Here's How We Get From A Single Point To A Four-Dimensional Hyperspace
Ideas of dimensions and higher dimensions show up frequently in popular culture and are extremely important in science.
But what, exactly, does it mean to have four dimensions?
Dimension is a surprisingly complicated mathematical concept. Here, we will look at the basics of how we go from zero dimensions to the kind of higher dimensional spaces prevalent in science and science fiction.
The Single Point
We start with a lowly, single point. A mathematical point has zero dimension - it has no length, area, or volume. We cannot say too much about a single point on its own. However, points are the fundamental building blocks of geometry - more interesting spaces and structures are made out of uncountably many points related to each other in some way.
One Dimension
Now we look at something more interesting. The basic one dimensional space is a line, extending out infinitely in both directions, as indicated by the arrows on the left and right ends:
Whenever we are working with a space like a line, or the higher dimensional spaces we will be looking at later, we want to have a way to describe where we are in the space. So, we need a system of coordinates describing the positions of all the points on our line.
Here is where the idea of a number line will be useful to us. We assign a fixed point in the middle of the line to be our origin, or zero, point, choose a fixed unit length, and then identify each point on the line with its distance in our fixed units from the zero point, with points to the right of the origin being positive, and points to the left being negative:
By turning our line into a number line like this, we can say where any point is on the line by referring to its number.
This gives us an idea of what it means for a line to be one-dimensional. We only need one number, or coordinate, to describe where we are.
Much of geometry depends on being able to find the distance between two points in a space. On a line, we can describe the distance between any two points using their numbers by subtracting the smaller number from the larger number. The points represented by 2 and 4 are two units away from each other - 4 - 2 = 2.
Number lines are perfect for representing quantities where we are considering only a single variable - time, incomes, temperatures, anything that can be represented by a scaled quantity.
Before we move on to higher dimensions, there are a couple important subsets of the number line to look at.
First, a set that looks somewhat boring in our one dimensional case, but that will be much cooler in higher dimensions: the set of points that are all one unit away from the origin. On a line, there are only two such points: -1 and 1:
This set is that is just two disconnected points. We can think of this set then as being zero-dimensional.
The second important subset is the unit interval, or the collection of numbers between 0 and 1:
The unit interval is a line segment - a small, bounded piece of the line with a finite length. The unit interval has a length of 1 unit.
Two Dimensions
Now we look at the basic two-dimensional space: the plane. We form a plane by placing two lines, or axes, at right angles to each other:
As with the line, we want to set up a system of coordinates so that we can describe where points are in the plane. To do this, we turn both of our axes into number lines, with the origin of our coordinate system being the point where the two lines intersect:
Now we can describe where any point in the plane is using two coordinates: one number for where the point is relative to the horizontal axis, and one number for where the point is relative to the vertical axis. These coordinates are usually written as an ordered pair - two numbers in parentheses, separated by a comma. The first number is the horizontal coordinate, and the second number is the vertical coordinate:
This is a big reason we think of the plane as being two-dimensional: I need two numbers, and only two numbers, to describe where a point lies in the plane.
As with the line, we want to find the distance between two points. On a line this was very simple - we just subtracted the smaller point from the larger point. On a plane, things are a little more complicated. We will try to find the distance between two points: (2,1) and (1,3). This distance is the length of the line segment connecting those points:
The trick to finding the length of that line is to realize that we can draw in two other lines, making a right triangle:
The advantage of this is that we can pretty easily measure the lengths of the two new sides of our triangle. The length of the horizontal side is the difference of the horizontal coordinates: 2 - 1 = 1. The length of the vertical side is similarly the difference of the vertical coordinates: 3 - 1 = 2.
Now we can find the length of the long side, and thus the distance between our two points, using the Pythagorean Theorem: for a right triangle, the length of the long side is the square root of the sum of the squares of the other two sides.
We take the squares of our two shorter sides: 12 = 1, and 22 = 4. Add these together, and we get 5. Then, the length of the long side of the triangle is the square root of 5. This is an irrational number, and its first few digits are 2.236... So, our two points are about two and a quarter units apart.
In general, we find the distance between two points in the plane by subtracting the coordinates of the points from each other and using the Pythagorean Theorem, as above.
Now that we have an idea of how to find the distance between two points in a plane, we can look at some special sets. On the line, the set of all points that were one unit away from the origin was pretty boring - just 1 and -1. In the plane, this set is somewhat more interesting: a circle centered at the origin, with radius of 1:
This circle is called the unit circle. We saw on the line that the set of points one unit away from the origin was zero-dimensional, one less than the dimensionality of the line. Similarly, we can think of the unit circle as being one-dimensional in a couple ways, again one less than the two-dimensional plane.
First, if we were a teeny tiny little ant crawling along the unit circle, we would not notice the curvature of the circle, and think we were walking in a straight line, making the circle somewhat line-like.
Secondly, we are developing the idea of a space or shape's dimension being how many coordinates one needs to describe a point in the space or shape. We only needed one number to describe a point on a line, so a line is one-dimensional, and we need an ordered pair - two numbers - to describe a point in the plane. We can describe points on the unit circle using just one coordinate - the angle formed by drawing a line from the point to the origin, relative to the right side of the horizontal axis:
This shows one cool aspect of dimensions - it is possible to have a one dimensional shape, our unit circle, "embedded" in a two dimensional plane.
On the one dimensional line, we also looked at the set of points between 0 and 1 - a line segment we called the unit interval. There is a two dimensional version of this set in the plane: the unit square. The unit square is the set of all points for which both coordinates are between 0 and 1:
The unit square is the two dimensional analogue of the unit interval. The area of the unit square is 1, so this represents the baseline for measuring area in the plane.
