Quote from: Anna Ogordova on September 05, 2007, 02:59:08 AMQuote from: electra on June 18, 2006, 08:29:14 PMelectra, couldn't you go to some fee image upload service to adjust your image for this borad???you meant free image upload service..

Quote from: electra on June 18, 2006, 08:29:14 PMelectra, couldn't you go to some fee image upload service to adjust your image for this borad???

Quote from: s e a m on September 18, 2007, 06:55:40 PMWell, it's really an interesting subject .. every so often as you're driving along there's just one shoe lying there on the road. There's never the other shoe in the pair, just that one shoe. Does someone throw their shoe out the window in disgust? Do kids throw their parents' shoes out the back of the station wagon? Do they sprout from seeds sewn by bird droppings in the pavement? This is a worldwide phenomenon: I've seen road shoes sit there, dusty and flattened, in India, Europe, and Mexico and on many highways and byways of North America. Many great and not-so-great minds have wrestled with this phenomenon without arriving at any firm conclusions. David Feldman devotes 7 pages to the topic in his book When Do Fish Sleep, in the course of which he elucidates 13 theories on lone shoe origin. Clearly, what Dave needs is find himself a date.There is disagreement on how widespread the phenomenon is. Some say it's confined to North America, and that you never see shoes on, say, the German autobahn. There is no single explanation for the lone shoes. One woman said she placed an extra pair of shoes on the roof of the car while she loaded some stuff, then forgot about them and pulled off. When she checked a while later they were gone. Another said a passenger had his feet up on the dash when the car hit a pothole, whereupon he became unshoed. Unshod. You know what I mean. Yet another claimed he personally had gone around the country strategically depositing shoes in order to sow panic amongst the populace. There's one in every crowd.None of this really gets at the heart of the matter, however. One dedicated research team, including two short and irrepressible members who several times came perilously close to contributing personally to the lost shoe population, recently conducted a 1,500-mile cross-country car trip, traveling on everything from interstates to gravel roads. En route they passed thousands of identifiable items of roadside debris, chiefly pieces of retread tire on the interstates (how anybody can stand to drive on those things you will never know) and food packaging (mostly cans and bottles) everywhere else. Total shoe count: 4, including one each in Knoxville, Tennessee, and Louisville, Kentucky, and 2 on the road into Chicago. Granted this was in May, not (to hear some tell it) the height of shoe season. And they probably missed a few, such as when one of their little researchers was screaming at the top of her lungs. Still, considering the vast quantity of roadside junk, we are talking about a tiny number of shoes. I would venture to say people have the idea that the highways are littered with shoes because (1) a roadside shoe is such an ineffably memorable sight, and (2) virtually all other trash on the road is either anonymous or numbingly commonplace. As to why you always see one shoe, never a pair, what do you expect? Assuming most of the shoes are lost by accident, the chances of two randomly ejected shoes landing together is vanishingly small.seam, is your avatar the depiction of Smale's paradox, saying that you can turn a sphere inside out in 3-space by passing the surface through itself without making any holes or creases (eversion)?http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/MORIN/VISUALIZATION/visualiz.htmhttp://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg

Well, it's really an interesting subject .. every so often as you're driving along there's just one shoe lying there on the road. There's never the other shoe in the pair, just that one shoe. Does someone throw their shoe out the window in disgust? Do kids throw their parents' shoes out the back of the station wagon? Do they sprout from seeds sewn by bird droppings in the pavement? This is a worldwide phenomenon: I've seen road shoes sit there, dusty and flattened, in India, Europe, and Mexico and on many highways and byways of North America. Many great and not-so-great minds have wrestled with this phenomenon without arriving at any firm conclusions. David Feldman devotes 7 pages to the topic in his book When Do Fish Sleep, in the course of which he elucidates 13 theories on lone shoe origin. Clearly, what Dave needs is find himself a date.There is disagreement on how widespread the phenomenon is. Some say it's confined to North America, and that you never see shoes on, say, the German autobahn. There is no single explanation for the lone shoes. One woman said she placed an extra pair of shoes on the roof of the car while she loaded some stuff, then forgot about them and pulled off. When she checked a while later they were gone. Another said a passenger had his feet up on the dash when the car hit a pothole, whereupon he became unshoed. Unshod. You know what I mean. Yet another claimed he personally had gone around the country strategically depositing shoes in order to sow panic amongst the populace. There's one in every crowd.None of this really gets at the heart of the matter, however. One dedicated research team, including two short and irrepressible members who several times came perilously close to contributing personally to the lost shoe population, recently conducted a 1,500-mile cross-country car trip, traveling on everything from interstates to gravel roads. En route they passed thousands of identifiable items of roadside debris, chiefly pieces of retread tire on the interstates (how anybody can stand to drive on those things you will never know) and food packaging (mostly cans and bottles) everywhere else. Total shoe count: 4, including one each in Knoxville, Tennessee, and Louisville, Kentucky, and 2 on the road into Chicago. Granted this was in May, not (to hear some tell it) the height of shoe season. And they probably missed a few, such as when one of their little researchers was screaming at the top of her lungs. Still, considering the vast quantity of roadside junk, we are talking about a tiny number of shoes. I would venture to say people have the idea that the highways are littered with shoes because (1) a roadside shoe is such an ineffably memorable sight, and (2) virtually all other trash on the road is either anonymous or numbingly commonplace. As to why you always see one shoe, never a pair, what do you expect? Assuming most of the shoes are lost by accident, the chances of two randomly ejected shoes landing together is vanishingly small.

