Gali's MTC Senior Project

Abstract: This paper delves into the mysterious world of higher dimensions using analytical comparisons between our perceived third dimension to two dimensions. Many thoughts are inspired by Edwin Abbott’s “Flatland”. Topics covered include various explanations of the term dimension; a mathematical treatment of objects in higher dimensions like hyperspheres and hypercubes and proofs for our very own existence in this world as we know it.

*This paper has benefited greatly from comments received from Prof. Andrew Rich and the intellectually stimulating work of Edwin Abbott. Any errors or omissions are the responsibility of the author.

The existence of the fourth spatial dimension has been a lively area of debate since the time of the Greeks, who dismissed the possibility of a fourth dimension. One of the Greek philosophers, Ptolemy even gave a proof that higher dimensions could not exist.[1] His reasoning was that the maximum number of straight lines one can draw that are mutually perpendicular to each other is three and thus the fourth dimension cannot exist. What he actually proved was that humans cannot biologically visualize a fourth dimension. Computers can routinely manipulate calculations in higher dimensions.

Some Greek philosophers like Plato had different views from Ptolemy about higher dimensions. In his famous book, The Republic (370 B.C.), Plato explained higher dimensions by analogy[2]. He gave an example of three prisoners who, ever since birth, had been chained to their seats in a cave. Behind them was a fire which reflected their shadows onto the wall in front of them. As such, their view of the world was confined to their two dimensional shadows and; therefore, they had no idea about their three-dimensionality. When one of them was freed into the free world, he was so perplexed by his three-dimensional surroundings that he found it difficult to explain this new dimension to his fellow prisoners, who in turn thought him to be crazy for even proposing the idea of higher dimensions. Similarly, since human beings have always been exposed to a three dimensional world, they find it difficult to comprehend the idea of higher dimensions.

Another significant contributor to the thought of higher dimensions was a great scholar and theologian of the Victorian era (1838 – 1926) called Edwin Abbott Abbott. In his novel Flatland, he also uses the analogy principle to describe life in a two-dimensional world. The main character, ASquare, is exposed to life in a higher dimension by a three-dimensional being, ASphere. Through this adventure epic, one can use the flatland to sphereland comparison to visualize a fourth dimensional land - hyperspace[3]. In this land, freedom of movement would be greater than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points can prevent all movement. In hyperspace, however, with 3 of its points fixed it could still rotate about the plane passing through those points. A rigid body would have 10 possible different movements in hyperspace, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points must be fixed to prevent all movement.

In hyperspace, a flexible sphere could without stretching or tearing be turned inside out. Two rings of a chain could be separated without breakage. Our knots would be useless since a knot could be unknotted without removing the fastened ends. Just as in our space a point can pass in and out of a circle without touching its circumference, so in hyperspace a body could pass in and out of a sphere (or any enclosed space) without going through the surface surrounding it. In short, all of our space including the interior of the densest solids is open to inspection and manipulation from the fourth dimension, which extends in an unimaginable direction from every point of space. Does hyperspace really exist physically? If so, our universe must be a mere abstraction or shadow cast by a more real four-dimensional world.

However, before undertaking the study of the fourth dimension, one needs to fully understand the basic concept of dimension. What is dimension? Can it mean more than one thing? Are there many aspects of dimension? Is dimension in mathematics different from dimension in economics? Let us examine the different definitions of dimension in order to get a solid understanding and basis for higher dimensions. Dimension, depending on what perspective it is defined from, can take on many meanings: physical dimension, vector dimension, fractal dimension, geometric dimension and superstrings. We shall proceed to examine each of these categories of dimension.

The word dimension is used many aspects of our everyday life. Usually people speak of degrees of freedom to signify the number of dimensions. A typical example is the Shower problem. If person A is in a shower with only one knob for running cold water, that could be seen as one degree of freedom. Person B in a shower with two knobs for hot and cold water would have two degrees of freedom, one being either on or off, and the other is either hot or cold. An even luckier person C would have an additional knob to modify the water pressure and thus he would have three degrees of freedom, water availability, water temperature, and water pressure. A fourth person D might have an additional option of controlling the pattern of the water spray, either jet or sprinkle. This would make him have four degrees of freedom. As you can see, we can build up onto these options to create infinitely many degrees of freedom. As such, the degrees of freedom can be defined as the number of choices to be made in order to uniquely specify the desired shower condition.

