What do we mean when we say something is infinite? Limitless? Immeasurable? Eternal? Or ∞?
The “lazy 8” symbol, now commonly known as the infinity symbol, was introduced in 1655 by John Wallis, and the hope is that by the end of this series of articles you’ll see infinity as much more than just the infamous symbol. We’ll start our discussion with something relatively easy to think about…
This is a dot •
These are two dots ••
What if there are many dots connected to each other?
This is a line!
So, what is a line, according to mathematics?
Remember the days, when your high-school teacher started to talk about lines, rays, and line segments, possibly drawing diagrams like the following:
“A line does not have ends; it goes on forever in both directions.”
“A ray has one end but goes on forever in the other direction.”
“A line segment has two ends.”
By “end”, we mean that if you draw the line you are going to stop at that point, and by “going on forever” we mean that you will not stop. So, if we think of measurements of length for these three examples, only a line segment would have a finite length, which we can in fact measure; both a ray and a line have an infinite length, and we cannot measure them. This is hopefully an intuitive way to capture the idea of infinity.
Analogously, we connect many lines together to form a plane.
In the 2-dimensional case, we have several possibilities, including: a plane, a half-plane, and a bounded 2-dimensional region.
An example of a plane is the Cartesian Plane (or x-y plane), which is considered to be infinite in 2-dimensions.
If we only consider the region where x>0 on the x-y plane then we have an example of a half plane.
For a bounded region, we have to ‘stop’ in both directions. So for example, a square bounded by the lines x=1, y=1, x=-1, and y=-1 is an example of a bounded 2-dimensional region.
Now, if we consider the measurements of area associated with the above, only the third example (square) has finite area, which could be calculated to be 4. This is because it is enclosed by intersecting line segments. The x-y plane is infinite in both the x- and y- directions and is not enclosed by any lines, and thus we cannot measure the area because it is infinite. The half-plane example has only one “edge” (of infinite length), namely x=0 (or the y-axis). Still, the area is infinite, because it is infinite in the y-direction, and “semi-infinite” in the x-direction.
Now we can try to generalise our working above for lines and planes to the 3-dimensional case. We could form a 3D-space by connecting many layers of planes, but what is an infinite 3D-space? If we assume the universe we live in is indeed infinite, then it is a perfect example (here is a discussion on whether or not the universe is infinite; here is another link to an interesting article about how the universe could be a dodecahedron). What is an analogous example of a half-plane in the 3D case? Just imagine cutting the universe into halves, and pick half of the universe to be an example. A bounded 3D region? A can of baked beans will suffice, and like with our square above we have a means to measure its volume, because it is finite!
Okay. Pretty basic stuff. Infinities in 1-, 2-, 3-dimensional spaces. But, did you notice that we actually built up the 3d-space starting only from points??
In fact, whenever “many” is used, “infinitely many” would be more mathematically accurate. A very subtle point is that even a line segment consists of infinitely many points. Why is this the case? Take the line segment between 0 and 1 as an example: 0.5 lies on the segment, as does 0.1.
If we zoom in, taking the segment between 0 and 0.1, 0.01 actually lies on the segment as well.
Assume we are very close to 0, at some point x, which is very small but greater than zero. After zooming in, we could still find a point (eg. x/10) lying between 0 and x.
This shows 1) no matter how close you are to a point you can still get even closer; 2) there are infinitely many points lying on any line segment because we could repeat the above procedure for as long as we want. At first sight it might seem contradictory to the fact that a line segment has finite length. The key message is that length is a different geometric concept from the idea of how a line is “made” from points. (If you are interested, at the very end there is a sidebar discussion on this).
his brings us to the end of our first glance at infinity so let me summarise the important bits for you to take away. We have talked about how infinity is used to describe the properties/behaviour of geometrical objects in different dimensions (how large), and how it is also used to construct a higher dimensional object from a collection of lower dimensional objects (how many).
The series continues here.
****Here is a sidebar discussion to blow your mind (or perhaps just to confuse you even more)****
Things can get bizarre if we try to understand the true mathematical definitions of points, lines, planes, and spaces. When we refer to points, they are dimensionless objects in the mathematical reality, but the dots we draw do have diameters. When we refer to lines, they are only 1-dimensional, so they do not have “width”. Yet, when we draw a line, it does have another dimension associated with width, though very small. Quoting from Paul Lockhart in his book Measurements: “Any measurement made in this universe is necessarily a rough approximation… The smallest speck is not a point, and the thinnest wire is not a line.” Neat, huh?