Yet Another Mini-Lecture
Feb. 2nd, 2005 02:41 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowOne concept that has become increasingly pervasive in modern mathematics is that of duality. Variations on the idea appear in every major branch of mathematics, but it's a little hard to describe what they all have in common. Loosely speaking, a duality establishes a correspondence of some sort which "reverses things", in some sense. One of the earliest kinds of duality to be noticed was logical negation, which reverses things in the following sense: if (P is true) implies (Q is true), then (Q is false) implies (P is false). It also interacts nicely with the other logical operations: to say that (P and Q) is false is to say that (P is false) or (Q is false), and to say that (P or Q) is false is to say that (P is false) and (Q is false). This sort of phenomenon turns out to be immensely valuable whenever it appears. What I'm going to talk about under the cut is a version of duality that is useful in geometry.
A triangle may be viewed in either of two ways; you can focus on the vertices, in which case the sidelines are secondary constructions, or you can focus on the sidelines, in which case the vertices are secondary. Let's call these "triangles" and "trilaterals" respectively, but keep in mind that they're two ways of thinking about the same object. Similarly we can talk about quadrangles and quadrilaterals, and so on up the line. We can even extend the notion to general curves. If we think of a curve as made up of infinitely many infinitely short line segments, then what are the sidelines? The tangent lines! So we can think of a curve as a locus - described by the points that lie on it - or as an envelope - described by the lines that are tangent to it.
How can we link these two ways of viewing things? Consider two planes. (They might be the same plane, and in the construction I'm going to give they are, but they don't have to be.) A duality between the planes establishes a correspondence: each point in the first plane is paired with a line in the second plane, and each line in the first with a point in the second. In each case, the point and the line are called "duals" of one another. There's just one condition: if the point P lies on the line L in the first plane, then the dual of L (a point in the second plane) must lie on the dual of P (a line). Under the duality, then, each triangle in the first plane corresponds to - is dual to - a trilateral in the second, each quadrilateral to a quadrangle, and each locus to an envelope.
The condition has a number of interesting consequences. For instance, if P and Q are distinct points and L is the line connecting them, then the dual of L is the point where the duals of P and Q intersect. Or, again, if three points are collinear - they all lie on the same line - then their duals are concurrent - they all pass through the same point.
There's a catch. Any two distinct points lie on exactly one line. However, two distinct lines may not pass through a single point - not if they're parallel. So, duality of this sort doesn't quite work if we use ordinary Euclidean planes. What we have to do is extend to the projective plane. In essence, we make up a bunch of new points, "points at infinity", one for each family of parallel lines, and we say that the lines in that family intersect there. The points at infinity are thought of as being infinitely far away, naturally, and together they lie on the "line at infinity". If we do this to each of our planes, then we can get away with constructing our duality. (If it seems as though we're just making up stuff, well, yes, that's exactly what we're doing; but it works, and it can be proven to work - at least, it works just as well as ordinary Euclidean geometry does. We mathematicians make stuff up all the time; as long as it works, we're fine with it. Sometimes the stuff we make up turns out to be real...)
Let's get down to an actual construction; I'll show you how to establish a duality in a single plane. First, draw a circle. We establish as a principle that the dual of any point on the circle is the tangent line at that point, and vice versa. It turns out that that, together with the condition that defines a duality, is all we need. Here we go.
Suppose we have a line that cuts the circle at two points P and Q. Since P and Q lie on the line, their duals - i.e., the tangents at P and Q - must pass through the dual of the line. So draw the tangents, and see where they intersect: that's the dual of the line. (Note that if the line passes through the center of the circle, P and Q are opposite each other and their tangents are parallel, so the dual of the line is a point at infinity.) Turn this around; suppose a point P lies outside the circle. Then there are exactly two tangents to the circle that pass through P, and the dual of P is the line connecting the contact points of those tangents. (Take a ruler and lay it down on P. Start rotating it around P until it just grazes the circle; that gives you one of the tangents. Rotate the other way, and you'll get the other one.)