Even in mathematics, dimension takes on several meanings depending on what branch of mathematics you are dealing with. In linear algebra, a vector space can have any positive, integral dimension, whether it is greater than three or not. The dimension of a vector space V is the number of vectors (cardinality) of a basis of V. A vector space V over a given field F is a set in which addition and multiplication operations are defined so that v + w (where v, w are in V) is defined, and a * v (where v is in V and a is in F) is defined[4] . We say that the vectors v₁, v₂, .., v_n are linearly independent, if and only if whenever a₁, a₂, ..., a_n are elements of field F such that a₁v₁ + a₂v₂ + ... + a_nv_n = 0, then a_i = 0 for all i=1,2,...,n. A more intuitive explanation is that we cannot express any one of these vectors v₁ .. v_n as linear combinations of one another. We define a basis as a subset of a vector space V which is both linearly independent and spans the vector space V (is a generating set of V). This means that any vector in V can be written uniquely as a linear combination of its basis elements. An important theorem follows from this, for any given vector space, any two bases have the same number of elements. This means that every vector space has one unique dimension. Having described all the technical terms, it is now clear how the description of dimension of a vector space comes about. A basis gives a coordinate system for a vector space and thus the number of choices for the coordinates to specify a point. This dimension of the vector space V is called the cardinality of the basis. For example, the vector space R³ has {(1,0,0), (0,1,0), (0,0,1)} as a basis, and therefore we have dimension dim_R(R³) = 3. The vector space R⁴ has {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)} as a basis, and therefore we have dim_R(R⁴) = 4. Here we see that we can go on infinitely and so more generally, dim_R(Rⁿ) = n. The dimensions in linear algebra are synonymous to dimensions in degrees of freedom. In a Vector Space, the number of its basis vectors determines its dimension likewise, in a shower, the number of its options determines its degrees of freedom hence its dimension. Therefore, mathematically the theory of higher dimensions already exists in linear algebra.

In scaling and measurement, dimension is viewed differently. If we take an object and scale it down by a linear factor k, we are still able to get m smaller copies which when fitted together, give the original object. The dimension d is then the power of k such that k^d = m. This can be similarly applied to various shapes and objects as shown in the table below. (In this example k=3)

Note that all dimensions in scaling and measurement are integral. Could there be dimensions that are not integral? This topic was pondered by Polish mathematician called Waclaw Sierpinski who used a Sierpinski gasket as shown below to develop fractal dimensions.

Figure SEQ Figure \* ARABIC 1Sierpinski's carpet obtained from http://www.jimloy.com/fractals/chaos0.gif

The square gasket above is formed by starting with a square A as shown in attachment. This square is similarly scaled down by a linear factor of 3 such that we need 9 of the smaller copies of the square to get the original one. Now if we take out the middle square as shown in figure B, we would be left with 8 small squares(1..8). If we take out the middle square from each of those squares, each one would be divided into 8 equal even smaller squares ( I_1.. I₈) . We can go on infinitely removing the middle of each smaller square. This is how we get the Sierpinki’s gasket. Using our method again, we see that our dimension f would not be integral but instead fractal since 3^f = 8. Solving for f gives f = (Ln 8) / (Ln 3).

From this, we can come up with a definition for fractal dimension. For any given object, if we scale it down by a linear factor of k; and we are still able to get m smaller copies of that original figure which when fit together, give the original object, then the fractal dimension f is the power such that k^f = m.

