Can you tie a knot in four dimensions? A mathematician explains.
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 bigger space? Is this “space-time“, the popular idea from Einstein’s book theory of relativity?
The almost insurmountable difficulty of visualizing the fourth dimension inspired mathematicians, physicists, writers and even some artists for centuries. But even though we can’t really imagine it, we can understand it.
What is dimension?
The dimension of a space captures the number of independent directions it contains.
A line is one-dimensional. We can move forward and backward in it, but they are opposite directions and not independent. You can also think of a rope or piece of rope as practically one-dimensional, because its thickness is negligible compared to its length.
A surface, like a football field or the skin of a ball, is two-dimensional. There are independent forward and sideways directions.
We can move diagonally across a surface, but this is not an independent direction because we can arrive at 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 spacetime is considered four-dimensional: you have the three dimensions of space, but moving forward or backward in time counts as a new direction.
One way to imagine a four-dimensional space is to create a three-dimensional immersive movie, where each “frame” is three-dimensional and you can also move quickly forward and backward in time.
Consider the cube
Analogies in lower dimensions are a powerful tool for understanding higher dimensions. An example of this technique is drawing cubes in more dimensions.
A “two-dimensional cube” is just a square. To draw a three-dimensional cube, we draw two squares and then connect them corner to corner to form 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 this to draw cubes in five or more dimensions. (You will need a large sheet of paper and will need to keep your lines neat!)
This experiment can help to accurately determine the number of corners and edges of a higher-dimensional cube. But for most of us, that won’t help us “see” any. Our brains will only interpret images as complex networks of lines in two or at most three dimensions.
Knots
We can tie knots in three dimensions because one-dimensional strings “cling to each other.” This is why a long rope wrapped around itself, if done well, will not come loose. We entrust our lives to knots when we navigate or climb.
But in four dimensions, the knots unravel instantly. We can understand why by using an example with fewer dimensions, like we did with cubes.
Imagine a two-dimensional ant colony living on a flat surface divided by a line. 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 become three-dimensional, this ant will easily cross the line. To step over, it must move slightly in the new vertical direction.
Now, instead of an ant and a line on a flat surface, imagine a horizontal and vertical piece of rope in three dimensions. These will snag if pulled in opposite directions.
But if space became four-dimensional, it would be enough for the horizontal piece of rope to move a little in the new fourth direction to completely avoid the other.
Thinking of four dimensions as a movie, the pieces of rope live in a unique three-dimensional framework. If the horizontal piece of rope moves slightly in a future frame, in that frame there is no vertical piece, so it can easily move to the other side of the vertical piece before moving back.
From our three-dimensional perspective, the strings appear to slide past each other like ghosts.
Larger knots
Is it then impossible to tie a rope in higher dimensions? Yes: any knot tied on a rope can be undone.
But all is not lost: in four-dimensional space, we can tie two-dimensional surfaceslike balloons, large picnic blankets or long tubes.
There is a mathematical formula that determines when knots can stay tied: take the size of the object you want to tie, double it, and add one more. According to the formula, this is the maximum dimension of a space where knotting is possible.
The formula implies, for example, that a (one-dimensional) rope can be knotted in at most three dimensions. A (two-dimensional) balloon surface can be knotted in up to five dimensions.
The study of knotted surfaces in four-dimensional space is a dynamic research topic, which provides mathematical overview in the mysteries still poorly understood in the subtleties of four-dimensional space.
This edited article is republished from The conversation under Creative Commons license. Read the original article.
