Dimensions and Perceptibility: Space, Time, and Hyper-Dimensions in Physics and Mathematics

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

17-06-2024

Abstract:

This study delves into fundamental concepts of dimensions across physics and mathematics. Space is initially defined as encompassing height, depth, and width, constituting a three-dimensional continuum where all physical entities reside and move. Classical physics traditionally perceives space through three linear dimensions.

In the disciplines of modern physics, various perspectives on space and time emerge, extending beyond the confines of relativistic physics. Relativistic physics introduces the concept of spacetime, describing a boundless four-dimensional continuum integrating space with time, pivotal in relativity. This concept reshapes our understanding of gravitational forces and the fabric of the universe.

Beyond relativistic physics, other disciplines within modern physics, such as quantum mechanics and statistical physics, offer distinct interpretations of space and time that may align with classical frameworks. These interpretations often diverge from the seamless integration portrayed in relativistic spacetime. Quantum mechanics explores discrete states and probabilistic behaviours, challenging classical notions of continuous space and time. Statistical physics examines macroscopic systems governed by statistical laws, where the concept of time evolves differently from deterministic frameworks.

Mathematically, dimensionality in a space signifies the minimum coordinates necessary to pinpoint any given point within it. For instance, a one-dimensional line necessitates a single coordinate, exemplified by the number line where each point corresponds to a real number. Points are conceptual entities without physical dimensions, rendering them imperceptible to human senses.

Expanding into higher dimensions, a two-dimensional plane consists of infinite lines, each potentially infinite in real numbers, while a three-dimensional space extends further to encompass infinite planes. Beyond these realms lie hyper-dimensions—dimensions surpassing the conventional three spatial dimensions—where human perceptibility ceases due to inherent physical constraints. Entities within our three-dimensional realm remain disconnected from hyper-dimensional spaces, illustrating their imperceptibility.

The study also addresses time as the fourth dimension, akin to hyper-dimensions in its imperceptibility. Time, crucial in the context of spacetime theory, is mathematically treated as a dimension but fundamentally experienced differently from spatial dimensions. This distinction underscores the challenge of representing time through mathematical models despite its difference in human experience compared to spatial dimensions.

Throughout, adherence to established scientific and mathematical principles ensures a coherent representation of these fundamental concepts, illuminating the nuanced interplay between spatial dimensions, time, and hyper-dimensions within the realms of physics and mathematics across different theoretical frameworks.

Dimensions and Perceptibility: Space, Time, and Hyper-Dimensions in Physics and Mathematics

Space encompasses the dimensions of height, depth, and width, within which all objects exist and move. Alternatively, it can be described as a three-dimensional continuum containing positions and directions. In classical physics, space is often considered in three linear dimensions. However, when considering space together with time within the disciplines of modern physics, only the relativistic context describes it as a boundless four-dimensional continuum known as spacetime.

In physics and mathematics, the dimension of a mathematical space is defined as the minimum number of coordinates needed to specify any point within it. For example, a line is one-dimensional because only one coordinate is needed to specify a point on it, such as the point at 5 on a number line.

A number line is a horizontal straight line that visually represents numbers. It is a pictorial representation of real numbers, with every point on the line corresponding to a real number and vice versa.

A point, represented by a dot '.', indicates an exact location in space but has no length, width, or height—no size or dimension. Thus, a point is a conceptual, rather than a physical, entity. Points are imperceptible to humans because they have no physical size.

Since every point on a number line corresponds to a real number, and every real number corresponds to a point, a one-dimensional number line contains an infinite number of points or real numbers. This one-dimensional line is perceptible to humans as it represents countable real numbers.

A two-dimensional plane contains an infinite number of lines, each containing an infinite number of real numbers. A plane with countable lines of real numbers is perceptible to humans, making it a physically perceptible dimension.

Similarly, a three-dimensional space contains an infinite number of planes, each with an infinite number of real numbers. A three-dimensional space with countable planes of real numbers is also perceptible to humans, making it a physically perceptible dimension.

Dimensions beyond the three spatial dimensions are known as hyper-dimensions. Hyper-dimensional spaces are imperceptible to humans because we do not have physical reach beyond the three-dimensional space. Entities within our three-dimensional existence are not physically interactable with hyper-dimensions or hyper-dimensional spaces.

Since hyper-dimensions are imperceptible, the fourth dimension, time, is also imperceptible to us. We represent the fourth dimension through mathematical or conceptual models, often manifested through physical frequencies.

Explanation of the consistency:

Definition of Space:

Classical and Modern Physics: Classical physics often considers space as having three dimensions (height, depth, and width). Modern physics, especially in the context of relativity, combines space and time into a four-dimensional continuum known as spacetime. This accurately reflects the transition from Newtonian to Einsteinian physics.

Mathematical Space:

Dimensions: In mathematics, the dimension of a space is defined by the minimum number of coordinates required to specify a point within it. This definition aligns with the standard mathematical concept of dimensions.

One-Dimensional Line: A line is one-dimensional because a single coordinate suffices to specify a point on it, e.g., the point at 5 on a number line.

Number Line:

Representation: A number line is described as a horizontal line that visually represents numbers, where each point corresponds to a real number. This accurately describes how number lines are used in mathematics.

Concept of a Point:

Properties: A point is represented by a dot and has no dimensions (length, width, height), making it a conceptual entity rather than a physical one. This is consistent with mathematical definitions.

Imperceptibility: Points are imperceptible because they lack physical size, aligning with the conceptual nature of points in geometry.

Infinite Points on a Line:

Infinity in One Dimension: A one-dimensional number line contains an infinite number of points or real numbers, consistent with the concept of continuity in mathematics.

Perceptibility: While individual points are imperceptible, the number line as a whole is perceptible because it represents countable real numbers.

Two-Dimensional Plane:

Infinite Lines: A plane contains an infinite number of lines, each with an infinite number of real numbers. This reflects the properties of two-dimensional spaces.

Perceptibility of Planes: A plane with countable lines of real numbers is perceptible to humans, making it a physically perceptible dimension.

Three-Dimensional Space:

Infinite Planes: A three-dimensional space contains an infinite number of planes, consistent with the properties of three-dimensional geometry.

Perceptibility: A three-dimensional space with countable planes of real numbers is perceptible to humans, aligning with our everyday experience of physical space.

Hyper-Dimensions:

Definition: Dimensions beyond the three spatial dimensions are known as hyper-dimensions. This term is consistent with terminology used in higher-dimensional mathematics and theoretical physics.

Imperceptibility: Hyper-dimensional spaces are imperceptible to humans because we lack physical reach beyond the three-dimensional space. This reflects the limitations of human perception and physical interaction.

Fourth Dimension (Time):

Imperceptibility: Time as a fourth dimension is imperceptible in the same way as hyper-dimensions and is often represented through mathematical or conceptual models. The use of spacetime in relativity theory is not consistent with this statement, where time is treated as a dimension but experienced as naturally as we experience events within spatial dimensions.

By adhering to these established scientific and mathematical principles, the study remains consistent and accurately represents the underlying concepts.