28-09-2012, 03:33 PM
4D Visualization
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Introduction
Higher-Dimensional Space
The world around us exists in 3-dimensional (3D) space. There are 3 pairs of cardinal directions: left and right, forward and backward, and up and down. All other directions are simply combinations of these fundamental directions. Mathematically, these pairs of directions correspond with three coordinate axes, which are conventionally labelled X, Y, and Z, respectively.
The arrows in the diagram indicate which directions are considered numerically positive and which are negative. By convention, right is positive X, left is negative X, forward is positive Y, backward is negative Y, and up is positive Z, and down is negative Z. We shall refer to these directions as +X, -X, +Y, -Y, +Z, and -Z, respectively. The point where the coordinate axes intersect is called the origin.
As far as we know, the space we inhabit consists of these 3 dimensions, and no more. We may think that space has to be 3-dimensional, that it can't possibly be anything else. Physically, this may be true, but mathematically, there is nothing special about the number 3 that makes it the only possible number of dimensions space can have. It is possible to have dimensions lower than 3: for example, 1D space consists of a single straight line stretching off to infinity at either end; and 2D space consists of a flat plane, extending in length and width indefinitely. However, nothing about geometry restricts us to 3 dimensions or less. It is quite possible—and mathematically straightforward—to deal with geometry in more than 3 spatial dimensions. In particular, we can have a 4th spatial dimension that lies perpendicular to all 3 of the familiar cardinal directions in our world. The space described by these 4 dimensions is called 4-dimensional space, or 4D space for short.
Why Bother?
Why bother trying to visualize a higher-dimensional space that we can neither experience nor access directly? Besides pure curiosity, 4D visualization has a wide variety of useful applications.
Mathematicians have long wondered how to visualize 4D space. In calculus, a very useful method of understanding functions is to graph them. We can plot a real-valued function of one variable on a piece of graph paper, which is 2D. We can also plot a real-valued function of two variables using a 3D graph. However, we run into trouble with even the simplest complex-valued function of 1 complex argument: every complex number has two parts, the real part and the imaginary part, and requires 2 dimensions to be fully depicted. This means that we need 4 dimensions to plot the graph of the complex function. But to see the resulting graph, one must be able to visualize 4D.
Einstein's theory of Special Relativity postulates that space and time are interrelated, forming a space-time continuum of 3 spatial dimensions and 1 temporal dimension. While it is possible to visualize space-time simply by treating time as time and examining “snapshots” of space-time objects at various points in time, it is also useful to treat space-time geometrically. For example, the distance between two events is the distance between two 4D points. The light-cone also has a particular shape that can only be adequately visualized as a 4D object.
Is it possible to visualize 4D
Some believe that it is impossible for us to visualize 4D, since we are confined to 3D and therefore cannot directly experience it. However, it is possible to develop a good idea of what 4D objects look like: the key lies in the fact that to see N dimensions, one only needs an (N-1)-dimensional retina.
Even though we are 3D beings who live in a 3D world, our eyes actually only see in 2D. Our retina has only a 2D surface area with which it can detect light coming into our eye. What our eye sees is in fact not 3D, but a 2D projection of the 3D world we are looking at.
Using Cross-sections
Since we are creatures confined to 3D, we have no way of directly exploring higher-dimensional objects. We can, however, employ various indirect means to study and understand them. One method is to intersect a higher-dimensional object with our world to see what its various cross-sections look like.
To illustrate this, let's apply dimensional analogy again. Suppose we are only 2D beings, living in a 2D world, and unable to see into the 3rd dimension. Suppose we're trying to understand what a cylinder is. We know what circles and squares are, because these objects exist in our 2D world and we can directly handle and see them. But we haven't the slightest idea what a cylinder might be. We have no way of directly seeing such an object, because our retina is only 1D, and a 2D retina is needed to adequately perceive a 3D object