My Time in Space, page 13
The standard proofs of Pascal’s theorem in today’s textbooks use analytic or projective techniques that were not available in his time, or that seem foreign to his ocular and intuitive methods. However there is one proof, discovered in 1826, that uses only the simplest and clearest consequences of a few basic and strikingly visual ideas. Its inventor was Germinal Dandelin, a military engineer in Napoleonic France, and later professor of mining engineering in Brussels. Hardy, in trying to locate the aesthetic qualities of a proof, as apart from the ‘seriousness’ of the result, writes:
… there is a high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail – one line of attack is enough….
Dandelin’s proof has all these narrative strengths. It is very dramatic; it sets off in an apparently irrelevant direction, appears momentarily to be lost in a thicket of lines, and suddenly confronts one with the result. I will expound it here, not for its own sake but, if I can, to convey the experience of understanding a proof, the rush of aesthetic adrenaline at its success. (To be realistic, perhaps that thrill will only be vouchsafed to those readers who have already trod similar ways, or are prepared to force their way through the thicket several times, to and fro, beating a path in which they will not be snagged by details. As for the others, I would rather they turn the page than abandon me completely!)
We will introduce a few preliminary ideas. (I note that mathematicians, including Pascal, frequently use this ‘we’, not because they count themselves kings of infinite space but that they are counting on, imploring, the reader to join with them in the venture, to share in the process of proof, which is nothing if it is not followed, and not in blind faith but with a critical mind ready to disconfirm or confirm it. A proof is a rational seduction. Also I should admit that I will glide over some little complications in the following, concerning lines that might be parallel rather than intersect; but anyone who has enough geometry to notice these elisions will also be able to fill them in. So, I beg you, follow me and if there are difficulties, jump over them without letting them perturb you.)
In Pascal’s definition of a cone we have already met the concept of a line that sweeps out a surface. Dandelin’s proof uses another form generated by lines, called a hyperboloid of one sheet. Imagine a sphere that can spin on a vertical axis; let a line be fixed to it, touching it at one point on its equator; then if the line is also vertical it will sweep out a cylinder when the sphere is rotated (Fig. 11). But if the line is askew it will sweep out the shape we need (Fig. 12). We will call its successive positions the generators; they are straight lines lying in this curved surface. (It is a common shape for lampshades because it can be made out of strings, playing the role of generators, stretched slantwise between two round loops of wire.) An equally askew line leaning in the other direction would sweep out the same surface, so there are two families of generators. No two generators of the same family intersect, but each generator of one family intersects all those of the other family.
Figs. 10–12. Curved surfaces generated by movement of a line. A few generators are shown in each case. (10) the cone; (11) the cylinder; (12) the hyperboloid of one sheet, with some generators of the first family shown as bold lines and one generator of the second family as a fine line. (If the generators were extended far enough, the one from the second family would eventually intersect with all those from the first family, with the exception of the one directly opposite it, to which it is parallel.)
Having introduced the hyperboloid and its generators all we need are some obvious facts about lines and planes, such as that two intersecting lines define just one plane in which they lie, that two planes intersect in a line, and three planes in a point; for instance, the ceiling and two walls meet at the corner of a room.
Fig. 13. The hyperboloid, the conic, and the six generators. (For convenience I have drawn the conic in the neck of the hyperboloid, but it could be in any position. Generators are shown by finer lines where they lie on the farther side of the hyperboloid. Generators of one family are marked X and those of the other family Y.)
Now we will take a plane cutting through the hyperboloid; if the plane is horizontal the intersection will be a circle, and if slanting, one of the other conic sections (this is easily proved, but we shall take it as established). In the conic we will inscribe a hexagon. Through one of its corners we draw a generator belonging to one of the families, through the next corner a generator from the other family, and so on all the way round (Fig. 13). That makes three generators from either family; each generator from one family meets the three from the other, giving nine points of intersection in all. Those first two generators and the first side of the hexagon make a triangle we will call the first triangle; each side of the hexagon has a corresponding triangle, so that the conic is encircled by six triangles (Fig. 14). (Is this a prefiguration of that eye crowned with thorns? If so it is a Pascalian ‘reason of the heart’ for believing that this proof is indeed the one used by Pascal.) Among the nine points of intersection mentioned are one corner from each of the six triangles; that leaves three more points of intersection, making a triangle we will call the seventh triangle.
