I am still stuck on inclined planes. In order to find the force that keeps the object on the surface of the plane, that is, the normal force, I am supposed to do a calculation based on the sine of the angle of the plane's inclination. Diagrams of this process are provided in all my physics texts which deal with inclined planes. However, I was mystified by these diagrams (as I said in my last posting). The closest I got to something that made sense was at The Physics Classroom on their Inclined Planes section where the vector triangles were drawn out. But why would you multiply by the sine of the inclined plane, when the triangle side you wanted wasn't even on the diagram?

You know by now that if something doesn't make sense to me, I will go at it until I either solve it or go crazy. This time, I sensed that a geometric proof lay behind this drawing, with a lot of assumptions about equal (and opposite, according to Newton) quantities and parallel lines. I lugged out my geometry and added another triangle superimposed on the original ones, made out of the vector lines that described the "normal force" and its perpendicular "parallel force." Then I cranked out a proof that showed me that the same angle of inclination reappears in that superimposed triangle where its sine will actually deliver the proper measurement for "normal force." It is just about impossible to demonstrate geometrical proofs in a written Webjournal like this one, so you'll have to take my word for it. Now I'm satisfied that this works, and I can go on to do as many vector and inclined plane problems as I can find.

I am still fascinated, and somewhat dubious, of the evident fact that abstract diagrams which express quantities in length of lines can say anything about the real behavior of mass in gravity, moving down an inclined plane or anywhere else. It all goes back to old Pythagoras, one of my favorite philosophers and scientists of all time. He was one of the first in the West to prove that the natural world can be described with mathematics. Every time I do vectors and geometry, I owe something to Pythagoras, his predecessors in the ancient Middle East, and his successors in the Mediterranean. The idea that the world has a mathematical order, in which things can be discovered by doing abstract calculations, is still a new concept to me, even if it is more than three thousand years old.

The Day the Music Died

Today, Sunday February 27, is the last day that my local public radio station will play classical music. It has been switched to an all-talk, all-news format. I am devastated by the change, a sell-out which was done purely for monetary gain. It is a betrayal of all that I expect from public radio. Because of this, I have decided never to give any money to that station again.

I will instead spend my money on gadgets and services which will allow me to connect to Internet radio, or download music files to a portable player, but it won't be the same. There is still one classical music radio station left in my area, but it chops classical music up into little bits to fit a pop music format with noisy, obnoxious ads. Now that classical music and commentary is gone from this radio station, I feel as though I have lost a dear friend or even a family, because their broadcasting was more than classical music. It was music-loving announcers with quiet, comforting voices, telling us interesting facts about music as well as weather reports and a few comments here and there. It is one less quiet haven in a world which is too often, for me, a place of screaming sonic assault.

Posted at 3:30 am | link

There has recently been a proliferation of online "test-your-mental-gender" psychological profile tests, and I took one last night at about 3 AM. It showed my profile to be at exactly the average female mental profile. Despite my almost 20 years of experience working with blueprints and architectural rendering, I didn't do well at the "rotate the shape in space" test which is said to prove your mind's manliness. But I was great at the "remember where everything is" test, which proves that I know how to keep house like a neat lady concerned with details. I also was good at the "find synonyms" test, which proves that I'm nice and verbal and chatty like a girl, rather than strong and silent like a guy.

From the beginning, I've viewed my mathematical and physics journey through the lens of gender. The whole noise over the remarks of Harvard president Lawrence Summers, which I mentioned in an earlier post, has finally brought this matter into the popular attention. No matter how many times the "reasonable" people try to tell us that gender differences don't (or shouldn't) matter in learning and practicing math and science, there are other voices, also citing scientific evidence, often involving testosterone, that say that they do. I feel trapped between these social forces, as if no matter what I do, I cannot help be defined mentally by my gender. As a female, I must struggle to achieve what a male presumably can do "naturally" with far less effort. And when I falter, I always wonder whether it's because I'm a female, trespassing in archetypal men's intellectual territory where my little brain was not meant to go.

I am currently working on vectors. This is the third time I have tried to understand this subject. I solved two-dimensional vector problems while I was doing trigonometry. I dutifully drew the diagrams and solved the triangles. The introductory vector problems set the vectors at right angles, so that the sums could be found with simple trigonometry. The later problems, which I also did, set the vectors to be summed, at non-right angles, so that you had to invoke the laws of sines and cosines, also known as the sine formula and the cosine formula, or even more arcane laws made up by Hellenistic mathematicians in ancient Alexandria. These problems I solved as well, if you remember my posts about the British trigonometry book and the Red-Spine trig book from February and March of last year.

