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KASTsystem

Tesseract

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After reading Tesseract's answer in the Explain Your Screenname thread (I quoted his post below), I became interested in finding out more about what a tesseract actually is. I found this really good link where you can rotate a tesseract through 4th dimensional space. It doesn't look exactly like the picture Tesseract posted, but it's pretty interesting.

 

Well, here is the link: http://pw1.netcom.com/~hjsmith/WireFrame4/.../tesseract.html

 

Originally posted by Tesseract

<---------- A great book and some geometry...

 

http://www.alexgarland.co.uk/tesseract/gifs/tesserct.gif'>

 

 

Definition

 

Take six cubes and arrange them into the shape of a crucifix. Take two more cubes and stick them either side of the crucifix, at the point where the cross is made. Now you have a tesseract. A tesseract is a three-dimensional object. A tesseract is also a four-dimensional object - a hypercube - unravelled.

 

A square unravels to a line. Two dimensions unravel to one.

 

A cube unravels to a cross. Three dimensions unravel to two.A hypercube unravels to a tesseract. Four dimensions unravel to three.

 

You exist in three spatial dimensions. In the same way that a one-dimensional boy could not visualize a two-dimensional square, or a two-dimensional boy could not visualize a three-dimensional cube, you cannot visualize a hypercube. A hypercube is a thing you are not equipped to understand. You can only understand the tesseract.

 

This means something.

 

We can see the thing unravelled, but not the thing itself.

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zesto you draw that shit? im feeling it....

 

 

for sure tess wins that screen name contest....my mind went into overdrive after reading that.....

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Guest --zeSto--

^ I only wish i could!

 

google image search baby !

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Guest imported_Tesseract

KAST, its cool that you brought this up...As mental said it kinda blows your mind and thats exaclty what happened when i read that book. The idea of the fourth dimension and its geometry-shapes that you can 'prove' in theory but cant visualise- was in my head for a while and after some 'investigation' i decided to make an animation about it...It is an animated geometrical 'view' of the novel where in the first part i explain what a Tesseract is and in the second assuming that the 4th dimension is time (einstein style) i give a visualised 'explanation' about the book and those things in general...You should know what the novel is about but the Tesseract folder is now opened :P and in my next reply i'll try to make things clearer...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

...so official, hehe...

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Guest imported_Tesseract

*This sums it up...

 

Four - Dimensional Intuition*

 

 

A LINE IS one-dimensional. A flat surface is two-dimensional. Solid objects are three-dimensional. But what is the fourth dimension? Sometimes people say that time is a fourth

dimension. In the physics of Einstein's relativity, a four-dimensional geometry is used in which a three-dimensional space and a one-dimensional time coordinate are merged into a single four-dimensional continuum. But we don't want to talk about relativity and space-time. We only want to know if it makes sense to take one more step in the 1ist of, geometrical dimensions. For instance, in two dimensions, we have the familiar figures of the circle and the square. Their three-dimensional analogs are the sphere and the cube. Can we talk about a four-dimensional hypersphere or hypercube, and make sense?

We can go from a single point up to a cube in three steps. In the first step we take two points, 1 inch apart, and join them. We get a line interval, a one-dimensional figure. Next, we take two l-inch line intervals, parallel to each other. 1 inch apart. Connect each pair of end-points and we get an l-inch square, a two-dimensional figure. Next, take two l-inch squares, parallel to each other. Say the first square is directly above the second, I inch away. Connect corresponding corners, and we get an l-inch cube.

So, to get a l-inch hypercube, we must take two l-inch cubes, parallel to each other, 1 inch apart, and connect vertices. In this way, we should get a l-inch hypercube, a four-dimensional figure.

The trouble is that we have to move in a new direction at each stage. The new direction has to be perpendicular to all the old directions. After we have moved back and forth, then right and left, and finally up and down, we have used up all the directions we have accessible to us.

http://www.freephoto-i.net/users/Hypercube/fig1.jpg'>

We are three-dimensional creatures, unable to escape from three-dimensional space into the fourth dimension. In fact, the idea of a fourth physical dimension may be a mere fantasy, a device for science fiction. The only argument for it is that we can conceive it; there is nothing illogical or inconsistent about our conception,

We can figure out many of the properties that a four-dimensional hypercube would have, if one existed. We can count the number of edges, vertices, and faces it would have, Since it would be constructed by , joining two cubes, each of which has 8 vertices, the hypercube must have 16 vertices. It will have all the edges the two cubes have: it will also have new edges, one for each pair of vertices that have to be connected. This gives 12 + 12 + 8 = 32 edges. With a little more work, one can see that it will have 24 square faces, and 8 cubical hyperfaces.

