Why you can’t tie knots in four dimensions
- Written by Zsuzsanna Dancso, Associate Professor of Mathematics, University of Sydney
We all know we live in three-dimensional space. But what does it mean when people talk about four dimensions?
Is it just a bigger kind of space? Is it “space-time”, the popular idea which emerged from Einstein’s theory of relativity?
If you have wondered what four dimensions really look like, you may have have come across drawings of a “four-dimensional cube”. But our brains are wired to interpret drawings on flat paper as two- or at most three-dimensional, not four-dimensional.
The almost insurmountable difficulty of visualising 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 the thickness is negligible compared with the length.
A surface, such as a soccer field or the skin of a balloon, is two-dimensional. There are independent directions forwards and sideways.
You can move diagonally on a surface, but this is not an independent direction because you can get to the same place by moving forwards, then sideways. The space we live in is three-dimensional: in addition to moving forwards 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 moving forward or backward in time counts as a new direction.
One way to imagine four-dimensional space is as an immersive three-dimensional movie, where each “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 more 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 need to keep your lines neat!)
This experiment can help accurately determine how many corners and edges a higher-dimensional cube has. But for most of us, it will not help us “see” one. Our brains 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 on each other”. This is why a long rope wound around itself, if done right, won’t come apart. We trust knots with our lives when we’re sailing or climbing.
Two ropes catch on each other if pulled in opposite directions. This is what makes knotting possible.
Zsuzsanna Dancso, CC BY
But in four dimensions, knots would instantly come apart. We can understand why by using an example in fewer dimensions, like we did with cubes.
Imagine a colony of two-dimensional ants living on a flat surface divided by a line. The ants can’t cross the line: it’s an impassable barrier for them, and they don’t even know the other side of the line exists.
A colony of flat ants in a two-dimensional world don’t even know that a world on the other side of the line exists.
Zsuzsanna Dancso, CC BY
But if one day an ant, and its world, becomes three-dimensional, that ant will step over the line with ease. To step over, it needs to move just a tiny bit in the new, vertical direction.
If one ant becomes three-dimensional, it can see across the line and step over it with ease.
Zsuzsanna Dancso, CC BY
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 on each other if pulled in opposite directions.
But if the space became four-dimensional, it would be enough for the horizontal piece of rope to move just a little bit in the new, fourth direction, to avoid the other entirely.
Thinking 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, in that frame there is no vertical piece, so it can easily move to the other side of the vertical piece before shifting back.
Imagine four-dimensional space as a movie of three-dimensional frames. The bottom left cube shows a horizontal piece of rope in front of a vertical piece, both in the ‘present’ frame. The horizontal piece can move into the future frame (second column), where it is able to slide towards the back (third column), then move back into the present frame, now behind the vertical piece.
Zsuzsanna Dancso, CC BY
From our three-dimensional perspective, the ropes would appear to slide through each other like ghosts.
Knots in more dimensions
Is it impossible, then, to knot a rope in higher dimensions? Yes: any knot tied on a rope will come apart.
But not all is lost: in four-dimensional space you can knot two-dimensional surfaces, such as balloons, large picnic blankets or long tubes.
There is a mathematical formula that determines when knots can stay knotted: 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 space where knotting is possible.
The formula implies, for example, that a rope (one-dimensional) can be knotted in at most three dimensions. A (two-dimensional) balloon surface can be knotted in at most five dimensions.
Studying knotted surfaces in four-dimensional space is a vibrant topic of research, which provides mathematical insight into the the still poorly understood mysteries into the intricacies of four-dimensional space.
Authors: Zsuzsanna Dancso, Associate Professor of Mathematics, University of Sydney
Read more https://theconversation.com/why-you-cant-tie-knots-in-four-dimensions-272445