Now I am doing vector problems about weights on inclined planes. These are in chapter 2 of my Barron's physics study book. Now there are not only vectors of gravitational forces, but there is the so-called "normal" force which is perpendicular to the inclined plane, and the "parallel" force which is, uh, parallel to the inclined plane. And there is the force of friction and the force of pulling or pushing the weight up the incline. I had no clue what they meant by "normal" force. What's so "normal" about it? Was there such a thing as an "abnormal" force?

I could not figure out the diagrams in the Barron's book. They drew a triangle representing the inclined plane and its vertical and horizontal dimensions, but I didn't get that those were not the verticals and horizontals they wanted me to figure out. Finally I went to the great oracle of our day, and revisited an excellent website out of the Chicago area called The Physics Classroom which is one of the best high school-level teaching sites I've found. Their unit on inclined planes gave me the very helpful information that "normal" means "perpendicular." Sure enough, when I looked up the word "normal" in my etymological dictionary, I found that it comes from the Latin word norma which means "carpenter's square" or "ruler." The square tool gives perpendicularity to a construction, or regularity, hence "normality."

Diagrams often confuse me. I don't know which gender they are supposed to confuse, but I think it's my visual artist's mind that gets misled. I expect these diagrams to somehow be pictorial representations, but they are not meant to be. The length of a vector line expresses magnitude, an abstract quantity, but I see it as a literal picture of something. With inclined planes, there is a starlike cluster of interacting forces diagrammed: forces perpendicular and parallel to the plane, as well as the up-down forces of gravity. A hypothetical block or wagon moves up and down an angled line. The gravitational forces acting on it have to be diagrammed in right-angle vectors to it, because since it's sitting on something that's holding it up (that angled line), it can't go straight down.

Force equals mass times acceleration, but which part is the mass and which is the acceleration? I tried re-drawing these diagrams in the margins of Barron's book, but I kept getting the concepts wrong and I had to use white-out to eliminate my scribbles. I should never annotate a math or physics book in an un-erasable medium. I have only tried to solve a few inclined plane problems, and so far I've gotten most of them wrong. I have to take into account the acceleration of gravity, the co-efficient of friction, the angle of incidence, and the mass of the block, and find the net force. I wish I had more net force. Right now I am struggling up that inclined plane, against a heavy force of gender friction and intellectual gravity.

Posted at 3:00 am | link

I have been attempting to remember my childhood interests and pastimes, to see whether I had any of the traits which would have made me into a scientist, if circumstances had been different. Usually, as I read the biographies (or hagiographies) of scientists, their interest and outstanding talent appears very early. They excel before they even get to high school. Mathematicians and physicists are particularly early sprouters, as in this astonishing character recently profiled in the New York Times, Erik Demaine, who got his PhD at age 20 and became the youngest professor ever at MIT. He's only 23 now and has co-authored or authored over 100 papers. What was I doing at age 23? I had just gotten back from a year in Rome and was suffering through my first year in graduate school doing Greek and Latin studies. But certainly no mathematics. No 100 papers, either.

As I have discussed exhaustively elsewhere, as a youngster I was wretched at mathematics. No precocious talent there at all. But I did have a certain "scientific" turn of mind in that I was much interested in birds, plants and fungi, and geology. And if something interesting happened in the sky, like a comet, I was very intent on astronomy related to it, and I remember reading whole books just about comets. I never saw Comet Ikeya-Seki in 1965-66, but I was much inspired by it. The first comet I really saw was Comet Bennett in 1970, and I drew a picture of it in my notebook journal.

Even in my youngest days I was a birdwatcher and my parents claim that I urged them to become birdwatchers too. We have all been enthusiastic birders ever since. As for plants, I learned to identify garden varieties when I was still in elementary school. Later on in my teen years I spent lots of time in the forests and meadows near my home, searching out wildflowers, and mushrooms (which were neither edible nor hallucinogenic, but often poisonous). I had, and still have, guides to lots of flora and fauna. I also had a tiny, low-powered, but working microscope, made by "Micronta" of Japan. Encouraged by high school biology classes, I scooped up pond water in jars and looked at the protozoa that hatched as I kept the covered water jar on my shelf. I spent hours watching paramecia and amoebas and rotifers under my microscope, as if they were tiny birds, complete with species and behaviors to observe. I thought of them somehow as my pets (I had no dogs or cats in my youth) but these pets, alas, didn't live very long and perished along with their pondwater.

