Fundamental Finite Unit Of The Fourth Dimension example essay topic

Imagine going to a magic show, where the world's top ranked magicians gather to dazzle their wide-eyed crowd. Some would walk through jet turbines, others would decapitate their assistants only to fuse them back together, and others would transform pearls into tigers. However, with each of these seemingly impossible stunts, there is always a catch. A curtain will fall momentarily; a door will shut; the lights will go out; a large cloud of smoke will fill the room, or a screen will hide what is truly going on. Then, a very different magician comes on, and performs stunts like entering a closed box without opening any doors, and placing a mouse in a sealed bottle without removing the cork. These do not seem very extravagant compared to the amazing feats other magicians pull off, but what leaves the crowd completely baffled is the fact that he does these tricks without placing a handkerchief over his hand, or doing it so fast the crowd misses what is going on.

To perform the mouse-in-the-bottle trick, he shows the mouse in his hand, slowly twists it in a strange manner, and right before your eyes, his hand completely disappears! A few instants later his hand reappears inside the bottle, holding the mouse. There seem to be two parts of his arm; one in the bottle, and one out. His arm looks severed, yet he has complete control of his fingers inside the bottle.

The hand lets go of the mouse, and again vanishes from inside the bottle, and reconstitutes itself on the magicians arm. He pulled it off candidly, without the smoke and mirrors. Everything that was seen actually happened. This magician, breaking the tradition of fooling the audience with illusions, used cutting edge knowledge of higher-dimensional science to perform this marvel.

He sent his arm outside of 3-D space, twisted it in the fourth dimension, and placed it back into the bottle. The fourth dimension is not time, but an extra direction, just like left, right, up, down, forward, and backwards. This magician has used the fourth dimension for entertainment purposes. However, the fourth dimension has other, more practical uses and applications in the realm of mathematics, geometry, as well as astrophysics, and holds the explanation to such natural phenomena as gravity and electromagnetism.

To this day, many scientists and other people accept time as being the fourth dimension. This notion is completely absurd. Time does play an important role in the description of an object, but it is incorrect to perceive it as a dimension. Mass, volume, color, state, and frequency are all components used to describe an object, be it matter, wave or energy, but they are not dimensions.

The three spatial dimensions known to us are used to describe where an object is in 3-D space, while mass, volume, color, etc., describe how it is. Describing when it is would be done using time, and saying time is a dimension would be like saying that mass is a dimension, which is incorrect. Dimensions are reserved to tell where an object is, and all other components of its description are entirely separate. Time has been confused as being the fourth dimension for several reasons. It seems to have first been referred to as such in H.G. Wells' The Time Machine, which came out in the late 19th Century.

Equivalents to the 2-D ordered pair (x, y) have been used to describe a point either in 2-D space (x, y, t), or in 3-D space (x, y, z, t). Strange inconsistency is that the 1st, 2nd, and 3rd dimensions all need the dimension below them, while time does not: a 3-D (3 axes) world cannot exist without first having a 2-Plane (2 axes), and a 2-D plane cannot exist without first having a 1-D (1 axis) line; but a point on a 1-D line can exist in time, which would make time 2-D. In this situation, time is the second dimension, the t-axis. If it is well accepted that time is the fourth dimension, the t-axis, how is it that in this situation time is the second dimension, which is well confirmed as being the y-axis? How can time simultaneously be the t-axis and the y-axis? It can't. They are two separate aspects of the object and cannot be the same.

Time is avery important factor of an object's description, but it cannot be considered a dimension. If time is not a dimension, and more specifically, not the fourth dimension, then what is? Understanding the fourth dimension to its full extent is beyond the power of the human mind, but we can infer what the fourth dimension might be by drawing connections between the three dimensions we are familiar with. When jumping from one dimension to the next, we add an extra axis, or two new directions. Let's examine the first dimension, consisting of the x-axis. It has two directions: left and right.

The basic infinite unit for the first dimension is a line, its basic finite unit is a segment. When jumping to the second dimension, we add another axis (y), thereby adding two new directions: up and down. The basic infinite unit for the second dimension is a plane, its basic finite unit is a square. Moving on to the third dimension, we add one more axis (z), creating two more directions: forward and backward. The basic infinite unit for the third dimension is space, and its basic finite unit is a cube. So far, the elements discussed have been easy for the human mind to understand, since the standard of the universe is in three dimensions, and concepts less than or equal to human capabilities can easily be understood; however, it is difficult to deal with anything greater.

