Author: Brian Kevin
Higher-dimensional modeling is a powerful approach in various fields, from physics and mathematics to computer science and economics, enabling the representation of complex systems and phenomena beyond the constraints of three-dimensional space. It allows for integrating multiple parametrizable characteristics, such as time and scale, as additional geometric dimensions, offering new possibilities for analysis and problem-solving across diverse disciplines. In physics, the theory of relativity changed how we think about space and time, demonstrating that measurements such as distance, velocity, and acceleration depend on the observer’s frame of reference. Herein, the thought experiment describes a way to conceptualize higher dimensions —dimensions beyond the three (length, width, and height) that humans can perceive—by treating their boundaries as equal to the ‘relative’ measurements derived from relationships in the dimensions we already understand, at infinitely increasing higher dimensions; for example, at infinitely higher dimensions, the length of the object at (n) dimensions equals the relative length/difference in point measurement between the (n)D and (n-1)D. It also describes why time should not be perceived as a dimension. This conceptualization is abstract and, hence tries to shy away from existing concepts, although it makes an effort to validate itself against the fundamental concepts of science, such as energy conservation.
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at infinitely higher dimensions, the length of the object at (n) dimensions equals the relative length/difference in point measurement between the (n)D and (n-1)D.
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Humans perceive (experience and understand) components of the world up to three dimensions because they exist as three-dimensional beings. Their perception of dimensions builds up progressively: a one-dimensional object, such as a line, is made of points connected in a row. A two-dimensional object, for example, a square, is formed by connected lines, each of which can also be conceived as a series of connected dots (the same is true for areas and volumes). A three-dimensional object, such as a cube, is built from connected squares.
But higher dimensions? These are difficult to imagine directly because they are outside our sensory experience.
Extending the described pattern above to higher dimensions, Imagine that a 4D cube (also called a tesseract) is formed by connecting 3D cubes in the same way that a cube is formed by connecting squares. In this higher-dimensional space, we could measure relationships between these 3D cubes—perhaps something analogous to ‘distance’ or ‘volume’- but these measurements would be new and unfamiliar to us because they occur in a dimension we cannot directly perceive.
What if predefined physical constraints do not bound higher dimensions, but instead have the bounds defined as ‘relative’ measurements? Similar to how acceleration is a relationship between changes in velocity. For instance, in three dimensions, we can measure how far apart two objects are (distance). In higher dimensions, the ‘distance’ might represent something more abstract—such as how a 3D object interacts with or relates to another in a way that may incorporate time, motion, or even forces. In a literal sense, getting the constraints between 3D objects will always yield 3D objects, hence necessitating the imaginary status of the constraints of higher dimensions. The space between two cubes will always be perceived as a cube or a cuboid. This does not get us anywhere. This texts argues against the dimensionality of time, in the example of velocity and acceleration, these parameters are factors of time. As will be clear later, they could be constraints of higher dimensions, but not time itself. Below, different ideas are described to lead to the general concept of relative measurements constituting the constraints of higher dimensions.
For the sequence from lower to higher orders described previously to hold, the joining of the dots, lines, or shapes must follow a certain rule that aligns with the laws of nature. If we are to form a line by joining dots, then they must be straightly aligned for the two extreme points to be linked. This is because two squares linked randomly can form an ‘open square’ or other skewed shape.
If forming a cube from existing squares, then one must appreciate that each line of the squares holds corresponding dots, and so joining them at any parallel lines to form an enclosure should produce boundaries that are twice as ‘thick’. This is not the case. We could assume, that for any of the squares, the probability of any two parallel sides is equal(internally, it’s the probability of a point particle existing in any space), and thus assume that the points(point particles) forming the lines are related by a factor. In the case where the higher dimensional would possess boundaries that have the same/equal measurement of constraints, then the factor is naturalized to one, and what determines which of the lines join will be the proximities, i.e the measured relative ‘length’ between the lower dimensional constraints.
The idea of convolved shapes claims that higher dimensions could be a series of convolved lower dimensional constraints and that the pattern of convolution is governed by laws consistent with those of nature. What then determined how dimensions can be convolved? And does it account for the equal thickness despite joining two lines? Quantum entanglement? Or just simple energy states like entropy? In quantum mechanics, when two particles become entangled, their quantum states are connected regardless of the distance between them. This quantum entanglement. At the same time, particles existing in the same energy state can be described as a single system. They are one of the same thing.
Time, which is considered the fourth dimension is an example of this. It is physically imperceptible, but when convolved with the effect of other parameters, it becomes so tangible. A force causes a body to change its position, and what humans observe is an object changing its position, by using the physical quantity of point measurements, we define a length, the distance the object moves. Since the cube can be defined as a system within a larger system, its effects on the larger system are relative, hence we define a displacement. But time is not entirely a dimension.