The table below shows the number of "parts" of the Interval, square, cube, and hypercube. It is a startling discovery that the sum of the parts is always a power of three!

http://www.freephoto-i.net/users/Hypercube/fig2.jpg'>

In a course on problem-solving for high-schoolteachers and education students, the gradual discovery of these facts about hypercubes takes a week or two. The fact that we can find out this much definite information about the hypercube seems to mean that it must exist in some sense, Of course, the hypercube is just a fiction in the sense of physical existence. When we ask how many vertices a hypercube has, we are asking, how many could it have, if there were such a thing, It's like the punch line of the old joke- "If you had a brother, would he like herring?" The difference is that the question about a nonexistent brother is a foolish question; the question about the vertices of a nonexistent cube is not so foolish, since it does have a definite answer .

In fact, by using algebraic methods, defining a hypercube by means of coordinates, we can answer (at least in principle) any question about the hypercube. At least, we can reduce it to algebra, just as ordinary analytic geometry reduces questions about two- or three-dimensional figures to algebra. Then, since algebra in four variables is not essentially more difficult than in two or three, we can answer questions about hypercubes as easily as questions about squares or cubes. In this way, the hypercube serves as a good example of what we mean by mathematical existence. It is a fictitious or imaginary object, but there is no doubt about how many vertices, edges, faces, and hyperfaces it has! (or would have, if one prefers the conditional mode of speaking about it.)

The objects of ordinary three- or two-dimensional geometry are also mathematical objects, which is to say, imaginary or fictitious; yet they are closer to physical reality, unlike the hypercube which we cannot construct.

The mathematical three-cube is an ideal object, but we can look at a wooden cube and use it to determine propertics of the three-cube. The number or edges of the three-cube is 12; so is the number of edges or a sugar cube 12. We can get a lot of information about it two- and three-dimensional geometry by drawing pictures or building models and then inspecting our pictures or models. While it is possible to go wrong by misusing a picture or model, it is rather difficult to do so. It takes ingenuity to invent a situation where one could go wrong in this way. As a general rule, the use of pictures and models is helpful, even essential in understanding two- or three-dimensional geometry. Reasoning based on models and figures, either actual ones or mental images of them, would be called intuitive reasoning, as opposed to formal or rigorous reasoning. When it comes to four-dimensional geometry, it might seem that since we ourselves are mere three-dimensional creatures, we are excluded by nature from the possibility of reasoning intuitively about four-dimensional objects. And yet, it is not so. Intuitive grasp of four-dimensional figures is not impossible.

At Brown University Thomas Banchoff, a mathematician, and Charles Strauss, a computer scientist, have made computer-generated motion pictures of a hypercube moving in and out of our three-dimensional space. To understand what they have done, imagine a flat, two-dimensional creature who lived at the surface of a pond and could see only other objects on the surface (not above or below). This flat fellow would be limited to two physical dimensions, just as we are limited to three. He could become aware of three dimensional objects only by way of their two-dimensional intersections with his Hat world. I f a solid cube passes from the air into the water. he sees the cross-sections that the cube makes with the surface as it enters the surface, passes through it, and finally leaves it.

If the cube passed through repeatedly, at many different angles and directions, he would eventually have enough information about the cube to "understand" it even if he couldn't escape from his two-dimensional world.

The Strauss-Banchoff movies show what we would see if a hypercube passed through our three-space, at one angle or another. We would see various more or less complex configurations of vertices and edges. It is one thing to describe what we would see by a mathematical formula. It is quite another to see a picture of it; and still better to see it in motion. When I saw the film presented by Banchoff and Strauss, I was impressed by their achievement, and by the sheer visual pleasure of watching it. But I felt a bit disappointed; I didn't gain any intuitive feeling for the hypercube.

A few days later, at the Brown University Computing Center, Strauss gave me a demonstration of the interactive graphic system which made it possible to produce such a film. The user sits at a control panel in front of a TV screen. Three knobs permit him to rotate a four-dimensional figure on any pair of axes in four-space. As he does so, he sees on the screen the different three-dimensional figures which would meet our three-dimensional space as the four-dimensional figure rotates through it.

Another manual control permits one to take this three-dimensional slice and to turn it around at will in three-space. Still another button permits one to enlarge or shrink the image; the effect is that the viewer seems to be flying away from the image, or else flying toward and actually into the image on the screen. (Some of the effects in Star Wars of flying through the battle-star were created in just this way. by computer graphics.)

At the computing center. Strauss showed me how all these controls could be used to get various views of three-dimensional projections of a hypercube. I watched, and tried my best to grasp what I was looking at. Then he stood up, and offered me the chair at the control.

I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it! The hypercube had leaped into palpable reality, as I learned how to manipulate it, feeling in my fingertips the power to change what I saw and change it back again. The active control at the computer console created a union of kinesthetics and visual thinking which brought the hypercube up to the level of intuitive understanding.

