Of all the mathematics I have yet done on my now seven-year journey, I have been least fond of trigonometry. And the part of trigonometry that I've disliked the most has been trigonometric identities. These are those arcane laws or perhaps descriptions of how one trigonometric feature is the same as two others in relationship. Or they describe the relationships between sines and cosines which are added or subtracted. There seems to be no end to them. I remember when I tried to work with them three years ago. The memory numbs the mind. I am sure that somewhere out there are math geeks who just LOVE trigonometric identities. But I'm not sure I want to have any contact with these people.

The Identities are placed in light blue boxes in the text, so that we students will notice them. The mild light blue does not convey the metallic, riveted quality of the math. Underneath these revealed windows are the proofs, which we are supposed to work through. That's right, it just isn't math without a proof. Show us the proof! In math, unlike in real life, there actually IS a proof, at least at this level. The atheist says, Show me the proof of heaven! Show me the proof of God. Is religion the same as math and science? Must there be proof? Is citing the Bible or the Koran a proof? Certainly not to the scientist!

Heaven, for a mathematician, is where all the proofs come out correctly. It is the domain of right answers, for all the legitimate functions that exist in the universe. There in mild blue mathematical heaven, where the light of those identity windows suffuses the bright sky, mathematicians inhabit the crystalline structures built from what were only concepts down below on old Earth. In heaven, calculus angels draw diagrams in shining sand.

And what of Hell? The mathematics of monotheism demand a reciprocal, no? Hell is where all the answers are wrong, but no one can find a way to disprove them. Hell is endlessly multiplying garish purple, green, and red fractals of complication with no way out of the labyrinth. Hell is a vast noisy neon-lit techno club where you are forced to take math exams while you are stoned. Some people probably would like this Hell.

How can I make myself want to study trigonometric identities again? By daring myself, as in "I DARE you to work through that proof that's in the book?" By shaming myself, as in "You managed to finally get to calculus but you can't go on because you can't hack high school math?" Should I perhaps cajole myself into doing the work, because I will get to learn new things if I manage to remember the old things? Or should I appeal to my peculiar pride and remind myself of the singular path that I have traveled now for almost seven years, as I have watched all my fellow aspirants drop their programs and disappear? I'm not going to get to math heaven, and I'm not going to get to physics heaven, which are attained only by the canons at the Accelerator Cathedrals, or the Academically Tenured. Why on earth should I study trigonometric identities again? Because somewhere, somehow, I will demonstrate the proof.

Posted at 2:32 am | link

On a Saturday afternoon in urban McLean, Virginia, the streets are deserted and hardly any cars pass along an avenue which in most days is thick with traffic. This is an affluent town, and most of the people have gone away to their other homes or some form of vacation getaway. Meanwhile, the stores are nearly unpopulated and the golden light of a solstice evening slants across clear parking lots.

It's a surrealistic summer of space in a place where none of it is usually in evidence. I have fantasies of open space the way other people have fantasies of consuming or performing. I drive slowly through a quiet neighborhood at twenty miles an hour, looking at gardens and houses, with no one impatiently honking their horn behind me. High clouds drift across the sun with no hint of rain or turbulence.

I gaze at porches and wish I were sitting on one with a drink in my hand. I wonder whether I will ever get my wish, to live in an open space in a house with a porch. I had a porch for many years in Cambridge, Mass. but there was no view at all there, and I could barely see the sky. I saw plenty of such houses and landscapes last year in Maryland and Pennsylvania, so I know they exist.

Here on a lawn in a modest neighborhood (though it is being invaded by destruction and the building of oversize gross houses on small lots) is a vision of summer in the evening light. The gazebo is covered with netting to keep the insects out, in a kind of "oriental" drapery, reminding me of some Indian palanquin of pleasure. But there is no one in the gazebo, neither pasha or houri.

The solstice has passed, and the days will now slowly get shorter. In the southern hemisphere, they are getting longer. If only I could be like the arctic tern, which migrates from one end of the earth to the other following the sun. Is there any place on earth where gentle sunlight and green trees last all year? Perhaps there isn't, and such a landscape exists only in the secret gazebo of the World of Imagination.

Posted at 4:02 am | link

I am still pondering radians. The word, which is beautiful in itself, is of fairly modern origin; it is not ancient Greek. It sounds like "radiance," which has the same linguistic root. Astrologically the Sun is represented by a dot in the center of a circle. Alchemically, this represents not only the Sun, but Gold, the ultimate material. That is the same diagram of the Unit Circle which gives us trigonometry and radians. My Pythagorean prayer is offered to the One in the Center whose radiance reaches everywhere.

