It would … be the number of the point at which Achilles overtakes the tortoise—if he does overtake him—by exhausting all the intervening points successively. Or it would be the number of the stars, in case their counting could not terminate. Or again it would be the number of miles away at which parallel lines meet—if they do meet. It is, in short, a ‘limit’ to the whole class of numbers that grow one by one, and like other limits, it proves a useful conceptual bridge for passing us from one range of facts to another.

-William James, Some Problems of Philosophy

This site is called ‘Point at Infinity’ in part because that is exactly what it attempts to do. Everything we experience directly in life is finite. Infinity fascinates us, but we cannot touch it. The best we can do is ‘point’ at it.

But the phrase ‘point at infinity’ also has technical mathematical meanings, and this week we will explore two of them. Today: points at infinity in projective geometry.

We start our story in Renaissance Italy with the development of the method of perspective by Brunelleschi and other painters and architects. In brief, the use of perspective allows for an accurate depiction of three-dimensional scenes on two-dimensional surfaces. The method can be imagined as follows: Consider the two-dimensional surface (a canvas, say) as a windowpane placed between the observer and the scene to be depicted. For every element of the scene, draw a line between that element and the observer’s eye. The point where that line passes through the windowpane is the point where it should appear in the two-dimensional image. (A woodcut by Albrecht Dürer depicting exactly this process is the title image to this post.)

One of the most important realizations of the early adopters of perspective is that, although this method of transferring three-dimensional scenes onto two-dimensional surfaces does preserve the straightness of straight lines, it does not preserve the parallelity of a pair of parallel lines. In particular, if, in the three-dimensional scene, you have a set of mutually parallel lines that are not themselves parallel to the two-dimensional surface, then, when they are transferred onto the surface, they all meet in a single point. Such a point is called a vanishing point.

The mathematics of perspective was studied on and off for the next few hundred years, but it wasn’t until the work of Gaspard Monge and his student, Jean-Victor Poncelet, in the late 18th and early 19th centuries, that it really took off and became the basis for a field known as projective geometry.

I cannot possibly give a proper introduction to projective geometry here, but that will not stop me from trying. One way to motivate the study of projective geometry is to point out some possible shortcomings of Euclidean geometry. One of these shortcomings is its occasional inelegance. Euclidean geometry proofs must often accommodate annoying ‘special cases.’ For instance, in Euclidean geometry, any two distinct lines meet in exactly one point unless the lines are parallel. A pair of distinct circles may intersect in zero, one, or two points depending on their relative size and position. And so on. Proofs involving consideration of these special cases are, of course, perfectly correct, but, for mathematicians, the elegance of a proof is often as important as its correctness. It was gradually realized that projective geometry offered certain advantages over Euclidean geometry, providing a more elegant, richer environment for investigation. Over the nineteenth century, projective geometry became the central focus of the growing field of algebraic geometry.

Like Euclidean geometry, projective geometry can be described by a list of axioms. One way to view these axioms is to think of them as the consequences of extending Euclidean geometry by requiring that every pair of distinct lines intersects in a single point. For example, the axioms for a projective plane are as follows.

1. Given any two distinct points, there is exactly one line containing them.
2. Given any two distinct lines, there is exactly on point on both lines.
3. There are four points such that no line contains more than two of them.

(The third axiom is essentially present to exclude degenerate cases.) An interesting fact is that there are finite models of the projective plane axioms. The smallest and most well-known is the Fano plane, which has seven points and seven lines, each line containing three points.

The study of finite projective planes is a fascinating and vibrant area of mathematics, but, this being a blog about infinity, we will not say more about them. Instead let us turn to the real projective plane, denoted by $\mathbb{RP}^2$, which can be thought of as the projective equivalent of two-dimensional Euclidean space. We will describe the real projective plane in two ways.

First, recall that $\mathbb{R}^3$ denotes three-dimensional Euclidean space and consists of all triples $(a,b,c)$ of real numbers. The points of $\mathbb{RP}^2$ can be thought of as the points $(a,b,c)$ in $\mathbb{R}^3$ such that at least one of $a, b, c$ is non-zero, under the additional requirement that, if $(a_0,b_0,c_0)$ and $(a_1, b_1, c_1)$ are two such points and there is a real number $r$ such that $(a_0, b_0, c_0) = (ra_1, rb_1, rc_1)$, then $(a_0, b_0, c_0)$ and $(a_1, b_1, c_1)$ actually denote the same point in $\mathbb{RP}^2$. Another way of saying this is that, if the line passing through $(a_0, b_0, c_0)$ and $(a_1, b_1, c_1)$ passes through the origin, then $(a_0, b_0, c_0)$ and $(a_1, b_1, c_1)$ are the same point in $\mathbb{RP}^2$. Put more elegantly, the points of $\mathbb{RP}^2$ are precisely the lines through the origin of $\mathbb{R}^3$.

