When thinking about dimensions, or dimensionality, it is helpful to think of each dimension as representing a degree of freedom (or variable) in a system. In M-Theory, these dimensions represent different possible modes of vibration of a string. The math demands extra dimensions in order for the equations to work out. When thinking about a “Theory of Everything”, we must conclude that we would need enough variables to describe absolutely EVERYTHING in human experience, both objective and subjective. It is in the mathematical constructs of extra dimensions in M-Theory that we may find a mathematical framework for understanding, or at least conceptualizing, how everything in our lives fits together.
When adding dimensions, each dimension above the previous one is the next logical extension to the system. It is easier to visualize using points, lines, planes and volumes, and this generally accepted method is derived from the point-line-plane postulate in Euclidian Geometry. Basically, you start with a point (Dimension 0). An infinite series of points defines the next dimension which would be a line of infinite length (Dimension 1). Another infinite line perpendicular to the first defines the subsequent dimension which would be a plane of infinite area (Dimension 2). An infinite series of planes stacked on top of each other (or, alternately, another infinite line perpendicular to the plane) defines the next dimension which would be a cube (or space) of infinite volume (Dimension 3). Now for Dimension 4 we represent Dimension 3 as a point in the next higher dimension and repeat the process. In this methodology, each dimension is considered orthogonal (90 degrees) to the one before it.
It is worth noting that in order to describe a “space” as we progress through the dimensions, we generally need three dimensions to do it. How we define them depends more on what we are trying to characterize than anything. For example, an infinite set of constants of infinite possible values could be defined as an infinite plane with the x-axis representing the values and the y-axis representing the constants (or vice versa). This works fine for constants (or variables) that have no direction, just magnitude. On the other hand, we could also represent the same set of constants with no direction by representing the x-axis as one variable and the y-axis as another variable to make a plane. Then the z-axis could represent an infinite stack of planes representing 2 variables each. But which is more correct? In the context of dimensionality this problem comes up when referring to “initial conditions” and “interaction variables” as we will see later. Now initial conditions could represent a very wide variety of things, including both constants with a direction (such as a force acting on a particle) and without direction (such as a mathematical constant like C or G). So a third axis is needed in order to define a “direction” for those variables that have a direction. So when representing initial conditions or interaction variables we really do need to use three dimensions to fully describe them: one for all the possible variables, one for their “magnitude”, and one for their “direction”.
Life in 12 Dimensions 