General questions
What is 4D?
4D stands for 4-dimensional or 4 dimensions. It commonly refers to either 4-dimensional space-time in Einstein's theory of General Relativity, or to 4-dimensional Euclidean space. On this site, we mainly focus on 4-dimensional Euclidean space.
What is Einstein's theory of General Relativity?
General relativity is a theory in physics which explains gravity in a geometric way, rather than as a force. You can find more information about general relativity on Wikipedia. On this site, we are more concerned with 4-dimensional Euclidean space rather than 4-dimensional space-time.
What is 4-dimensional Euclidean space?
Euclidean space is the generalization of 2- and 3-dimensional spaces studied by the Greek mathematician Euclid, to any finite number of dimensions. It encompasses the concepts of length and angles.
Two-dimensional Euclidean space is like a flat plane, where every point is located by two coordinates. There is only width and height in 2-dimensional space. There are two perpendicular directions in 2D space. Three-dimensional space is like the space we live in, where every point is located by three coordinates. There is height, width, and also depth in 3-dimensional space. There are 3 mutually perpendicular directions in 3D space.
Likewise, 4-dimensional Euclidean space is one where every point is located by four coordinates. Not only there is height, width, and depth, there is also a fourth quantity that describes 4D thickness. There are 4 mutually perpendicular directions in 4D space.
How can there be 4 mutually perpendicular directions?
The space we live in is only 3-dimensional, and so it is impossible to have 4 mutually perpendicular directions. However, we are interested in 4D space, which has another dimension in addition to the familiar 3. This additional dimension makes it possible to have 4 mutually perpendicular directions.
Isn't the 4th dimension just time?
In Einstein's theory of General Relativity, the 4th dimension is indeed time. However, what we are interested in here is not merely understanding 4D space-time as 3D slices along various points in time. What we are interested in is to understand the 4th dimension as a spatial dimension. If we wanted to add time into the picture, we could say time is the 5th dimension.
However, labels such as “4th dimension” are just that: mere labels. In General Relativity, it refers to time; but mathematically-speaking, it can equally validly refer to another spatial dimension. That is what we are interested in here.
Nevertheless, this does not mean that what we learn here is useless in dealing with 4D space-time. On the contrary, developing a geometric understanding of 4D space can help a lot in understanding 4D space-time, because mathematically-speaking, both deal with vectors having 4 components. Much of what we learn here can be applied to 4D space-time as well. For example, the shape of a light cone is a 4-dimensional spherical cone. Understanding how to visualize this object geometrically helps us see what a light cone really looks like.
I have trouble visualizing 4 mutually perpendicular directions.
So do most people. The space we live in is only 3-dimensional; so it is not surprising that we have trouble understanding 4-dimensional space. We have no direct experience of 4D. To understand 4D requires some help. That is what this site is about: to teach you various methods of visualizing 4D and to help you develop an idea of what 4D space is like.
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.
There is more to this answer than meets the eye. In reality, our eyes 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 are the pros and cons of 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.
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 are the pros and cons of 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 projecting light from the scene onto the film. By studying such images, we try to reconstruct a mental model of what the 4D object is like.
The projection method is possibly 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 is not the answer to everything. Sometimes, projection images 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.
The Projection Method
What's so good about projections? I saw a Java applet that does 4D projections, but it only confused me more.
Firstly, the projection method is not a silver bullet; you still need to learn how to interpret the projection images properly before you can understand it. Staring at it without understanding what is happening will not magically make you understand 4D.
Secondly, many Java applets that does 4D projections do not perform hidden surface removal. That is to say, it's like being in a room where everything is transparent glass with no opaque surfaces. You can't tell what is in front and what is behind. Only if you already know what to expect, or if you grope around, will you be able to understand what you see. Unfortunately, we have no prior experience of 4D and cannot directly explore it, so this doesn't help.
Thirdly, even with applets that do perform hidden surface removal, the result is often still very confusing, because (almost?) all applets use line-drawings to represent the projections. While this makes it more feasible to program, the resulting images can be highly ambiguous and suffer heavily from illusions. This is because line-drawings do not convey enough information to unambiguously define the 3D shape of the projection, and so our minds would often interpret it wrongly.
What are some of the ways to improve this situation?
The underlying problem behind all these issues is that when we use the computer screen to display the images, we are actually performing a further projection from the 3D image to the 2D screen. This means we're losing information from two of the dimensions of the original 4D object. It is very difficult to reconstruct the 4D object from such deficient information.
The solution, therefore, lies in minimizing the loss of information by using better methods of conveying the 3D depth of the image. These methods include:
- Use some form of depth cueing, such as dotted lines for edges that lie on the far side of the projection. While helpful to a certain extent, this method is still prone to ambiguity.
- Use a red/cyan scheme with 3D glasses in order to see 3D depth correctly. This method relies on the fact that our mind is much better at inferring depth when there is sufficient parallax. It works much better in experience than merely using depth-cueing such as dotted lines.
- Use parallel or cross-eyed stereo projections. This is essentially the same idea as the previous method in that it tries to make use of parallax.
- Exaggerate the thickness of lines and/or vertices, so that the viewer can infer, based on which lines are obscuring which other lines, where exactly each line lies in 3D space. One could even use 3D tools such as ray-tracers, like POV-Ray, to create realistic lighting, shading and shadows, so that the 3D shape of the image is more immediately obvious.
A superior approach to all these is to discard the line-drawing method altogether, and use 2D surfaces to represent the 3D image. This makes the ridges of the 4D object much clearer, and makes the bounding volumes unambiguous. However, this is harder to program and more difficult to pull off successfully, because one must use shading to convey the shape of each surface, yet at the same time make it semi-transparent so that other surfaces behind it would still be visible.
