I finished most of the limit problems in chapter 2.5 of Anton's book. There were a few left at the end which were more or less beyond me. I've noticed that a lot of my math and physics books include problems like this at the end of sets, which are meant to teach new material and lead into the next chapter. In a classroom or tutorial I would imagine that the teacher and the students worked these out together, but I can't do that. Instead I have the teacher's manual, with not only the answers but the documentation for how the problem was solved. I can usually work it out using this manual book, which is as large as the textbook itself. I have saved some of the last problems for later, in case I actually can meet with one of my Friendly Mathematicians.

Now I have started Chapter 2.6, which is titled: "Limits: A Rigorous Approach." Now I don't know whether any of my math or physics studies have been rigorous, but given my situation I do what I can. There's a kind of dark thrill about "rigorous," almost an S&M tinge of mathematical bondage and discipline. This chapter will introduce me to some of the customs and ways of the mathematical scene. It uses Greek letters, which so far I have understood mostly in the context of Greek language. I know that "epsilon" means "a very small quantity" because I read about it in Paul Hoffman's wonderful biography of the great mathematician Paul Erdos, The Man Who Loved Only Numbers. Erdos used to refer to children as "epsilons" because they were small. Chapter 2.6 also uses a small "d" or delta. Big "D," the triangular delta, is already somewhat familiar to me and will appear in the next chapter about derivatives and differentiation.

The book refers to the Rigorous Approach chapter as "optional." Does that mean that the students in a calculus course skipped it? Or did the professor assign it if there was time? Will I need it later on? I will read through the chapter and try to follow the material anyway, with the comforting thoughts that I do not have to assimilate this all at once, that there will not be a test, and that my future physics career in another universe does not depend on it. There are problems for this chapter, too, and I'll try to solve them if I get through the Rigorous Approach, but the problems are "optional," too. After all, for me, all of calculus is optional.

Posted at 3:24 am | link

As summer winds down, I have reached yet another round number on this Electron Weblog. This is post number 300. That's a lot of posts, over a period of two and a half years. As I do when I reach a round number, I will present a progress report.

I started ELECTRON BLUE in February of 2004. Back then, I was slogging my way through trigonometry. I finished, or rather gave up on, trigonometry later in 2004, when I studied logarithms for a few months. Finally, in late 2004, I turned my attention to introductory physics, that is, classical mechanics. I spent all of 2005 working on first year physics. This meant mass, weight, gravity, Newton's laws, falling objects, rolling objects, sliding blocks, acceleration, friction, torque, tension, vectors, acceleration, orbits, and all those other things which make up the heavy world which we must struggle through. By the end of 2005, I had more or less covered what a high school sophomore or junior might study. It was 35 years late, but at least I did it.

At the beginning of 2006 I finally started what I had been promising to do for two years: calculus. I began with a review of functions, which was helpful because I never quite understood them or their notation. Then I moved into instantaneous velocity, and after that, limits. I am doing well on limits, so soon I will move to the next chapter of the book, and start on derivatives. I am pleased that so far I have not hit the proverbial "wall" which drove other math students away from calculus. Either I am not there yet, or there is no wall, or I have recognized it and found my way through or over it. I will continue with calculus all through 2006, and probably well into 2007.

So far, I'm enjoying calculus, even though I have not yet been told what it is good for in the "real" (engineering, physics) world. I expect to find out more not only through my books but through the calculus course on DVD, taught by Professor Starbird (more on this later). The good thing about Professor Starbird is that he can teach me any time of day or night, since he is only a "virtual" presence on a DVD. But the bad thing about the virtual professor is that I can't ask him any questions.

I don't know how many of my "Friendly Scientists" and "Friendly Mathematicians" are still reading this Weblog. After a period of self-promoting enthusiasm when I tried to alert various Real Physicists to my presence on the Electron Weblog and my quest to learn physics, I have given up trying to contact them, other than the ones I already know. Physicists and their world continue to be, for me, a society as distant and enchanted as "Harry Potter"s author J.K. Rowling's "Wizarding World," a universe where people who know and do amazing things travel around the globe the way I go to the Starbucks around the corner. These enchanted folk gather in impressive places and share their arcane knowledge, while I am off gazing at herds of cows in Pennsylvania. It is not my business to try to crash their party.

