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Since time immemorial, humans perceive the world as one of three dimensions. The human eye can look at an object in the real world as a three-dimensional object, but when it appears on a flat screen in turns into a two-dimensional object.
Hosny (2014) states that the human eye sees the world around as a three-dimensional plane since it can process the width, the height, and the depth. In another context, these three characteristics of three-dimensional objects referred as the latitude, the longitude, and the altitude respectively.
But in contrary, the eyes can still perceive the depths of the images in a flat screen like computer screen or televisions even though it is rendered on a two- dimensional plane that only resembles the width and height.
For humans to perceive a three-dimensional object in a two-dimensional plane, it must have depth cues. Depth cues are information telling the human brain the distance of the things relative to the eye. Depth is how far things are from humans.
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“ She followed all my directions. It was really easy to contact her and respond very fast as well. ”
Aside from depth cues, there are also motion in the images to tell inform the brain about the location of an object. And lastly, to perceive three-dimensional objects in two-dimensional plane, viewer must have physiological depth cues Hosny, (2014) added.
It is the changes that the body undergoes when seeing objects and processing information just like watching a 3-dimensional movie in the cinema portraying the different distances of the objects and the human body.
Similar to this, Strickland (2010) thoroughly explains that three-dimensional object appears because of the way a human eye focuses on objects.
The brain interprets the light that passes through the eyes when the light reflects from an object to the eye. The brain then interprets this information and forms an image based on the information gathered by the eye. If an object is far from the eye, the light traveling from the object with one eye is parallel to the light traveling from the object to the other eye.
The secret behind three-dimensional cinema is the illusion by showing the same images in each eye with different locations which tricking the brain into thinking that the flat image has a depth making it a three-dimensional object.
However, physicists and mathematicians have ventured beyond the possibilities of the three-dimensional plane, conceptualizing the idea of hyperspaces of four and even more dimensions.
Kaku (2014) starts his introductions in four-dimensional space by presenting three methods of presenting the hypercube: through their shadows (projection), through their unraveling, and their cross section with 3d space.
Among these three methods, the easiest method to choose in presenting the hypercube is through their shadows in a 2-dimensional plane.
Coxet (2014) suggests following the analogy presented by Abbott in his satirical novel “Flatland: A Romance of Many Dimensions”. If a two-dimensional being (represented by a Flatlander) wanted to see a cube, then the three-dimensional being must present a cube’s shadow in a plane where the Flatlander lives. It means that a shadow of a cube must present two-dimensional characteristics such as length and width.
Given the situation, Abbott (1884) suggests that if a four-dimensional being wants to present a hypercube to a three-dimensional being, it must present a cube’s shadow which is a three-dimensional object to the three-dimensional being. The projected image could be a cube inside a cube as suggest by Kaku, (2014).
This research only covers the creation of a working model of a hypercube that can cast a three-dimensional cube in a three-dimensional plane. Apart from this, the research only focuses on creating a theoretical formulation of the hyper cube’s volume and surface area.
This research upon succeeding will be beneficial, especially to future researchers with the same research that serves as a basis and a related literature with according to the topic.
According to Conway, (2018), there are different ways that higher dimensions, including the four-dimensional hypercube come into play with modern math and physics. In math, there is a topology that may not have the real-world application, but ultimately gives us a small glimpse of the higher dimension and how higher dimensional objects such as the hypercube behave and look like which interests the human mind.
Studying hypothetical objects and planes which is far all know that does not exist in a three-dimensional model of the universe, yet can be defined mathematically.
Also, the study of higher dimensions in physics can make a great contribution in the formulation of a grand unified theory that explains the whole concept of the universe from the general relativity, a theory that explains the large scale structures like galaxies and stars to the quantum mechanics which explains the smallest structures ranging from atoms to subatomic particles but mathematically contradicts the general relativity.
The study anchored entirely on the theory of multiple universes suggested by the Superstring Theory proposed by GabrieleVeneziano, an Italian theoretical physicist and one of the pioneers of the early model of the Superstring Theory. Superstring theory suggests that each elementary particle is smallest version of a string, just like in guitar, oscillating in different ways to create different types of particles (Hamer, 2019).
The major problem the Superstring theory faces is that it requires the universe to have at least 10 dimensions to work, but other version of the theory proposes as many as 26 of these dimensions and humans only perceive the universe as four dimensions. These dimensions allowing humans to perceive up and down, left and right, forward and backward, and the time.
One explanation is the compactification, a notion that suggests that the other dimensions are folded down in ways humans cannot perceive. It is force of the universe imprisoning three-dimensional being in three-dimensional space. It is like looking at a paper directly from a side that suggesting only it is just a line even though the paper has a lot of surface area you are not just seeing (Hammer, 2019).
To break it down, dimensions are facets of reality humans perceive in a daily basis. The awareness of the dimensions in the surrounding defines a three-dimensional being capable of understanding and interpreting the length, width, and depth of all objects in the universe -the x, y, and z axes respectively (Williams, 2014).