In this example. we can start with abstract or algebraic understanding alone. This can be used to design a computer system which can simulate for the hypercube the kinds of experiences of handling, moving and seeing real cubes that give us our three-dimensional intuition. So four-dimensional intuition is available, for those who want it or need it.

The existence of this possibility opens up new prospects for research on mathematical intuition.

http://www.freephoto-i.net/users/Hypercube/fig3.jpg'>

Instead of working with children or with ethnographic or historical material, as we must do to study the genesis of elementary geometric intuition (the school of Piaget), one could work with adults, either trained mathematically or naive, and attempt to document by objective psychological tests the development of four-dimensional intuition, possibly sorting out the roles played by the visual (passive observation) and the kinesthetic (active manipulation.) With such study, our understanding of mathematical intuition should increase. There would be less of an excuse to use intuition as a catchall term to explain anything mysterious or problematical. Looking back at the epistemological question, one wonders whether there really ever was a difference in principle between four-dimensional and three-dimensional. We can develop the intuition to go with the four-dimensional imaginary object. Once that is done, it does not seem that much more imaginary than "real" things like plane curves and surfaces in space. These are all ideal objects which we are able to grasp both visually (intuitively) and logically.

 

Further Readings. See Bibliography

H. Freudenthal [1978]; J. Piaget. [1970, 71]; T. Banchoff alJd c. M. Strauss

 

*Taken from Linda Henderson's book "THE FOURTH DIMENSION AND NON-EUCLIDEAN GEOMETRY IN MODERN ART" Princetown university press, new jersey.

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Guest imported_El Mamerro

Awesome stuff... I was also really interested in the whole "I understand, but can't visualize it" deal that happens in physics (and there's also the fact that I'm a physics freak), and decided to write a paper on it for my visual perception class last year. Hypercubes are mentioned in there too... my bad for not putting the images I used, those were deleted. Enjoy or ignore:

 

 

 

 

 

Seeing the Unseeable:

The Role of Vision in our Understanding of the Universe

 

 

 

The human imagination, as vivid and unlimited as it might seem, is still hindered by what information it has physically received through the senses. A person might have a dream where he or she can fly through walls, seemingly defying the laws of physics, but a person cannot dream about existing and moving in additional dimensions, or what it feels like to move at the speed of light. All of these conditions are essential parts of the universe we live in, yet our brains and bodies are incapable of perceiving such things. We can discuss these phenomena and mentally grasp them, but it is impossible to visualize and fully understand them, since the brain has no physical evidence or information with which to construct the appropriate mental images.

 

With the modern day advances and discoveries of theoretical physics, we are starting to understand that the world is much more complex and illogical than our brains can visualize. The possible existence of up to seven additional space dimensions, the space/time theory postulated by General Relativity, and non-local quantum phenomena are a few examples of building blocks of the universe we are incapable of visualizing, but fully capable of discussing and elaborating upon. How can we discuss something that simply doesn’t fit inside our heads? How can we seem to understand the scope of such things, when the main ingredient our brain has to work with, physical data obtained through the senses, is absent from these abstract concepts?

 

Throughout the years, scientists have had to devise elaborate visual metaphors and diagrams that attempt to translate these complex ideas into a format our senses can digest and process. Obviously, a lot of the information is lost in the translation, and there is always room for misinterpretation, but very often these metaphors are clever, easy to understand, and convey enough meaning for the human mind to visually understand what’s visually impossible. This investigation will deal with some of the most effective visual translations physicists have used to communicate their ideas, and how the brain can use this visual information to produce non-visual abstract thought.

 

Early in the history of modern civilization, human beings relied on strictly visual observation to formulate their hypotheses about the world we live in. The sun and the stars moved across the sky, therefore it was believed that they revolved around an immobile Earth. They reasoned the horizon appeared to be flat, so the Earth must be flat as well. The ancient Greeks may have come up with the concept of atoms as the smallest indivisible quantity of matter, but their list of elements was limited to four: water, air, earth, and fire. These were the four most distinct materials in the world, and therefore they supposed that all things are made of mixtures of these elements.

 

It was Aristotle who first started conceiving of gravity and the Earth’s spherical shape through the use of visual information. Instead of believing that the Earth was the center of the universe, he believed the Earth just happened to be at the center of the universe. He noticed that substances and materials tend to move towards a central area, just as pouring water or dust into a bowl makes it settle at the center of the bottom. In this way, he reasoned that all parts of Earth, being made of the heaviest element, had settled around the center of the universe, and that the Earth must therefore be spherical. He also noted that the night sky appeared to have substantial changes when viewed from different cities, and that these changes were large enough to suggest the Earth was not really that big when compared to the rest of the universe and the stars. These discoveries, mainly obtained through visual observations, shaped the study of physics and astronomy for almost two thousand years.