In Geometry, dimension of an object is the number of coordinates needed to specify a point on that object. An intuitive explanation for this definition would be best captured by following a sequence of n-hypercubes starting with the zeroth dimension and progressing up to the fourth dimension. An n-hypercube is the generalization of the cube within n dimensions, with a 3-hypercube being the traditional cube. By seeing each n-hypercube build up from the previous one, we shall get a better understanding of the fourth dimension.

a) A point – Assume a point in space which is 0 dimensional because of no width, length or height. It is infinitely small. Every point is has exactly the same measurements because it has no dimension. It represents the zeroth dimension.[1]

b) A line segment – From a) extend our point in any direction. This will create a line segment which is one – dimensional. This is because we need only one coordinate in order to determine any point on the line segment. Therefore, if you expanded the line segment infinitely, it would cover one-dimensional space.

c) Square – Likewise from b) if we extrude the line segment in any perpendicular direction to the first, we create a square. Similarly, all squares are two dimensional since they require two coordinates in order to determine any point on them. These coordinates lie on the length and width of the square. If you expanded the square infinitely, it would cover two-dimensional space.

d) Third dimension – from c) we can further extend the square in a third direction to get a three dimensional cube with length, width and height. All of the edges within a single cube are the same length, and all of the angles are right angles. A cube, once expanded, covers a three dimensional space because one requires at least three coordinates in order to determine any point on the cube.

e) Fourth dimension. From d) we take the finite cube and extend it in yet another direction that is perpendicular to all its edges. However, it is difficult to visualize this within the restrictions of the third dimension. Thus we need another dimension called tetraspace[1] in order to achieve this. As a result of this extension, we get a hypercube. The hypercube is a generalization of a 3-cube to 4 dimensions. It is also called a measure polytope. It is a regular polytope with mutually perpendicular sides.[2]

Likewise if we expand the hypercube infinitely, it will span a four dimensional space.

There are several ways to view the hypercube as shown by these three diagrams; inner projection and parallel projection of a skewed hypercube respectively. We assume that the arrows are not edges in themselves but merely projections which are perpendicular to all three edges that meet at the corner of the original cube. According to Garret Jones, the fourth dimension is all space that one can get to by traveling in a direction perpendicular to three-dimensional space.[3] From the table below we can use our building up principle to calculate dimensions of higher dimensional objects.

N-cubes	Vertices	Edges	Faces	Number of cubes	Number of HyperCubes	Number of 5-dimensional cubes
Point	1	0	0	0	0	0
Line segment	2	1	0	0	0	0
Square	4	4	1	0	0	0
Cube	8	12	6	1	0	0
HyperCube	16	32	24	8	1	0
PentaCube	32	80	80	40	10	1

On careful analysis, we can see that given an object of certain dimension n, if we know its corresponding dimensional aspects like vertices, edges, faces, number of cubes and hypercubes, then we can figure out the dimensional aspects of the next object with dimension n+1. Looking at the table above, we simply multiply the value of the cell by two and add it to the value in the cell left of it in order to get the value of the cell below it.

How does this formula come about?

In this diagram, we move line segment i through a distance perpendicular to its edge to form square iii. Therefore, the formula below the diagram shows that if we add the shape properties of the line segment i and ii together with the translation, we get the shape properties for the square. This is how we fill the table above.

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PART TWO

HIGHER DIMENSIONS

I. Fold-Outs and Duals

According to Mathworld, a regular polygon is an n-sided polygon in which the sides are all the same length and are symmetrically placed about a common center. Therefore, it looks exactly the same at every vertex. In three dimensions, a solid which is a combination of polygons is called a polyhedron. In four dimensions, instead of polyhedron, we have a polytope. There are only five regular polyhedra in three-dimensions; cube, tetrahedron, octahedron, dodecahedron and icosahedron[4].

However, in higher dimensions, there could be more. Euclid’s proof for the existence of no more than five regular polyhedra is important when finding how many more are in higher dimensions.

These five regular polyhedra fall into three groups: two dual pairs and one self-dual polyhedron.

What is the dual of a regular polyhedron?

If we have polyhedron a1, its dual, polyhedron b1, is formed by replacing if all the vertices of a1 with faces and all faces with vertices. The edges of a1 are rotated through 90 degrees. The number of vertices and faces of two duals are swapped such that if a1 has n-faces and m-vertices, its dual, b1, would have m-faces and n-vertices.