I was not averse to high school science classes, then, as long as they did not have the dreaded mathematics to deal with. I have mentioned the beloved general science teacher in my old junior high, Mr. Cinkosky. The science class I loved the most in junior high, though, was taught by a young man called Mr. Wassell. He spent an entire semester on geology, and especially the glacial geology of the local New England terrain. We learned about moraines and drumlins and eskers and sinkholes, and why the soil of New England was so thin and poor. This was totally fascinating to me, because it was rooted in the very land that I could see and touch every day. Why do all the long hills in my native region point the same way? Because the glaciers of the last Ice Age shoved them into their parallel places. I hope that Mr. Wassell is still with us somewhere; I would love to let him know that his glaciology unit back in 1966 is still remembered by at least one of his students.

I love to observe, and I love to classify; often to excess, as many of my friends and co-workers know. I don't know whether that counts as scientific curiosity. I did not ask as many "why's" as a budding scientist should, at least that I remember; I was satisfied to observe, note, and classify things. The idea of ecosystems or evolution was beyond me; I just liked to find things and identify them.

So it seems I could have become some sort of biologist, if things had been different; but in my youth, back in the '60s, it was unheard of for a girl to want to be a scientist of any sort, at least in my school. Nor were girls encouraged to be scientists; that was also unheard of. At least, I never heard it. And if I had wanted to be a scientist, there would have been that impenetrable barrier….MATH.

I remember my junior high school chemistry classes mainly because we were assigned to make artistic representations of atoms of the different elements. Each kid in the class got his or her own element and was responsible for how he or she depicted it, but the depiction had to show the arrangements of electrons in their "shells" and a reasonable facsimile of the nucleus. These atoms, of course, were known according to '60s physics, not current physics, so we were still taught to conceive of the electrons orbiting the nucleus like little tiny moons orbiting a nuclear Jupiter.

My element was number 69 (don't laugh; I had no clue back then and no one else in my class did either). Element 69 is a "rare earth" called THULIUM, which has very little use to science or industry, other than as a source of X-rays for portable medical machines. My portrayal was a rather tedious black and white pencil rendering of lots of round orbits and little round electrons and a big pomegranate-like nucleus full of particles. But one other girl had rendered her element (which one, I have forgotten, but it was a fairly high number) into a glorious three-dimensional sculpture, using painted styrofoam balls threaded through piano wire. Piano wire has the virtue of bending into a perfect circle, so it was a beautiful mobile made of nested wire circles. I suspect now, looking back, that she had had parental help making this. I was so jealous. I wanted to grab her atom from where it was hanging from the ceiling and run off with it. She wouldn't let me have it after the class assignment was done, either. This was perhaps an early and cautionary experience, for me, of competitive science.

But physics, the hard stuff, was beyond me. I could identify birds. I could find mushrooms. I could raise protozoa. But I couldn't do physics, because it had math. So I missed the prime experience of primal science. I missed having that probing curiosity that asks why and how things move, or accelerate, or heat up, or light up, or blow up. That was, frankly, for boys. The die was already cast before I even got into high school. I never did those young physicist things which set a youngster on the path to the laboratory.

But now things are different. Even though I am physically older, I feel as though the clock has been re-set for me ever since my year 2000 experience at Fermilab. As I painstakingly poke my way through the physics I should have learned along with my geology and biology in junior high school, I can now experience the curiosity of a young physicist and ask those questions I never thought to ask before. Like: why does light reflect off raindrops or mist in a circle around the source? Why does a spinning jar lid seem to go faster as it loses its momentum? (I actually have been pondering that one for many years, long before Fermilab.) Why do xenon and other fluorescent lights flicker rather than give a steady light? These are things that I am now entitled to ponder. I have been given back my curiosity. It's never too late to have a scientific childhood.