 

Similar observations were made by Nicholas Copernicus, where he elaborated on Aristotle’s ideas of the spherical shape of the Earth. He went so far as to announce that the Earth was not the center of the universe, and that it revolved around the sun. However, it wasn’t until the telescope age, brought about by Galileo Galilei, that visual information from the stars and planets began playing a more important role in physics. Finally, Isaac Newton began to make use of mathematics to explain the world we see around us, and did nothing less than almost completely redefine it. His discoveries, when seen by the average person living today, may seem as nothing more than just using common sense, but in fact, before Newton, the world we live in was perceived to be a much different place. He was the first scientist to show that the world is ruled by forces that don’t necessarily appear obvious to naked eye, and he did it so clearly that they are now part of the human psyche. When visually describing the world we live in, it is impossible to not make use of Newton’s ideas at some point.

 

Newton’s best quality was the clarity of his ideas and the relatively simple mathematics involved in them. To describe his discoveries, Newton could make use of simple examples, such as the cliché apple falling from a tree. These were easy to visualize by the average non-mathematic person, since the forces described dealt with real-world matters and within the scope of our perceptive ability. Since the brain deals much better when given concrete data from the senses to comprehend a concept, Newton’s ideas are easily understood due to the fact that we see them at work constantly through our lives.

 

Physics remained on a real-world level for some time, until the discovery of atoms and the particles that make them up created a rift in what was observable directly and what wasn’t. Mathematics had taken over physics, and leading scientists understood that the actual workings of the universe are not exactly what we perceive from it; the real building blocks of the world were outside the human perceptive ability and impossible to study visually. Even as technology raced to make these “unseeable” particles visible, it was clear enough at some point that even the most powerful optic microscope would never be able to probe the depths of mass at the atomic level. Instead, scientists began relying more on mathematical equations and indirect observations to get accurate conclusions about atoms. To this day, optical microscopes have very limited magnification abilities. This lead to the invention of the electron scanning microscope, which gathers visual subject data in an indirect process that does with electrons what the human eye does with photons, and then produces a visual image with amazing clarity. Even this microscope has its limits, which lead to the invention of mammoth devices such as particle accelerators, which study particles in an even more indirect fashion. Scientists began having trouble explaining to the world their findings, since these were increasingly esoteric and non-visual. The mathematics involved were also becoming more complicated and went beyond Euclidean geometry (the term used to describe “normal” mathematics), making the concepts even harder to explain mathematically.

 

And then came Einstein. Completely revolutionizing the world of physics and redefining the universe, his two major contributions, the theories of special and general relativity, seem to defy logic and are only observable at distances and speeds impossible for human beings to experience. His visual analogies have been increasingly refined through the years, and thus today, knowledge that used to be limited to leading theoretical physicists can now be taught to teenagers in high school. Following is a series of visual explanations of these two theories, to give an example of how visual aids can aid in the understanding of abstract concepts.

 

The concept of special relativity, which tells us that motion affects the passage of time, rests on two simple facts: 1) There is no “absolute” notion of motion, that is, you can only measure a person in motion if you first state that you are stationary. Likewise, the said person can claim he/she was stationary and that YOU were in motion. 2) The speed of light (670 million miles an hour) is a universal constant; light never accelerates or slows down, nor is it possible to catch up to it. If you were to somehow run at the speed of light and attempt to catch up to a photon, you would find that the photon still travels away from you at the speed of light. With these two facts in mind, let’s look at special relativity at work.

 

Suppose you were to build an atomic clock (See figure 1.1 in the pictures appendix at the end). Such a clock would consist of two mirrors, one above and facing the other, with a photon bouncing back and forth between them. Every time the photon strikes one mirror, bounces to other one, and strikes the first one again, the clock registers a tick. That said, imagine you were to observe a stationary light clock, and one next to it which is moving to the right (figure 1.2). All logic and Newtonian thought would dictate that both clocks will tick with the same frequency. However, the photon in the stationary clock only has to travel up and down. On the other clock, the photon not only goes up and down, but also to the right. The result, from your perspective, is that the photon in the moving clock travels in diagonal paths (figure 1.3), which means it had to travel a greater distance to achieve a single tick. Special relativity dictates that both photons are moving at the exact same speed, which means that the photon in the moving clock, having to travel a greater distance, takes longer to complete a tick that the photon in the stationary clock. This means that objects in motion experience a slower passage of time. At human scales, even the fastest attainable speeds are way too slow for major differences in passage of time to be detected; therefore, we never perceive the effect of special relativity on our lives. However, it does exist, and by “slowing” a photon down through this analogy we can easily visualize how it works.