As can be noticed, there are very distinct patterns and steps that are constant when increasing the dimensional value: basically it is adding an axis that is mutually perpendicular to all previous axes. By adding a z-axis, all three lines join together at a single point, all forming right angles to each other. With this template, describing the fourth dimension becomes easier. When progressing to the fourth dimension, one more axis would be added (call it "w"); this will create two new directions (call these w+ and w-), which are impossible for a 3-D mind to visualize. The basic infinite unit of the fourth dimension is hyperspace (4-D space), and it's basic finite unit is a hypercube (a 4-D cube).

In hyperspace, it is possible to have four axes joining at a single point, all forming right angles to each other. This seems absolutely incredulous; four axes can never meet perpendicularly! This is a 3-D mind speaking again. Two perpendicular axes are impossible obtain on a line, and three perpendicular axes are impossible to obtain on a plane.

Four perpendicular axes are impossible to obtain in 3-D space, which is why it can't be visualized; but it is easily obtained in four-dimensional hyperspace. Hyperspace seems extremely theoretical, without many solid facts with which to back it up. But it is surprising how many factors and phenomena lean towards the fourth dimension for an explanation. Mathematically, geometrically, and physically, hyperspace mysteriously connects into a radiant harmony of completeness. Geometrically, hyperspace makes sense; it all fits together. Going back to the basic finite unite of the fourth dimension, the hypercube, let's draw some connections with the lower dimensions.

To better understand the following paragraph, refer to appendix For a visualization of these concepts. Moving even before the first dimension, let's examine the zeroth: A point. It has no directions, meaning it has no infinite unit, just a finite one: the point. To convert a point into a segment, (1-D finite unit) you would duplicate the point (0-D unit) and project it into the added x-axis. Then, connect the vertices; you get a segment, a 1-D finite unit. To convert a segment into a square, (2-Dfinite unit) you would duplicate the segment (1-D unit) and project it into the added-axis.

Then, connect the 4 vertices; you get a square, a 2-D finite unit, composed of four segments all sharing common vertices (points) with their 2 perpendicular segments. To convert a square into a cube, (3-D finite unit) you would duplicate the square (2-D unit) and project it into the added z-axis. Then, connect the 8 vertices; you get a cube, a 3-Dfinite unit composed of six squares all sharing common edges (segments) with their 4 perpendicular squares. Making the jump to the hypercube is no different.

To convert a cube into a hypercube, (4-D finite unit) you would duplicate the cube (3-D unit) and project it into the added w-axis. Then, connect the 16 vertices; you get a hypercube, a 4-D finite unit composed of eight cubes all sharing common faces (squares) with their 6 perpendicular squares (Newbold). This boggles the mind. No 3-D human could ever see a hypercube, because a hypercube cannot exist in a 3-D world just as a cube cannot exist on a 2-D plane; a plane is missing two directions necessary to allow the cube to exist. Our 3-D world is missing two directions necessary to allow a hypercube to exist.

Another way to attempt to visualize the hypercube is by using tessera cts. Figure 1 in the diagram depicts six two-dimensional squares, arranged in a cross-shaped alignment. The two outer squares can be folded up via the third dimension; next, the other squares can also fold up, forming the fundamental finite unit of the third dimension: the cube. Similarly, Figure 2 depicts the three-dimensional version of the cross, the tessera ct, which consists of eight cubes forming a cross-like object. Just like the cross was an unfolded cube, the tessera ct is an unfolded hypercube. The two outer cubes can be folded up via the fourth dimension; next, the other cubes also fold up, forming the fundamental finite unit of the fourth dimension: the hypercube (Kaku 71).

This is of course impossible to visualize, even imagine, with a three dimension mind. Imagine a two-dimensional person living on a plane. He could see the six squares that form the cross, but he could never even fathom having the squares fold up into a dimension greater than his own. It is impossible for him to even imagine it. Visualizing this fold-up is very easy for us, with 3-D minds. However, visualizing a tessera ct folding up into a hypercube defies human comprehension.

