting's done.

This acrylic gouache was originally developed for craft painting, on wooden, cardboard, or plaster items. When I first used it, I had problems with fading and yellowing as the paint aged. But the types available now are professional artists' quality, although they're still marketed to crafters as well. One of the nicest things about this paint is that because it is also craft paint, it is packaged in pre-blended colors, so that the non-professional crafters won't have to make a mess mixing colors. This makes my job a lot easier, too. They have blended beautiful, natural vegetation greens, fruity gold, misty distance blue, sandy beige, weathered wood brown, and warm white. There are plenty of bright spectrum colors, too, in case I want to paint pictures of flowers, or Trader Joe tomato signs.

There are quite a number of brands of acrylic gouache now, from many different countries. My favorite for pre-mixed colors is Jo Sonya's Artist Colors which are made in, of all places, Australia. These are marketed as craft paints but I have found that they are as good as any professional paint. My other mainstay of this type is Polycolor acrylics which are made in Italy. (Site is in Italian.) These do not come pre-mixed, but in brilliant pure colors, and they are expensive. The top of the line in the acrylic gouache department are Golden Matte Fluid Acrylics but they are very expensive. A new, and less costly, entry in the field is Turner Acryl Gouache, from Japan, which features some exquisite pre-mixed colors, as well as unusual textures. This is probably more than you would ever want to know about acrylic paint.

I wondered whether painting these landscapes would be a diversion from the more "serious" geometric and space abstractions I've been producing. But I don't have to worry about losing my "edge." It turns out that the countryside has its own geometry, with interesting intersections of angles and forms made with planted fields of vegetation, buildings, and dirt rather than mathematical graphs.

Posted at 3:27 am | link

My passage through the "Rigorous Approach to Limits" chapter is going very slowly. I can sort of figure out what the book is trying to say. Something about an arbitrary interval of possible outputs of a function, that will always be smaller than the interval of outputs dictated by another arbitrary quantity at some distance from the limit. What I think it's trying to say is, that there is always some place (marked by a number) closer to the limit, as long as you don't actually get to the limit.

I don't "get" proofs yet, even though I've now been doing mathematics for six years. I'm not talking about geometric proofs, which I enjoy. These are mathematical proofs with no pictures or congruent triangles. They seem to hinge on one quantity being proven equal to another quantity and substituted for it. I know this is really simple math, but somehow I haven't gotten the knack yet. The calculus proofs, and the methods I've learned to find instantaneous velocity and limits, all seem like cranking clockwork to me. Maybe that's why the Enlightenment natural philosophers, fueled by Leibniz' and Newton's calculus, believed in a clockwork universe.

Today is September 7, which marks the sixth anniversary of my epiphany at Fermilab. Once I had set myself on that path to math and physics, I knew that someday, I would actually learn calculus. I delayed it for at least one of those years while I spent time learning first year classical mechanics. Now I am well into it. Could I have looked ahead at my life from my time in college, and said, for instance, in 1974, "Do you know that in 2006 you will be studying calculus and not classics?" I would laugh and totally disbelieve it. But after September 7, 2000 (a very different anniversary than the one coming up in four days) I could see my calculus destiny over the luminous horizon of Fermilab's beamline.

Something was born on that day, a math and science entity coexisting with the artist person. I am not particularly "talented" in mathematics or science, not like those brilliant youths who teach themselves calculus in a few weeks. As I continue to say, what I have going for me is perseverance and desire. However, when it comes to the Rigorous Approach, I think I am going to ponder it for only a few more pages. I consulted with my Friendly Scientists and they said that I didn't need the rigorous part right now. Since it is in a book and not a short-lived natural phenomenon (like a supernova or a volcanic eruption), I can put it away and come back to it when I need to. At this point I would like to move on to derivatives, as well as watch Mount Etna pour lava from its craters.

September 7 is also the birthday of Andrei, the husband of my co-worker Kim. He's only in his early twenties, lucky boy, and already an engineer. Happy birthday, Andrei!