This should, with good reason, remind one of our description of the method of perspective. Indeed, suppose that the observer’s eye is placed at the origin. Now consider two points in the three-dimensional scene. If the line containing these points also passes through the observer’s eye, then these two points will be drawn at the same place on the canvas and therefore should be thought of as the same point in $\mathbb{RP}^2$.

I have so far described the points but not the lines of the real projective plane. Essentially, lines in $\mathbb{RP}^2$ correspond to planes through the origin in $\mathbb{R}^3$. More precisely, if $(a,b,c)$ is a point in $\mathbb{RP}^2$, then the set of all points $(x,y,z)$ satisfying the equation $ax + by + cz = 0$ describes a line in $\mathbb{RP}^2$, and all of the lines are described in this way. Moreover, as one would hope, if $(a_0, b_0, c_0)$ and $(a_1, b_1, c_1)$ describe the same point in $\mathbb{RP}^2$, then they also determine the same line. The interested reader is left to verify that this construction satisfies the projective plane axioms.

We next offer a more intuitive description of $\mathbb{RP}^2$. Start with the two-dimensional Euclidean plane, $\mathbb{R}^2$, consisting of all pairs of real numbers. Now, for each class of parallel lines in $\mathbb{R}^2$ (i.e. for each possible ‘slope’, where the slope of the vertical line is considered to be $\infty$), add a new point at infinity. $\mathbb{RP}^2$ then consists of all of the points of $\mathbb{R}^2$ plus all of the new points at infinity. The lines of $\mathbb{RP}^2$ are the lines of $\mathbb{R}^2$ (where each such line now includes the point at infinity associated with its slope) plus a single new line that contains all of the points at infinity. The points at infinity thus represent the points where the parallel lines of Euclidean space meet. They are the vanishing points of perspective drawing.

Why are these two descriptions the same? To illustrate, I will provide a translation from the first description to the second. First, consider a point $(a,b,c)$ from our first characterization, and suppose that $c \neq 0$. The line in $\mathbb{R}^3$ containing the origin and $(a,b,c)$ intersects the horizontal plane lying one unit above the origin at the point $(\frac{a}{c}, \frac{b}{c}, 1)$. We then identify the point $(a,b,c)$ from the first description with the real point $(\frac{a}{c}, \frac{b}{c})$ from the second description.

What if $c = 0$? In this case, the line in $\mathbb{R}^3$ passing through the origin and $(a,b,c)$ does not meet the horizontal plane one unit above the origin; the two are parallel. But parallel objects meet at infinity, so $(a,b,c)$ will correspond to a point at infinity from our second description, namely the point at infinity corresponding to the slope $\frac{b}{a}$ (or $\infty$, if $a = 0$).

And the lines? Suppose $(a,b,c)$ determines a line in our first description. If it is not the case that $a=b=0$, then this corresponds to one of the lines in $\mathbb{R}^2$ (if my hasty calculations are correct, the line $y=-\frac{a}{b}x-\frac{c}{b}$ or, if $b = 0$, the vertical line $x = -\frac{c}{a}$). If $a=b=0$, then this line corresponds to the line containing the points at infinity in the second description.

I should note here that, although the second description of $\mathbb{RP}^2$ seems to make a distinction between the ‘real’ points of $\mathbb{R}^2$ and the ‘points at infinity,’ this distinction is at its core illusory. Indeed, in my translation from the first to the second description of the real projective plane, I could just have easily have decided that the points $(a,b,c)$ satisfying $a = 0$ should correspond to the points at infinity instead of the points satisfying $c = 0$. There are in fact infinitely many ways I could translate the first description to the second, each of which would identify a different set of points from the first description with the set of points at infinity from the second. The real projective plane exhibits a profound symmetry which I have only been able to begin to hint at here.

I do not expect that, after this hurried introduction, the reader has developed an intuitive feel and appreciation for projective geometry. But I extend an invitation to spend some time in the real projective plane, to get to know it, to recognize its power, the unity created merely by the addition to Euclidean space of points at infinity. And then I invite the reader to go to an art museum, to revisit masterpieces of perspective with potentially new eyes, to ponder the violent breaking of perspective that happened in the early 20th century with the Cubists, who rejected the assumption that a painting should represent the viewpoint of a single observer at a single point in space, to wonder if perhaps this seemingly chaotic art is in fact at heart an illustration of the elegant symmetries of projective geometry.

P.S. As a bonus, here’s a third (and probably unhelpful) description of the real projective plane. Start with a Möbius strip. Now glue the Möbius strip’s single edge to itself, preserving direction (of course, this cannot be done in three dimensions, so first move to a four-dimensional space). Now you have a real projective plane!

P.P.S. If I were a different writer, I might have ended this post by going back to considering ‘Point at Infinity’ as a name for this blog in light of what we have learned about projective geometry. I might have made the case that this blog, or perhaps the idea of infinity, is itself a ‘point at infinity’ where the parallel pursuits of art and mathematics meet. I might have tried to obscure the fact that, on closer inspection, this analogy doesn’t hold up very well at all. But I am not a different writer, and I will refrain from this indulgence.