New Dimensions. A Study on the 4th Dimensional Figures in Euclidean Space

Introduction

My exploration deals the creation of figures, 0-dimensional to 5-dimensional, within the Euclidean Space (space in which we create 3-D figures) and Cartesian Space (space in which we create 2-D figures). I have scrutinized the concepts of figures from 0-D to 3-D and I have developed patterns on how each dimension proceeds into the next. I plan to use these patterns to create 4-D and possibly 5-D figures if not in reality, then in computer designs. I plan to use the cube as my basis of developing patterns and hypotheses because I believe it is the most understandable and most iconic figure in geometry. I realize that I am limited by research sources as there are no humans, or possibly few humans, that can see the world in dimensions beyond the 3rd; thus, many of my sources will be based on hypotheses and problems that lack proof. I will be using much of my own deduction and speculation as well, which may also degrade the reliability of my conclusion. Still, I plan to find out whether there are examples of “super-dimensions” (4th – n th) and if these “super-dimensions” follow the rules of geometry or if I am following a fool’s path.

Why I Started the Investigation

I have long been fascinated by mathematical concepts and structure that math follows. Most math concepts I have learned throughout my academic career have come to me with ease, because they make sense to me and there is always a definite answer or range of answers (ex: Finding the intersection of two lines or parabolas). However, I began thinking of how mathematicians have been wrong in the past, performing slight errors in calculations which, at the time, seemed negligible, but in later years, proved to be a roadblock to further develop mathematics. I also began thinking of how a number of theorists, mathematicians, physicists, chemists, and other experimentalists throughout history have purposefully or accidently based an entire idea on a mistakenly confirmed theory that reshaped the mindset of people in later generations. Development in these math-science based fields would then be furthering a flawed idea. Eventually, experimentalists in these math-science fields would realize that the past developments have not been set on the right course, but has been thrown off on a tangent either from the introduction of the field or from some time along its development.

Thus, I began to think of not only individual mathematical concepts, but also what mathematical concepts are based on. I began to think whether I had the potential to discover something that people in the past have not. I realized that I am still a teenager in high school though, so even in the event that I do discover something extraordinary, my findings would most likely be premature.

From my adolescent point of view, much, if not most, of all math, in the past as well as the present, is based on geometry and the Euclidean Space (the space that we use to map coordinates, create planes, draw vectors, etc.) As a result, I came to an interesting topic: Could mathematics be utilized in dimensions beyond the 3rd dimension? In the past, I have seen representations of a 4-dimensional cube, called a tesseract, but I did not understand mathematics enough to comprehend it. Perhaps if we understand more about the tesseract, mathematics could develop to an entirely new level.

Creating the Rules

I know that without patterns or rules to go by, I would be going in circles trying to pursue new higher dimensions. I did not want to look up anything on the internet at this point because I want to create a set of rules on my own, with the mathematical understanding I currently have. I think that if I search up “how to create 4th dimensional cubes”, I may actually see ideas and subconsciously believe that the ideas were original to me, resulting in plagiarism. Only after I develop my own set of rules will I search up if the rules had been mentioned before.

Dimensions (each dimension can also be named n -cube for n th dimension) (the mathematical terms for the figures refer to the maximum number of edges that can be seen from the figure – ex: at a certain angle, the tesseract appears to have 8 edges [ Octa choron]:

- 0-D: Point
- 1-D: Line
- 2-D: Square (Tetragon)
- 3-D: Cube (Hexahedron)
- 4-D: Tesseract (Octachoron)
- 5-D: Penteract (Decateron)

Rules

1. Every time there is an increase in dimensions, a new axis is introduced (0-D: no axis 1-D:

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x-axis 2-D: x-axis, y-axis 3-D: x-axis, y-axis, z-axis …)

(Figure 1)

http://www.highend.com/support/digital_lighting/dl2supportguide/3D.asp

2. All lines of the same axis must be parallel to one another (one x-axis line must be parallel to all other x-axis lines; y-axis line to other y-axis lines; etc.)

