Imagine going to a magic show, where the worlds top ranked magicians gather todazzle their wide-eyed crowd. Some would walk through jet turbines, others woulddecapitate their assistants only to fuse them back together, and others would transformpearls into tigers. However, with each of these seemingly impossible stunts, there isalways a catch.
A curtain will fall momentarily; a door will shut; the lights will go out; alarge 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 withoutopening 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 pulloff, but what leaves the crowd completely baffled is the fact that he does these trickswithout placing a handkerchief over his hand, or doing it so fast the crowd misses what isgoing 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 completelydisappears! 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.Order now
His arm lookssevered, yet he has complete control of his fingers inside the bottle. The hand lets go ofthe mouse, and again vanishes from inside the bottle, and reconstitutes itself on themagicians arm. He pulled it off candidly, without the smoke and mirrors. Everything thatwas seen actually happened. This magician, breaking the tradition of fooling the audiencewith illusions, used cutting edge knowledge of higher-dimensional science to perform thismarvel.
He sent his arm outside of 3-D space, twisted it in the fourth dimension, andplaced it back into the bottle. The fourth dimension is not time, but an extra direction, justlike left, right, up, down, forward, and backwards. This magician has used the fourthdimension for entertainment purposes. However, the fourth dimension has other, morepractical uses and applications in the realm of mathematics, geometry, as well asastrophysics, and holds the explanation to such natural phenomena as gravity andelectromagnetism. To this day, many scientists and other people accept time as being the fourthdimension.
This notion is completely absurd. Time does play an important role in thedescription 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 usare 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 adimension would be like saying that mass is a dimension, which is incorrect.
Dimensionsare reserved to tell where an object is, and all other components of its description areentirely separate. Time has been confused as being the fourth dimension for severalreasons. 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) havebeen used to describe a point either in 2-D space (x,y,t), or in 3-D space (x,y,z,t). Astrange inconsistency is that the 1st, 2nd, and 3rd dimensions all need the dimension belowthem, while time does not: a 3-D (3 axes) world cannot exist without first having a 2-Dplane (2 axes), and a 2-D plane cannot exist without first having a 1-D (1 axis) line; but apoint on a 1-D line can exist in time, which would make time 2-D. In this situation, time isthe 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 wellconfirmed as being the y-axis? How can time simultaneously be the t-axis and the y-axis? It cant. They are two separate aspects of the object and cannot be the same. Time is avery important factor of an objects description, but it cannot be considered a dimension.
If time is not a dimension, and more specifically, not the fourth dimension, thenwhat is? Understanding the fourth dimension to its full extent is beyond the power of thehuman mind, but we can infer what the fourth dimension might be by drawing connectionsbetween the three dimensions we are familiar with. When jumping from one dimension tothe next, we add an extra axis, or two new directions. Lets examine the first dimension,consisting of the x-axis. It has two directions: left and right. The basic infinite unit for thefirst dimension is a line, its basic finite unit is a segment. When jumping to the seconddimension, 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 moredirections: 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 thehuman mind to understand, since the standard of the universe is in three dimensions, andconcepts less than or equal to human capabilities can easily be understood; however, it isdifficult to deal with anything greater. As can be noticed, there are very distinct patternsand steps that are constant when increasing the dimensional value: basically it is adding anaxis that is mutually perpendicular to all previous axes.
By adding a z-axis, all three linesjoin 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 fourthdimension, 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 infiniteunit of the fourth dimension is hyperspace (4-D space), and its basic finite unit is ahypercube (a 4-D cube). In hyperspace, it is possible to have four axes joining at a singlepoint, all forming right angles to each other.
This seems absolutely incredulous; four axescan never meet perpendicularly! This is a 3-D mind speaking again. Two perpendicularaxes are impossible obtain on a line, and three perpendicular axes are impossible to obtainon a plane. Four perpendicular axes are impossible to obtain in 3-D space, which is why itcant be visualized; but it is easily obtained in four-dimensional hyperspace. Hyperspace seems extremely theoretical, without many solid facts with which toback it up.