Let the unit circle represent the circle of the seasons, Earth's orbit around the Origin Point Sun. The year is thus 2Pi, and half a year is Pi. We are almost at the summer solstice here in the northern hemisphere, which is, after all, Pi radians around the earth from our antipodes in the southern hemisphere. Count Pi radians from the winter solstice six months ago. (All time measurements are very approximate.) The summer is the radiant time for us in the north, and the weather is just as I like it, hot and simmering and steamy.

Monotheistic time as measured in Zoroastrianism, Judaism, Christianity, and Islam is linear and progressive, an arrow of divine justice which goes only in one direction. The time of Radians is cyclical and repetitive, spinning through its multiples and divisions of Pi in number-mystical serenity. Reach 2 Pi, and you're back where you started. Reach 4 Pi, and you're back where you started when you started. Someday I may advance far enough to calculate in extra dimensions. In that case, are there other unit circles at angles to our own unit circle, each with their own Pi? Are there unit circles where Pi isn't 3.14159265…? Or is Pi, as the Divine Single Sun suggests, the one constant in all the worlds?

Radiant with daimonic light, the sacred spheres of solar unit and planets spin in their appointed paths. The reason for calculus is because the world, as Pythagoras first declared, is mathematically describable. If there were another world which was not mathematically describable, could we even conceive of it, let alone witness it? If we were in a solar system with multiple suns, would our philosophy and sacred geometry be different? The greatest gift that the One has given to sentient beings is Imagination. We can imagine things which cannot exist. Can the Omniscience do that? Let the summer sun stand still in the sky, at least until next week.

Posted at 2:46 am | link

We've all heard of the phenomenon called deja vu, where something we experience seems to be familiar, even though we know we have never seen it before. Esotericists like to think of this as evidence of reincarnation or at least some form of time-warping perception, but there are numerous neuroscientific theories for why this happens. The opposite of deja vu also happens, called jamais vu, where you are in familiar territory or see something you know well, but have the feeling that you have never seen it before. I have recently had it driving through the town at twilight on a clear June evening, where the deep green leaves cover the street signs and I feel as though I am anywhere in America, in some Ray Bradbury dreamworld lit by fireflies, where ghosts sit talking comfortably with the living on the quiet porches.

When I once again open my calculus book to trigonometry review after the flurry of distractions from the art show and maternal visit, is it deja vu or jamais vu I experience when looking at the mechanical orbits of the Unit Circle? During my bout with trigonometry back in 2004, I was concerned almost entirely with angles in degrees, not radians. Now, for calculus, I must learn them in radians. I lull myself to sleep by recounting to myself the formulas for how to transform measurements in degrees to radians (multiply by pi/180) or radians to degrees (multiply by 180/pi). I have to keep reminding myself that Pi is not just a Greek letter but a specific number, even though it spins its digits off into a random-like infinity. You'd think I'd be used to Pi the number by now, but I'm not, probably because I was a Greek scholar and think of Pi as a letter, for instance the first letter of "Pyracantha." (Pi, Upsilon, Rho, Alpha, …)

I've seen radians before. I know that they are expressed in some fraction or multiplication of Pi. It's an interesting idea, to wrap the radius around the circumference like a thread. I wouldn't have thought of it. I am told that in calculus trigonometry, you have to use radians rather than degrees, and the reason was explained to me by a patient Friendly Mathematician at the recent Baltimore convention. But the convention was a bustling, noisy place, and I confess that I have entirely forgotten the explanation. As with a lot of my math, I have to take it for granted and memorize it, whether I falsely feel that I have seen it before or whether I have forgotten something that once was familiar.

Posted at 4:24 am | link

Mother got home to the Boston area after three days here in the DC area. I managed to get her safely through all the objectives that I had planned for her, and she was well-fed and well-watered and well-rested. I made sure she had goodies at all time as well as a place to sit. She met my friends and presented my deserving Webmistress with a beautiful strand of beads. Everybody she met treated her well. So she had a great time and went home happy.

Meanwhile I have at least 6 loads of laundry to do, none of it Mother's, but I didn't get to do any of mine while she was here. I have de-cluttering to do and I haven't done any art product in weeks it seems, ever since my show opened. Mother saw the show and regretted that my architectural pieces were not larger, though she did respect the technique and the precision. I sold one piece on opening night but don't know if I will sell any more, although I know that at least two of them will eventually be bought by the owners of the businesses depicted there.

I think I am relieved, I finally don't have a deadline or a Momline to deal with. I turn round and round in my overstuffed dwelling and think, what productive thing should I do first? Laundry laundry laundry. I'm still tired. I really want to get working on graphics and sequential art. Trader Joe's has me doing nothing but lettering. I want to give some stuff away, just to make room for more stuff. What should I do next? Fold the laundry. This load was all orange.