Now look at the first triangle and the one associated with the opposite side of the hexagon, i.e. the fourth triangle (Fig. 15). Each of these triangles lies in a plane defined by two generators, and those two planes meet in a line, which is a side of the seventh triangle. That line meets the plane of the original hexagon, in a point we will call the first crucial point. But the two sides of the hexagon we have been concerned with, lying in the planes of the first and fourth triangle, must meet where those two planes and the plane of the hexagon all meet; that is, in the first crucial point. Similarly, each pair of opposite sides of the hexagon gives rise to a crucial point. These three points all lie in the plane of the hexagon and in the plane of the seventh triangle; so they must lie on the line in which those two planes intersect (Fig. 17). Hence, ‘La Pascale’: the intersections of opposite sides of a hexagon inscribed in a conic lie on a line.
Figs. 14–17. Dandelin’s proof. (Having established the configuration of six lines and their nine points of intersection, we can throw away the hyperboloid.) (14) The crown of thorns and the seventh triangle. (15) The planes of the first, fourth and seventh triangles all hinge on one side of the latter. (The conic is omitted for the moment.) (16) The crucial point where the first and fourth sides of the hexagon meet lies on the side of the seventh triangle. (17) All three crucial points lie in the plane of the seventh triangle, and in the plane of the hexagon; therefore they lie on the intersection of these two planes.
It was not until the mid-nineteenth century that projective geometry’s independence of and logical anteriority to our familiar schoolbook geometry was established. Euclid’s geometry is ‘metric’; the concept of distance is basic to it, and is left undefined within it. Projective geometry starts from hardly more than points, lines and planes, of which all we know is that two points define one line passing through them, two lines (lying in the same plane) intersect in one point, two planes intersect in a line, and so on. Obviously, since Pascal’s definition of a cone mentions the circle it has a secret dependence on the idea of distance, a circle being the locus of a point at a fixed distance from another point, its centre. The same is true of Dandelin’s proof above, since it utilizes the hyperboloid which was introduced with the help of a sphere. So these are not purely projective proofs, and it was some time before projective geometry found its own way to these theorems without such borrowings from metric geometry. But once the concepts of projective geometry have been sufficiently elaborated through theorem after theorem, the idea of distance can be defined in terms of them. Stranger still, the details of the definition can be chosen so that the metric geometry that arises is not the familiar one of blackboards; it can be the geometry that describes the properties of figures drawn on a curved surface such as that of a saddle or (with some massaging of difficulties) a sphere. These are all two-dimensional geometries – but three-dimensional projective geometry similarly gives rise to three-dimensional metric geometries analogous to those of the saddle-shape, the sphere, or the intermediate case of the flat surface. The three-dimensional equivalent of this last is the one we know, or think we know, as the Euclidean geometry of everyday life. But the others are logically consistent, if difficult to visualize; these strange ideas of curved spaces were given definite content by geometers of the nineteenth century building on the work of Pascal and his contemporaries. So how do we know what sort of space we actually live in? Perhaps if we could see more of it or measure shapes in it more accurately, we would find that on a cosmic scale it is spherical and returns upon itself like a circle or the surface of the globe. Or it may be an infinite space of the sort that terrified Pascal, in which case it is either flat (Euclidean) or hyperbolic (which is quite unimaginable in three dimensions, but a hyperboloid surface gives a two-dimensional inkling of its twisted nature).
What shape is space? And, as it is known to be expanding, will it continue to do so, or will it someday begin to contract again, perhaps into the state it started from, something hardly more than a point? These are the most enormous, if least pressing, of empirical questions, and I am exhilarated by the fact that our age is the first with the technical wit to answer them. In 1999 astronomers analyzing the motions of distant galaxies and the brightness of supernovas announced (with due tentativeness) their long-awaited findings: spacetime is either flat or very nearly so, and will expand at an accelerating rate for ever. Consider what this means: since the universe came into existence a finite length of time ago, its past is as nothing, quite literally, in mathematical terms, in comparison with what is to come. More marvellous yet, this will always be the case. Even if we ourselves perish in its cold, and Pascal’s lamplight fades, the world is eternal dawn.
The Battleground of Form
But, should the universe prove the astronomers wrong by bringing itself to a full stop and disappearing, and even had it never existed at all, Pascal’s theorem would still be true. Had no mind ever conceived a conic and no hand ever drawn a hexagon, it would still be the case that, ‘If a hexagon is inscribed in a conic, the points of intersection of opposite sides will lie on a line.’ The truths of mathematics transcend the existent, and by their own account of it mathematicians have to make strange journeys to bring those truths home to us; they talk of dream, revelation, intuition, and they claim to waste little time trying to prove results that later turn out to be false; they have a nose for the genuine article of faith. The logical processes of proof may have little to do with this primary capture of truth; after the event, it may be the labour of years to construct a proof others can follow.
