Another one of my cacti is in bloom. I couldn't identify the species of the first one that bloomed earlier this year, but I do know the species of this one. It is a Mammillaria hahniana, also known as the "Old Lady Cactus," distinguished by its white hair. These cacti are common in cactus gardens, and this one has bloomed before during its tenure in my collection. I put this plant out on the terrace with the other cacti and succulents and it had a good summer of sunlight and desert-equivalent conditions, so it was lively enough to flower once I brought it inside. The buds appeared some weeks ago as a ring of white fuzzy cones around the top, and when ready, the flowers emerged one by one, though not in a complete ring the way it did the last time it bloomed. Here are two pictures of my hahniana cactus, one in cloudy light and one in sunlight.

You may also notice, although this isn't entirely clear in the photograph, that the areoles, the cone-shaped points on the surface of the cactus from whose apexes the spines emerge, form neat spiraling lines that move in towards the center of the top of the cactus, which is where new growth forms. This spiral structure of cactus areoles forms what is called a Fibonacci spiral, after the mathematical series discovered by the twelfth century Italian mathematician Leonardo Fibonacci. The Fibonacci series, in which the next number is the sum of the two previous numbers, has been found to describe all sorts of growth processes, especially in living things. The series, when translated into a series of squares of proportional sizes, generates the Fibonacci spiral, which can be seen in flower heads, seed pods, pinecones, the famous chambered nautilus shell (seen on the Fibonacci spiral webpage) and the areolae of my hahniana cactus.

Why should this spiral appear so often in nature? Why not other spirals? The science writer Michael S. Schneider has written a wonderful if somewhat New Agey book, called "A Beginner's Guide to Constructing the Universe," in which he talks about why the world we see follows mathematical rules and the patterns generated by the first nine integers. In chapter five, on "fives" in nature, Schneider explores the Fibonacci series and spiral. The spiral appears in growth patterns, he explains, because:

….Although different in size, every segment (of the Fibonacci spiral) has the same curvature. If a microphotograph into the eye (of the spiral) were enlarged, it would find another part of the spiral upon which it would fit exactly. No other spiral will do this, nor will any other spiral accommodate the dynamic balance that nature values.….These characteristics are the open secret of balance of animal horns, seashells, plants, and galaxies.….the tree that puts out branches and leaves in spiral "staircases" around their respective "eyes" can get enormously large; yet the tree always balances no matter how massive it grows to be." (Schneider, pg. 148-149)

This also holds true for my little cactus. As the new growth and new areolae appear at the top, they expand proportionally at the same curvature, forming a concentric cluster of Fibonacci spirals at the top. To see what kind of wild, psychedelic images computers can generate from Fibonacci spirals, visit this site, maintained by artist Edward S. May. But somehow, I prefer to look at my non-computer-generated, hairy, spiny cactus with its little magenta flowers.

Posted at 3:52 am | link

Hardly any math is getting done here. I haven't had the time. I have been holding down the day job as well as getting a number of private art jobs done, such as a portrait, a T-shirt logo, and preparing for a small show this weekend. I don't know what I spend the rest of my time doing. I seem to spend a lot of time cleaning things, and pushing papers around my living quarters. The papers come in the mail, stacks and stacks of glitzy holiday catalogs and advertisements for bright colored things. The more papers I throw out or recycle, the more I have. I'm not sure how that works. Is this extra paper mass coming in from another dimension?

I also spend plenty of time in Starbucks Coffee shops. Some of this is because I do decorative signs for them. They pay me in coffee and food items. This is a time-honored artist thing. I have drunk many a "free" double tall cappuccino while putting art nouveau designs on the "Daily Offerings" board of Starbucks. You can see a couple of my Starbucks images here and here. They are done in water-based paint markers on smooth-textured chalkboards. When it's time to change the theme and coffee blend, I just wash it all off with detergent and water. I photograph all the designs, so that when the original is gone, I still have the image.

If Starbucks Coffee did not exist, someone would have to invent it, so that people could sit and read newspapers, sketch in their sketchbooks, study their college textbooks and write weblogs on their laptop computers. I have not gotten to the point of writing this Weblog on a laptop at Starbucks, but I know it will happen sooner or later. Sometimes I can be disgustingly trendy. But you will never catch me wearing a bare-belly crop top or a ring through my eyebrow.