Posted at 3:19 am | link

I justified my existence today. How did I do this? By creating an art work. By doing work on something meaningful, both to me and to other people. I am a "works-oriented" person. Just "being" isn't enough for me. "Doing" and working are what make life worth living. I suppose I am somewhat of a "Protestant work ethic" type, although I am anything but a Protestant. I believe that in order to, as it were, earn my right to exist, I have to keep making and doing good things. The meaningful work for today was a coffee sign board at Starbucks. I currently decorate sign boards for five different Starbuckses. I don't get paid in money, because Starbucks policy doesn't allow paid outside contractors. But at certain Starbucks, I can get all my coffee for free, and some merchandise items as well. Barter works for me.

My designs this September are all based on paisley designs and stylized plants, fruits, and leaves. I use warm autumn colors, especially my favorite orange, which is also the current Starbucks graphics theme color for autumn. The design I did on Tuesday is based on a printed Persian tablecloth from Isfahan, which was given to me (also in barter for services) by a Persian friend.

The image is slanted because I had to stand somewhat sideways from the board so that the flash of my camera wouldn't reflect off the smooth surface. The sign is about two and a half feet squarish, (actually a trapezoid), and it is painted in opaque acrylic markers.

This elaborate design will stay up for about two months, and in November, I will erase it to make way for the winter holiday-theme display board. A work of justification is not necessarily permanent.

Posted at 2:57 am | link

I visted "Mount Aetna" while traveling in Maryland, but I have come home to view renewed volcanic activity from the real Mount Etna in Sicily. Mount Etna is constantly active. It erupted spectacularly from 1999 to 2002, but after that it had not shown visible fire until recently this year. A couple of months ago, a fissure opened high on the mountain slope and poured out some streams of lava for a few weeks. That eruption ceased, but now the mountain is flinging forth fountains of sparks and fire from an old crater which had not erupted since 2000, the so-called "Southeast Crater" or more descriptively, the "Southeast Cone."

The Internet, source of so much good and bad, brought me back to the great volcano, whose distant fires I had seen with fascination in my childhood when I was touring in Sicily with my parents. There has been a series of Web cameras trained on the mountain for at least ten years. In 1998, I discovered the Web camera and quickly became a devoted volcano fan all over again. It was my first re-contact with the world of any science at all. I read up on anything volcanic and volcanological that I could find. That year, which was a rough year for me in many ways, was made better by my re-introduction to volcanology. And this contact, initiated through an Internet novelty, was my re-introduction to science in general and the scientific way of life led by the geologists I read about. It was volcanoes that first gave me a sense that I might have some scientific potential myself. My next entry here will mark the anniversary of a day which took me beyond volcanoes into the world of physics and mathematics.

I quickly initiated my friends into watching the Volcano-Cam too. We became volcano fans, and the mountain did not cease putting on wonderful shows. In 1999, Etna erupted on a regular schedule from the Southeast Cone, and every blast was delivered to us via the Web-cam. We used to call it the "Saturday Night Volcano," because for a number of months it erupted on a surprisingly regular schedule, every Saturday night. That volcano was definitely the life of the party.

In 2002, the mountain began a more general eruption, and it blew out a brand new crater very near Southeast Cone. We watched in amazement, but our viewing was to be short-lived. Not only did the ash and cinders from this new cone bury the old tourist observatory which we could see from our internet vantage point, but ultimately, the Web camera itself was burnt up by volcanic blasts. That's what happens when you put a camera too close to the volcanic action.

It wasn't till about 2005 that the cameras were reliably back online, but nothing happened onscreen other than puffs of steam and occasionally ash. But Etna would not be quiet for long. Now we can see the Saturday Night Volcano again. For the moment, it seems to be emitting steady, but low-intensity, bursts of flame and glowing rocks, known as "Strombolian" activity after Etna's smaller cousin Stromboli, rather than excitedly popping off its cork and gushing at regular intervals. It may fizzle out, or it may intensify into another blazing lava-fest. No one knows. But my volcano-fan friends are back watching, enthusiastically hoping that the mountain will put on some more shows.

You, too, can be a volcano fan! View the Etnaean fires at the "Etna Trekking" Web Cam. Remember that Italy is six hours ahead of the East Coast of the USA. The fountains of fire are not visible by daylight; you will only see smoke and steam then. Check in when it is night over there for the best effects. The camera updates its image every thirty seconds.

Posted at 3:02 am | link