Today I feel like explaining a little bit of algebraic geometry, namely a theorem by Max Noether and Castelnuovo on the so-called Cremona group of the plane. (My intention is to explain what the theorem states, not to prove it, which would take far more space than an entry here would allow. I hope I can make what follows clear to people with a relatively modest knowledge of mathematics—perhaps an undergraduate level.) Incidentally, let me mention that Max Noether is the father of Emmy Noether, who was certainly the most remarkable woman mathematician ever (and far more important to mathematics than Max Noether).
We concern ourselves with birational transformations of the plane, so let us first explain what this means. A rational map of the plane to itself is given, by definition, by two expressions, (x′,y′), giving the coordinates of an image point in function of those, (x, y), of the original point, in which we impose that x′ and y′ be rational functions of x and y, in other words quotients of two polynomials in these two variables (which we can assume, after elimination, to have no common factor), say, with real coefficients, the denominator being—of course—not identically zero. For example, letting x′=0 and y′=0 maps the entire point to the origin (0,0); letting x′=−x and y′=y defines the symmetry with respect to the vertical axis; letting x′=x and y′=x² gives a rational map that projects the entire plane onto the parabola with equation y=x², and so on. Note that a rational map is not always defined everywhere, since we have allowed polynomials in the denominator: for example letting x′=1/x and y′=1/y gives a rational map that is defined only so long as x and y are both nonzero (i.e. away from the coordinate axes), but we still call this a rational map of the plane to itself; since the denominator cannot be identically zero, there are always some points in the plane (many points, in fact: in a topological sense they are “dense”) for which the mapping is defined. Now we can generally compose two rational maps of the plane to itself, merely by replacing the x and y variables in the “outer” map by the x′ and y′ given by the “inner” map; for example, composing the former map (x′=1/x and y′=1/y) with itself gives the identity map (x′=x and y′=y), simply because 1/(1/t)=t. (Note that we cannot always compose two rational maps, because the composition might make sense nowhere, in other words it might end up giving an identically zero denominator, something we have excluded.)
When a rational map of the plane to itself is such that there exists another map which, when composed with it, (makes sense and) gives the identity map (viz. x′=x and y′=y), we say that the map (either of the two maps, actually) are birational transformations of the plane. For example, our previous example (x′=1/x and y′=1/y) is a birational transformation of the plane, since, as we have explained, when composed with itself, it gives the identity map. On the other hand, the constant rational map (x′=0 and y′=0) is not a birational transformation of the plane, since composing it with anything (in any order) gives a constant map if it makes sense at all. In perhaps more intuitive terms, a birational transformation of the plane is a rational map of the plane to itself which can be inverted by another rational map. (It is not exactly a bijection in the function sense, because it might not be defined everywhere. However, it is “mostly” a bijection.) The misfortune of composition not always being defined does not happen for birational transformations: the composition of two birational transformations of the plane always exists (makes sense) and is always itself a birational transformation; and, of course, by definition, any birational tarnsformation has an inverse which is also a birational transformation. (For those who know what that means, birational transformation of the plane form a group, the Cremona group of the plane. Note that it does not suffice for a rational map to be bijective in order for it to be a birational transformation: for example, x′=x³ and y′=y³ defines a rational map which is bijective because cube roots always exist and are unique, but which is not a birational transformation because cube roots are not rational functions.)
Now let us consider some particular birational transformatins. First of all, we have those of the following form: x′ = (a1·x+b1·y+c1) / (a0·x+b0·y+c0) and y′ = (a2·x+b2·y+c2) / (a0·x+b0·y+c0), in which the ai, bi and ci are real numbers: in other words, x′ and y′ can be expressed as quotients of first degree (aka affine) polynomials, with equal denominator; I'm not claiming that any such form defines a birational transformation, but it turns out that almost all do (all one needs to impose is the nonvanishing of the determinant of the 3×3 matrix of coefficients, and then the inverse transformation is given in the same form by the inverse matrix); those that are of this form are called projective transformations of the plane. For example, all plane symetries, rotations, translations, and much else, are all projective transformations. The composition of two projective transformations of the plane is a projective transformation, and the inverse of a projective transformation is a projective transformation (in technical terms, this means that projective transformations form a subgroup of the Cremona group). Basically, the projective transformations are the “uninteresting” birational transformations: they take lines into lines, so they are geometrically boring. (Another fact worthy of note is that given four distinct points in the plane, no three of which are aligned, and given another quadruplet of points satisfying the same condition, there is a unique projective transformation taking one set in the other in the prescribed order.)
Now let us consider two very simple examples of birational transformations which are not projective transformations: the first one, which has x′=1/x and y′=1/y, we have already mentioned; the second one is x′ = x/(x²+y²) and y′ = y/(x²+y²). Both give the identity when composed with themselves, as is easily checked, so both are, indeed, birational transformations of the plane. The first one corresponds to inverting the two (cartesian) coordinates independently, and the second corresponds to inverting the radial coordinate of polar coordinates (in other words, the distance to the origin) while keeping the angular coordinate fixed. Call these two transformations the basic Cremona transformations of the plane.
Well, the remarkable fact, which is the statement of Max Noether's transformation theorem, is that these two basic Cremona transformations are essentially the “only” ones we can form, or, rather, they are the building blocks for all other birational transformations; the precise statement is: any birational transformation of the plane can be written as the composition of projective transformations and the two basic Cremona transformations which we have given (of course it may be necessary to use either of them multiple times in the composition). So any birational plane transformation, no matter how complex, can be reduced to these basic transformations. (In reality, the theorem is somewhat simpler if we work over the complex numbers instead of the reals as we have done: then we need only one basic Cremona transformation, and either one will do. Over the reals we need two. But I did not want to make this entry more complicated by considering the plane with complex coordinates.)
To give an idea of why this is remarkable, we can consider the situation in other dimensions: is is straightforward to define birational transformations of the line, of three-dimensional space, and in fact, of n-dimensional space for any n, and among these are the obviously defined projective transformations. Now it turns out that in dimension one, the only birational transformations are the projective transformations (also called homographies): there are simply no others (and there is no Cremona transformation). In dimension three or more, it is not possible to find a finite number of particular birational transformations which, when composed with the projective transformations, will span all other birational transformations: the Cremona group is simply too complicated, and it is hardly possible to say anything useful about it.
Well, I don't know whether this is all very enlightening; at least, it is a little but elegant bit of algebraic geometry.