 

General relativity was devised as a way to integrate gravity with special relativity. Gravity was thought to work instantaneously; that is, if a star were to explode in outer space, it’s gravitational force would be exerted immediately upon the surrounding stars. But since special relativity announced that light is the fastest thing in the universe, how could gravity work faster? Einstein then discovered that gravity was not a “beam” force, like light. Instead, gravity is the result of curvatures in space. Space, time, and motion, as demonstrated by special relativity, are interwoven in an expansive “fabric” that defines the universe. When mass is placed on this fabric, its weight causes the fabric to stretch. Space’s invisibility and its three-dimensionality makes it hard for us to envision how it can be curved, but it can be easily visualized by representing space as a flat, two-dimensional grid (figure 2.1). An object resting on this grid causes it to be weighted down, stretching and curving the area around it (figure 2.2). If we were to duplicate this grid at a 90 degree angle, and place the object where both grids met, we begin to get a better idea of how an object bends three-dimensional space (figure 2.3). Finally, Einstein explained that the “stretching” of space caused by an exploding star would occur at the speed of light, no faster. But, just as in special relativity, the scales at which the stretching of space would be noticeable are planetary and beyond, leaving us with a very limited perception of gravity as that force which makes things fall down (which is exactly what Newton described it as).

 

Another important discovery in physics was that of the possibility of additional space dimensions. While non-Euclidean geometry made it easy for mathematicians to study the geometry of space with more than three dimensions, the fact that human beings can only perceive three dimensions makes it extremely difficult for us to imagine what additional dimensions must be like. After all, length, width, and breadth are enough to pretty much describe every object in the visible universe, and if we toss time in as an additional dimension, we can record all of the universe’s events in four dimensions. Through the study of additional space dimensions, however, our three dimensional world can appear to be astonishingly simple. Just as the weather and seasons were a mystery to ancient civilizations who thought they lived in a flat, two-dimensional world, our abilities to probe the three-dimensional space around the Earth has shown us how clouds form and affect each other in the spherical atmosphere, and how the tilt in the Earth’s axis causes differences in warmth over the year.

 

The easiest way to imagine what these additional dimensions might be like is to study how different life would be like in a lower-dimensional universe, and then imagining ourselves as being studied by higher dimensional beings in this same manner. Scientists have coined the term Flatland to describe a two-dimensional universe with length and width but no breadth, and Hyperspace to describe a universe with more than three-dimensions. In here, the anatomy of lifeforms would be essentially different. For example, a Flatland being would not have a digestive tract, since such a system would divide a being into two distinct objects (figure 3.1). They would probably eat in a fashion similar to an amoeba, which encircles its food with appendages and then encloses it within its body. In the same way, as beings living in three dimensions, our bodies are built with systems specifically adapted to take advantage of these dimensions, which would be unthinkable to a Flatlander. Similarly, a being living in Hyperspace would probably eat in a wildly inconceivable way to us. Also, we can look at a Flatlander and see all of his entirety, represented by an outline. We can see his left side, right side, top, and bottom. All a Flatlander can see of a another Flatlander would be only one of these sides. In the same way, a Hyperspacer would be able to look at us humans from all angles simultaneously. He would see our top, bottom, left, right, front, and back. The closest human beings have come to expressing what this must look like was through the artistic movement known as Cubism, where multiple angles of a subject were portrayed simultaneously in a painting. Furthermore, we can start to imagine what the visual system of a Hyperspacer would be like. Our two separate eyes and visual cortex are built to help us define three-dimensional space by receiving photons bouncing from objects and making complex calculations in our brain. It would be safe to say that two eyes would not suffice for Hyperspacer, and that their “vision” would certainly not be photon-based.

 

To place a Flatlander in jail, fellow Flatlanders would simply construct a circle or square around him, and he would be unable to escape. For humans, placing someone in jail means constructing a cube or sphere to surround said person. However, a Flatlander can easily be rescued by a three-dimensional being, by just plucking him out of Flatland and into the three-dimensional universe, and placing him back in Flatland outside of the circle (figure 3.2). If such an event were to happen, Flatlanders would see their prisoner vanish into thin air, and suddenly reappear outside of the jail. A similar event would happen in our world, if a four dimensional being were to rescue a jailed prisoner. We would see the person simply disappear, and then materialize outside the cube. We would be baffled at how such a seemingly impossible event could occur, but through understanding how we can affect a two-dimensional universe, the event becomes a lot more plausible.

 

Another easy way to imagine higher dimensions is through the concept of a hypercube, or tesseract. To us human beings, a cube is the simplest graphical representation of 3D space. To a Flatlander, a square would be the simplest example of 2D space. A hypercube would serve a similar function to a Hyperspacer. To visualize what a hypercube looks like, we should first imagine how we could teach a Flatlander to visualize a cube. One way to do so is by unfolding a cube into its two-dimensional components, which would look like six squares arranged in a cross (figure 3.3). A Flatlander might not be able to envision how these squares can be folded together to form a cube, but at least he can conceptualize its structure by looking at the squares that make it up. In the same way, we can conceptualize the structure of hypercube by looking at a three-dimensional unraveled version of it. But even then, the shape of an unraveled hypercube looks like there is absolutely no way to fold it back up (figure 3.4).