Posted at 2:29 am | link

My mother has arrived in the DC area for a few days' visit. The main reason she made the journey is to see me and my art show. She's staying with me in my studio dwelling, where there is visible dust and clutter. I did not have time to get this visible dust and clutter out of the way. Mother noticed this. But it is not too much of a problem. On Thursday June 7 we accomplished the important mission and she visited Falls Church Art and Frame to see the show. She made complimentary remarks to the gallery owner and to me about my art. That day, I also brought my mom into Trader Joe's to meet my co-workers. Lots of nice things were said, and I don't think too many people were making things up to please my mother.

There has been a lot of fine dining, starting with luxury treatment and gourmet snacks aboard the first-class train she took from Boston. Naturally there are goodies from Trader Joe's and Starbucks as well, and a dinner at my favorite local restaurant. On Friday we expect to visit the National Gallery for more important artistic experiences, and if all goes well Mother will meet more of my friends. I hope that I am a good host and that I can continue to do well by Mom until she rides back to Boston. Mama Mia!

Posted at 2:12 am | link

"First Friday" is an old-fashioned Catholic devotional practice, but it is also the monthly arts festival held by Falls Church Arts and other places and people around the downtown area. One of the centerpieces of the Arts First Friday is the opening of each month's show at Art and Frame of Falls Church. June, then, is my month, and June 1 conveniently fell on a Friday, thus giving me the entire month of thirty days to display my work. My birthday is also in June, so I get a gallery art show for my birthday.

I have been working on this show, as faithful readers of this Electron know, for almost a year. I negotiated the June month with the gallery owner back last September. Since then I have been producing my architectural portraits in preparation for the show. The last in the series, the one with Bedo's Leatherworks, (see the May 30 entry) was only finished a few days before the show opened. I came to the gallery some hours before the show opened on Friday, to arrange the art on the wall and put the identification and price tags up. At the last moment, we decided to put two of my de-commissioned Trader Joe's dairy signs in the show, though they are not for sale due to copyright reasons. This is what the whole show looked like, with a separate row for my geometric abstractions.

I was very touched to find a splendid bouquet of flowers already there at the gallery for me. They had been sent by my wonderful Kansas friends, one of whom you met when we went to New England together.

We placed the flowers in their vase on colorful cloths which were set up for the refreshments. People started coming in even before the official opening time of 6 PM. The catering was done by the "Two Sisters" whose coffee stand I depicted in one of my paintings. They brought not only coffee but sandwiches and pastries. Other people brought more food, as if it were a potluck supper. The gallery provided wine (from Trader Joe's, of course).

In a half hour the gallery was filled with people. I was delighted that two of my ambient musician friends from the Stillstream.com community came to the show. It was the first time I had ever met anyone from this group face to face. My loyal friends from my own circles came as well, including one who made a two-hour drive from her home in south central Pennsylvania. Some of my co-workers from Trader Joe's also came to the show. The rest of the crowd were local folk who love art and come to the festive events every month, as well as other artists and members of Falls Church Arts. I got to meet some of these people, many of whom have lived in the area for many decades. They told me fascinating stories about the buildings I had portrayed for the show. Each one of them has a history, and for some of these sites, that history soon will only exist on paper or in memory.

Here's a picture of the group in the gallery space.

Some of the people you see there are my dear friends who not only socialize with me but also buy my art. They're not only friends, they're "Patrons." I was also happy to see young folk there (such as the girl in the turquoise shirt looking intently at my art wall). Around 7:30 or so I gave a short talk about the art on the wall, with my wine glass in my hand. I never have a problem talking about anything in public, but it was quite noisy and I had to raise my voice to be heard over the active crowd. But I got my message through. Sometime during the show evening, a Falls Church worthy bought my Bedo's picture. That's a fast turnaround—from studio to sold in four days. I hope more will sell throughout the month.

By eight PM, the party was declared over and we all dispersed into the balmy night to further social engagements. I went with a bunch of my friends to a nearby Japanese restaurant. I was greatly relieved to get the gallery show up with everything going smoothly and no opening night disasters. I am still enjoying the sense of freedom now that the deadline is over. But there is plenty more that I have planned that was put aside because of the preparations for the show, and now I must go to it. You will hear those plans as I accomplish them. Today, Saturday, I was back in a Starbucks doing a coffee sign, and on Sunday it's back to work among the artisan bread and organic romaine lettuce at Trader Joe's. And there's also my trigonometry review, which emerges from the architectural perspectives and my trusty T square and triangle. Electron readers who attended the show, thanks for being there!

Posted at 3:27 am | link