Orbital Explains: The Line At Infinity

Simon Williams breaks down the mathematical theory of geometric infinity.

You’ve probably never stood on a train track, and looked as it disappeared off into the distance. But let’s imagine you have. 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 along side 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. This idea upsets modern day mathematicians, who don’t like the word ‘unless’; they believe it’s not elegant. So if every other pair of lines meet, why can’t parallel lines meet too? This is the premise behind geometric infinity.

So if every other pair of lines meet, why can’t parallel lines meet too? This is the premise behind geometric infinity.

This idea is concreted by the following thought experiment: imagine you are stood on a very small island and all around you is ocean. Now imagine this ocean extends out to infinity, and so you are at the centre of an infinite plane. The line from your eye to the horizon is consequently parallel to the ocean, and also extends for infinity. 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, because the ocean is infinite. The line would only go above the horizon when the boat passed over it, which it cannot do as the horizon is at infinity.

The line from your eye to the horizon is consequently parallel to the ocean, and also extends for infinity

We can see the horizon line is the point where the sky meets the ocean. The parallel lines once again meet at a point. 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 just 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, if the plane had a boundary. Walk in any direction across the plane forever, and you’ll get there!

Back to the island, 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 across the plane will only ever meet any other line once. However, this line will meet the line at infinity twice, once at either side of the circle; this is a problem. Therefore, we must alter the meaning of a point. Let’s 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 centre of the infinite plane 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; so why not let A and B be the same? Thus, the line at infinity is in fact semi-circular. Similarly, 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. Thus, 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 three dimensional space, but at infinity – the end of the line. •

Back to the island, 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 across the plane will only ever meet any other line once. However, this line will meet the line at infinity twice, once at either side of the circle; this is a problem. Therefore, we must alter the meaning of a point. Let’s 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 centre of the infinite plane 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; so why not let A and B be the same? Thus, the line at infinity is in fact semi-circular. Similarly, 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. Thus, 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 three dimensional space, but at infinity – the end of the line. •