But it is surprising how many factors and phenomena lean towards the fourthdimension for an explanation. Mathematically, geometrically, and physically, hyperspacemysteriously connects into a radiant harmony of completeness. Geometrically, hyperspace makes sense; it all fits together. Going back to thebasic finite unite of the fourth dimension, the hypercube, lets draw some connections withthe lower dimensions. To better understand the following paragraph, refer to appendix Afor a visualization of these concepts. Moving even before the first dimension, letsexamine the zeroth: A point.
It has no directions, meaning it has no infinite unit, just afinite one: the point. To convert a point into a segment, (1-D finite unit) you wouldduplicate the point (0-D unit) and project it into the added x-axis. Then, connect thevertices; 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 addedy-axis.
Then, connect the 4 vertices; you get a square, a 2-D finite unit, composed of foursegments all sharing common vertices (points) with their 2 perpendicular segments. Toconvert 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 4perpendicular squares. Making the jump to the hypercube is no different. To convert acube into a hypercube, (4-D finite unit) you would duplicate the cube (3-D unit) andproject it into the added w-axis.
Then, connect the 16 vertices; you get a hypercube, a4-D finite unit composed of eight cubes all sharing common faces (squares) with their 6perpendicular squares (Newbold). This boggles the mind. No 3-D human could ever seea hypercube, because a hypercube cannot exist in a 3-D world just as a cube cannot existon a 2-D plane; a plane is missing two directions necessary to allow the cube to exist. Our3-D world is missing two directions necessary to allow a hypercube to exist.
Another way to attempt to visualize the hypercube is by using tesseracts. Figure 1in 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 squarescan 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 tesseract, whichconsists of eight cubes forming a cross-like object. Just like the cross was an unfoldedcube, the tesseract is an unfolded hypercube. The two outer cubes can be folded up viathe fourth dimension; next, the other cubes also fold up, forming the fundamental finiteunit of the fourth dimension: the hypercube (Kaku 71).
This is of course impossible tovisualize, even imagine, with a three dimension mind. Imagine a two-dimensional personliving on a plane. He could see the six squares that form the cross, but he could nevereven fathom having the squares fold up into a dimension greater than his own. It isimpossible for him to even imagine it. Visualizing this fold-up is very easy for us, with3-D minds. However, visualizing a tesseract folding up into a hypercube defies humancomprehension.
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 uppossibilities for scores of new shapes and forms, that were never possible on a plane or inspace (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 objectsin todays world. On a two-dimensional plane, a square and a circle must always beseparate.
A merger of the two is impossible. Looking a step higher, throughthree-dimensional eyes, combining a square and a circle is simple: the result is a threedimensional cylinder. Thus we see that different two dimensional objects can combine inthe third dimension to create a unified shape. Other examples of merging shapes are: acircle and a triangle form a cone, a triangle and a square form a pyramid, inversely, thesquare and the triangle form a prism, the triangle and the circle form a three-cornereddome, and the square and the circle form a four-cornered dome. From these examplesseveral 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-axisalone, but they share positions on the z-axis. If this is true, then three two-dimensionalshapes can merge in the fourth dimension, or one 3-D object and one 2-D object can. Forexample, a 3-D sphere and a 2-D triangle can merge in the fourth dimension, making it ahypercone. It is simultaneously a sphere and a triangle, just as a cone is simultaneously acircle and a triangle. Another aspect of the fourth dimension is found in geometrys roots: mathematics.
Using exponents, we can raise the dimensional value of a number. Take the number 3, forexample. The number 3, like any other number, is one-dimensional. It be madetwo-dimensional by squaring it; 32 = 9. Thus we see that 9 is the one-dimensional valuefor two-dimensional 3. A one-dimensional value can not only be squared (raised to thesecond power), but it could just as easily be cubed.
33 = 27. From this we infer that 27 isthe one-dimensional value of three-dimensional 3. Any number can also be raised to thefourth power; it would make just as much sense to call it hypercubing a number, just asraising to the second or third powers is squaring or cubing. In math, multidimensionalreasoning is very easy and simple, since it doesnt 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 twovariables, and x and a y.
This draws a line on the graph, on which every points x and yvalue 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 tovisualize it, using x, y, and z coordinates. Using this model, an equation sporting fourvariables can easily be obtained (Guarino). It would only make sense to be able to visuallyexpress 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 thefourth axis to exist. Fortunately, there is a way to represent the fourth dimension usingjust three. This is done by faking the fourth dimension using what is available in threedimensions. To explain this, lets 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-dimensionalsurface.