Posted at 3:23 am | link

About fifteen years ago, some clever music producers got the idea of mixing Gregorian chant with a pop disco or dance beat. The resulting album, MCMXC AD, by the German band "Enigma," was a massive international hit. About half of the early music enthusiasts who heard it probably dropped dead of shock at the blasphemy of the thing. The rest of the early music purists sensibly ignored it, but a daring, slummy few of them may have even…. liked it.

Well, for those whose ultra-pure musical taste was not violated enough by the subsequent commercialization and vulgarization of Gregorian chant by hit records, even by monks as in the 1994 mega-hit Chant, by the monks of Santo Domingo de Silos, here's yet another sign, if any be needed, of the death of pure culture. And I ….like it.

Boy choirs have been piping their heavenly harmonies into the European air for centuries; think of the Vienna Choir Boys or any number of cathedral kiddies. England is particularly fond of them and great composers even into the twentieth century, such as Gustav Mahler and Benjamin Britten, have written compositions for them. It was only a matter of time before the same notion that made Enigma so popular came to the boy choir world. In 1999, under the direction of composer and choir-director Robert Prizeman, a London-area boy choir took the name of "Libera" and recorded an album under the same title. They followed this up with "Luminosa" in 2001 and "Free" in 2004. Along the way they have also done film music and commercials.

"Libera" in Latin has lots of meanings. It means "children," but it also means "free," as in "liberation." Prizeman, a British film and TV composer, composed pieces for the youths in lush, skwushy, big chords which are far too sweet and old-fashioned to be used in "mainstream" modern classical composition, and accompanied them with synthesizers, organ, and some orchestral instruments. And yes, on some of the songs, there are electronic disco or dance beats. Some of the pieces are adaptations of well-known plainsong or classical tunes; others are original Prizeman compositions as well as Prizeman words. All of this is wrapped in digital reverb which sends the sound into a dizzying pool of ultra-reflected shimmer.

The result is an enchanting flight of globalized electronic angels floating over the ruins of Western civilization. They are the only singing group ever to make the "Dies Irae" sound cute. They also have real moments of beauty and reverence, as well as some catchy plainsong-inspired songs and moving adaptations of traditional religious and folk tunes. "Libera" brings me a restful hour of charm, sweetness, and aural warmth in a dark, cold world.

These heavenly boys are one of my guilty musical pleasures. Sometimes it's just too hard to maintain my uncompromising, skeptical, unsentimental, ultraserious, relentlessly ironic cultural stance. I grow weary of dissonance and abstraction. I grow even wearier of apologizing for my cultural impurity. It's Lent, a time of penance and introspection, but this is one sin I am not sorry for.

Visit, if you wish, the Libera website and click on their sample links to hear snippets of their heavenly sound.

Posted at 3:41 am | link

I mentioned my love of gardening and plants in a previous post, and those few hardy perennials who actually read this Blog have seen my cactus blooms. I have had many more flowers than these, all winter long, because I garden under lights. I have not had the chance, since I moved to an apartment in a big city, to garden in a real plot of dirt. But I have an array of artificial plant lights as well as some sunny windows, and I'm able to raise and keep quite a variety of vegetation. Recently I did an inventory of my plants so here's a look at my plant collection.

Pothos ivy, Rhaphidophora aurea, three pots.

Climbing philodendron, Philodendron scandens, three pots.

Peace lily, Spathyphyllium wallisii, four pots. These leafy clusters are descendants of a single bunch which I bought back in 1977.

Dracaena palm, one pot. I inherited this plant in 1990 from one of my previous workplaces.

Aloe plants, Aloe variegata, seven pots, five large (12" diameter, 12" high) and two small. These aloes are descendants of plants I grew from seed, starting in 1984.

One pot of sansevieria, Sansevieria trifasciata, which needs almost no maintenance.

Under the lights: one pot mint, one pot oregano. (I don't grow any of that other kind of herb.)
Six pots of geraniums, Pelargonium zonale, four red blossoms, two pink. The red blossoms are an orangey-red color, known to artists as "cadmium red light."
Six pots of african violets, Saintpaulia ionantha, three lavender blossoms, one purple blossom with white edging (picotee), one white blossom, and one pink. All are in bloom. The word ionantha means "purple flower" in Greek.

Nine cactus pots of different varieties, including the two which flowered. Some of these I grew from seed. One of them is a white hairy pillar cactus which I acquired when it was the size of an egg, and it is now more than a foot tall. Cacti and succulents do well in the dryness of my dwelling.