My own capabilities in this regard are modest; Cambridge judged correctly in awarding me a second-class degree, and since I left the field I have forgotten most of the standard results and ready-to-hand techniques that are second nature to the professional. But I have had glimmers of illumination; in the days when I was a visual artist now and again an odd little theorem would turn up in my preliminary workings for an abstract painting, like a freebie toy in a cornflakes packet. And though my head is bloodied against the bars of my limitations I delight in following the creative mathematician so far as I can into that glorious unknown, which I picture not as a chill Platonic heaven but an Aladdin’s cave, a darkness full of dazzlement. But are there cracked bowls and counterfeit jewels among its treasures? Are the mathematicians right in supposing that the goods to which they have their mysterious access are all sound? Sometimes it takes generations to come to a judgement on particular cases.
During my London years, since no one bought my artworks, I had to help support myself and my painting habit by drawing diagrams for technical books. One day in 1968 a typescript by a G. Spencer Brown*arrived from the publishers Messrs George Allen & Unwin, with the author’s rough sketches to guide me in illustrating it. Laws of Form was the take-it-or-leave-it title. The meat of the text began with:
THE FORM
We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form….
and it culminated with what appeared to be a derivation of the basic rules of logic from two drastically simple formulae. The steps in between looked like no mathematics I had ever come across; for instance:
according to which something rather complicated equals a blank. I was intrigued, and read it closely, following through the argument line by line, and came to believe that despite the Age-of-Aquarius tone of some of its pronouncements (‘A recognisable aspect of the advancement of mathematics consists in the advancement of the consciousness of what we are doing, whereby the covert becomes the overt. Mathematics is in this respect psychedelic.’) this was hard-hat mathematics, with genuine applications to electrical engineering, and that it did actually found logic on the formal properties of the elementary act of drawing a distinction.
Laws of Form uses the symbol to mark a distinction; one could use a closed box just as well since it makes a distinction between its inside and its outside, but Spencer-Brown’s mark is typographically handier. An innovative if unnerving move is to use a blank to represent the state of affairs before a distinction is drawn. A distinction allows us to name the states or values distinguished. Drawing the same distinction twice is no more informative than drawing it once; if a name is called and then called again, it still indicates the same state; so:
The mark can be understood as an instruction to cross the boundary, say from inside to outside. Crossing the boundary again takes you back to your first state, or, ‘to recross is not to cross’; Spencer-Brown writes this as:
(The two marks are drawn slightly separated for clarity, but should be thought of as occupying exactly the same position.) These two equations are the ‘primitive equations’ of a very simple arithmetic, in which there are only two elements, a mark and the absence of a mark. By applying these equations repeatedly it is possible to work out whether more complicated arrangements of marks within marks, such as that on the previous page, are equivalent to a single mark or to the absence of a mark.
So a calculus – that is, a symbolic system within which calculations can be made – is born. Within a few pages it flowers into theorems of some complexity. Very soon one can draw conclusions about unspecified arrangements of marks; if the letter p stands for such an arrangement, it is easily proved that
(This follows from the fact that any arrangement is equivalent to or to.) Not quite so simple, but provable in the same way, is the equation:
Already the arithmetic has developed into an algebra, with these two equations as its primitive propositions. Now we jump to another level of considerations with endless arrangements such as
where the string of dots indicated that the same pattern continues indefinitely. We can write this as
and say that f is re-entered into itself. Equations of this sort can have two solutions; for instance
whether f =
An obstacle, i.e. a spur to invention, then arises:
does not work for either of the standard values of f. Whatever the value of f, the equation contradicts itself; it has the structure of a paradox. Spencer-Brown proposes to call the value of f imaginary in such cases. (Analogously, in the algebra of numbers, when we come to an equation like -1/x = x, which is not satisfied by any real number, we invent the so-called ‘imaginary’ number i, defined as a solution of that equation. It is routine in both pure and applied mathematics to use such imaginary numbers in calculating results represented by real numbers.) Or we can imagine a tunnel undermining the distinction which up to now has been inviolate (Fig. 18), and picture the value of f flowing outwards, changing from ‘marked’ to ‘unmarked’ or vice versa as it crosses the boundary, and re-entering through the tunnel. The value at any fixed point P would then flip between ‘marked’ and ‘unmarked’ continually as if a train of square waves were passing it.
Fig. 18. An undermined distinction, and successive stages in the outward propagation of a value and its re-entry through the tunnel.
This introduction of time permits the use of the calculus in the design of switching circuits, and the main text ends with a diagram of an arrangement that flips from one value to the other every second time a variable within it flips; i.e. it is a counting device, and it has practical applications, being ‘the first use, in a switching circuit, of imaginary Boolean values in the course of the construction of a real answer’. Finally, in two appendices Spencer-Brown uses his calculus to prove the axioms of Boolean logic, and gives it an interpretation by identifying the marked state with ‘true’ and the blank space or unmarked state with ‘false’.