What if we have a line that doesn't cut the circle? Well, just pick two points on the line, P and Q. Both of them lie outside the circle, and we know how to construct their duals. Do so; the point where they intersect is the dual of the line. Similarly, if a point P lies inside the circle, draw two lines through it. Both of them cut the circle, so we can construct their duals as above; the dual of P is the line connecting the duals of the two lines. (The dual of the center of the circle is the line at infinity.)
Okay, so we know how to construct the duals of points and lines. What can we do with this construction? Stay tuned...
A triangle may be viewed in either of two ways; you can focus on the vertices, in which case the sidelines are secondary constructions, or you can focus on the sidelines, in which case the vertices are secondary. Let's call these "triangles" and "trilaterals" respectively, but keep in mind that they're two ways of thinking about the same object. Similarly we can talk about quadrangles and quadrilaterals, and so on up the line. We can even extend the notion to general curves. If we think of a curve as made up of infinitely many infinitely short line segments, then what are the sidelines? The tangent lines! So we can think of a curve as a locus - described by the points that lie on it - or as an envelope - described by the lines that are tangent to it.
How can we link these two ways of viewing things? Consider two planes. (They might be the same plane, and in the construction I'm going to give they are, but they don't have to be.) A duality between the planes establishes a correspondence: each point in the first plane is paired with a line in the second plane, and each line in the first with a point in the second. In each case, the point and the line are called "duals" of one another. There's just one condition: if the point P lies on the line L in the first plane, then the dual of L (a point in the second plane) must lie on the dual of P (a line). Under the duality, then, each triangle in the first plane corresponds to - is dual to - a trilateral in the second, each quadrilateral to a quadrangle, and each locus to an envelope.
The condition has a number of interesting consequences. For instance, if P and Q are distinct points and L is the line connecting them, then the dual of L is the point where the duals of P and Q intersect. Or, again, if three points are collinear - they all lie on the same line - then their duals are concurrent - they all pass through the same point.
There's a catch. Any two distinct points lie on exactly one line. However, two distinct lines may not pass through a single point - not if they're parallel. So, duality of this sort doesn't quite work if we use ordinary Euclidean planes. What we have to do is extend to the projective plane. In essence, we make up a bunch of new points, "points at infinity", one for each family of parallel lines, and we say that the lines in that family intersect there. The points at infinity are thought of as being infinitely far away, naturally, and together they lie on the "line at infinity". If we do this to each of our planes, then we can get away with constructing our duality. (If it seems as though we're just making up stuff, well, yes, that's exactly what we're doing; but it works, and it can be proven to work - at least, it works just as well as ordinary Euclidean geometry does. We mathematicians make stuff up all the time; as long as it works, we're fine with it. Sometimes the stuff we make up turns out to be real...)
Let's get down to an actual construction; I'll show you how to establish a duality in a single plane. First, draw a circle. We establish as a principle that the dual of any point on the circle is the tangent line at that point, and vice versa. It turns out that that, together with the condition that defines a duality, is all we need. Here we go.
Suppose we have a line that cuts the circle at two points P and Q. Since P and Q lie on the line, their duals - i.e., the tangents at P and Q - must pass through the dual of the line. So draw the tangents, and see where they intersect: that's the dual of the line. (Note that if the line passes through the center of the circle, P and Q are opposite each other and their tangents are parallel, so the dual of the line is a point at infinity.) Turn this around; suppose a point P lies outside the circle. Then there are exactly two tangents to the circle that pass through P, and the dual of P is the line connecting the contact points of those tangents. (Take a ruler and lay it down on P. Start rotating it around P until it just grazes the circle; that gives you one of the tangents. Rotate the other way, and you'll get the other one.)
What if we have a line that doesn't cut the circle? Well, just pick two points on the line, P and Q. Both of them lie outside the circle, and we know how to construct their duals. Do so; the point where they intersect is the dual of the line. Similarly, if a point P lies inside the circle, draw two lines through it. Both of them cut the circle, so we can construct their duals as above; the dual of P is the line connecting the duals of the two lines. (The dual of the center of the circle is the line at infinity.)
Okay, so we know how to construct the duals of points and lines. What can we do with this construction? Stay tuned...
