Imagine you’re stood on a train track and look along the tracks as the they disappear off into the distance. The two mutually parallel tracks seem to meet just as they venture over the crest of the horizon. A strange illusion. You know that the two tracks can never meet, because a train has to be able to move over them with a fixed distance between its wheels. Perhaps it is just an illusion, created by the optical receptors in your eye…or perhaps it isn’t.
In Euclidean mathematics two parallel lines will exist alongside each other at a fixed length and never meet. Euclid also explained that two lines will meet at exactly one point, unless they are parallel. Exceptions upset modern day mathematicians, who don’t like the word ‘unless’: exceptions are not elegant. If every other pair of lines meet, why can’t parallel lines meet too? This is the premise behind geometric infinity.
Let us motivate the idea using the following thought experiment: imagine you are stood on a very small island, on a very calm day and all around you is ocean. Imagine this ocean extends out to infinity, never-ending: you are at the centre of an infinite plane. The line from your eye to the horizon is consequently parallel to the ocean and is itself also infinite. If a boat sets off from your island and you draw a line from your eye to the bottom of the boat, this line will never be above the horizon line despite how far away the boat sails. The line would only go above the horizon when the boat passes over the horizon, which it cannot do as the horizon in this scenario is at infinity.
We can see the horizon line as the point where the sky meets the ocean. Looking at the horizon and drawing a line to your eye, two parallel lines (the line from your eye to the horizon and the surface of the ocean) once again seem to meet at a point far off in the distance. However, our last thought experiment confirms that this point is no ordinary point that lies on the infinite plane, but in fact is an abstract point that exists at infinity. This point is what mathematicians call the ‘line at infinity’, and is simply an extension of Euclid’s idea of an infinite plane.
The line at infinity is an example of a mathematician’s desire to ‘plug’ a hole in mathematical inconsistencies. Intuitively, the line acts as the boundary to the infinite plane. Walk in any direction across the plane forever and you’ll get there!
If we stand on the island again, now imagine the infinite plane (the ocean) as the interior of a circle and the circle itself is the line at infinity, bounding the plane. Any line that travels directly across the plane will only ever meet any other line once. However, each line travelling across the plane will meet the line at infinity twice, once at either side of the circle. This poses a problem. If we want for our theory to be constant for every line, then we must alter the meaning of a point.
Lets say the first point where the line meets the circle is called A and the second point is called B. If I walked from the island at the centre of the infinite plane and I walk forever, I’d reach A. Then, if I followed the line at infinity clockwise forever, I’d reach B. If I then carried on from B, walking clockwise forever, I’d once again reach A. I would have walked the same distance, in the same direction, to end up at two different points. This is counter intuitive and is the crux of the problem. How can I walk in the same direction for the same amount of time and end up somewhere different. Answer, A and B must be the same point. Thus, the line at infinity is in fact semi-circular and by extension A and B are in fact the same point in space.
Consequently, the line at infinity is a semi-circle that wraps around itself to make a circle. As a result, every line in Euclidean space in turn wraps around itself to make a circle. A depressing result of this idea of infinity is that one can walk forever and return to exactly the same place. So what does this have to do with the train tracks from earlier? The parallel lines now do meet, but only at the line at infinity. All parallel lines meet at the point where the line at infinity meets itself: point A. So at the end of it all, everything in existence on this infinite plane revolves not around the centre, but in fact around point A; a point that exists not in perceivable space, but at infinity. The end of the line.
This article was first published in The Orbital magazine, the official student publication of Royal Holloway, University of London. It has been edited for release on this platform.