Now the dreadful "Holiday Season" has arrived, with its commercial and cultural flogging. I stare dimly at the heaps of brightly lit sweets and gift items, and the only thing I really want to do is retire to a woodsy den somewhere and hibernate like a bear. Wake me in April. It is very noisy here. Every store and public place has a soundtrack playing, and this year they seem to have all picked "soul hits" as their default. (It may be a regional thing. You folks in California or Boston might not have to listen to it.) This is a kind of pop music involving a lot of howling, screeching, and belting. "Traditional" Christmas tunes are subjected to a "soul" arrangement, resulting in abominations such as Aretha Franklin wailing her way through "O Tannenbaum." The Queen of Soul has had far better moments than this. I have also heard a hip-hop version of "White Christmas." I know it is coming, somewhere in between the aisles at Giant Foods or Staples. No! Not again!!

This weekend, after the turkey feast, I am off to a small fantasy convention, where I will show a few pieces of art. This convention, which has been held in the same place and same time each year for the last twenty-one years, is mostly a social get-together. At least two of my Friendly Mathematicians will be there. But there will probably be a pop-music Christmas soundtrack at the hotel.

What I have managed to read in Sawyer's calculus introduction book is a discussion of "negative velocity," in which the direction of movement might be "backwards," but still measurable, although in negative numbers. At this moment I feel as though my math and physics study is going at negative velocity.

Posted at 4:22 am | link

(Note: This is an adaptation of an essay that I wrote last year, before I started ELECTRON BLUE. The spam situation is still the same as it was when I wrote it. I was inspired to re-write it by a news item.)

According to a recent news article, Bill Gates receives four MILLION e-mails every day and that there is a whole department at Microsoft set up just to handle the immensity of messages directed at him. The vast majority of this is spam, that is, unsolicited commercial e-mails, blasted into the Internet by the trillions every day.

I get spams too, but not as many as Bill. That's rather an understatement. I get about a hundred a day, which are caught in my ISP's righteous spam filter. No matter what they try, they can't get through this. Hail to Earthlink! I can open up the "repository" and look at what it caught, in fact I have to do this because occasionally it captures something which is legitimately addressed to me from a previously unknown sender.

What impresses me about the lineup of spams was that most of them are attributed to a person's name, cited in the "sender" field. Bogus of course, and paired with a bogus address, but still a personal name. This is adopted as a strategy to get through the filters, though since they were caught by my filter, it wasn't a very good strategy. Yet one after another, each spam mail has a person's name attached to it. I looked at a week's worth of spam (yes, I have too much time on my hands) and found hundreds and hundreds of these names. I was receiving drug ads and mortgage re-financing ads (the penis enlargement ads, so popular in 2003, seem to have passed out of fashion) from a whole crowd of people who were only names. A crowd of ghosts.

It is said that if you name something, then it has a kind of existence; it is made at least virtually real. Here on my screen was an endless roster of virtually real people: Mel Nolan. Timmy Clark. Rachael Fair. Theresa Schultz. Goldie Serrano. Debbie Cooper. Jed K. Burgos. Blair Harden. Kerry Cardenas. Anita Barone. Kitty Langston. Ruthie Yu. Irvin Flynn. Ellen Stephens. Dalton Avery. Tyrone Garces. Patricia Watkins. Michael Dominguez. Darin Sutherland….the list goes on and on and on. I collected more than three hundred names, all different. I have no idea how many other people received mail from these phantom names, but for me, every spam had a name, and none of them repeated. There were no repetitions of common names like "Smith" or "Jones" or "Martinez."

I started to wonder: who were these people? A kind of sociological curiosity arose in me, and I started taking notes. I noticed that the ethnic range of these names was rather limited. Apart from a couple of Chinese, Vietnamese, or Jewish names, almost all of the names were either "Anglo" or Latin, and even the Latin names often had an "Anglo" first name. There were also numerous names which seemed to fit the stereotype of an "African-American" or black-sounding name. There were not too many names that were clearly "European" such as German, Italian, or French names. And, more subtly, there seemed to be no "British" sounding names either, no one named Trevor or Nigel or, for that matter, Camilla.