One dendrobium orchid, which I got as a give-away from a friend a few years ago. It bloomed in 2002 but has fallen on hard times. Nevertheless, it is not dead yet.

I plan to add more plants as spring approaches. In the summer, I move all my cacti and succulents out onto my terrace, which also is like a desert during those months.

See my garden in a sunny window opening onto the terrace.

Fluxions of ivy

I have been reading an inspiring book called "Great Physicists: the life and times of leading physicists from Galileo to Hawking," by William H. Cropper. I've read the chapters on Galileo and Newton, and as I continue studying physics, I'll re-visit the book to read more biographies and meet more discoveries. Cropper's account of Newton contains the best short introduction to calculus I have read so far. I did not know, before I read this enlightening chapter, that I have already been working with derivatives for some time. Calculus is like a tough plant that grows even in the desert environment of my study.

On another note, many congratulations to New England Patriots fans (myself and my Massachusetts family included) on the Super Bowl win. Both the Red Sox and the Patriots in less than a year…how can we stand all this success?

Posted at 11:58 pm | link

I did not choose to study math and physics at my advanced age for the "right" reasons. You'd think that I'd be doing this for the joy of learning, for the mental hygiene of it, or because I was curious about the way the Universe works. Well, I'm not that nice.

If I were curious about the way the Universe works, I would just read the non-mathematical books about physics etc. which have no nasty equations in them, written for layfolk, and be satisfied that Stephen Hawking or that cute Brian Greene has done all the hard work, so I didn't have to. If I wanted just the joy of learning I'd study horticulture, which is another one of my major interests. And I'd even get to work with the real thing (plants and gardens) rather than just read about it. As for mental hygiene, too late, I'm already mentally polluted.

What keeps me going in math and physics is my drive for power. It's disgusting, but it's true. I live an ultra-quiet, bohemian life in which I must understate everything, dress in grungewear (I work in messy circumstances), stay humble and self-effacing and not intrude myself on people, not interrupt conversations, never display anything except my art, and be as unobtrusive as possible. I am a proper lady. No fur coats, fancy car, hot s**t bling-bling for me! But hidden behind this proper face is someone who is constantly concerned with power and wants to acquire and exercise it.

I once appalled a friend of mine by saying this. She was afraid I was going to go over to the Dark Side. She is now busy knitting decorative socks in the rural countryside, which is where I am afraid I will end up if I don't get beyond Newton's laws. I am just about to start chapter 2, on "Forces: Push and Pull." I will need to do some mental pushing and pulling for this, but I am ready for it. This will be the third time around for me on this, and three times usually is what it takes for me to really learn it.

There's no need for me to trouble my curly head with mathematics when I could read those nice easy books about physics with no math in them. But for me, learning mathematics and physics is acquiring power. It's like grabbing the steering wheel of the universe, which has been made into a fast car. What thrills me about physics is that scientists can find out the mathematics that makes the universe turn. And they have a certain bravado and arrogance about them, at least some of them do, which I would like to experience at least sometime in my life for myself. I would like to swagger, even just once. What? I want to be arrogant? That's SO superficial and childish! Shame on the Electron, I should remain small and negative! But that desire for power is part of what has kept me going for four years now.

The trouble is, I can learn whatever I want, even if it takes me twenty years, but I will never be able to practice it, nor do experiments or research. Everything I learn will be already discovered and well-documented. This is one of those things I didn't think about back in 2000. Pure knowledge is nice, but it doesn't do anything. For pure knowledge, I can study philosophy. But that power thrill isn't there. I'm always reminded that I can do art with physics themes. Frankly, that's rather like knitting decorative socks. Not that there's anything wrong with that. But it doesn't turn the wheel of the speeding universe, or shine the beams of its headlights into the darkness.

