This article explores the importance of multidimensional theories in modern physics, tracing the history of dimensions from the ancient Euclidean concept of dimensions, through the mathematical developments of Descartes and Riemann, to Einstein’s theory of relativity.
Euclid is credited with using the term “dimension” to give mathematical meaning to the properties of objects: length, width, and depth. In Euclidean geometry, a straight line is defined as a typical one-dimensional object because it has only one property: length. In the same way, a plane with the properties of length and width is typical of a two-dimensional object, while a solid with length, width, and depth is typical of a three-dimensional object. In this way, the mathematics of Euclid’s time provided mathematical support for the ancient Greeks’ idea of a three-dimensional world. Ancient Greek philosophers of the time sought to understand the fundamental structure of the material world through mathematical geometry, and Euclid’s geometry became an important tool in this philosophical quest.
For generations after Euclid, the world continued to be perceived as three-dimensional. Any idea of a fourth dimension was dismissed as mathematically absurd. Even the great astronomer Tollemi didn’t believe in the idea of a fourth dimension. His explanation was that it was possible to draw three straight lines perpendicular to each other in space, but it was impossible to draw a fourth such axis. This was because the physics and philosophy of the time viewed space as absolute, meaning that it existed only within measurable limits. As a result, concepts beyond four dimensions were considered abstract and far removed from reality, and many scholars did not take them seriously.
In modern times, the French mathematician Descartes approached geometry in a different way than Euclid. He introduced an abstract numerical system called “coordinates” rather than the length, width, and depth of an object. According to him, the dimensions of an object correlate to the number of coordinates needed to represent it. For example, a line is one-dimensional because it can be represented using only one coordinate, while a plane, which can be represented using two coordinates, is two-dimensional. In the same way, a solid is three-dimensional because it requires three coordinates to represent it. While Euclid’s dimensions were qualitative in the sense that they were based on the properties of sensory objects, Descartes’ dimensions were quantitative in the sense that they were based on abstract numbers. Descartes’ approach paved the way for geometry to expand its scope beyond observation of the physical world and into mathematical reasoning and logic. However, he was unable to overcome the resistance of the mathematicians of his time, who were unwilling to acknowledge the possibility of the existence of something that could not be seen.
It wasn’t until the 19th century German mathematician Riemann that the concept of a fourth dimension was recognized. He utilized Descartes’ definition of coordinates to demonstrate that it was possible to describe dimensions from zero to infinity. Riemann’s work opened up a new paradigm in geometry and required a completely new way of thinking for mathematicians. He used mathematical abstractions to go beyond the concept of space, and in doing so, he broke down traditional notions of space and dimensionality. According to him, there is no need to refer to mathematical dimensions only in perceptible space. It is enough to be able to refer to a purely logical conceptual space, which he encompassed in the concept of a manifold. A manifold has as many dimensions as the number of factors that determine it. If an object or domain is made up of an immeasurable number of factors, it is a manifold of near infinite dimensions. Riemann’s theory pushed the limits of the mathematical imagination and had a profound impact on our understanding of space in physics.
Thanks to Riemann’s liberal definition of dimensions, Einstein was able to conclude that the universe is a four-dimensional manifold. The three dimensions of space plus one more dimension, time, could explain the motion of the universe. Einstein’s theory of relativity led to the understanding of time and space as a continuum, which further expanded the concept of dimensionality in modern physics. Today, the study of multidimensional space continues, and it remains an important topic of ever-evolving scientific inquiry.