 

Perhaps a better way to solve this problem is by examining a hypercube’s shadow. If we were to shine a flashlight through a wireframe cube so as to project its shadow on Flatland, a Flatlander would see a square within a square, with diagonal lines connecting the corners (figure 3.5). If we rotated the cube under the flashlight, the shadow would execute motions that appear to be impossible to a Flatlander. If a Hyperspacer shone a light through a hypercube, we would see its shadow as a cube within a cube, and if he were to rotate it, the motions would seem impossible to our three-dimensional brains (figure 3.6).

 

With the advent of superstring theory, the leading theory in physics today, the matter became more complicated, when up to ten space dimensions were suggested to make up our universe. While mathematics is the logical tool to use to study such complex landscapes, theorists have nonetheless attempted to explain what this space looks like in layman’s terms. A good example is the video game analogy. In certain video games, a spaceship is piloted across a square, flat TV screen, but it is not limited by its four sides. If you drive the spaceship through the left side, it appears on the right side and vice versa; in the same fashion, piloting it over the top edge makes it reappear on the bottom. In a way, this video game is contained within its own universe. If we were to analyze the topology of the universe as a three-dimensional shape, we would find that it is shaped like a doughnut. The top and bottom edge join together, turning the flat screen into a cylinder, and then the cylinder’s left and right side are joined to form a doughnut (figure 4.1). This technique is called orbifolding, and is extensively used by superstring theorists to explain the topology of the additional space dimensions.

 

Another example of an orbifold is a cone. If a Flatlander were to walk around a cone in the same way the spaceship traveled across the screen, he would eventually come to his starting position. If you were to unfold the cone into a flat shape and show the Flatlander the distance he walked, he’d be shocked to see that he arrived at his starting point by walking less than 360 degrees around its apex (figure 4.2). If we were to orbifold our own three dimensional world, it would take us less than 360 degrees to travel around a pole and reach our starting point. Superstring theory postulates that each of our additional dimensions are orbifolds of the previous dimension, but are all compressed so much we don’t notice we are traveling through them. If they were to be accidentally uncompressed for any reason, the universe would become an utterly confusing place to be, where all of the previous Newtonian rules cease to work and our 3D-based bodies and senses become useless.

 

Scientists have produced visual images of what this orbifolded multidimensional part of the universe “looks” like (figure 4.3). They are called Calabi-Yau shapes, and while they contain a high degree of inaccuracy due to the fact that its a two-dimensional image of a six-dimensional shape, they nonetheless present enough visual information to make the whole concept less alienating and easier to comprehend for non-mathematicians.

 

Our visual sensory system is extremely complex and elaborate, and provides us with a wealth of information to understand the world around us. Unlike other animals, like dogs, which rely heavily on their sense of smell, or bats, which navigate using their auditory system, the human being navigates and understands the universe by what he can see. Even though the current concept of the universe is extremely non-visual, we began exploring it through visual means, and to this day, visual analogies are a crucial component of explaining these “abstract” concepts. However, understanding the workings of the universe also shows us how limited our perception is, and how we can only perceive an infinitesimally small fraction of the world. Yet, this limited perception, combined with a brain capable of conceptualizing incredibly complex information, has been enough to help us decipher the mysteries of the universe, and to speak of non-visual matters as if they were in full sight. It is interesting to ponder if our concept of the cosmos would differ if vision wasn’t our primary sense. Could we question the vastness of the universe if we were unable to see the stars? Would time and motion still appear to be directly related if we couldn’t perceive the latter? The fact is, as visual beings, our mental image of the universe, however abstract and non-visual as it might be, still calls upon visual perception to help define itself.

 

 

 

 

 

 

Bibliography:

 

1) Arnheim, Rudolph Visual Thinking, California: University of California Press, 1989

 

2) Danielson, Dennis Richard The Book of the Cosmos: Imagining the Universe from Heraclitus to Hawking, Cambridge, Mass: Perseus Publishing, 2000.

 

3) Greene, Brian The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, New York: Vintage Books, 2000

 

4) Kaku, Michio Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension, New york: Anchor Books, 1994

 

5) Hawking, Stephen A Brief History of Time, New York: Bantam Books, 1988

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Mamerro, nice stuff bro...i didnt read it yet, i hate reading big stuff on 12oz, i'll print and read later..I'm pretty sure it'll be good. Here are some frames from my animation work i mentioned before, i wish i could upload the whole thing somewhere but i dont so enjoy:

http://www.freephoto-i.net/users/Hypercube/Frames.jpg'>

...I have a thing for geomerty:o

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Tesser and Mamerro:

 

Thank you very much for all that information. I found it to be fairly easy to understand the text, whereas most writing on Physics leaves me pretty confused.