3. All lines segments must have the same length

4. Lines from different axes may intersect only at the vertices (Each vertex should contain every axis that exists within the dimension) (Ex: Cube [3-D] all axes (x,y,z) pass through each vertex)

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5. The figure must have rotational, reflection, and point symmetry

6. Each dimension is created by connecting the corresponding parts of 2 parallel figures from the previous dimension (0-D: a point, 1-D: 2 parallel points connected by a line [line], 2-D: 2 parallel lines; corresponding endpoints are connected by lines [square], 3-D: 2 parallel squares; corresponding vertices are connected by lines [cube], 4-D: 2 parallel cubes; corresponding vertices are connected by lines [tesseract], 5-D: 2 parallel tesseracts; corresponding vertices are connect by lines [penteract])

7. When a figure from a dimension (n -dimension) is rotated so that it is facing you, the viewer, the figure from the previous dimension ([ n -1]-dimension) should appear. (If a line rotates to the z-axis (axis pointing towards the viewer), the viewer can see a point. If a square rotates to the z-axis, the viewer can see a line. If a cube rotates to the z-axis, the viewer can see a square. If a tesseract rotates to the z-axis, the viewer can see a cube. If a penteract rotates to the z-axis, the viewer can see a tesseract.)

8. The sides of a figure are made from the figures from the previous dimension. (Sides of a square are lines, sides of a cube are squares, sides of a tesseract are cubes)

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(Figure 2)

http://fleischfilm.com/?cat=8

9. A figure from any dimension can display at least two figures at the same time, figures from the previous dimension. (From a line, the viewer can see at least 2 points. From a square, the viewer can see at least 2 lines. From a cube, the viewer can see at least 2 squares. From a tesseract, the viewer can see at least 2 cubes. From a penteract, the viewer can see at least 2 tesseracts.)

10. A pattern is seen when increasing in dimensions. The pattern only starts for the dimension when the dimension’s figure is introduced (ex: the pattern for 1-D starts when lines start to appear on the table). The pattern is used to find how many figures of each dimension there are in a single dimension. (ex: how many lines and squares there are in a tesseract)

Dimension of the cube – top row across; Figures seen – left column down (ex: In a 3-D cube, there are 8 vertices, 12 lines, and 6 squares)

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(Table 1)

How I found the values for Table 1: I first tried to find out the number of figures within each dimension by individually counting each individual figure, but I soon realized that doing so is futile because I had no way of keep tracking which ones I counted and which ones I need to count. Instead, I found out all the “easier” numbers first: how many of the figures of the dimension are in the dimension (how many cubes within a cube [1]) and how many of the figures from the previous dimension are in the dimension (how many squares are in a cube [6]). For the higher dimensional figures, I first visualized all the different versions of the same figures there would be in a dimension (there are 3 different squares in a cube: “x and y”, “x and z”, “y and z”; there are 4 different cubes in a tesseract: “x, y, and z”, “x, y, and w”, “x, w, and z”, “y, w, and z”) After I found out the all the different version, I visualized how many of one version of the figure there would be in the dimension, and then I multiplied the number of one version of the figure by the number of different versions (ex: To find out how many squares there were in a tesseract, I visualized that there are 6 different types of squares: “x and y”, “x and z”, “x and w”, “y and z”, “y and w”, “z and w”. Then I visualized how many of the “x and y” squares there are in a tesseract, in which there were 4 “x and y” squares. Afterwards, I multiplied 4 by 6 because there are 6 different versions of squares within the tesseract).

The following patterns are both found from Table 1

Pattern 1 (Recursive pattern) The variable “n” represents the dimension, and the variable “An” represents the number of figures within the dimension. The equations follow:

- Points/Vertices f{x} follow An=2n
- Lines g{x} follow An=2(An-1) + f{x}
- Squares h{x} follow An=2(An-1) + g{x}
- Cubes i{x} follow An=2(An-1) + h{x}
- Tesseracts j{x} follow An=2(An-1) + i(x)
- Penteracts k{x} follow An=2(An-1) + j{x}

For Points/Vertices, the equation is a simple geometric sequence equation that has a common ratio of 2. For each consecutive equation after Points/Vertices, the equation becomes recursive, as the equation for the previous dimension is needed to complete the equation for the current dimension. I gave names to the equations of each dimension to make the notation simpler. (Points/Vertices: f{x}, Lines: g{x}, Squares: h{x}, Cubes: i{x}, Tesseracts: j{x}, Penteracts: k{x})