I find super-competence fascinating, even compelling. The physicists and mathematicians whose Weblogs or books I read are heroes to me, not just because they are dealing with fundamental realities and looking into the secrets of the universe, but because they have mastered something which I am just beginning. I really appreciate mastery. Not the "power over" type of mastery, the "power to do" type. It is almost 6 whole years since I went to the cathedral of physics, Fermilab, on September 7, 2000. That was where my current journey began, with, as it were, a "baptism of particles." I don't know how far I will get. Relativity? Quantum mechanics? Particle physics theory? M-theory? I'm just doing this on my own, so I have no idea how I would learn any of these. But I will find a way if I can.

Some things about my life have improved since Electron 200. In my art work, I have figured out what the focus of my efforts should be, that is, fine art to be shown in galleries. I have to start on a small scale, but I have been told that there is a market for "fine art" of a realistic sort in local, small-town galleries, including my own Falls Church. I am doing two "lines," to use marketing language, of art. One is the geometric space abstractions you have been seeing on this Weblog so far. Now, I am starting another line of paintings of idealized countryside scenes. I probably won't show the two types together, which would confuse collectors. But as an artist with a commercial illustration background, I am not much into "self-expression" or any political or social message. I just want to create good-looking pictures which will appeal to people and be bought. The fun for me is solving the problems in making the image: how to render things in paint or pencil, how to get the perspective right, how to duplicate just the right color of light on trees.

Another thing that has improved is my ongoing health situation in regard to, uh, the female "change of life." I have been tormented by hot flashes, as longterm Electron readers know. But recently, I have had success with a combination of soy extract and a prescription drug, which suppresses the hot flashes (except, unfortunately, on very hot days such as we have had this August). In moderate weather, the regime works great and I am hot flash-free, at least for now. With biological systems, you can never tell. The combination tends to upset my stomach, but I can keep this under control by eating judiciously and not taking the pills without food.

I continue to enjoy my day job at Trader Joe's, which has allowed me to do all this non-paying "fine art." Though I still exhibit at some science fiction conventions, this isn't my main focus any more. That culture, in my view, has lost a lot of its creativity and excitement anyway, as much of it has been outsourced to Japan and books have given way to film and video games. I still continue on my own "sequential art" project, when I have the time.

When I first started ELECTRON BLUE, there were a few other bloggers and private contacts who were trying to do things similar to my project. There was an engineer guy in New Jersey who was trying to learn quantum mechanics in his middle years. There was a lady in Australia who was actually going back to school in her middle years, trying to get a degree in mathematics. There were others, too. But I regret that I have not heard anything from these people in a long while, and their Weblogs have been discontinued. I cannot possibly be the only older math and physics student in the world, but it often feels that way. I have learned, or rather, re-affirmed one thing about myself during the six years I have been working on the Physics Project and the three hundred postings at ELECTRON BLUE. Which is, that I may not be super-competent, but I am super-persistent. I don't give up. And I hope that I never will. Therefore I stay in the driver's seat of my Electron Car and the road lies ahead.

Posted at 3:06 am | link

When I came home from vacation, I found spiders in their webs at the bases of my apartment walls and dust on the floor. Fortunately, a friend had come in to water my indoor and outdoor plants during my absence, since there had not been more than a few drops of rain here while I was away. This place endured three days of deluge just a couple of months ago, but August has been very hot and dry. There's always some extreme of weather to complain about.

I found five dried, spent blossoms on "Bucky" the cactus. I knew that I would probably miss one of Bucky's two flowerings, which occur in late summer. My plant-watering friend reported that on the night he had come in to water the plants, he had found Bucky in bloom. His timing was good, since those blooms only last for one night. At least someone was there to witness it.