However, using an artists trick called perspective, the third dimension canfaked 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. Inanalyzing its structure, we note that a cube is composed of six squares. However, thereare not six squares on figure 3s cube. There are only two: square ABDC and squareEFHG (see fig 4 A).
The other four shapes that comprise this cube are actuallyparallelograms that are representing full squares skewed through three-dimensionalperspective (see fig 4 B). In 3-space, angle EAB is 90o , however, in two-space, on thisflat representation, angle EAB is about 135o. Therefore, if a three-dimensional object canbe represented by faking in the second dimension, it would only be right that afour-dimensional object could be faked in our 3-D world. This is done by first havingthree lines joining at point all forming right angles to each other, then adding another linegoing through that point. It wouldnt really matter at what angle, either way it would beright, or rather, wrong, since it is only faking an extra axis (see appendix B for a look atfaking the fourth dimension). With this, four-variable equations could be graphed on arotating four-dimensional graph emitting the same qualities as a two or three-dimensionalgraph.
All points on the graph would be expressed in terms of (x, y, z, w), meaning everypoint has a four-dimensional value. One might think about the fourth dimension, agree it is a good theoretical idea, andacknowledge its practical use in math and geometry, but might wonder whether it exists inthe real world. Hyperspace makes sense in math, the numbers match up, so where is thisextra axis? Can we walk through it? Can we travel in hyperspace? How? Is it just apointless theory? Surprisingly, the four natural forces in the universe: gravity,electromagnetism, and the nuclear forces strong and weak can only be explainedthrough the idea of hyperspace. At a recent lecture, Kip Thorne, physics professor at Cal Tech and renownedphysical theorist, explained the nature of black holes.
To give a visual idea, he held in hishands a black rubber ball, a sphere. He announced that the circumference of the spherewas about 30 cm. From this, you would expect that the radius of the sphere would be30/p or about 10 cm. He continued to explain that it is not 10 cm, but that it was manymiles long. This seems impossible! To explain this, he made his audience imagine theywere blind ants living on the surface of a trampoline.
By counting their steps, the antswalk 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 antsattempt to discover the trampolines radius, the are surprised to find out that it is not 20/pmeters, but much more (see fig 5). In this situation we see that a two-dimensional circlecan 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 trampolinescenter had a greater z-axis value than its outer edge (Thorne Lecture).
It was a 2-D circleoccupying 3-D space. If the ball that Thorne was holding had a radius more than itsdiameter divided by p, then that 3-D sphere must be occupying multiple coordinates on anextra axis: the fourth dimension. The center of the sphere would have a greaterfour-dimensional value that its surface. This would mean that a black hole issimultaneously 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 hypercone. No longer is this fuzzy numbers and twisted math; it is an actual documented phenomenonthat can only be explained through the introduction of a new, four-dimensional axis. Thisphenomenon of curving space is called space-time warpage.
Einstein said that space-timewas warped by the presence of matter (Rothman 217). The density of the matter woulddetermine the degree of subsequent warpage. This means that large amounts of mass likeplanets 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 toslide toward the center, where the rock is. If a marble is on the trampoline, it is making aslight dent in on the surface, but it is so small it is practically negligible. It will naturallyflow towards the rock, since the rock is creating the greater warpage.
In this instance, theattraction between the two objects is two-dimensional. Objects on the surface would slidetoward the rock, however, an object underneath it or hanging above it would feel no forceattracting 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 its four-dimensionalwarpage. This proves that the only way gravity can be explained is with the fourthdimension. Einstein also stated that the greater gravity is in a field of reference, the slowertime will run (Encarta General. .
. ). As previously stated, large amounts of dense mass havea greater gravitational pull, meaning the four-dimensional warpage is proportional to theobjects gravity and mass (Gribbin 41). If this is true, than the speed of time in a givengravitational reference is equal to the slope of space-times warpage (see figure 6), whichin turn can be measured by the specific objects density.
