The Biography of a Dangerous Idea

[PITH]

electra, couldn't you go to some fee image upload service to adjust your image for this borad???

you meant free image upload service..

LOL internet

[s u n d a y]

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.htm

http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg

[s u n d a y]

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.htm

http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg

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

[thegossipgirl]

...(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.

[...]

An application,

http://gc.kls2.com/

[imam]

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

You don't appear to be the original "s u n d a y" -- your username is "s  u n d a y," isn't it? Or is it that you just forgot your password or smth and set up another account?

[game it up]

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.

Mathematically, "dimension" refers to the number of coordinates needed to describe a point, or equivalently the degrees of freedom of motion in a space. A line is one-dimensional because a point in the line needs only one coordinate for its description. You could say "dang, there's a hundred people in front of me," which describes very well your sorry position in line. As another example, the volume of sound is a one-dimensional concept. A particular volume needs only one number to describe it, possibly from the scientific decibel scale, or maybe on the stereo knob "turn it up to 10" scale. A two-dimensional space needs two numbers for each point. The flat, infinite plane from high-school geometry is the prime example, with each point given an x and a y coordinate. The surface of a sphere is also two-dimensional; for example, points on the Earth are described by longitude and latitude. Though we spend most of our days wandering the two-dimensional surface of the Earth, our space is in fact three-dimensional, which means we can move on three axes, North-South, East-West, and Up-Down. Describing points in space requires three coordinates: to spot an airplane, you need longitude and latitude, plus elevation.

The next step is the fourth dimension. Mathematically, it's no problem to define four-dimensional space, or "hyperspace." It's just an abstract space that needs four coordinates to describe each of its points, which works very well for computations, but is not much help in visualization. Trying to think in 4 dimensions is a serious challenge, and requires a complicated collection of mental crutches to make any progress. The most effective crutch is the analogy with one lower dimension, a trick perfected in the novel "Flatland," written by the 19th century minister E. A. Abbott. The book is the story of A. Square, who lives in a 2-dimensional world. Mr. Square describes his world, with some not-so-subtle criticism of Victorian society, and then is visited by a sphere from the third dimension. You can imagine a 2-dimensional being as an amoeba trapped in a microscope slide, or as an ink spot moving on a piece of paper. Often it's easier to picture him as very flat and living on the surface of a table.



Let's use this analogy to explain Slade's feats of four-dimensional dexterity. Consider a challenge for a two-dimensional spiritual medium. We present him with a rubber band and a penny, and challenge him to put the penny inside the rubber band. You too, can play this game, but as a 2-dimensional being you'll need to keep the penny and rubber band flat on a table at all times. Clearly, it can't be done. However, using the third dimension you can pick part of the rubber band off the table, slide it over the penny, and set it back down. The 2-dimensional being would see part of the rubber band mysteriously disappear, then reappear on the other side of the penny.

Now, we have some of the tools to help us understand Slade's challenges. In further sittings, Slade's "spirits" caused rings of wood to disappear from a tabletop and reappear encircling the table's leg, caused burns to appear on pig intestines held below the table, and caused snail shells to teleport from table to floor. he rubber-band-and-penny thought experiment shows us exactly how Slade's rings-around-the-table trick could work, if the medium had access to the fourth dimension. He simply lifts "up" the ring into the fourth dimension, and sets it back "down" around the table. But putting rings around a table is not what Zöllner had challenged Slade to do! In fact, Slade was to link the two wooden rings to each other. The rings were of different woods, each carved from a single piece. Two such linked rings are physically impossible to create, so their existence alone would provide excellent evidence for the fourth dimension. Linking the table, though impressive, is possible to fake.

So are Slade's other feats. His initial feat, tying knots in a closed loop of rope, could also be done with four dimensions: move part of the rope out of our three-dimensional space, move it across the other part of rope, then bring it back to this world. But Zöllner was obviously suspicious of the rope trick, because his second challenge to Slade was to tie a knot in a closed loop cut from a pig's bladder. Unlike the sealed loop of rope, which could be switched or tampered with, Slade had no way to create a knot in any continuous piece of pork. He had three choices: cut the loop and risk exposure, actually use the fourth dimension, or claim that the spirits weren't in the mood. Not surprisingly, he chose the latter. Slade's final feat was to teleport some snail shells. Again, the fourth dimension is a good way to do this sort of thing. You move the shell into the fourth dimension, move it where you want it to go, then drop it back into our prosaic three-space. The two-dimensional analogy should help make this clear, as a third-dimensional being could lift an object out of the plane, move it, and set it back down. But again, this was not what Zöllner had asked for. In fact, Zöllner's challenge to Slade was to take the snail shells, which had clockwise spirals, and turn them into snail shells with counterclockwise spirals.

[game it up]

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. The four-dimensional version works the same way. You could use the fourth dimension, for example, to turn a right shoe into a left shoe. In 1909, Scientific American held an essay contest to explain the fourth dimension, and many of the essays focused on mirror reversals. Isomeric chemicals such as dextrose and levulose (literally right- and left-handed sugars) were presented as evidence for the existence of the fourth dimension on the molecular scale, and one Zöllner enthusiast claimed that clockwise and counterclockwise snails are produced by a hyperspace reversal, right down to their "juices."