There are so many household tasks to do after a return from a trip: laundry, bill paying, dusting, spider removal, unpacking, trinket management, and studio preparation. I re-assembled the desktop fountain, filled it, and activated it again, for its eleventh year of trickling. I ordered a lot of new art materials, especially landscape green paints, and I will have to re-arrange a part of my studio area to accommodate them. They have now arrived, and I am in the re-arrangement process. My art studio, as well as yours, can benefit greatly from the wonderful stuff at The Container Store. I will actually give away extra things that I have not used in years. I intend to be artistically efficient, as well as pastoral and astronomical.

The Electron Car needed a bit of work and maintenance after the Pennsylvania-Maryland trip, and that has now been done as well. What remains is the rest of the laundry, which is never-ending, and the dust, which is also never-ending.

Finally I opened my calculus book and got back to the important stuff. I went back to the pages detailing the cumbersome process by which instantaneous velocity is found at any point in a function's output. I wrote the process down in order complete with illustrations of how the substitutions of various quantities work out so that it can be simplified at the end. I wondered whether the orderly Amish did any calculus. After reviewing instantaneous velocity, I will return to the set of limit problems and finish those. Then I will turn the page to the next subject, which is derivatives.

Posted at 11:43 am | link

They're starting the harvest of corn and apples in Pennsylvania, but I have returned home with a harvest not of produce but of images. I returned safely home to my Northern Virginia dwelling with a folder full of 434 (as Windows counted them) digital camera photographs. My vacation mission is accomplished. I now have an extensive library of country imagery: fields, hills, farm buildings, cows, horses, winding tree-shaded lanes, front porches, weathered wood, small towns, and a couple of cats. I managed to avoid taking any pictures of people. I also have sketchbook pages of colored pencil drawings done on site or from very recent memory.

I now have the resources to create a pleasant rural world in art, in the tradition of those landscape painters in Europe and America who portrayed the countryside as benign and friendly, full of picturesque things which gave city people good feelings. I know that this will be a work of the imagination, rather than any "accurate" depiction of reality. Remember from my philosophical essays that myth is reality, too. There will be no manure piles or broken-down trailers in my country-themed art. I plan to paint a series of works using these photographs, as an alternative style of art to my usual geometric hard-edged abstractions. I am interested to see which ones will be more popular, and where. And maybe at some point, if I get a bit daring, I might combine the two styles.

Meanwhile, I'm back to work, and my more or less routine city existence. What did I miss during my vacation? I missed doing math! I opened up my calculus book to find that all my instantaneous velocity problems, and many of my limit problems, had unsolved themselves, so I must work through a couple of them again to get back up to, uh, speed. And I can always think about the vectors of country roads and the curved geometry of rolling hills. "Harvest Home" as a holiday refers to the Autumn Equinox, which is still a month away, but here for you is one of my photographs of a cornfield soon to be harvested, in a world which in memory is already ideal.

Posted at 11:43 am | link

I have passed from Pennsylvania into Maryland, and am currently touring the Hagerstown area. This is in northwest Maryland, just as the state gets very thin on the map. The terrain is hilly, with some higher mountain ridges, part of the northern reach of the Appalachians. This area was one of my destinations when I planned my trip, because east of Hagerstown is a little town called Mount Aetna. As a volcano fan, I just had to see a place in geologically stable Maryland bearing the name (spelled the Classical Greco-Roman way, rather than the familiar "Etna") of the famous volcano in Sicily.

Mount Aetna is reached by a country road, appropriately titled "Mount Aetna Road." I drove southeast from Hagerstown, through the new housing developments which are eating up farmland all over the country, and past some big ugly "McMansions" which are also proliferating in the area. After that, there is still countryside, with old houses and barns and fields and cattle. The road is quite twisty and hilly, and it became less and less traveled as I went further into the countryside. It was a hot, dry day, as all my days have been on this journey, accompanied by the ambient soundtrack of cicadas and crickets.

I kept on going as the road climbed up a hill and into the forest. Finally I reached an intersection on the hillside, with a fire station and a handful of houses. This was Mount Aetna. It wasn't even big enough to be a village, just a place in the road. But there was a sign to identify it.