Apart from these dimensions lies what scientist and mathematicians believe. In the framework of Superstring theory, the universe extends up to 10 different dimensions. These dimensions are the key to unlock the secret of the universe, and all the elementary particles and fundamental forces of nature that lies within these boundaries.
The first dimension, as already noted, is what defines length (x- axis), and just basically a straight line. Add to it the y-axis (the second dimension) or the height and we can produce two-dimensional shape such as a square with width, and length.
Depth involves in perceiving three-dimensional object (the z-axis) and gives object a sense of area and a cross-section. The common three-dimensional object is the cube which has length, width, height, and hence volume (William, 2014).
Modern scientist believe that time is what makes the fourth-dimension which governs of all known matter at some point in time. It is essential in plotting an object’s position in the universe by determining the time.
According to the Superstring theory, the fifth and sixth dimensions are the idea of multiple possible universes. When seeing in fifth-dimension, the similarities and differences of different world in different point in time are slightly from one another.
When seeing in the sixth-dimension, it is like seeing a plane of possible world, just like seeing in fifth-dimension, but the difference is the ability to compare different worlds and position universes according to a given point in time (i.e. the Big Bang). In theory, if one can master the fifth and the sixth dimension, they could travel back in time or jump into the future.
Access to all possible worlds with different initial condition is that theoretically perceive to gain in the seventh-dimension. The eight- dimension suggests a plane of possible universe history branching from different initial condition infinitely (hence why are they are called infinities).
The ninth-dimension gives the ability to compare different possible universe histories, starting from all different possible laws of physics and initial conditions. The final and tenth-dimension suggests that everything possible and imaginable is covered in this dimension.
The existence of the Superstring theory and the additional six dimensions which humans cannot perceive is necessary for the Superstring theory to be consistent in nature. It is also necessary to the study by defining what the different dimensions are and how it interacts with each other.
The concept of Superstring theory and Superstring theory suggests how objects, shapes, and planes behaves in different conditions, thus, allowing the research to take shape by studying the behavior of a cube with time in a four-dimensional plane and creating an analogue version of a four-dimensional hypercube with three-dimensional shadow and theoretically formulate the surface area and volume.
The Superstring theory or string theory in general is essential in making this research possible because it explains the behavior of objects and axes in different planes and dimensions, hence, supporting the notion of the proponent’s idea of formulating theoretical surface area and volume equation for a four-dimensional hypercube given that the model is correct.
The Superstring theory suggests that elementary particles acts like a small version of Superstrings vibrating at different behavior to produce different types of particles. By studying these Superstrings, it is necessary to establish higher dimensions and add an additional six higher dimension to establish consistency in nature.
The independent variables are the dimensions which are the w, x, y, and z axes assuming that these axes are not on a same plane. The dependent variable is the measurement of the object along planes and how it behaves.
This study was conducted to create a working model of a four-dimensional hypercube and to formulate the theoretical equation for the surface area and the volume.
Specifically, the study attempted to answer the following questions:
The researcher hypothesizes that at the end of the research, the proponent will be able to create a working model of the four-dimensional hypercube. It is expected that the proponent can formulate a working theoretical equation for the volume and the surface area of the hypercube. The researcher also hypothesizes that the working model of the hypercube can show a three-dimensional shadow at a three-dimensional plane to a three-dimensional being.
In conducting this study, the following assumptions are made regarding the study:
The study will be conducted with the aim of formulating a theoretical equation of the surface area and the volume of a four-dimensional hypercube given that the model is working.
It only focuses on building a working model that will be use to formulate the desired theoretical equation.
Also, the research’s only focus of presenting the model of the hypercube is through the use of one of the method suggests by Kaku (2014) in presenting the four-dimensional model of the hypercube to the three-dimensional being which is the use of shadows and lights.
Apart from this, the proponent will theorize an equation for the surface area and volume of the four-dimensional hypercube given that the model is working by examining the surface of the tesseract by presenting a model of unraveled tesseract and apply the general surface area formula in order to derive another surface area formula from the sides of the model.
The proposed equation is limited only on answering the surface area and volume of the four-dimensional hypercube. Any four-dimensional object except for the hypercube will be not satisfying the given theoretical equation. It served as another big step towards a better understanding of four and higher dimensions by examining how objects in four dimensions look like and behave.
The study will be valuable and significant to the following entities.
Dimensions - a measurable extent of some kind, such as length, breadth, depth, or height; a plane or measurement of an object
Fourth- dimension- time regarded as analogous to linear dimensions; a dimension in which time is essential in determining the location of an object
Third-dimension- a geometric setting in which three values are required to determine the position of an element; the physical world
Flatland- a region in which the land is predominantly flat; a plane consists of only x, and y axes (width and length respectively)
Unlocking Dimensions: Hypercube in Four-Dimensional Space. (2024, Feb 06). Retrieved from https://studymoose.com/document/unlocking-dimensions-hypercube-in-four-dimensional-space
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