Fig SEQ Figure \* ARABIC 1: The cube and the octahedron are duals. Diagram obtained from http://www.cms.wisc.edu

In trying to determine how many regular polytopes existed in higher dimensions, mathematician William Stringham proved that at most there can be one regular polytope with cubical faces – the hypercube[5].

Regular Polytopes

William Stringham’s managed to sketch diagrams of 6 regular polytopes. However, his proof that these were all the regular polytopes was incomplete and he was not sure that he had found all the regular polytopes. [6]

In order to determine the regularity of polytopes, we need to examine the properties of the polyhedra at each vertex. Considering the hypercube, it has 16 vertices, with four edges at each vertex. Any three of these edges determine an ordinary cube; hence four cubes at each vertex.

Theory: There can only be one regular polytope with cubical faces, namely the hypercube.

Proof: In three dimensions, we determine the possible regular polyhedra by examining the possible ways in which a collection of polygons can fit around a vertex. Therefore, in four-dimensions, instead of determining how many possible polyhedra can fit around a vertex, we consider the number that can fit around an edge. If the angle sum of the polyhedra around an edge does not fill the space around the edge, then the figure can be folded up into the fourth dimension[7].

Therefore in the case of the hypercube, it is clear that at most three cubes can fit around an edge without filling the region. It follows that there can at most be one regular polytope with cubical faces, namely the hypercube.

Fig 2: three cubes around an edge in three space --http://www.cs.brown.edu/people/dla/polytope/polytope.html

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What about the dual of the hypercube? [8]

In order to construct the dual, we choose the vertex in the center of each three-dimensional face, and we connect vertices from faces that share a common boundary polygon. The vertices coming from the three-dimensional faces sharing a vertex determine a three-dimensional polyhedron, the dual cell of the vertex. It is this collection of the dual cells that form the dual polytope for that regular polytope.

Therefore, in the case of the hypercube, we choose a point in the center of each of the eight cubical faces. Around each of the 16 vertices are four cubes, each set of four contributing a tetrahedron. This shall later form a collection to create the polytope, 16-cell[9]. The 16-cell is analogous to the octahedron in three-space.

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The Regular Polytopes: Properties and relation to polyhedra[10]

Polytope	Other names	Polyhedron equivalent	Dual	Properties	Diagram
Simplex	Pentachoron, pentatope	tetrahedron	Self-dual	5 tetrahedral cells, 10 triangular faces, 10 edges, 5 vertices
Hypercube	Tesseract	cube	16 cell	8 cubic cells, 24 square faces, 32 edges, 16 vertices
16 cell	Cross-polytope hexadecachoron	Octahedron	hypercube	16 tetrahedral cells, 32 triangular faces, 24 edges, 8 vertices
24 cell	icositetrachoron	none	Self-dual	24 octahedral cells, 96 triangular faces, 96 edges, 24 vertices
120 cell	hecatonicosachoron	dodecahedron	600 cell	120 dodecahedral cells, 720 five sided faces, 1200 edges, 600 vertices
600 cell	hexacosichoron	icosahedron	120 cell	600 tetrahedral cells, 1200 triangular faces, 720 edges, 120 vertices

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Fold-out Patterns in Different Dimensions

a) Line segment to Square

We can construct the square by folding the line segments that is divided into 4 equal segments through an angle of 90 degrees.

b) Square to Cube

As shown in the diagram below, we can fold 6 squares in a cross arrangement into a cube by joining the edges joined by the arrows. The second diagram shows the folding in process.

Fig 4: Figures of Cube fold-out from http://www.cs.brown.edu/people/dla/polytope/polytope.html

c) Cube to Hypercube

Just like flatlanders cannot actually put three squares together because of their inaccessibility to the third dimension, we cannot put the four cubes together to form part of a hypercube in four-space because we do not have access to it. However, we can use analogy to determine that the fold-out of the hypercube would look like this in three-space.