The forest was gold-green and peaceful, as yet uninvaded by developers and ugly mansions, but there was no Sicilian volcano. What I did discover, down the road, was the very pleasant and inviting Mount Aetna Camp and Retreat Center, which had not only a nature study center, a frog pond, and a small group of horses, but luxury rooms for woodsy getaways. I hope to return there, either by myself or perhaps on retreat with my companions in the Order of St. Michael.

While I was there I met with one of the administrators, who gave me information not only about the retreat camp but about the village and site of Mount Aetna. No one knows quite how it got its name, but he said that the area was known in the past for its foundries and furnaces. The location of the fire station, in fact, had once been a metalworks. The area was a manufacturing site for cannons. I could imagine that some classically trained individual, inspired by the smoke, sparks, and molten metal of the forges, might have named (or re-named) the spot after the fiery, lava-pouring mountain of the distant Mediterranean.

Posted at 11:42 am | link

I'm not an Impressionist. I have no desire or intention to paint like Monet or Renoir. But those Impressionists have so influenced art since their time that it is now considered the height of artistic honesty to paint outdoors, as Monet and the other Impressionist landscapists did. Before the invention of photography, and after the development of "modern" painting in the eighteenth century, painters did sketches outdoors, maybe in oil or watercolor, but they finished their paintings indoors. The Impressionists did the whole thing outdoors, which was supposed to make it look bright and fresh and immediate, rather than studied and academic like the brownish, slick art they were rebelling against.

And then that wicked photography came along, which captured reality too well and violated the Biblical injunction against graven images of ensouled beings (which is why the Amish don't like having their pictures taken). By the middle of the nineteenth century, only a few years after its invention, artists already were using photography as reference for their work. Even famous ones such as Cezanne, Courbet, and Delacroix used photographs, and Degas depended heavily on photography to catch those "naturally unposed" moments in his images. By the twentieth century, art and photography were inseparable, and one of my favorite American painters, Charles Sheeler, was both an artist and a photographer. I haven't found out whether another of my favorites, Edward Hopper, used photographs, but I suspect he must have. And yet….there still seems to be something slightly dishonest about using photographs as reference for art.

I'm still roaming through central Pennsylvania. Wednesday was a very bright, hot day, with a hard sunlight beating down on dry fields. I would have liked to sit outside and sketch, but I would have been fried. Instead, as I have been doing all through my trip, I've been snapping dozens and dozens of effortless digital photographs. I am documenting the landscapes and buildings of Pennsylvania for later use in my studio. Since I don't have a whole lot of time, I can't spend hours painting or drawing outdoors even if I had shade to do it in. I did manage to get a couple of colored pencil sketches of vegetation in evening sunlight, after the heat of the day had passed.

Sheeler loved to paint and photograph farm buildings, including many in Pennsylvania, because their geometries are so simple and abstract. I am gathering Sheeler-esque images to work with, as well as more idyllic, pastoral landscapes full of August's golden light. My artistic intent is not to create a fresh, immediate impression. I know, from my twenty-first century perspective, that as soon as I have photographed that landscape, or as soon as I have set it down on paper with pencil or watercolor, I have created a false image. I crop and alter the photograph. I selectively remove things from the painting that don't fit in a harmonious composition. I omit telephone poles, power lines, signposts, trash, or cars which would get in the way of the ideal landscape I want to create. In a painting, I use color tricks and enhancement to make things look better than real. Even if I never took any photographs and did all my sketching outdoors, I would still be creating green lies in the name of art.

Posted at 2:34 am | link

Oley Valley, near Oley, Pennsylvania, August 14, 2006.

I've been enjoying my summer tour around Pennsylvania so far, and I've made a number of on-site drawings. I've also taken lots of photographs for later reference. The corn is mature, and the land is wrapped in golden August haze and light. The fields and forests resound with the buzzing of cicadas and the chirping and zipping of crickets. Twittering swallows and goldfinches fly above the fields. Today I visited Oley, a historic town near Reading, which is the place that Wallace Stevens mentioned in his poem "Credences of Summer." Stevens was born in Reading, but he would have been disappointed to see how poorly it is doing nowadays.