The hypercube is probably the most easy four-dimensional concept to understand. Yet it is not alone in 4-D geometry. In fact, discovering the fourth dimension opens up possibilities for scores of new shapes and forms, that were never possible on a plane or in space (Koch). The circle, triangle, and square are very familiar to us. They form nice, simple equations when expressed mathematically, and are the basis of many natural objects in today's world.

On a two-dimensional plane, a square and a circle must always be separate. A merger of the two is impossible. Looking a step higher, through three-dimensional eyes, combining a square and a circle is simple: the result is a three dimensional cylinder. Thus we see that different two dimensional objects can combine in the third dimension to create a unified shape. Other examples of merging shapes are: a circle and a triangle form a cone, a triangle and a square form a pyramid, inversely, the square and the triangle form a prism, the triangle and the circle form a three-cornered dome, and the square and the circle form a four-cornered dome. From these examples several conclusions can be drawn.

Every two-dimensional shape needs two axes to exist. By merging these shapes, one of them occupies the x-axis alone, one occupies the y-axis alone, but they share positions on the z-axis. If this is true, then three two-dimensional shapes can merge in the fourth dimension, or one 3-D object and one 2-D object can. For example, a 3-D sphere and a 2-D triangle can merge in the fourth dimension, making it. It is simultaneously a sphere and a triangle, just as a cone is simultaneously a circle and a triangle. Another aspect of the fourth dimension is found in geometry's roots: mathematics.

Using exponents, we can raise the dimensional value of a number. Take the number 3, for example. The number 3, like any other number, is one-dimensional. It be made two-dimensional by squaring it; 32 = 9. Thus we see that 9 is the one-dimensional value for two-dimensional 3. A one-dimensional value can not only be squared (raised to the second power), but it could just as easily be cubed.

33 = 27. From this we infer that 27 is the one-dimensional value of three-dimensional 3. Any number can also be raised to the fourth power; it would make just as much sense to call it "hyper cubing" a number, just as raising to the second or third powers is "squaring" or "cubing". In math, multidimensional reasoning is very easy and simple, since it doesn't require visualization. However, every mathematical equation can be expressed visually using a graph. Most commonly, a two-dimensional graph is used to express equations that include two variables, and x and a y.

This draws a line onthe graph, on which every points x and y value can be inserted into the equation, and have both sides of the equation balance out. For equations dealing with three variables, a three-dimensional graph can be used to visualize it, using x, y, and z coordinates. Using this model, an equation sporting four variables can easily be obtained (Guarino). It would only make sense to be able to visually express this equation using a four-dimensional graph. But this leads to a great problem. This is a three-dimensional world, and it lacks the two directions necessary to allow the fourth axis to exist.

Fortunately, there is a way to represent the fourth dimension using just three. This is done by "faking" the fourth dimension using what is available in three dimensions. To explain this, let's have a look at the dimensions that we can understand. just as a hypercube cannot exist in space, a cube cannot exist on a flat, two-dimensional surface. However, using an artist's trick called perspective, the third dimension can " faked" on a flat piece of paper. Note the cube in figure 3.

It appears very normal to us, as we are used to seeing three-dimensional objects shown on two-dimensional medium. In analyzing its structure, we note that a cube is composed of six squares. However, there are not six squares on figure 3's cube. There are only two: square A BDC and squareEFHG (see fig 4 A).

The other four shapes that comprise this cube are actually parallelograms that are representing full squares skewed through three-dimensional perspective (see fig 4 B). In 3-space, angle EAB is 90 o, however, in two-space, on this flat representation, angle EAB is about 135 o. Therefore, if a three-dimensional object can be represented by faking in the second dimension, it would only be right that a four-dimensional object could be faked in our 3-D world. This is done by first having three lines joining at point all forming right angles to each other, then adding another line going through that point. It wouldn't really matter at what angle, either way it would be right, or rather, wrong, since it is only faking an extra axis (see appendix B for a look at faking the fourth dimension). With this, four-variable equations could be graphed on a rotating four-dimensional graph emitting the same qualities as a two or three-dimensional graph.