Pattern 2 (Combinations pattern) The variable “n” represents the dimension, the variable “r” represents the figure relative to its dimension (point=0, line=1, square=2, cube=3, tesseract=4, penteract=5), and the variable “An” represents the number of figures within the dimension:

A_n=_nC_r(2^n-r )

I realized that some of the numbers in the table were very familiar, but I did not remember where or when I have seen them. After tinkering around with my calculator, I recalled the sequence, 1,4,12,32,80, as being some kind of permutation or combination sequence, so I tried to apply nPr and nCr to the table. After much trial and error, I found that the number of figures, from highest to lowest dimensional figure, in a dimension is actually a combination (nCr) with a multiplying factor (2n-r); thus I derived the equation, An=nCr(2n-r). Each time I try to find a figure from a previous dimension in the current dimension, the multiplying factor exponentially increases by 2. This multiplying factor is applied to the formula after the combination is found. The equation I found, Pattern 2, An=nCr(2n-r), is different from the recursive equations from Pattern 1 in that the Pattern 2 allows me to find the number any figure I want from any dimension without the need to find the number of every figure before the figure I want. (I do not need to find number of cubes in a penteract in order to find number of tesseracts in a penteract).

Ex: Finding number of lines, squares, cubes, and tesseracts within a tesseract. [A_n=_nC_r(2^n-r )]

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(Table 2)

Drawing a Tesseract

Before I began to draw my sketches of what a 4th dimensional figure may look like, I looked up whether there were already examples of 4th dimensional figures created. The most widely-accepted “4th dimensional” figure was indeed the tesseract, or 4-cube, also known as the hypercube.

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http://en.wikipedia.org/wiki/Tesseract

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https://www.flickr.com/photos/ezorigami/7444523446/

(Figure 3) (Figure 4)

However, I realized that the tesseract did not follow all of my rules when I tried to visualize in the way it is presented. The tesseract displayed online seemed to be a cube completely inside of another cube, with lines connecting all corresponding points of the cube. The connection between the corresponding points was a part of my procedure of creating higher dimensions, but the lines of the new axis, the w-axis, are not parallel to one another.

Parallel lines W-axis lines from Figure 3

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It makes sense to say that when the w-axis is introduced in the 4-cube, all of the lines from the new axis should be parallel because from the previous dimensions, all lines of the same axis were parallel (see Figure 2). In the squares, the 2 lines of the x-axis are parallel and the 2 lines of the y-axis are parallel. The cube itself follows the same rule – the 4 lines of the x-axis are parallel, the 4 lines of the y-axis are parallel, and the 4 lines of the z-axis are parallel.

Another flaw I see in online figures, Figure 3 and 4, is that the sides of the 4-cube are trapezoidal prisms, not cubic prisms. Cubes are the figures in the dimension before the 4-D tesseract, so the sides should be made of cubes, not trapezoidal prisms.

However, I needed a starting place, so I began my sketches with sketches of the online representation of the tesseract.

The progression of my sketches goes as follows (Sketches contain both 4-D and 5-D representations):

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(Figure 5) (Figure 6)

My first drawing [Figure 5] is direct copy of the online image of the 4-cube. I first pictured the 4-cube as it looks on a flat piece of paper as a cube within another cube. Later, I further developed the drawing to a spatial representation as a cube behind another cube on a completely different set of x, y, and z axes. I thought the spatial representation is relatable to the online image as the line connecting the 2 cubes [w-axis] can be seen closing into a focus point if the viewer sees the spatial representation from a diagonal angle. Then, I used my spatial representation of a 4-cube as the basis for a spatial representation of the 5-cube [Figure 6])

I know that to increase from one dimension to the next, I must duplicate the figure and connect their corresponding points. However, I realized that my spatial representation of the 5-cube did not “look” 5-dimensional; it just looked like 4 different 3-cubes on parallel planes that are connected to one another by lines from a new axis. Thus, I thought to design the 5-cube from the online model of a 4-cube.