I am avoiding the cities as much as possible, because I want to spend time in the countryside taking in pastoral images. This part of Pennsylvania is a very orderly place. There seems to be no wildness at all; every farm is neatly maintained, the cornfields sharply bordered. The soyfields are all weedless and green, the vegetables grow in neat rows. I see very few buildings in disrepair.

My current stopover is in Amish country. I have seen them in their farms and on their porches, and have passed their horse-drawn buggies and carts on the roads. I have never seen real Amish before. It is forbidden to photograph the people, but things, buildings and animals are OK to photograph. Their farms are the most orderly of all.

According to the (web) sources I consulted, the Amish live by an unwritten set of social and moral rules called the "Ordnung," not easily translated from the German to English. It's the "order" of how things are if you are Amish. To me, and probably to most other visitors and tourists, they are like humanoid aliens, or perhaps refugees who have passed through a time machine to set their colonies in our modern world. I am trying not to pay too much attention to them, as it is landscape and buildings which concern me. But they are part of the land; this is their home, not mine. And they built many of these buildings. All of it is created by people who value order, and the idea of order, just as the poet did as he contemplated the geometric straightness of Pennsylvania farmlands.

Posted at 12:25 am | link

I am packing my things for vacation. I am planning to follow the example of poet Wallace Stevens and study calculus by graphing the patterns of haystacks in rural Pennsylvania. Well, not quite, but I do expect to go to central Pennsylvania and find beautiful landscapes to photograph and draw and paint. I'll be wandering around there and also in central Maryland for about ten days, and hope to return on August 20.

I have been doing limit problems, without too much difficulty. I will be taking a vacation from doing math as well, but I will take my calculus lecture DVD's along in case I have a moment to watch a program. I'll have Internet connectivity for at least some of my journey, but I don't know whether I will post an entry to this Electron or not. I don't have any specific plans after the first couple of days.

Things have been kind of suspended for me anyway regarding art and math in the last few days, because I've been involved in reading "Harry Potter and the Half-Blood Prince," the sixth book in J.K. Rowling's compelling series about the boy wizard. This was a thrilling tale even at over 600 pages, full of intrigue, romance, comedy, and tragedy. It's not just a kids' story any more, Rowling has now made this series worthwhile for people of all ages. It's not High Literature but it has the elements of it nevertheless, with genuine moral and emotional conflicts and the familiar neo-Zoroastrian background of the Great Battle between Good and Evil. If you read carefully, there are only lightly concealed references to current political and global realities in the tale as well. I love the magical world that Rowling has made up, which is much less pretentious than Tolkien's. And even though I am not much of a fiction reader, this fiction kept me turning pages when I should really have been doing something else, like calculus or art. Book 7, the last of the series, will appear next year.

I'm off, if all goes well, on Thursday August tenth, hoping for pastoral peace and green meadows, soft breezes and forested hills, and a vacation from limits.

Posted at 2:09 am | link

With news of destruction raining down on Biblical lands, and tastes of heat, drought, and dry thunderbolts here, my thoughts turn, as many other people's, to intimations of Apocalypse. Yet August is the warm quiet doze of the year, the sleepy afternoons of sizzling insect noises and dry open skies and golden sunlight, and at least in my own little circle it seems that nothing will happen, at least nothing has happened yet. Every day I say that. Nothing has happened….yet.

I look up into the sky, seeing the Great Summer Triangle of Vega, Deneb, and Altair directly overhead. This geometry precedes the rising of Sirius, the Dog Star just before dawn, which gives its name to the "dog days" of summer. I look up into the clear, dark August night, and so far I have seen only the familiar Moon and planets and stars. But what if, just over the horizon, some phenomenon of terrifying brilliance impends, ready to surprise me and shatter the familiar sky?