The phone book of any American city is filled with a global variety of ethnicities and ethnic names. Plundering the phonebook to randomly generate names would not yield such an exclusive selection. Where did these names come from? They were contemporary names, not ones from the past with their Biblical and historical allusions. And they were American names, this earnest mixture of Anglo and Latin. Where were these people? Were they really real somewhere? They were not cute internet nicknames, they were given names. Were they plundered from mailing lists, hacked from office rosters of companies and firms or insurance listings or internet providers' memberships? I have heard that names and addresses of "real" people are commonly stripped and used as the origin of spams, in a scam known as a "joe job." Were all of these hundreds (thousands!) victimized this way? Was there really a Kirk Perry, a Frederic Beal, a Hugh Sandoval, a Stanley Godwin, an Arline Booth, a Carrie McIntyre, or a Jay Rodriguez, now cursing and frantically changing his or her e-mail address because his name and identity have been stolen in order to sell sex enhancement drugs?

As I often do, I had surrealistic thoughts. The spams, as thick as falling leaves in autumn, fill the virtual air, each with its own "realistic" sender name: Danielle Corcoran, Cassandra Pace, Coleman Lamb, William Fisher, Cathy Ann McNeil, Mariana Perez, Toby Bruce, Benjamin Bailey, Daniel Knowles, Jennifer Shore. Billions of spams, every day. One day's worth of spam has far exceeded the population of the entire Earth. Each day, astronomical numbers of virtual people are being named and thus, in a ghostly way, created. Do they have souls? Do they have lives, working in their offices somewhere in another dimension created by the unrelenting machinations of spam coders? Whole cities full of people, working tirelessly to send you and mortgage scams and prescription drugs, drugs, drugs, an endless supply of Viagra and amateur teen girl webcam pictures?

The answer was here, revealed by a bit of search engine revving. The Kleimo Random Name Generator can create as many virtual identities as you need. It uses data from the U.S. Census to combine names, as many as you want. It even regulates the "obscurity factor" of the names so you can get names from very ordinary John Smith-types to highly obscure ethnic or fantastic ones. And yet every name belongs to someone somewhere.

The creation of the trillions of names from lists is an example of a kind of math called combinatorics which I have not yet studied. Given any fixed number of first names and last names, combinatoric math can tell us just how many different ghost identities can be created from the lists.

Recently, a comic element has entered into the name generators, as spam-names have been created not from "realistic" names but from unusual words, seemingly picked at random, joined by a "middle initial." I've received spams from hilarious entities such as "Firmament V. Explosive," "Fiercely M. Rocketing," "Electroplate A. Obligingly," "Mailman L. Artistry," "Scapulae P. Yelping," "Instructor T. Bulbs," and other similar characters. "Primly R. Spree" and "Origin C. Fleshy" might appear in British bedroom farce, while "Psoriasis H. Lifeworks," "Tattled C. Senselessness," and "Outfitters M. Obnoxious" might come from down-home American satire. But beware of characters like "Dissolving P. Cynic," "Reproaching T. Aplenty," "Psycho I. Masculinity," or, for Gawd's sake, "Madwomen R. Imbalance." I would much prefer to receive my spam e-mail from "Exotically J. Budgeting," "Ebullient Anathema," or my favorite so far, "Cocktail A. Beatitude." And how is Mrs. Beatitude?

But most of the spams come from the "realistic" name generator. I have read speculations by imaginative physicists who posit that there are an infinite number of universes, parallel but unreachable from ours, and in at least some of them are Earths just like ours, with people just like us. In fact, somewhere out there, supposedly, is our exact duplicate. Is this particle shower of personalized junk e-mail really the whisper across the dimensions of other universes, filled with uncounted but named beings, whose only trace on our consciousness is a subject line with a name that sounds tantalizingly real: Clark Bernard, Robert Allen, Shawnda Ivan, Merrill Nichols, Averyl Sanders, Dwayne Negron, Nadia Duran, Janet Filson Davis, J.D. Martinsen, Florence Siegel, Margaret Madden…?

Posted at 3:43 am | link

W.W. Sawyer's book on calculus begins with some simple graphs which unfortunately missed their mark with me, at first. There is such a thing as being too visual and too literal when studying mathematics, and this is one of my problems. The first two graphs attempt to show "snapshots" of an object moving vertically (on the plane of the page, perpendicular to the top and moving towards the top of the page) at regularly sequential times. The two quantities being measured are movement in space, and time elapsed. However, the sequence of dots at different times looks like a diagonal line if you join them, which I was tempted to do. If I joined the dots, it made me think that the object was moving in a diagonal line, when it was not. It took me quite a while to figure this out.

If the hypothetical object (represented only by a dot) was indeed moving in a diagonal line, it wouldn't have made much difference in the graph, since it was still moving at a steady pace. But if the graph were depicting the steady, unchanging pace rather than the movement in space, then it wouldn't look the same. The line would be flat, or horizontal.

