The 4D FAQ
Methods of Visualizing 4D
How can we 3D beings possibly visualize 4D?
Some people believe that it is impossible to visualize 4D, because we are confined to 3D, and have no direct experience of 4D, and probably will never be able to directly see 4D. However, most of us have never seen the earth from space either, and most of us probably never will; yet we have a very good idea of how the earth is a very large sphere. How is this possible? The answer is, photographs.
Although we live in 3D space, our eyes actually cannot see 3D directly. The retina of our eyes are only 2-dimensional arrays of light-sensitive cells; so what our eyes see are really only 2D “photographs” of the 3D world around us! Yet we have no trouble at all inferring the 3rd dimension from the 2D images that our eyes see. Likewise, a hypothetical 4D person would not see 4D directly, but only 3-dimensional images of the 4D world. But it would be able to infer the 4th dimension from those 3D images.
Now, the key here is that what the 4D being sees are 3D images, and we understand 3D very well, so it is perfectly possible for us to “see” the images in the 4D being's eyes. All we need to do is to learn how to infer the 4th dimension from these 3D images.
The 4D visualization document on this site discusses this idea in much more detail.
What are some of the methods of 4D visualization?
There are various ways of exploring higher-dimensional spaces. The 4D visualization document describes these methods in more detail. As a quick overview, 4D visualization methods include: dimensional analogy, cross-sections, and projections.
What is dimensional analogy and how is it useful?
Dimensional analogy is a method of inferring things about higher dimensions by seeing how lower dimensions relate to our 3D world.
First, we study how things in lower-dimensional spaces such as 2D behave. Then we study the equivalent things in 3D, and compare the two, finding out how something in 2D generalizes to 3D. Then we apply the same principle to infer what would happen if we generalized from 3D to 4D.
Dimensional analogy is a very powerful tool that enables us to understand things that happen in 4D by comparing them to analogous things in 3D.
What is the cross-section method?
The cross-section method takes a 4-dimensional object and intersects it with our 3D world, similar to how we might take a 3D object and intersect it with the 2D plane to see what is the shape of its intersection. By taking many such intersections of the same object, we can derive some useful information about it.
What are the pros and cons of the cross-section method?
The cross-section method has its place in our 4D visualization toolbox, but unfortunately the information it gives can be rather hard to understand. It is like collecting many pieces of steak and trying to imagine what a cow looks like just based on the shape of the slices.
What is the projection method?
The projection method takes a 4-dimensional object and projects it onto a 3D hyperplane, similar to how a camera captures an image of a scene by capturing light projected from the scene onto the film. By studying such images, we try to reconstruct a mental model of what the 4D object is like.
What are the pros and cons of the projection method?
The projection method is the easiest way to develop an intuitive grasp of higher-dimensional space, because it most closely parallels how our own eyes see. Our eyes do not directly see 3D, but only 2D images of it—in particular, 2D projections of it. Our mind then reconstructs a 3D model of the world based on these images. Since this is something so innate to us, the projection method is relatively easy to understand.
However, the projection method has its own disadvantages. Projection images often suffer from illusions, which arise because the image may be ambiguous—it can be interpreted in more than one way. This may lead to an inconsistent mental model of the 4D object, resulting in confusion. Illusions are unavoidable in projections, because projection tries to represent an n-dimensional object with only n-1 dimensions: ambiguity is bound to arise.