I am always looking up at the sky, watching clouds by day, stars or clouds by night. I wait for lightning to the west, rain to the east. And on many occasions, I feel the nearness of that "non-ordinary reality" I was talking about in my set of essays last month. I will go out onto my terrace, or peer out the window, and instead of seeing the usual, familiar, friendly things in the sky, I might see a huge, strange moon, the size of a grapefruit at arm's length, hanging in the sky where nothing was before. Or I could see a supernova suddenly blaze out nearby, brighter than the full moon. Or I might see a comet stretching across half the sky, or a flaming bolide exploding across the heavens. I imagine seeing these things. I have not actually seen them, at least in "ordinary reality."

Yet I have seen, on rare occasions, things that come close. I've seen more than one larger meteor rip silently across the sky, often in bright green fire. One such apparition was so brilliant I thought for sure that this was It, that a missile was falling. But nothing happened, no explosion, not even a landing; it was just a momentary celestial firework. I searched for confirmation of a fireball at that time on the Web, but no one seemed to have noticed it but myself, since it took place at around 2:30 AM when normal people are asleep.

The inexorable seasonal cycles do not reassure me; they tempt me to expect more of the same, in a peaceful, circular round of events. The late summer crickets sing, the harvests are ripening, the goldenrod is in bloom. The catastrophe, the cataclysms, the apocalypses, interrupt that cycle, and punch a hole in it which will not quickly be patched over. Today, August 6, is the day the atomic bomb was dropped on Hiroshima, out of a clear, dry, sunny blue sky.

Posted at 2:53 am | link

It's hard to stay philosophical and mathematical when the temperature has been hovering around 100 degrees F., as it has been for the last couple of days. There won't be a respite from the heat till the end of the week, it is said. The air in my area has been declared unbreathable ("Code Red") with an "Excessive Heat Warning" and I can attest to that. I am lucky to hide in my air-conditioned workplace and my air-conditioned home.

The air is so full of haze that the colors of the trees have faded to a shimmering greenish grey, much like the now-faded landscape paintings of the French painter Jean-Baptiste Camille Corot. The effect is called atmospheric perspective, and is much used by artists, including myself, to portray distance when the usual straight lines and converging vistas are not in the picture. Atmospheric perspective runs on the same principle that makes the (clear) sky blue: the scattering of light from atoms of gas and particulates in the atmosphere. As you look through more and more atmosphere, or a thicker atmosphere, colors become bluer, and sunlight becomes redder as more and more light is absorbed. In an atmosphere as thick as the smoggy soup currently over my city, most of the color is bleached out, showing a white sky and blue-grey vegetation. The sun set in a blaze of murky orange, sinking into purple-grey mists.

It is August in the northern hemisphere of our planet, as summer edges toward its precipitous downward fall. A planet orbits a moderate star in the moderate suburbs of our galaxy, all of it circling round and following the expansion of the universe in various directions that some great celestial physics student could compile into a single vector, perhaps. Or if there are more than four dimensions, then someone would have to work really hard to figure out that vector. The planet (Earth, that is,) is inhabited with people who have attained enough technological know-how to establish a global communications network so that people as far away as the East Coast of the USA and as close as Kuala Lumpur, Malaysia, can share stories and pictures of their fellow creatures. (Check out the July 24 entry there for a familiar picture and link.)

I have no doubt that in other galaxies, other intelligent creatures are experiencing similar things, doing science, cultivating other creatures as pets, making art, and listening to ambient music. We will never know of their existence, but I know (have faith?) that they are out there. And on some other planet, it is a misty, sweltering August, where gases sizzle towards entropy just as they are doing here. And there may be as many other universes as there are galaxies in this one, quintillions of suns and Augusts, mists and creatures. And even farther out….some creative scientists are wondering, as is written in this fantastic article on the COSMIC VARIANCE blog, whether our universe is merely the result of a momentary fluctuation in the entropy of what would otherwise be a mythologically vast homogeneous unity of everythingnothingness. (Parmenides, check your e-mail.)

Fortunately, our weather system is dynamic, always changing, and soon the energy of the same sun that is steaming us will send the bad air away with a blast of storm clouds, and we'll see the atmospheric perspective of blue skies and cooler temperatures.

Posted at 2:41 am | link