Remember that a couple of entries ago, I realized that the graph of a linear equation, a straight line, is also the graph of an arithmetic progression in which a quantity increases by the same amount over and over again. The slope of the diagonal line represents the amount of the arithmetic increase. If the slope is 3, for instance, then the graph depicts a progression which is always increasing by adding 3. It's counting up by threes. But the slanted line that results from y = 3x also looks like the trajectory of something moving. When I first encountered the acceleration of gravity, I just could not get it through my head what "thirty-two feet per second per second" meant. Why did they repeat the "per second" bit? Finally I was able to understand (after working with progressions) that the peculiar repeated phrase really meant "thirty-two feet per second FASTER EACH second." If it's free-falling in the ideal abstract Newtonian universe, it's going 32 feet (or 9.8 meters) faster each second.

So that straight line is the line of something which is accelerating at a steady pace, adding the same amount of speed each second as it goes faster. But the steep graph of y = 3x looks like it's going up, not down. Well, I suppose it could look like something going down the slope if I looked at it another way. But in this graph, nothing is going either up or down. The only thing that it's measuring is how fast its rate of whatever is changing.

It gets even more confusing with graphs that depict things that have varying rates of change. If something starts off slow, then adds speed as it accelerates at a faster rate each second, the curve will start wide and then get vertically steep quickly. To graphic-visual me, it looks like a rocket-plane taking off, and could depict that if you used the curve as a visual depiction of motion in space rather than rate of change. But the same curve could just as easily depict some other varying quantity which had nothing to do with moving in space. I have to stop making pictures out of graphs, which is hard to do because if I am in artist mode, everything I see is a picture.

Posted at 1:41 am | link

I've been quite busy these days updating the price tags at work. As my more regular readers know, I work at Trader Joe's, a gourmet supermarket which is familiar to Californians but which also has many stores in the Midwest and Northeast. Each Trader Joe's has its own crew of signmakers who create all the signage for the store by hand. Everything from price tags to small ads to bigger "billboards" is all done by real people right there in the store, rather than printed out impersonally the way the big chains do. I am one of three hard-working artisans who work for my local Trader Joe's as signmakers. The entire store is full of our work. Not only are we paid for our work, we are also invited to sample the goodies and wines day by day as they are offered to us. Yes, they pay us to drink wine and sample gourmet food.

Every few months we need to generally update our price tags, more than the ones that get updated every week due to price changes. Tags get messed up, or fade, or the product changes or is discontinued; any number of things happen to make them obsolete. This month has been an update month, and so I have been especially busy switching out hundreds and hundreds of hand-written price tags. Our backgrounds or color borders might be done by computer and/or reproduced by color copying, but all our writing is hand-calligraphed, thus giving Trader Joe's a "humanized," friendly look that appeals to customers.

Imagine a Trader Joe's store as a three-dimensional structure in which the local conditions of the general function "gourmet food" are specified by a single price on a calligraphed price tag. The thousands of price tags are the "infinitesimals" (an old word once used for the innumerable points which made up the curves that calculus analyzes). My job is to create the fullness of a well-signed, information-rich space by making the infinitesimals which comprise that three-dimensional structure.

As my manager explains, the updating of the price tags is a never-ending activity, since the criteria for the tags as well as the prices are always changing. He refers to signmaking as "trying to hit a moving target." Since one of the practical functions of calculus was improving the accuracy of military artillery, then I can say that learning calculus will in some way, if only metaphorically, help me in my work as a signmaker and professional gourmet.

Posted at 2:09 am | link

I've finished my review of sequences and progressions, with both the White Algebra Book and the 1958 algebra book. I was encouraged that I was able to remember the material and solve the problems without the bewilderment and anxiety that accompanied my first encounter with them. Sequences and progressions are one of my favorite bits of math so far. (Trigonometric identities have been my least favorite.) There is a kind of dancelike quality to working with them, even though I don't actually do any dancing. It also feels like sports such as tennis or baseball, where your hits go in sequence after sequence. I am personally very un-athletic, unlike the physical marvels I so often find among the science and math types who regularly run marathons, practice martial arts, and climb mountains. The only exercise I get is on paper. But I can feel the affinity between math, science, and manly athletic achievement.