Fig 5: Hypercube fold-out from http://www.cs.brown.edu/people/dla/polytope/polytope.html

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II. Shadows and Projections

We can use shadows and projections to examine higher dimensions and their projections into our space. Plato was one of the first people to use the allegory of the cave to challenge readers about the dimensionality of shadows[1].

Projection from 3D to 2D

We can project three-dimensional objects like the cube onto a two-dimensional surface using shadows. This is done by holding up the cube made of sticks in three-dimensional space in the sunlight so that it casts its shadow lines on the two-dimensional paper. The image of the cube can then be traced out as shown in the diagram[2].

A more practical application used by computers is to provide the coordinates of the three edges coming from a vertex a cube denoted by pairs of numbers. As such, with four vertices, we get 4 pairs of numbers which allows the computer to display three edges and subsequently compute the other edges of the cube. A practical application for this is used in architecture like in the construction of the Johnson Art Museum[3].

Using the geometric definition to describe how to project a 3D object onto the 2Dscreen:

Figure: projection from an arbitrary 3D point (x₁ ,y₁ ,z₁) to a 2D point on the monitor (looking down from above the monitor) [4].

Assumptions:

1. A three-dimensional brown hexagon is to be projected onto the monitor.

2. The notation above is in the (x,y,z) plane such that x1 is the x-coordinate and z1 is the z-coordinate.

3. The center of the monitor is at the origin of our coordinate system (0,0,0). The monitor surface is lies in the x-y plane such that all points on it have a z-coordinate of zero.

4. The right eye shown above is located at (d,0,z₀). This means that the left eye is at (-d,0,z₀).

5. The eyes are at a distance z₀ from the monitor and vertically centered (y = 0).

6. A blue ray of light which originates at an unknown point on the monitor surface (x₂,y₂,0), passes through a vertex (x₁, y₁, z₁) of the hexagon, and enters the right eye.

7. We are assuming that points (x₁, y₁, z₁) and (d, 0, z₀) lie in a straight line and we want to find the point on the monitor surface where this ray originates from.

We shall only go into the detail of proving the last assumption since the rest are trivial.

Proof: We need to show part a) and b) below in order to prove assumption 7 above.
a) Prove that the two points (x₁, y₁, z₁) and (d, 0, z₀) lie on a straight line.

b) Use the result to find the coordinates for the point on the monitor where the ray originates from.

Part a): We shall apply our geometry knowledge on coordinates and parametric equations to find this. Using the parametric equation of a line segment, we know that given two points (x₁, y₁, z₁) and (x₂, y₂, z₂), the point (x, y,z) is on the line segment determined by (x₁, y₁, z₁) and (x₂, y₂, z₂) if and only if there is a real number t such that

x = (1 - t)x₁ + tx₂,
y = (1 - t)y₁ + ty_2,

and

z = (1 - t)z₁ + tz_2,

Therefore, applying this to our above example, we get these three equations:

i) x = tx₁+ (1 - t)d

ii) y = ty₁+ (1 - t)0

iii) z = tz₁ + (1 - t) z₀

If t = 1, then we see that parametric equation for (x,y,z) = (x₁, y₁, z₁)

If t = 0, the parametric representation gives (x,y,z) = (d,0, z₀).

Therefore, the line segment must pass through two points (x₁, y₁, z₁) and (d,0, z₀).

Part b): Now in order to find the originating point of the ray in the monitor, we need to find that value of t which gives a z-coordinate of zero. We set z = 0 in equation iii) above such that
0 = tz₁ + (1 - t) z₀

We then solve for "t" to get;
t = z₀ / (z₀ - z₁)

Plugging this value of "t" into the equations i) and ii), we get the intersection (x₂, y₂,0) of the line with the screen.

x₂ = (dz₀(x₁ - z_{1) )} / (z₀ - z₁)

y₂ = z₀y₁ / (z₀ – z₁)

This is the point on the monitor at which the ray originated.

