We all know that we live in three-dimensional space. But what does it mean when people talk about four dimensions?
Is it just a larger type of space? is it”space-time,” the popular idea that emerged from Einstein’s theory of relativity?
The almost insurmountable difficulty of visualizing the fourth dimension has inspired mathematicians, physicists, writers and even some artists for centuries. But even if we can’t quite imagine it, we can understand it.
What is dimension?
The dimension of a space captures the number of independent directions in it.
A line is one-dimensional. We can move along it forwards and backwards, but these are opposite, not independent, directions. You can also think of a string or piece of rope as practically one-dimensional, as its thickness is negligible compared to its length.
A surface, such as a football field or the skin of a balloon, is two-dimensional. There are independent forward and sideways directions.
You can move diagonally on a surface, but this is not an independent direction because you can get to the same place by moving forward, then sideways. The space we live in is three-dimensional: in addition to moving forward and sideways, we can also jump up and down.
Four-dimensional space has yet another independent direction. This is why space-time is considered four-dimensional: you have the three dimensions of space, but move forward or backward in time counts as a new direction.
One way to imagine four-dimensional space is like an immersive three-dimensional movie, where every “frame” is three-dimensional and you can also fast forward and rewind in time.
Consider the cube
A powerful tool for understanding higher dimensions is through analogies in lower dimensions. An example of this technique is drawing cubes in several dimensions.
A “two-dimensional cube” is just a square. To draw a three-dimensional cube, we draw two squares, then connect them corner to corner to make a cube.
So to draw a four-dimensional cube, start by drawing two three-dimensional cubes, then connect them corner to corner. You can even continue doing this to draw cubes in five or more dimensions. (You will need a large piece of paper and must keep the lines neat!)
This experiment can help you determine exactly how many vertices and edges a higher dimensional cube has. But for most of us, it won’t help us to “see” one. Our brain will only interpret the images as complex webs of lines in two or at most three dimensions.
Knots
We can tie knots in three dimensions because one-dimensional ropes “catch each other”. This is why a long rope wrapped around itself, if done correctly, will not come apart. We trust knots with our lives when sailing or climbing.
But in four dimensions, knots would immediately come apart. We can understand why by using an example in fewer dimensions, as we did with cubes.
Imagine a colony of two-dimensional ants living on a flat surface divided by a line. The ants cannot cross the line: it is an impassable barrier for them, and they do not even know that the other side of the line exists.
But if one day an ant, and its world, becomes three-dimensional, that the ant will cross the line with ease. To cross, it needs to move just a tiny bit in the new, vertical direction.
Now, instead of an ant and a line on a flat surface, imagine a horizontal and a vertical piece of rope in three dimensions. These will catch each other if pulled in opposite directions.
But if space became four-dimensional, it would be enough for the horizontal piece of rope to move just slightly in the new, fourth direction, to avoid the other altogether.
When we think of four dimensions as a movie, the pieces of rope live in a single three-dimensional frame. If the horizontal piece of rope shifts just slightly into a future frame, there is no vertical piece in that frame, so it can easily move to the other side of the vertical piece before shifting back.
From our three-dimensional perspective, the ropes seem to slide through each other like ghosts.
Knots in several dimensions
Is it then impossible to tie a rope in higher dimensions? Yes: any knot tied to a rope will come loose.
But all is not lost: in four-dimensional space you can knot two-dimensional surfacesfor example, balloons, large picnic blankets or long tubes.
There is a mathematical formula that determines when knots can remain tied: take the dimension of the object you want to knot, double it and add one. According to the formula, this is the maximum dimension of a room where it is possible to tie.
The formula implies, for example, that a (one-dimensional) rope can be tied in a maximum of three dimensions. A (two-dimensional) balloon surface can be linked in a maximum of five dimensions.
Studying nodal surfaces in four-dimensional space is a lively research topic, which gives mathematical insight into it still poorly understood mysteries into the intricacies of four-dimensional space.
This edited article has been republished from The conversation under a Creative Commons license. Read original article.