All points on the graph would be expressed in terms of (x, y, z, w), meaning every point has a four-dimensional value. One might think about the fourth dimension, agree it is a good theoretical idea, and acknowledge its practical use in math and geometry, but might wonder whether it exists in the real world. Hyperspace makes sense in math, the numbers match up, so where is this extra axis? Can we walk through it?

Can we travel in hyperspace? How? Is it just a pointless theory? Surprisingly, the four natural forces in the universe: gravity, electromagnetism, and the nuclear forces "strong" and "weak" can only be explained through the idea of hyperspace. At a recent lecture, Kip Thorne, physics professor at Cal Tech and renowned physical theorist, explained the nature of black holes. To give a visual idea, he held in his hands a black rubber ball, a sphere.

He announced that the circumference of the sphere was about 30 cm. From this, you would expect that the radius of the sphere would be 30/p or about 10 cm. He continued to explain that it is not 10 cm, but that it was many miles long. This seems impossible!

To explain this, he made his audience imagine they were blind ants living on the surface of a trampoline. By counting their steps, the ants walk around the trampoline and determine that the circumference is about 20 meters. Unknown to them, there is an extremely heavy rock lying in the center of the trampoline, causing its surface to bend down to a great degree. Because of this, when the ants attempt to discover the trampoline's radius, the are surprised to find out that it is not 20/p meters, but much more (see fig 5).

In this situation we see that a two-dimensional circle can have a radius more than diameter divided by p if and only if the circle is warped, making occupy multiple coordinated on an extra axis, just like the curved trampoline's center had a greater z-axis value than its outer edge (Thorne Lecture). It was a 2-D circle occupying 3-D space. If the ball that Thorne was holding had a radius more than its diameter divided by p, then that 3-D sphere must be occupying multiple coordinates on an extra axis: the fourth dimension. The center of the sphere would have a greater four-dimensional value that its surface.

This would mean that a black hole is simultaneously a sphere and funnel shaped object, which will be simplified into a triangle; and, just as a cone is a circle and a triangle, a black hole is a four-dimensional hyper cone. No longer is this fuzzy numbers and twisted math; it is an actual documented phenomenon that can only be explained through the introduction of a new, four-dimensional axis. This phenomenon of curving space is called space-time warpage. Einstein said that space-time was warped by the presence of matter (Rothman 217). The density of the matter would determine the degree of subsequent warpage. This means that large amounts of mass like planets and stars warp space more so than a lost electron randomly drifting through space.

Back to the example of the trampoline, all objects on its surface would have a tendency to slide toward the center, where the rock is. If a marble is on the trampoline, it is making a slight dent in on the surface, but it is so small it is practically negligible. It will naturally flow towards the rock, since the rock is creating the greater warpage. In this instance, the attraction between the two objects is two-dimensional. Objects on the surface would slide toward the rock, however, an object underneath it or hanging above it would feel no force attracting it to the rock. On a planet, however, the attraction is three-dimensional, meaning any object in 3-D space is attracted to the planet, because of it's four-dimensional warpage.

This proves that the only way gravity can be explained is with the fourth dimension. Einstein also stated that the greater gravity is in a field of reference, the slower time will run (Encarta General... ). As previously stated, large amounts of dense mass have greater gravitational pull, meaning the four-dimensional warpage is proportional to the object's gravity and mass (Gribbin 41). If this is true, than the speed of time in a given gravitational reference is equal to the slope of space-time's warpage (see figure 6), which in turn can be measured by the specific object's density. This raises two perplexing questions: What happens when the slope is vertical?

What happens when it is horizontal? Einstein explained that time cannot exist without matter, and vice versa. If matter can be expressed in amount of space-time warpage, the absence of matter would equate no warpage, meaning no time. Time would completely stop when warpage's slope was zero. Strangely, when the most minute amount of matter is placed in space, and warpage's slope is infinitely close to zero, time would be running at maximum speed! As more mass is added, warpage would increase, time would slow down, and come almost completely to a stop, then, when warpage reaches no slope, or a vertical line, time would either run at an infinitely fast rate, or it would cease to exist entirely.