Projection from 4D to 3D

Since we cannot visualize higher dimensions, computer programs are used to project a hypercube onto a two dimensional screen. A good example of such programs can be found at this website - http://dogfeathers.com/java/hyprcube.html. The java program shows a four-dimensional hypercube and its rotations. This hypercube is projected onto a two-dimensional screen[5].

How is a hypercube projected onto a screen?
Using a similar method as shown above, a hypercube can be projected onto a two-dimensional screen. Each vertex of the hypercube is represented by 4 coordinates. Rays are traced in 4-space from a viewpoint at (0,0,0,w), through each vertex of the object (which is centered at (0,0,0,0)) to a 3-space located at w=0. We can then continue to use the same method above to project the object from 3-dimensions to 2-dimensions.

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III. Slicing

i) Slicing the Square

In Flatland, there was a time when A Square dreamt of visiting Lineland and passing through the King’s world as a way of explaining higher dimensions. When the square was slicing through Lineland, all the King could see was a point of the line segment and then suddenly nothing.

ii) Slicing the Cube

Similarly, we can analyze the shapes (visualizations) that are formed as a cube slices through a two dimensional plane. This wonderful website shows all this.

http://www.math.union.edu/~dpvc/talks/2000-11-22.funchal/cube-slice-edge.html

a) Slicing a cube square first: As shown in the diagram below, the blue square represents the plane as the cube slices through it. Therefore, it is clear that with this format, we shall observe a square in the plane.

b) Slicing a cube edge first[3]: This shows what shapes would be formed from this process. The image on the right for each diagram shows the map view.

http://www.math.union.edu/~dpvc/talks/2000-11-22.funchal/cube-slice-edge.html

c) Slicing a Cube Vertex first

iii) Slicing the Hypercube

We can already visualize how the hypercube would look in our three-dimensional universe if it penetrated face first. First we would see an ordinary cube with 8 bright vertices, then a dull cube and finally another cube with 8 bright vertices[4].

What about if the hypercube penetrated vertex first, square first or edge first?

The shapes created are comparable to slicing the cube, except that in this case we have to raise each shape up by one dimension. Modern computers can exhibit these slices as shown below:

a) Slices of a Hypercube starting with a square[5]

The image formed starts with a square which grows into a cube, rectangular box (they all have square bases) and later shrinks back to a square and finally disappears

b) Slices of a Hypercube starting with an edge

c) Slices of a Hypercube starting with a vertex

The image formed starts with a point and grows into a small triangular pyramid. This expands to the four vertices of the hypercube, a point at which the corners are truncated off to form a semi-regular polyhedron as shown in the diagram. As the slicing continues, a regular octahedron is formed beyond which, is a reverse process back to a point.

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IV. Space time

Time has often been speculated to be the fourth dimension. The topic of space time is still under investigation and has not been formally proved. Rudy Rucker makes analogy, ‘if it wasn’t for time, I would live forever, and therefore, if it wasn’t for space, I would be everywhere’[6]. If time did not pass, I would be typing this paper always, at this moment.

Rucker presents the world as a block universe [7] . As such that space and time put together to make a block called spacetime. (3 dimensions of space and one dimension of time). This implies that eternity is right now.

Usually we assume that the past is non-existence and can never be retrieved, while the future is yet to come; therefore the 7:00am college union is non-existent but at the current moment, the union exists because it is in the now.

According to the spacetime theory, the present union does not exist by the fact that we can not sense it. It is not in our reality. Eternity is right now[8]. Instead of spreading out time in a continuum, the theory compresses time into the present.

Does this imply time the fourth dimension?

No! Just like we cannot proclaim that width is the second dimension, there is no need to be so rigid and proclaim time to be strictly the fourth dimension. Instead, since we discuss width and length as space dimensions, we can group time under higher dimensions[9]. Therefore, even if time could be one of there higher dimensions, there could be other possible dimensions like curved space times. Such block diagrams where first thought of by Russian mathematician – Hermann Minkowski (1864 – 1909).