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Guest imported_Tesseract

...END

 

http://lilt.ilstu.edu/bekurtz/dali/crucifixion.jpg'> The Expanding Uses of Spacial Dimensions in Art

 

Most aspects of art change over time. Mediums and technique change from artist to artist. Popular subjects change with each generation. Interpretations change every day depending on the whim of the viewers. One major and more slow changing aspect of art is the use of dimensions and the slow progression to the use of more spatial dimensions.

 

We live in a world with three spatial dimensions. This can be seen by looking at the top of any corner where the ceiling joins with the two walls. There are three and only three ways to go; up-down, left-right, and front-back. It is experienced any time you go to visit someone. Lets say your friend broke his leg and is in the hospital in our imaginary town. To get to your friend you must know three things to know where the hospital is. You must know what streets it is on and what floor you friend is on. If it is on the corner of 5th and Elm St. and he is on the third floor then 5th is you front-back, Elm St. is the left-right, and the third floor is the up-down dimension.

 

One aspect of having three dimensions is that we can see everything about a two dimensional world. There is nothing that they can hide from us 3-D beings. What is a sealed box to them is easily accessed from the third dimension. This apply to all dimensions. Anyone with (N) dimensions can see anything in a universe with (N-1) dimensions.

 

In Medieval art, flat two dimensional art was the norm. The Bayeux Tapestry is a well know example of this. All of the figures are flat and of about equal height. Only those of higher rank are taller. Even so, they are all flat and two dimensional figures. This supposedly was done to show that God was above all else and could see everything. He was the observer. The actual piece of art was meant to be like looking through the eyes of God. Because we are not God and cannot see as He does there is no difference between the foreground and the background. If a figure is pictured then it is going to be a more or less uniform height.

 

This changed with the coming of the Renaissance. New ideas abounded in this period, many of which questioned the foundations of the Church and God. Luther separated himself from the Church. Science destroyed the idea of an Earth centered universe. This moved humans form their special place in Gods work and separated us from Him. Art was affected in that artists abandoned the old two dimensional forms. Linear perspective was invented to give pictures the appearance of depth, the third dimension. This allowed parts of the picture, such as parts of people, building and even entire objects, to be hidden from view. This went directly in the face of the Medieval style which showed everything. Renaissance art did not deny that God was omnipotent, it merely shifted the point of view from that of God to that of an everyday man.

 

The painting "View of an Ideal City" by Piero della Francesca is supposed to show just that, the ideal city. It is a perfect example of linear perspective. Every thing gets smaller as it recedes from the viewer. There are two interesting things about this painting that show what is meant by the Renaissance being less centered on God. For one, the ideal city is apparently one here on Earth. It is not Heaven, nor is it the Second Jerusalem described in the Bible. It is a normal city, for normal men. It is ideal. There are also many parts of the city hidden from view. For most buildings only the face is seen. There is no hint at the content, which God would be able to see. It is most definitely the view of man.

 

With all three dimensions that we as human inhabit now covered, one would think that there is nothing left to discuss on the matter. How can an three dimensional artist use two dimensions to show four dimensions. It seems impossible. It is not, however impossible. Difficult, yes, but not impossible. Two people did it quite well, in fact.

 

Pablo Picassos works are not understood by most people. They seem strange at best, horrible and inartistic at the worst. His work seems like a hodge-podge of different views, barely forming a single picture. This is in fact the truth of the matter. Take his picture "Portrait de Nusch Eluard" for example. There are too many view points to count. Instead of having just one vanishing point there could easily be 20. For example, the ladys left eye seems to hint at a left profile as the angle of view. But he right eye suggests a frontal point of view. Even stranger are the four breasts he has given her. This gives way to at least two different views on this alone. Either each group of breasts, left pointing and right pointing, represent both breasts or each is really two views of the same breast. The Former requires two different view points, the latter requires four. Also curious is the table to her left. Not only can we see the front part of it but we can also see the back. This, again, requires more than one view point.

 

These affects can only be achieved using more views than any three dimensional observer can have. This requires the use of four dimensional viewing. Only someone who can see every aspect at once could see all of these views. Very little is hidden from the viewer of Picassos paintings, much like Bayeux Tapestry. (This is only a way to interpret the picture though. It is not the intent of the artist himself. Picasso stated that he never studied math or science and did not use it in his art. He remained adamant about this until his death so any similarities between his work and dimensional theories are interpretive.)

