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The Complex Projective Plane

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ApPENDIX 1 The Complex Projective Plane Many books have been written on a more elaborate geometry where the coordinates of points and lines, instead of being real numbers, are complex numbers. * A large part of the development of Chapter 12 can be carried over. Even the criterion v/Il < 0 for separation (§ 12'4) re­ mains significant, provided we remember that a number cannot be neg­ ative without being real. (Thus 'v/Il < 0' no longer means that 11 and v have 'opposite signs'; they might both be imaginary.) The general point collinear with (a) and (b) is (a + vb), where v is any complex number. Three collinear points (a), (b), (a+llb) determine a so-called chain of points (a + vb) where v/Il is real. Thus the line contains infinitely many chains, and instead of 3·21 we have two points decomposing a chain into two segments. Actually all but one of the axioms of real projective geometry apply also to complex projec­ tive geometry. The single exception is 3'14, which is false if the four points do not all belong to the same chain. In place of that axiom we have the synthetic definition for a chain, as consisting of three col­ linear points A, B, C along with the three segments BC/A, CA/B, AB/C. In this terminology, 3·14 states that the chain covers the whole line, which is just what happens in real geometry but not in complex geometry. Thus everything in Chapter 2 remains valid, but those later results which depend on 3·14 (directly or indirectly) are liable to be contra­ dicted. An interesting example is 3'31, which is contradicted by the following configuration: The nine points defined by the equations xi + x~ + x~ = X 1 X 2 X3 = 0 * For a good exposition of Hamilton's approach to complex numbers, see Robinson 1940, pp. 83-4. It is hardly necessary to add that the words real and imaginary are picturesque relics of an age when the nature of complex num­ bers was not yet clearly understood. 200 THE COMPLEX PROJECTIVE PLANE 201 VIZ. (0,1, -1), (0,1, -w), (0,1, _w2 ), (-1,0,1), (-w, 0,1), (_w2 , 0,1), (1, -1, 0), (1, - w, 0), (1, - W 2, 0), where W = e21ti/3, lie by threes on 12 distinct lines in such a way that any two of the points lie on one of the lines. These nine points are the common inflexions of the cubic curves xt + x~ + x~ + AXt X 2 X 3 = o. The classification of projectivities is actually simpler in complex than in real geometry. For now every projectivity has one or two in­ variant points, and a line cannot fail to meet a conic. On the other hand, besides the projectivity relating (a + vb) to (a' + vb') there is also an antiprojectivity relating (a + vb) to (a' + vb'), where v is the complex conjugates of v. Both these transformations preserve the har­ monic relation. Similarly, there is not only a projective collineation 12·51 but also an antiprojective collineation The projective polarity 12·61 always determines a conic (xx) = 0; but there is also an antiprojective polarity Xi = r.aijxj (aji = iii) whose self-conjugate points form an 'anticonic' (xx) = 0. Many results in real geometry are most easily obtained by regarding the real plane as part of the complex plane, so that a real line appears as a chain on a complex line. This method is especially valuable in the theory of circles; for the absolute involution has two invariant points (the circular points at infinity, discovered by Poncelet in 1813), and a circle is most simply defined as a conic that passes through these two points.* * Salmon 1879, p. 1. ApPENDIX 2 How to Use Mathematica® by George Beck This Mathematica program allows the user to show the scenes of two movie scripts written by H.S.M. Coxeter: The Arithmetic of Points on a Conic and Projectivities. Accompanying narrations can be found in the files Arithmetic and Projectivities. Start up Mathematica with some way of reading the narrations. En­ ter either «Arithmetic.m or «Projectivities.m to load the necessary files. If you do not have a colour monitor, lines and circles will appear too sketchy. Entering «blacken.m will correct this. If you run out ofmem­ ory, quit Mathematica, start again, and then continue from where you left off. There are 52 still scenes for The Arithmetic of Points on a Conic. To see the nth scene, enter Show[Scene[n]], where n is anyone of the numbers from 1 to 52. In the narrations [n] indicates the nth scene. You can