The 4D FAQ
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. This is different from the geometry of space-time in General Relativity, because we are dealing with 4-dimensional Euclidean space, but the geometry of space-time in General Relativity is not Euclidean.
Regardless, 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.
However, 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 understand what kind of shape a light cone really is.
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.