This raises two perplexingquestions: 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 beexpressed in amount of space-time warpage, the absence of matter would equate nowarpage, meaning no time. Time would completely stop when warpages slope was zero. Strangely, when the most minute amount of matter is placed in space, and warpages slopeis infinitely close to zero, time would be running at maximum speed! As more mass isadded, warpage would increase, time would slow down, and come almost completely to astop, then, when warpage reaches no slope, or a vertical line, time would either run at aninfinitely fast rate, or it would cease to exist entirely. This eerie paradox is one of theunsolved components of the four-dimensional explanation, along with one other: with thetrampoline example, the component that made the marble attracted to the rock was a) theslope of the curvature and b) the force of gravity pulling it down. If space-time is warpedvia the fourth dimension with the presence of mass, where is the four-dimensional forcethat is actually causing the attraction? The warpage is merely funneling the direction ofthe 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, wehave many examples to with which to relate. Waves create ripples in water, and compressand decompress air molecules, creating sound. Almost all waves we know about needmatter to exist. A water wave cannot exist without water, and sound cannot exist withoutair.
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 knownlaws! No other wave can exist in a vacuum, but somehow, electromagnetism can! Therehave been several theories to explain this, such as the suggestion of aether, which fillsthe vacuum and acts as a medium for light (Kaku 8). This gives a shady explanation ofhow 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 andparadoxes, and eventually was proven wrong in laboratories.
In the early twenties, theKaluza-Klein theory was born, suggesting that electromagnetic waves were actuallyvibrations in 3-D space itself (Kaku 8). This defies imagination, as this is only possiblethrough the acceptance of the fourth spatial dimension. Just like the two-dimensionalsurface of water can ripple, causing it to occupy multiple coordinates in three-space,three-dimensional space can ripple, causing it to occupy multiple coordinates infour-space. Another strange possibility opened with fourth dimension is the existence ofparallel universes. Using the third dimension, several two-dimensional planes can co-existin a parallel manner. Similarly, there could be multiple universes (3-D spaces) co-existingin four-dimensional hyperspace.
This of course is extremely theoretical, and could neverbe proven. It can only be explained through thought experiments. Imagine an occurrenceof extreme space-time warpage happening in two parallel universes at identical XYZcoordinates. They could possibly merge, creating a tunnel, or wormhole connectingparallel universes via the fourth dimension (see fig. 7 B).
If multiple universes do notexist, or a trans-universal wormhole is impossible to obtain, there is still the possibility of auniverse connecting with itself (see fig. 7 A). Science fiction writers have often romancedwith the idea of shortcuts through space. The fourth dimension turns these dreams intoreality.
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 holethat takes a shortcut through the fourth dimension. With this information, keep your minds open about things that perhaps you cannotfully understand. Furthering the research of higher dimensional science will surely amountto many practical uses in our lives. Speaking of its uses, just how did that magician pulloff the mouse-in-the-bottle trick? Its 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 thethird dimension (see fig.
8). Using this same concept, except one dimension higher, threedimensional objects can be placed and removed into and from closed spaces by lifting itacross the fourth dimension. So how did the magician twist his arm and make it penetratethe fourth dimension? Well, a good magician never tells his secret. Works CitedKaku, Michio.
Hyperspace. New York, New York: Oxford University Press, 1994. Thorne, Kip. Black Holes and Time Warps: Einsteins 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: DoverPublications, 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 Availablehttp://www.
dogfeathers. com/java/hyprcube. html, April 2, 2001. Koch, Richard, Department of Mathematics, University of Oregon.
Java Examples of3-D and 4-D Objects. Online Availablehttp://darkwing. uoregon. edu/~koch/java/FourD.
html, April 2, 2001. Guarino, Michael, Physicist, Bachelor in Physics, Teacher. Personal Interview. March30th, 2001. Rothman, Tony, Ph.
D. Instant Physics, From Aristotle to Einstein, And Beyond. NeyYork, New York, Byron Preiss Visual Publications, Inc, Ballantine Books, adivision of Random House, Inc. 1995.
Microsoft Encarta. Einteins Special Relativity. Online Availablehttp://encarta. msn. com/find/Concise. asp?z=1&pg=2&ti=761562147#s3 April 2, 2001.
Microsoft Encarta. Einteins General Relativity. Online Availablehttp://encarta. msn. com/find/Concise.
asp?z=1&pg=2&ti=761562147#s5April 2, 2001.