The Minkowski diagram[10] is analogous to the Flatland story when A Square tries to understand the concept of death and A Sphere lets him know that his entire life is in one continuum. There is no motion in spacetime. One’s mind extends the length of one’s span. Life can be thought of as a book that one’s soul reads and when the soul runs out of spacetime pattern to read, one dies[11].

Special Relativity Theorem

The Relativity theory was developed by Albert Einstein and it is subdivided into Special Relativity and General Relativity theorems. The Special Relativity theorem assumes the existence of a four dimensional continuum and it implies that nothing can travel faster than the speed of light because the closer one gets to the speed of light, the more mass one acquires, thus making it harder to accelerate. Therefore, it would require infinite supply of energy to accelerate to the speed of light. Consequently, the only events that are available in one’s spacetime are what one describes as his or her future. There lies a problem with the relativity of simultaneity because faster than light travel can lead to time-travel which is impossible as we know it.

This theory is explained by the twin paradox[12]. According to the paradox, if one twin stayed on earth doing nothing while the other accelerated to the moon and back; their relative ages would differ.

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V. STRING THEORY[13]

Higher dimensions take on an important role in String Theory in Physics even though this topic is still undergoing research. The term string theory properly refers to both the 26-dimensional string theory and to the 10-dimensional superstring theories that were discovered by adding supersymmetry. String theory is a science in progress; with new discoveries everyday. The goal of modern theoretical physics has been to explain the universe. A superstring is a hypothetical particle consisting of a very short one-dimensional string existing in ten dimensions. It is the elementary particle in a theory of space-time incorporating supersymmetry. The superstring theory, being a completely finite theory, gives us deeper insight into the era before the Big Bang. The theory states that at the instant of creation, the universe was actually an infinitesimal ten dimensional bubble which split to form our universe through the standard Big Bang. The quantum theory explains three of the four forces (the weak, strong, and electromagnetic forces) by assuming the exchange of tiny packets of energy, called "quanta." However, many critics dispute the validity of string theory and claim that it is not practical.

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VI. CONCLUSION

The topic of higher dimension is cause for much speculation triggered by the difficulty in visualizing these higher dimensional objects. However, we should remember to take into account that even the physical figures that we talk about everyday like the square or the cube do not actually exist. No one can claim to have seen a perfect square; however, we routinely do calculations with squares. Therefore, we should take this solace with us as we target higher dimensions and endeavor to understand as much as we can about higher dimensions, because the fact remains that they do exist.

[1] http://mathforum.org/mam/00/master/essays/B3D/1/JPG/figure03.jpg

[2] Obtained from http://www.math.union.edu/~dpvc/math/4D/cube-slices/cube-face.html[2]

[3] The slices for a cube (edge and vertex) were obtained from http://www.math.union.edu/~dpvc/talks/2000-11-22.funchal/cube-slice-edge.html

[4] Banchoff, p47

[5] Images of Hypercube Slices obtained from Banchoff p47-49

[6] Rucker, fourth dimensions, p134

[7] Rucker, fourth dimensions, p135

[8] Rucker, fourth dimensions, p136

[9] Rucker, fourth dimensions, p139

[10] Minkowski diagram obtained from http://www.brown.edu/Students/OHJC/ma8/papers/simultan.htm

[11] Rucker, fourth dimensions, p144

[12] http://www.brown.edu/Students/OHJC/ma8/papers/spam.htm

[13] Information on Superstrings obtained from http://superstringtheory.com/basics/index.html

[1] Banchoff p63, refer back to introduction for the cave allegory

[2] Banchoff p64

[3] Banchoff p65

[4] NewBold, Geometry of Stereo 3D Projection

[5] NewBold, Geometry of Stereo 3D Projection

[1] According to Garrett Jones, tetraspace is the technical term for fourth dimension.