Three Dimensions
There is no reason to stop at two dimensions. To get a third dimension, allowing us to model space, we repeat what we did to make a plane. We take our plane, and now add a third number line axis through the origin at a right angle to the plane:
We want the new axis, in blue, to be at a right angle to the plane. In this case, we can imagine the blue axis coming out of the screen towards us.
As before, we set up a coordinate system by making our axes into number lines. This lets us describe the location of a point using three numbers - two to describe the point's position relative to the plane as before, and then a third number to describe the point's position relative to the new axis:
So, for our point hovering in space, if we follow the blue line back to the plane, we get a horizontal axis coordinate of 2, a vertical axis coordinate of 3, and the length of that blue line - our third, spatial coordinate - is 5.
We represent points in three dimensional space with ordered triples - our point above has coordinates (2, 3, 5). Three dimensions; three coordinates.
To measure the distance between two points in three dimensional space, we extend the Pythagorean theorem. In the plane, to find the distance between two points, we subtracted the horizontal coordinates from each other, and the vertical coordinates from each other. Then, we used the Pythagorean theorem: we found the squares of these differences, added them up, and took the square root.
Here, we do something similar, but with all three of our coordinates. To find the distance of our point above, (2, 3, 5), from another point, say (0, 2, 2), we start by subtracting the two points' coordinates:
2 - 0 = 2
3 - 2 = 1
5 - 2 = 3
Then, we square each of these differences:
22 = 4
12 = 1
32 = 9
Add these up, and we get 4 + 1 + 9 = 14. Finally, the distance between the two points will be the square root of this sum: the square root of 14 is irrational, and is approximately 3.74. So, these two points are about three and three quarters units apart from each other.
We can also consider the three dimensional versions of our interesting sets. The set of all points in three dimensional space that are one unit away from the origin forms a sphere, named the unit sphere:
Just as the unit circle was a one-dimensional shape embedded in a two-dimensional plane, we can think of the unit sphere as being a two-dimensional surface that lives in a three-dimensional space. We live on the surface of a sphere, and we usually use two coordinates - longitude and latitude - to describe where we are, making a sphere two-dimensional. Further, just as our ant crawling along the unit circle thought it looked like a line, the surface of the earth looks a lot like a flat plane to us humans, being very close up to that surface.
We also have a three dimensional version of the unit interval and unit square - the set of all points whose coordinates are all between 0 and 1. This forms the unit cube:
In this sense, a cube is the three dimensional version of a square. Just as the unit interval had a length of 1, and the unit square has an area of 1, the unit cube has a volume of 1.
A big thing that has been happening as we have gone from one to two to three dimensions is that all we are really doing when we add a dimension is just adding an extra coordinate to describe our points. This is how we get to higher dimensional spaces - we can keep on throwing in new coordinates as much as we like.
Four Dimensions
To get a four-dimensional hyperspace, we take our three dimensional space and throw in a new axis, somehow at right angles to everything. This is super-hard to visualize, but this is why we have math.
All that I've done from a mathematical perspective is put in another coordinate. Points in a four-dimensional space can be described by an ordered quadruple - my origin point would be (0, 0, 0, 0), some other point might be (1, -3, 4, 7).
We can even define the distance between two points in the same way we did for the plane and for three-dimensional space by continuing to extend the Pythagorean Theorem. Take the differences of the two points' coordinates, square the differences, add up the squares, and take the square root of that sum.
This lets us define a unit hypersphere - the set of all points in 4-space that are one unit away from the origin. Just as the circle was a one-dimensional shape embedded in the two-dimensional plane, and the sphere a two-dimensional shape embedded in three-dimensional space, the hypersphere is a three-dimensional space, embedded in in our four-dimensional hyperspace.
It is actually possible that, on an unimaginably vast scale, the universe itself is such a hypersphere.
We can also make a four-dimensional version of a cube. The unit hypercube is defined as the set of all points in four dimensional space such that all four coordinates of the point fall between 0 and 1.
As with anything involving higher dimensions, it is hard to visualize a hypercube. One method for trying to get an idea of what a four-dimensional cube looks like is a Schlegel diagram:
This is a type of "perspective projection" of a four-dimensional hypercube into a three-dimensional space. It is the higher dimensional version of painting a three-dimensional room on a two-dimensional canvas using the rules of perspective.
More Dimensions!
There is no reason for us to stop at four dimensions. We can construct five-, or six-, or seven- dimensional spaces by throwing in yet more axes and representing points as five-tuples, six-tuples or seven-tuples. For any number of dimensions we want, we can just put in that many axes and consider that many coordinates.
While wrapping our heads around the geometry of higher dimensional spaces is both extremely trippy and interesting in its own right, these spaces are crucial in both pure and applied mathematics. The most basic way we use higher dimensional spaces is in representing functions that have more than one input variable. If I want to represent a certain company's profits as a function of its sales and labor costs, I need a three dimensional space - two dimensions for my two inputs, and then a third dimension for the function's output.
If I want to add in an input variable representing capital expenditures as well, I am now working in four-dimensional space - three input dimensions, and a fourth dimension representing the output of my profit function.
How we construct higher dimensional spaces and shapes is an example of a very common theme in mathematics. We start with the fairly straightforward idea of finding a way to describe where a point is on a line, or a plane, or in space - a coordinate system. Then, we see that it is possible to extend this idea further - keep throwing in more and more coordinates - with the extension leading to strange and fascinating results, like higher dimensional spaces.