H. G. Wells used the mirroring phenomenon in "The Plattner Story" of 1896, which is about a man who accidentally blasts himself a short distance into the fourth dimension. The man finds himself in a greenish world populated by spirits of departed humans, and can see faint images of the earthly realm overlaid on his new reality. After a week he manages to return home, but has become his own mirror image, as evidenced by photographs, his writing, and most impressively his heart, which now beats on the right side of his chest. "The Plattner Story" was not the only appearance of the fourth dimension in literature of the period. It is the science behind The Time Machine, and also the home for the angel that falls to Earth in A Wonderful Visit, Wells' first two novels. It is jokingly referred to in Oscar Wilde's "Canterville Ghost" of 1887, about an English spirit who is snubbed by the new American owners of his ancestral manse. And Joseph Conrad's The Inheritors of 1901 is about four-dimensional humans, devoid of conscience, who assume control of the earth. In Madeleine L'Engle's "A Wrinkle In Time," Charles Wallace, Meg, and Calvin "tesser" between worlds, traveling through the fourth dimension. The word "tesser" means four, and shows up in the word "tesseract," which is the four-dimensional analog of the cube.

Here are two fine examples of this technique, one by Picasso, who never explicitly acknowledged the influence of the fourth dimension, and one by Jean Metzinger, who clearly stated it as his goal.


J. Metzinger, Le Gouter/Teatime (1911)


P. Picasso, Portrait of Ambrose Vollard (1910)

In both, you can see the similarity between the faceted figures and the angled planes of the hypercube, and the teacup in "Le Gouter" is a perfect demonstration of multiple viewpoints combined to give a full impression of an object. As another example of four-dimensional cubism, look at Marcel Duchamp's "Nude Descending A Staircase, No. 2." There's a somewhat robotic figure shown in various stages of descent, as if we're seeing multiple exposures. In this picture, Duchamp (who was the greatest advocate of the fourth dimension in the art world) considers the fourth dimension as time.


M. Duchamp, Nude Descending a Staircase, No. 2 (1912)

Interestingly, this idea was one of the triumphs of Einstein's theory of relativity, but the relativity papers were published in 1916, four years after "Nude Descending"! Duchamp, though a brilliant artist, wasn't anticipating modern physics. He was simply following the lead of scientists who, from the mid 1800s, used time as another mental crutch towards understanding hyperspace.

The time crutch works as follows: Take your four dimensional object and cut it into a succession of three dimensional slices. Duchamp explains,

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.

Now imagine the slices played back as a movie, using the flow of time to "bind them to one another." The classic example of this, used in Abbott's Flatland, is to imagine a ball passing upwards through a plane. A being in the plane would first see a tiny dot, the top "slice" of the ball. As the ball moves up, the two-D observer sees the dot grow into a larger and larger circle. When the ball is halfway through the plane, the circle will be as large as possible, and then the observer will see it shrink to a point and disappear.



Just as we can't play kickball with a frisbee, a four-dimensional athlete would need a "hypersphere" in lieu of a ball. And if she kicked it through your room, you'd first see a pea-sized object, which would quickly grow to a melon, hover, shrink back to a pea, and disappear.

Ironically, as Einstein's theory of relativity was accepted in the early 1920s, its elegant definition of four-dimensional spacetime killed the romance between the public and the fourth dimension of space. Now that physicists were treating plain old time as a fourth dimension, speculations about mysterious "other" directions seemed ludicrous, and the fourth dimension disappeared from art and literature. The surrealist art movement was one of the few reappearances of hyperspace. The spiritual associations and irrationality of the traditional fourth dimension must have appealed to Salvador Dali, who used many images and allusions to the fourth dimension, for example in the "Crucifixion (Corpus Hypercubicus)" and "In Search Of The Fourth Dimension."



The fourth dimension in this crucifixion is the cubical "cross." We know from grade school that a flat paper cross can be folded into a cube, so we should be able to fold a three-dimensional collection of cubes into a tesseract. It seems that it would be impossible to "fold" two stuck-together cubes, but it's not, and if you can visualize this maneuver you're well on your way to higher dimensions. At the very least, take some solace from the plight of a two-dimensional being faced with two squares attached along an edge. He would assure you that folding along the edge is an absurd idea -- the two squares would surely rip apart.

Even relativity theory didn't answer the big question: does a fourth dimension of space exist? Physics says time is a fourth dimension, and modern string theories suggest a whole bunch of dimensions on the sub-atomic scale. But none of this precludes another direction, perpendicular to space, in which we could move if we only knew how. We are like the men in Plato's Republic, chained in a cave and illuminated from behind. Their entire world consists of their own shadows, thrown on the cave's wall. Shadows are all they have ever seen, shadows are all they know, and shadows are their reality. To tell these men that they are solid beings living in space is impossible, and it could be that way with us and the fourth dimension. If it's there, it's in a direction for which we have no conception and no way to look.

[Glee Client]

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.

this might actually have some sense in it. It will be difficult to understand, but I'll try ... This diagram represents a single string at a single point in time.



The blue represents a positive charge, and the red represents a negative charge, they both coexist within two seperate dimentions. The charges travel at c, 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. With the creation of two more physical dimentions, two more physical forces appear, these forces are electricity and magnetism. This theory also agrees with special relativity, because the speed of the charges traveling within the string is that of c, therefore when a string is accelerated to relative speeds it obeys the SR laws.

[#]

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.

http://www.youtube.com/watch?v=HOkAagw6iug&feature=related

[O]

[...] 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 [...]

Indeed, with the Mobius Strip the inside becomes the outside and the outside becomes the inside. It is an expression of non-duality. It reveals the Unity of all polarities, creating a state of Oneness, joining the whole and the part, the masculine and the feminine, expansion and contraction, spirit and matter, etc. Everything is One and nothing can be separated from anything else. All is completely intertwined, infinitely. The Mobius Strip is a spiritually significant symbol of balance and union. The Buddhist philosophy of Tantrism also is expressed by the Mobius Strip shape. "Tantra" is continuity; the word derived from the root 'tan', meaning to extend... extend continuously, to flow, to weave. The continuum is descriptive of the Nature of Reality, by contemporary physicist David Bohm... "a single unbroken wholeness in flowing movement."