The fourth dimension is a complex concept that deals with abstract reasoning, our sense of perception, and our imagination. Mathematics posits that a four-dimensional space is a geometric space with four dimensions. For many the fourth dimension is the element of time added to the three parameters of length, height and depth. How does a geometer incorporates time in the description of a structure, or a visual artist integrates time in a two dimensional flat surface image, when they both rely on well-defined principles that are a tangible descriptive of our reality? This chapter gives a brief overview of the different schools of thought in the Humanities and in Science, offers a possible definition of this elusive element needed to anchor the fourth dimension in our larger abstract reasoning consensus, and focuses on the specific of Mathematics and visual imaging to illustrate the particular benefit of collaborating on a simple, usable descriptive to create a sound outcome.
TopIntroduction
There is something I ‘know,’ which is that spatial dimensions beyond the Big 3 exist. I can even construct a tesseract or a hypercube out of cardboard. (David Foster Wallace, W. W. Norton & Company, 2004)
Scientists and philosophers have studied the concept of dimension since the beginning of time. Greek philosopher Aristotle and later a mathematician Ptolemy argued that no more than three spatial dimensions were possible. Inversely, Plato in the Allegory of the Cave tells us that concept and perception are two distinct things: “Relying on your physical senses alone is a sure way to make yourself effectively blind,” said Socrates to Glaucon. (Asscher & Widger, 2008).
It was not until much later that philosophers and scientists looked again in the possibility of additional dimensions in our immediate environment. French physicist Jean-Baptiste le Rond d’Alembert introduced the notion that one should consider time as a fourth dimension time in the late 1700s as ”an idea that may be contested but having some merit if only because of its novelty” (d' Alembert, 1751). According to a mathematician and writer E. T. Bell, a mathematician J. L. Lagrange (1811) said a few years later in Mécanique Analytique that the science of mechanics could be considered as the geometry of a space of four dimensions – three Cartesian coordinates with one time coordinate. Although many differ about the meaning and exactitude of this translation, most agree that the fourth element refers to the idea of time, as both a material and a spiritual dimension. This concept paved the way for later studies on a fourth dimension in physics and humanitie
Between 1826 and 1829 mathematicians Nicolai Lobachevsky and János Bolyai constructed a self-consistent system of geometry in hyperbolic space that permits an infinite number of straight lines to pass through a point parallel to a given line. Their theory for the first time in two thousand years questioned the fundamental principle of Euclidian geometry, which stated that, given any straight line and a point not on the line, there is only one other straight line passing through the point, which is parallel to that line (Casey, 2007). It led a German mathematician George Riemann to develop a new kind of differential geometry of space with any numbers or dimensions and curvature, and opened the door to the Einstein’s general theory of relativity where the fourth dimension is associated to the idea of space-time (Overduin, 2007).
TopThe Concept Of A Fourth Dimension
The Philosophical Order
Philosophers and scientist have been well aware of the possibility of higher dimensions in our universe. However they always had difficulties to identify and demonstrate in a rational way where and how these new elements could fit in the larger construct of our physical reality. Scientists, in particular physicists have been studying time since antiquity to assign it an objective quantity of measurement (Figure 1).
Figure 1. “Fleeting time”. Medieval coin, circa 1500, Bargelo museum, Firenze, IT
Three-dimensional space is represented by three coordinate axes x, y, and z. Each one defines a unique parameter of height, width, and depth. Each axis is orthogonal to the other two. A 4-dimensional space then should have an extra coordinate axis, orthogonal to the other three. Aristotelian logic tells us that time does not have a beginning and it is impossible to define the first moment of time, since it would have to be identified from an earlier period of time – which is inconsistent with it being the first moment of time (Barnes, 1984).
German philosopher Immanuel Kant asserts that since we are dependent on our limited senses, all we can know is the way things appear as they are represented to us through our senses and cognition.
The world of ‘things-‐in-themselves' is beyond the reach of our sensory-‐cognitive faculties and hence cannot be known to us. However he admits that the law, which determines that space “has the property of threefold dimension ... is arbitrary, and that God could have chosen another law [from which] an extension with other properties and dimensions would have arisen (Kemp Smith, 1933).