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(Figure 7) (Figure 8)

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(Figure 9) (Figure 10)

In Figure 7, I tried to connect two parallel 4-cubes to one another to create a 5-cube. Then, in Figure 8, I tried to draw the spatial representation of the 4-cube connected to a 4-cube in Figure 7. Still, the figure did not look convincing. Thus, In Figure 9, I tried to draw what a 5-cube would look like if it were one figure, not two of the figures from the previous dimension connected to one another, a 4-cube inside another 4-cube. Figure 9 ended up looking like 4 3-cubes inside one another in a consecutive series. In Figure 10, I tried to draw the spatial representation of Figure 9. I thought I was overlooking something, an obvious detail that in apparent in cubes.

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(Figure 11)

This is my first successful drawing of the 4-cube. I realized that the small detail, I was missing was Rule 3 (all lines must have equal length). In all preceding drawings, the length of the lines of the x, y, and z axis were equal, but the lines connecting the figures to their parallel counterparts [w-axis] were not. In Figure 5, my spatial representation of the online 4-cube would have been accurate if I had just shortened the length of the w-axis line to that of the x, y, and z-axis lines – in a way, clash the two cubes together so that one corner on both cubes are inside one another. All of the figures following Figure 5, until Figure 11, had the same flaw of having incongruent line lengths. Also, it was in this drawing, Figure 11, which I discovered Rules 7 & 9, the original 2 rules I had. I visualized how a real-life model of the hand-drawn 4-cube in Figure 11 would look like and I visualized ways of rotating the model. It was then I realized that at a certain angle, the 4-cube would look like 2 separate, but congruent, 3-cubes connected to one another, one 3-cube in the front and another 3-cube in the back.

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(Figure 12) (Figure 13)

After drawing my first successful representation of a 4-cube, I made a chart with the progression of figures going from 0-D to 5-D (Figure 12) and I visualized rotating each figure to see if it fit the rules. I soon realized that all of the cubes I drew followed the rule, so I decided to extend my discovery to figures not based on cubes and squares, such as an equilateral triangular prism (Figure 13). The transition to a non-cubic figure seemed to work, as the triangular prism seemed to fit all of my rules, but without a real-life model of a 4-D triangular prism, I was uncertain. I decided to go even further as to create a chart of sphere-based figures. However, as I tried to visualize how to create a 4-D sphere (4-sphere), I discovered several interesting hypotheses. The first is that spheres and perhaps all circular figures cannot exist in dimensions higher than the 3rd. I realized that in a sphere, there is an indefinite amount of different lines that intersect at a single point and an infinite amount of lines tangent to the circumference of a sphere. I believe it is impossible, at least as far as we know, to create a parallel sphere because no matter where I duplicate the sphere, there is a line that already intersects with the figure. Thus, it seems impossible to create n-spheres greater than 3-spheres, because in order to increase in dimension, there must be a parallel and congruent counterpart to the 3-sphere. This indefinite intersecting phenomenon did not occur with my cubic and triangular drawings. Those had a definite number of lines [axis], but with the sphere, each different line represents a different axis, which means there are infinite axes in a sphere. Thus, I create a second hypothesis that perhaps the indefinite number of axes may mean that the sphere is an inter-dimensional figure, a figure that exists throughout all the dimensions. Something else I noticed was that a 3-sphere is more or less the same figure as a 0-D point in Euclidean Space; if this were true, it would support my hypothesis that the sphere is inter-dimensional. However, I also noticed that in the universe, most things are made to be round, such as stars, planets, and quasars. Therefore, I hypothesize that perhaps there is no spherical shape higher than the 3rd dimension because spheres only exist in our 3-D world. Perhaps that is why all natural phenomena (tornadoes, earthquake [shockwaves], whirlpools, gravitational force, supernovas) in the world seem to have circular shapes. Perhaps if we lived in the 4th or 5th dimension, these phenomena and the objects in space would look different, as they would be seen as some higher dimensional figure. But, since we do not live in higher dimensional worlds, we cannot comprehend 4-spheres or 5-spheres.

Hypotheses for the n-sphere:

1. Spheres cannot exist beyond the 3rd dimension because it is impossible to duplicate a separate, parallel, congruent sphere
2. Spheres exist throughout all dimensions because spheres contain all possible axes [x, y, z, w, v, etc.]
3. There exists some form of a n -sphere for n -dimension, but one must exist in the n -dimension to perceive the n -sphere [we live in the 3rd dimension; therefore, we see all real life spheres as 3-spheres, instead of 2-spheres or 4-spheres]
4. N-spheres and circular figures beyond 3-D are possible to create, but follow a completely different set of rules than my rules for creating the tesseract [my rules only work for creating figures with definite vertices and lines].