 

Dali used the fourth dimension in a more conventional way. He was a great fan of science, mainly physics. After the atomic bomb was dropped, Dali was quoted as saying, My father today is Dr. Heisenberg, the one of the most brilliant quantum theorists ever. Dali tried to use science in many of his paintings. His best success at using higher dimensions the painting Corpus Hypercubus.

 

The Crucifixion is one of the most popular themes in art. Some are quietly reverent, such as Raphael's "Christ on the Cross With the Virgin." Others portray the horror of being crucified, thus showing how much Christ really suffered for us, such as Grunewald's "Small Crucifixion." Dali's stands out as an original concept, though. The only background features are a tile floor and beyond that a dark, rocky landscape. You can just barely see the sun breaking though the clouds over these rocks in the distance, leaving the rest of the sky dark. The only figures present are Jesus and Mary. Mary is on a raised platform before the cross. Jesus is gigantic in comparison to his mother, almost twice as long in actual size, not counting the fact that Mary stands in the foreground.

 

The most noticeable thing about the painting is that Jesus is crucified on an unfolded hypercube, a tesseract. The tesseract cannot be a hypercube until it is in hyperspace. While in three-space, it remains unfolded. This is a direct analogy to Jesus himself. While on earth He was a man, yet was also God. He is the tesseract, God "unfolded" so that we could better understand Him. Dali, having studied physics, used one of its most mystical aspects, hyperspace, to give a newness to one of the most popular themes in art.

 

In conclusion, there has been a definite trend towards adding dimension to art. It started with the two dimensional Medieval art work and it slowly progressed to linear perspective. Form there artists have pushed the bounds of artistic ability and of scientific theory through the use of the forth dimension.

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Kast, Thats about it as seen from different angles...The first essey i posted made things real clear for me when i was first interested in that stuff...This whole book is written in an 'easy' language and if you're interested i highly suggest it...The novel i posted, which is responsible for my interest in the first place, is also highly recommended...its a thriller set in Mannila...really dope.

 

Mamerro, i read the text...cool beans bro! I'd really like to see the pictures to...Its funny that you mention shadow cause that was exactly what i used in the animation to 'prove' my point.

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Tesseract,

 

I'm definitely going to check out that book and the novel too.

Question: can you hook up the animation somewhere on the web, or maybe post it as an animated gif?

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Kast, I really hope i could...right now the animation is in avi format 800x600 and about 120MB, its about 2 minutes>1sec=25 frames> so the animated gif is unfortunately out of the question...It was in my plans to make a 'smaller' and lighter version of it for the net...so we'll see how that goes and if i find some free webspace somewhere i'll hook it...I'll keep you posted

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M-theory

Apart from the fact that instead of one there are five different, healthy theories of strings (three superstrings and two heterotic strings) there was another difficulty in studying these theories: we did not have tools to explore the theory over all possible values of the parameters in the theory. Each theory was like a large planet of which we only knew a small island somewhere on the planet. But over the last four years, techniques were developed to explore the theories more thoroughly, in other words, to travel around the seas in each of those planets and find new islands. And only then it was realized that those five string theories are actually islands on the same planet, not different ones! Thus there is an underlying theory of which all string theories are only different aspects. This was called M-theory. The M might stand for Mother of all theories or Mystery, because the planet we call M-theory is still largely unexplored.

 

There is still a third possibility for the M in M-theory. One of the islands that was found on the M-theory planet corresponds to a theory that lives not in 10 but in 11 dimensions. This seems to be telling us that M-theory should be viewed as an 11 dimensional theory that looks 10 dimensional at some points in its space of parameters. Such a theory could have as a fundamental object a Membrane, as opposed to a string. Like a drinking straw seen at a distance, the membranes would look like strings when we curl the 11th dimension into a small circle.

 

http://www.damtp.cam.ac.uk/user/gr/public/qg_ss.html

 

F-theory

Vafa recently added a strange twist to this when he introduced yet another mega-theory, this time a 12 dimensional theory called F-theory (F for "father") which explains the self-duality of the IIb string. (Unfortunately, this 12 dimensional theory is rather strange: it has two time co-ordinates, not one, and actually violates 12 dimensional relativity. Imagine trying to live in a world with two times! It would put an episode of Twilight Zone to shame.)

So is the final theory 10, 11, or 12 dimensional?

 

Schwarz, for one, feels that the final version of M-theory may not even have any fixed dimension. He feels that the true theory may be independent of any dimensionality of space-time, and that 11 dimensions only emerges once one tries to solve it. Townsend seems to agree, saying "the whole notion of dimensionality is an approximate one that only emerges in some semiclassical context."

 

http://www.mkaku.org/m_theory.htm

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I'm sorry, rotating that 4th dimensional object made me lose some brain cells just now....I feel all stupid and confused. Shit like I saw a flying saucer.

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