The last thing I did in the 1958 book was finding the sums of infinite geometric progressions where the common ratio is numerically less than 1. I believe that this put me over the limit from algebra into calculus, so here I am. Now 1958 is back on the shelf (with the slide rule) and I have finally gotten out my calculus books.

I'll be starting with two introductory texts. The first, which was given to me by one of my Friendly Mathematicians almost a year ago, is called WHAT IS CALCULUS ABOUT? by W.W. Sawyer. My Friendly Mathematician has been patiently waiting all this time for me to stop dithering with college algebra and read this book. Well, the time has come. The second book was given to me by another friend who is not a mathematician but who knows plenty of them. This one is called HOW TO ENJOY CALCULUS by Eli S. Pine. The Pine book bears this dedication from the author:

"This book is dedicated to all those budding students who wanted to be Scientists, Engineers, Physicians, Chemists, Biologists, and even Mathematicians, who have had their dreams and careers destroyed because Calculus stopped them dead in their tracks.
I hope this book has an important part in saving the dreams of so many dedicated and hopeful students from now on!"

With that kind of sentiment, how could I refuse? Even if I have no dreams to save or career to destroy, I welcome Dr. Pine's invitation. I will also be working from a group of college texts which were given to me by another one of my Friendly Mathematicians, a college professor who had more textbooks than he could ever use. These books contain all the problems which I will want to solve in endless ranks and rows. That's the only way I know to learn this material, solve and solve and solve some more.

Along with Pine and Sawyer (who will generate quite a lot of mathematical planks and sawdust) I will be reviewing my earlier attempts at basic (junior high) physics. This means Newton's laws again, and mass and momentum and vectors and acceleration. I did quite a lot with acceleration and changing speeds some time ago, and it is time I reviewed it. In fact, some of the work I did with figuring out the rate of acceleration could be considered introductory calculus, so I have actually already done some of it.

Posted at 3:09 am | link

I wish I could say that my advancement in mathematics could be described with a geometric progression, but it is much more like the stodgy arithmetic progression than the soaring geometric. I am spending much time with these processes since the geometric progressions, and the limit of infinite geometric progressions, are one of the gateways to calculus.

My White Paul Klee book's problem sets are nicely set up to reinforce the teaching of the information pages. They lure me into a sense of confidence with lots of easy initial problems before introducing me to harder material later on. In general the White Book's problems are easier than the problems in the 1958 book, which must be evidence of the "dumbing down" of American education in the last decades of the twentieth century. Also, the White Book does not include the finding of arithmetic and geometric means, which may be useful so I will review it using 1958, where I learned it in the first place.

There are plenty of Big Sigs to sum up in the White Book's problem sets, as well as somewhat realistic word problems, complete with helpful graphs. Each problem is titled with its theme, so that they form mini-scenarios: "Depreciation," or "Population Increase," or "Investment Return." I also enjoy putting numbers into graphs, so that I can view the majestic curves of exponential and geometric functions. I am filled with respect for this fairly simple mathematics that can describe the multiplication of living things. I love Order wherever I find it, especially self-organizing Order. In a culture where chaos and wildness are hip and sexy, I stubbornly, and squarely, and un-hip-ly stand by my defense of the order of the world, including a fondness for housework and cleaning large numbers of small plastic objects (sign clips used in price tag displays at work). But that brings me back to the stodginess part. It will take me longer to get where I'm going, but I still intend to get there. Even if there is no "there" there (as Gertrude Stein said about Oakland, California) by the time I get there.

Breaking my Rule about Politics just once

It is part of my policy here at ELECTRON BLUE to have NO POLITICS. But on this day of history, where a contest even more important than the World Series will be decided, I need to simply say that I will be voting for Kerry and Edwards in this national election. I'm sure this is not a great surprise to anyone who reads this. I don't talk about politics on the Electron because there is nothing I could say that would be either more original or more insightful than the writings of the myriad of other bloggers. And I just don't know that much about civic affairs anyway. Quoting statistics or factoids doesn't convince me, and probably won't convince you either; they're too easily manipulated. But the events of the last few years, and the proliferation of violence, lies and religious fundamentalist anti-intellectualism, has roused even me to political concern.

I am sure that most, if not all of my handful of readers (at least those who are US citizens) will also be voting in the US election. Most likely they have their minds made up already. But just in case they don't, I hope they will vote for the Democratic candidates. Since my readers will probably have voted by the time they read this, I hope they did vote for Kerry and Edwards. It is going to be a long day, and possibly many long years as well.

Posted at 2:27 am | link