Posted at 3:19 am | link

I can't get away from the screaming. Everywhere I go, it's the same soundtrack: "soul" music. Howling voices and blaring horns, from decades gone by. LET ME TAKE YOU HIIIIIIIIIGHER! YOW! UH! It's in the big chain stores like Staples and Old Navy. It's in the mall. It's in the supermarket. PAAAAAAARTY! HEAR ME NOW! It's at the hairdresser's. It's in fancy gourmet Whole Foods. "Soul Hits" me everywhere I go. C'MON C'MON GIT ON UP NOW!!! YOOOOW! It's in the upscale rotisserie chicken restaurant. I asked the restaurant management to change it, and they said they couldn't, the soundtrack was piped in directly from corporate headquarters. GIMME GIMME GIMME THAT THANG Jeez, it's even in my beloved Starbucks sometimes. "Soul Hits." The same songs, in so many different places. There must be some conspiracy of marketing executives somewhere, which has decided that for my area, "soul" music is what will energize the customers and make them buy more. I have actually walked out of places without buying anything, because they play this soundtrack, but I don't count. I don't count as a radio listener, either. I'm a regular contributor to my local public radio station which plays classical music, and there are plenty more like me who are willing to give their hard-earned cash to keep classical music on the air. But we don't give enough. The new management has decided that in order to bring in more contributions they must drop the classical music and go to an all-news, all-talk format. I just don't count, and neither does anyone else in my area who likes non-popular music. (That includes fans of not only classical, but folk, jazz, blues, indie rock, world, experimental, or ambient.) Instead, I am forced, if I want to be in public places, to listen to this squalling, shrieking, caterwauling, "soul" cacophony.

Posted at 8:38 pm | link

"Project 911" is done, and I have photographed it, so that you can see it for yourselves. I am pleased with how it has come out. I am not going to attach much explanatory text to this posting, because I believe a work of art has to stand alone and convey its meaning without a whole bunch of words getting in the way.

The picture is officially called "The Geometry of Remembrance" and commemorates the 9/11 terrorist attacks. I concentrated not on any political or religious or social aspects, but just on the geometrical forms of the buildings that were destroyed or damaged. The picture is painted with acrylic on black-painted illustration board, all by hand without any computer enhancement. The dimensions are 20" x 16".

You are hereby invited to view my latest painting "The Geometry of Remembrance."

Posted at 3:45 am | link

Today is February 1, and this day marks a year since I began this Weblog, in a time of winter and trigonometric dreariness. Here I am in winter again, but trigonometry has turned into vectors (where it belongs) and I am finally accelerating for real. I did 20 physics problems about vectors and acceleration, both in straight lines and circles. I'll do the remaining 3 in the set soon. Not only did I do these problems, but I got them right, for the most part. The ones I got wrong were due to inattention in copying the given data or carrying out an algebraic operation. (It's harder to do math when I'm listening to music at the same time, which may interest neuroscientific types.) Much of this beginning physics depends on memorized formulas and plugging numbers in, which is not very creative and probably not proper educational protocol, but it beats re-deriving the thing from scratch each time. Here's a sample problem, number 19:

19. A car will skid if its acceleration, as it makes a turn, is more than 3.5 m/s². If the car is traveling at 20 m/s, what is the radius of the smallest circle it can travel in without skidding?

It turns out (so to speak) to be about 114 meters. Except if you are driving in my current conditions, in which case you are driving on ice and you have no guarantee you won't skid at any speed.

I have also finished "Project 911." It has been difficult, as usual, to push thick, sticky paint around highly precise geometric areas. I could have done this on the computer, sure, but it just wouldn't be the same, wouldn't have texture or special effects or weight or reflectivity or frame-ability. I'm still devoted to good old archaic painting by hand. When I have done the final touches and varnished the thing, I will photograph it and put it up here for all of you to see.

There is nothing like finishing a painting, AND solving a lot of physics problems correctly, to boost my confidence. I know that I am still at the most basic level, a kindergarten reader as it were. Those of us of a certain age, or even older, for Ghod's sake, will remember the famous children's first readers with "Dick and Jane." There is a fascinating and entertaining book about these readers, called "Growing Up With Dick and Jane" by Carole Kismaric and Marvin Heiferman. The original printings of these books are now collectors' items. Though they have been parodied and mocked, they are still remembered fondly and they remain in print. Some of us will recognize the ultra-simple texts that helped youngsters learn to read. Well, since I am at the "Dick and Jane" stage of learning physics, I will offer my own version of the classic text.

"Look up, Pyracantha.
You can see something.
It is Physics. It is Electron Blue.
It can go up, up, up.
It can go away."
Pyracantha said, "I want to go up.
I want to go up in it.
I want to go up, up, up.
I want to go up and away."

Posted at 3:16 am | link