[2] Polytope – A solid figure in higher dimensions composed of polyhedra. Analogous to polygon in two-dimensions

[3] Jones ,Garrett- “Fourth Dimension: Tetraspace”

[4] Banchoff, p90

[5] Banchoff, p95

[6] Banchoff, 95

[7] Banchoff, 95

[8] Banchoff, 98

[9] Figure of a 16-Cell Obtained from Mathworld.com

[10] Obtained from http://astronomy.swin.edu.au/~pbourke/polyhedra/platonic4d

[1] All the diagrams in this hypercube explanation were downloaded from tetraspace.alkaline.org

[2] According to Garrett Jones, tetraspace is the technical term for fourth dimension.

[3] Polytope – A solid figure in higher dimensions composed of polyhedra. Analogous to polygon in two-dimensions

[1] http://fusionanomaly.net/quantummechanics.html

[2] Fourth Dimension by Analogy- oldest reasoning that assumes that the fourth dimension is to three-dimensional space what the third dimension is to two-dimensional space.

[3] Hyperspace is a space having dimension n > 3

[4] http://mathworld.wolfram.com/VectorSpace.html

[1] http://fusionanomaly.net/quantummechanics.html

[2] Fourth Dimension by Analogy- oldest reasoning that assumes that the fourth dimension is to three-dimensional space what the third dimension is to two-dimensional space.

[3] Hyperspace is a space having dimension n > 3

[4] http://mathworld.wolfram.com/VectorSpace.html

[5] All the diagrams in this hypercube explanation were downloaded from tetraspace.alkaline.org

[6] According to Garrett Jones, tetraspace is the technical term for fourth dimension.

[7] Polytope – A solid figure in higher dimensions composed of polyhedra. Analogous to polygon in two-dimensions

[8] Jones ,Garrett- “Fourth Dimension: Tetraspace”

[9] Banchoff, p90

[10] Banchoff, p95

[11] Banchoff, 95

[12] Banchoff, 95

[13] Banchoff, 98

[14] Figure of a 16-Cell Obtained from Mathworld.com

[15] Obtained from http://astronomy.swin.edu.au/~pbourke/polyhedra/platonic4d

[16] Banchoff p63, refer back to introduction for the cave allegory

[17] Banchoff p64

[18] Banchoff p65

[19] NewBold, Geometry of Stereo 3D Projection

[20] NewBold, Geometry of Stereo 3D Projection

[22] Obtained from http://www.math.union.edu/~dpvc/math/4D/cube-slices/cube-face.html[22]

[24] Banchoff, p47

[25] Images of Hypercube Slices obtained from Banchoff p47-49

[26] Rucker, fourth dimensions, p134

[27] Rucker, fourth dimensions, p135

[28] Rucker, fourth dimensions, p136

[29] Rucker, fourth dimensions, p139

[30] Minkowski diagram obtained from http://www.brown.edu/Students/OHJC/ma8/papers/simultan.htm

[31] Rucker, fourth dimensions, p144

[32] http://www.brown.edu/Students/OHJC/ma8/papers/spam.htm

[33] Information on Superstrings obtained from http://superstringtheory.com/basics/index.html

BIBLIOGRAPHY

[1] Rucker, Rudy, “The Fourth Dimension: A Guided Tour of the Higher Universes”,1984.

[2] Abbott, Edwin, 1838-1926 “Flatland: a romance of many dimensions” Fifth edition revised, 1963.

[3] Burger, Dionys, “Sphereland: a fantasy about curved spaces and an expanding universe” 1983.

[4] Manning, Henry P, “The Fourth Dimension Simply Explained”
<http://etext.lib.virginia.edu/toc/modeng/public/ManFour.html>

[5] Banchoff, Thomas F, “Beyond the third dimension”. Scientific American Library. New York. 1990.

[6] Eric W. Weisstein. "Dimension." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Dimension.html >

[7] Eric W. Weisstein. "Cube." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Cube.html> (Nice demonstration of cube rotating)

[8] Eric W. Weisstein. "Hypercube." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Hypercube.html

[9] Jones ,Garrett. “Fourth Dimension: Tetraspace”
<http://tetraspace.alkaline.org/introduction.htm>

[10] <http://fusionanomaly.net/superstrings.html>

[11] Rucker, Rudy. “Mind tools: The Five Levels of Mathematical Reality” 1987.