seam, is your avatar the depiction of Smale's paradox, saying that you can turn a sphere inside out in 3-space by passing the surface through itself without making any holes or creases (eversion)?http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/MORIN/VISUALIZATION/visualiz.htmhttp://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg

...(BTW, A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two equal hemispheres = a circle on the sphere's surface whose center is the same as the center of the sphere = the intersection of a sphere with a plane going through its center = largest circle that can be drawn on a given sphere. Great circles serve as the analog of "straight lines" in spherical geometry) Thus, Great Circle of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that it obeys the axioms required of a projective plane:- any pair of distinct great circles meet at a pair of antipodal points; - and any two distinct pairs of antipodal points lie on a single great circle. [...]

Well, to begin at the beginning, Consider a sphere, and let the great circles (BTW, A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two equal hemispheres = a circle on the sphere's surface whose center is the same as the center of the sphere = the intersection of a sphere with a plane going through its center = largest circle that can be drawn on a given sphere. Great circles serve as the analog of "straight lines" in spherical geometry) Thus, Great Circle of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that it obeys the axioms required of a projective plane:- any pair of distinct great circles meet at a pair of antipodal points; - and any two distinct pairs of antipodal points lie on a single great circle. This is the real projective plane. If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. The resulting surface, a 2-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself.The projective plane cannot be embedded (that is without intersection) in three-dimensional space. However, it can be immersed (local neighbourhoods do not have self-intersections). Boy's surface is an example of an immersion. The Roman surface is another interesting example, but this contains cross-caps so it is not an immersion. The same goes for a sphere with a cross-cap. A cross-cap has a plane of symmetry which passes through its line segment of double points. In Figure 1 the cross-cap is seen from above its plane of symmetry z = 0, but it would look the same if seen from below. A cross-cap can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points.Once this exception is made, it will be seen that the sliced cross-cap is homeomorphic to a self-intersecting disk

2-dimensional renderings (i.e., flat drawings) of a 0-dimensional point, a 1-dimensional line segment, a 2-dimensional square, a 3-dimensional cube, and a 4-dimensional tesseract.

The shadow cast by a four-dimensional figure on our space is a three-dimensional shadow . . . by analogy with the method by which architects depict the plan of each story of a house, a four-dimensional figure can be represented (in each one of its stories) by three-dimensional sections. These different sections will be bound to one another by the fourth dimension.

Way back in 1827, the mathematician Möbius, of "Möbius strip" fame, realized that a trip through the fourth dimension could turn an object into its own mirror image. To understand, we return to the two-dimensional analogy. Take a symbol which looks wrong in a mirror, such as an N, and cut it out of a piece of paper. If you set it down on a table, you'll find there's no way to turn the N into the backwards N just by sliding the paper around the tabletop. But if you allow yourself a third dimension, you can simply lift up the N, flip it over, and place it back on the table.

In the first seconds after the Big Bang, there was no matter, scientists suspect. Just energy. As the universe expanded and cooled, particles of regular matter and antimatter were formed in almost equal amounts. But, theory holds, a slightly higher percentage of regular matter developed -- perhaps just one part in a million -- for unknown reasons. That was all the edge needed for regular matter to win the longest running war in the cosmos. When the matter and antimatter came into contact they annihilated, and only the residual amount of matter was left to form our current universe.

[...] and therefore the "mobius strip" string can have two sides at any point in time. The reason that the string forms into a "mobius strip" is because of the opposites attract law, and the string must twist in order for the opposite charges that are on oppostite sides of the string to attract; this causes there to be two more dimentions of space, and because of the nature of the energy also imposses time into a third additional dimention [...]