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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
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.
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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). A
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-D
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 a
very 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 A
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-D
finite unit) you would duplicate the segment (1-D unit) and project it into the added
y-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-D
finite 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 tesseracts. 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 tesseract, which
consists of eight cubes forming a cross-like object. Just like the cross was an unfolded
cube, the tesseract 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 tesseract folding up into a hypercube defies human
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 a
hypercone. 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 “hypercubing” 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 on the 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 ABDC and square
EFHG (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 90o , however, in two-space, on this
flat representation, angle EAB is about 135o. 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 hypercone.
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
a 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, the
Kaluza-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
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 light
year 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.
Kaku, Michio. 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
http://www.dogfeathers.com/java/hyprcube.html, April 2, 2001.
Koch, Richard, Department of Mathematics, University of Oregon. “Java Examples of
3-D and 4-D Objects”. [Online] Available
http://darkwing.uoregon.edu/~koch/java/FourD.html, April 2, 2001.
Guarino, Michael, Physicist, Bachelor in Physics, Teacher. Personal Interview. March
Rothman, Tony, Ph.D. Instant Physics, From Aristotle to Einstein, And Beyond. Ney
York, New York, Byron Preiss Visual Publications, Inc, Ballantine Books, a
division of Random House, Inc. 1995.
Microsoft Encarta. “Eintein’s Special Relativity”. [Online] Available
April 2, 2001.
Microsoft Encarta. “Eintein’s General Relativity”. [Online] Available
April 2, 2001.