4-Cubes and 5-Cubes in Color

For the next set of sketches, I decided to color code my drawings. I showed various people my original pencil drawings of the tesseract and penteract, but none of them understood what my drawing was or how my drawing represented the 4th and 5th dimensions. Some of the people recommended I color code my axes and cubes so that the figure would be easier to comprehend, so I took the advice and drew tesseracts and penteracts with color. The first set of drawings is color-coded by axis, and the last drawing is color-coded by cubes.

Red = x-axis; Green = y-axis; Blue = z-axis; Light Blue = w-axis; Purple = v-axis

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(Figure 14) (Figure 15)

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(Figure 16)

Figure 14 is a color-coded replica of my first successful drawing of the tesseract (Figure 11). Figure 15 shows all the different types of 3-cubes that could be seen in the 4-cube in Figure 14 (there should be 2 of each type of 3-cube within the 4-cube – total: 8 cubes) (see Table 1). Figure 16 is a color-coded replica of the 5-cube I drew in Figure 12.

The following sketch shows the 5-cube in its component parts.

Green = First Cube Red = Second Cube Blue = Third Cube Purple = Fourth Cube Light Blue = w-axis (creating first tesseract) Brown = v-axis (creating second tesseract)

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(Figure 17)

This drawing is the last of my dimensional drawings. My first color-coded 5-cube, Figure 16, is color-coded by axes, which is easy to understand, but difficult to comprehend because of the many different intersecting lines. Figure 17 displays the 5-cube in its component form of 4 different 3-cubes, which is easier to see than the 3 different axes making a 3-cube.

Creating a 3-D model of a 4-D and 5-D figure

After I had drawn out my sketches, I decided to create a real-life model, if even possible. I knew that the world we live in, as far as we know, is the 3-dimensional world. I wondered if I could even build a real-life representation of a 4-D and 5-D model if material in real life was only in 3-D. Nevertheless, I built a 4-D with paper and 5-D model with flashcards.

The models were based off my rules and as mentioned in the rules, cubes can be seen from the tesseract, and tesseracts can be seen from the penteract. Here are some of the interesting figures I found in each model by viewing the model from different angles.

The Tesseract (4-cube):

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(Figure 18) (Figure 19) (Figure 20)

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(Figure 21) (Figure 22) (Figure 23)

The Penteract (5-cube):

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(Figure 24) (Figure 25) (Figure 26) (Figure 27)

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(Figure 28) (Figure 29) (Figure 30) (Figure 31)

The tesseract (Figures 18-23) can be seen as a cube from the front and back sides, right and left sides, top and bottom sides, and diagonally front right bottom and back left top sides. The same applies to the penteract except that the figure seen is the tesseract. Also, with the penteract (Figure 24-31), there is another added side in which the viewer can see a tesseract, the front top left – diagonally upper left – side and the back bottom right – diagonally lower right – side. As mentioned in Rule 9, when the model is rotated to a side where the previous dimensional figure can be seen, there are always 2 of the figures. Also, when rotating the model, the previous dimensional figure seems to duplicate itself and stretch out towards the direction of the new axis (Figure 27 & 31).

Symmetry

Rule 5 (figure must have rotational, reflection, and point symmetry) did not occur to me until after I created the real-life models of the 4-cube and 5-cube. I realized the 4-cube only has rotational symmetry if rotated along the w-axis (when rotating on w-axis, the 4-cube appears as Figure 23). If rotated only the x, y, or z axis, one of the two 3-cubes will always be the opposite side. The same thing applies to the reflection symmetry of the 4-cube in that there is only symmetry if the 4-cube is reflected over the w-axis. The last type of symmetry is point symmetry, which requires that all parts from one point to the other point are symmetrical. It seems that from any vertex on the outer perimeter of the 4-cube to the vertex on the opposite side of the 4-cube, there is point symmetry (see Figure 22 – ex: far upper right vertex and far lower left vertex) For the 5-cube, the rotational, reflection, and point symmetry are the same as that of the 4-cube, except that the line of symmetry is now the v-axis instead of the w-axis. This makes sense because every time a cube increases in dimension, the cube needs to be duplicated in a direction and then connected by a new axis. Since the cube is duplicated off to the side, the line of symmetry is shifted. However, the 3-cube seems to have the ability of rotational, reflection, and point symmetry along all 3 of its axes, so why does the 4-cube and 5-cube differ? I believe that the 3-cube has several lines of symmetry because the z-axis, the axis introduced when increasing to the 3rd dimension, is ambiguous. X, y and z are all exactly 90 degrees apart from each other, creating equal perpendicular lines. The w-axis is 45 degrees off of x, y, and z; therefore, the introduction of the w-axis creates an angle that does not complement the perpendicular axes, and the introduction of the v-axis creates an even more distorted angle that renders the w-axis obsolete as a line of symmetry. I base this hypothesis on my observations on my real-life models, which could have slight errors in measurements and calculations.