animate the scenes numbered 2, 8, 17, 24, 30, 31, 42, 48, or 49. In the narration these are indicated by [2A], [8A], and so on. To create an animation, enter PreAnimateScene[n, T], where n is one of the numbers just given and T is the number of frames (try 20 at first). The result is a table of graphics; how actually to run the animation will vary from one type of machine to another. In the script Projectivities there are 33 scenes numbered from 61 to 93, anyone of which can be animated except for 84 (this is indicated by [84-A] in the narrations). As before, try 20 frames at first, except for the scenes 81, 88, 89 where 4 frames are enough. A representative sam­ pling of the animations is 67, 72, 80, 83, 88, 91. The file named Contents lists all the files (except the scenes) with the functions they contain. Conversely, the file named Index lists each function with the file that contains it. To save memory, functions directly associated with a particular scene are only held in RAM temporarily; so to see how Scene 12 is coded, or to change its default values, use an editor to look at the file Scene12.m. 202 HOW TO USE MATHEMATICA 203 This program was written using Version I, but no changes are needed for Version 2. AbsolutePointSize and AbsoluteThickness would keep line colours constant when resizing windows (see globals.m); using Stubs might help to simplify the memory management (see scene.m); and perhaps Block should be changed to Module throughout. The arithmetic of points on a conic. [1] The example of a ther­ mometer makes it easy to see how the real numbers (positive, zero, and negative) can be represented by the points of a straight line. [2A] On the x-axis of ordinary analytic geometry, the number x is represented by the point (x, 0). [3] Given any two such numbers, a and b, we can set up geometrical constructions for their sum, ... [4] ... difference, ... [5] ... product, ... [6] ... and quotient. However, these constructions require a scaffolding of extra points and lines. It is by no means obvious that a different choice of scaffold­ ing would yield the same final results. The object of the present program is to make use of a circle (or any other conic) instead of the line, so that the constructions can all be performed with a straight edge, and the only arbitrariness is in the choice of the positions of three of the numbers (for instance, 0, 1, and 2). [7] Although this is strictly a chapter in projective geometry, let us begin with a prologue in which the scale of the abscissa on the x-axis is transferred to a circle by the familiar process of stereographic projection. A circle of any radius (say 1, for convenience) rests on the x-axis at the origin 0, and the numbers are transferred from this axis to the circle by lines drawn through the opposite point. [8A] That is, the point at the top. In this manner, a definite number is assigned to every point on the circle except the topmost point itself. [9] Since the numbers 10, 100, 1000, ... come closer and closer to this point on one side and the numbers -10, -100, -1000, ... come closer and closer on the other side, ... [10] ... it is natural to assign the special symbol 00 to this excep­ tional point: the only point for which no proper number is available. The tangent at this exceptional point is, of course, parallel to the x-axis; that is, parallel to the tangent at the point o. [11] Having transferred all the numbers to the circle, we can forget about the x-axis; but the tangent at the point 00 will play an important role in the construction of sums. [12] For instance, there is one point on this tangent which lies on the line joining points 1 and 2, also on the line joining 0 and 3, and on 204 THE REAL PROJECTIVE PLANE the line joining -1 and 4. We notice that these pairs of numbers all have the same sum: 1 + 2 = °+ 3 = - 1 + 4 = 3. [13] Similarly, the tangent at 1 meets the tangent at 00 in a point which lies on the lines joining °and 2, -1 and 3, - 2 and 4, in accor­ dance with the equations 1 + 1 = °+ 2 = - 1 + 3 = - 2 + 4. These results could all be verified by elementary analytic geometry, but there is no need to do this, because we shall see later that a general principle is involved. [14] Having finished the Euclidean prologue, let us see how far, we can go with the methods of projective geometry (see § 11'2). Let symbols 0, 1, 00 be assigned to any three distinct points on a given conic.
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