This eerie paradox is one of the unsolved components of the four-dimensional explanation, along with one other: with the trampoline example, the component that made the marble attracted to the rock was a) the slope of the curvature and b) the force of gravity pulling it down. If space-time is warped via the fourth dimension with the presence of mass, where is the four-dimensional force that is actually causing the attraction? The warpage is merely funneling the direction of the bond, but the original source of the force is yet to be discovered. Along with gravity, other forces can be explained.

When it comes to waves, we have many examples to with which to relate. Waves create ripples in water, and compress and decompress air molecules, creating sound. Almost all waves we know about need matter to exist. A water wave cannot exist without water, and sound cannot exist without air.

But strangely, waves on the electromagnetic spectrum (including light, radio waves, and X-rays) can travel through a vacuum: the absence of matter. This is breaks all known laws! No other wave can exist in a vacuum, but somehow, electromagnetism can! There have been several theories to explain this, such as the suggestion of aether, "which fill [s] the vacuum and act [s] as a medium for light" (Kaku 8).

This gives a shady explanation of how light, proposed to be simultaneously a wave and a particle, can vibrate its own matter, allowing it to travel through empty space. This theory, however, had many gaps and paradoxes, and eventually was proven wrong in laboratories. In the early twenties, theKaluza-Klein theory was born, suggesting that electromagnetic waves were actually vibrations in 3-D space itself (Kaku 8). This defies imagination, as this is only possible through the acceptance of the fourth spatial dimension. Just like the two-dimensional surface of water can ripple, causing it to occupy multiple coordinates in three-space, three-dimensional space can ripple, causing it to occupy multiple coordinates in four-space.

Another strange possibility opened with fourth dimension is the existence of parallel universes. Using the third dimension, several two-dimensional planes can co-exist in a parallel manner. Similarly, there could be multiple universes (3-D spaces) co-existing in four-dimensional hyperspace. This of course is extremely theoretical, and could never be proven. It can only be explained through thought experiments. Imagine an occurrence of extreme space-time warpage happening in two parallel universes at identical XYZ coordinates.

They could possibly merge, creating a tunnel, or wormhole connecting parallel universes via the fourth dimension (see fig. 7 B). If multiple universes do not exist, or a trans-universal wormhole is impossible to obtain, there is still the possibility of a universe connecting with itself (see fig. 7 A). Science fiction writers have often romanced with the idea of shortcuts through space. The fourth dimension turns these dreams into reality.

It is impossible to exceed the speed of light, but it is possible to travel one lightyear in less than one year (Encarta Special... ). How? By traveling through a worm hole that takes a shortcut through the fourth dimension. With this information, keep your minds open about things that perhaps you cannot fully understand. Furthering the research of higher dimensional science will surely amount to many practical uses in our lives.

Speaking of its uses, just how did that magician pull off the mouse-in-the-bottle trick? It's quite simple actually. In a two dimensional world, an object can be placed an removed into and from a closed area by lifting it across the third dimension (see fig. 8). Using this same concept, except one dimension higher, three dimensional objects can be placed and removed into and from closed spaces by lifting it across the fourth dimension. So how did the magician twist his arm and make it penetrate the fourth dimension? Well, a good magician never tells his secret.

Bibliography

Kaku, Michi o. Hyperspace. New York, New York: Oxford University Press, 1994.
Thorne, Kip. Black Holes and Time Warps: Einstein's Outrageous Legacy. New York, New York: W.W. Norton & Company, Inc, 1994.
Thorne, Kip. "Black Holes and Time Warps". Lecture. University of Utah, Utah, February 26, 2001.
Reichenbach, Hans. From Copernicus to Einstein. New York, New York: Dover Publications, Inc., 1970.
Gribbin, Mary and John. Time and Space (Eyewitness Books). New York, New York: Dorling Kindersley Limited, 2000.
Newbold, Mark. "Stereoscopic Animated Hypercube". [Online] Available web April 2, 2001.
Koch, Richard, Department of Mathematics, University of Oregon. "Java Examples of 3-D and 4-D Objects". Guarino, Michael, Physicist, Bachelor in Physics, Teacher. Personal Interview. March 30th, 2001.
Rothman, Tony, Ph. D. Instant Physics, From Aristotle to Einstein, And Beyond. Ney York, New York, Byron Press Visual Publications, Inc, Ballantine Books, a division of Random House, Inc. 1995.