Conclusion and Evaluation

I personally enjoyed exploring the world of dimensions and geometry in a new scope. I have created real-life models of things I did not believe could exist in our world prior to this investigation. My findings with the tesseract and penteract are impressive, but still quite amateurish as I am not an expert in this field. I created my own rules to help me establish order and principle, so that I will not go off on tangents or create objects based completely on abstract thinking. However, I am most likely not the first to come up with the 10 rules I used to create the 4-cube and 5-cube and there is always room for improvement on my real-life models.

Based on my observations on the real-life models of the tesseract and penteract, I came up with several implications:

1. As a cube increases in dimension, a cube will gradually begin to represent a sphere more than a cube. (Figure 12) I realized that with each new dimension, the cube seems to have a more spherical shape. Even the mathematical terms for the tesseract and penteract (octachoron and decateron, respectively) suggest that the figure is approaching an infinite number of vertices, lines, etc. However, the cube would never actually become a sphere because the cube has a definite number of axes and the edges and vertices would not perfectly line up to a rounded surface.
2. Cubes in our modern world are misrepresented. When we think of a 3-cube, we generally think of the figure shaped like a box (Figure 18, 19, 21). However, Figure 21 is different in that the figure appears to be resting on a surface while Figures 18 and 19 seem to be floating. When we draw a 3-cube on a piece of paper, we must draw a diagonal line to represent the z-axis; otherwise, the drawing would look like a square. Drawing the z-axis in this way would be inaccurate because as we see in Figures 18 and 19, the diagonal line should actually be the w-axis. The z-axis should be the axis hidden behind the first cube. If we try to draw the 3-cube using the correct way to represent the z-axis, the resulting drawing would just be a square because the z-axis is hidden behind the square. Perhaps this means that every time we draw a 3-cube on a piece of paper, we are not actually drawing a 3-cube, but a 4-cube; the drawing may look only like a 3-cube because one of the axes is hidden.
3. For full understanding of a figure from one dimension, the next dimension must be explored. This implication extends from the previous implication. Personally, I had never thought of a 3-cube that we draw on a piece of paper as a 4-cube in disguise. I always thought that the 3-cube with a diagonal z-axis is accurate without question, but after I created the real-life model of the 4-cube, I realized that perhaps our 2-D representation (paper drawing) of the 3-D cube is wrong. And I soon wondered if my 3-D representation (real-life model) of the 4-D tesseract is wrong. Thus, I built the 3-D representation of the 5-D penteract, which unfortunately did not extend my understanding of the tesseract very much. Perhaps the shortage for understanding is due to our existence in the 3-D world. No matter what n-dimensional real-life models I want to create, I can only create a 3-D representation of a higher dimensional figure, which is insufficient for complete understanding of higher dimensions.
4. Math is a naturally inextinguishable source of knowledge and creativity. As there are things that drive our everyday ambitions and actions, such as literature and art, there are none that are as vast as the mathematical realm. Through math, we create abstract ideas, we create hypotheses which could one day become theories, and we create concrete examples to support the abstract ideas. With such creative manipulation of such strict, rigid laws, there is close to nothing that mathematics cannot do. If measurements and calculations are done correctly, creations based on mathematical concepts could have the potential to change not only our world, but possibly other worlds. The only question is when will the mathematical concept become recognized and explored?

WORD COUNT: 5,119 (Introduction to “…recognized and explored?”)