Math II (Несколько текстов для зачёта), страница 7

perhaps having to retract and being exposed to humiliation.

It may be that such a fear acts largely subconsciously

and inhibits one from making a bold step forward.

A man may get close to a great discovery and fail to

make the last vital step.

Possibly it is such a fear that blocks this step.(n46)

In these highly inflected lines, Dirac explicitly touched on his own terror of the humiliating failure that abutted any chance of success, a terror expressed in an ambivalence at once drawn toward risk and success (in the form of the quantum theory he helped create) and yet recoiling with fear from possible failure and "sticking his neck" out from his own place of security. There is here a psychological story of the ambivalence of leaving home, a "home" that is conjointly familial, social, and epistemic--Merchant Venturers' was the workplace of his father, his training ground in engineering, and the place of his first encounter with the projective geometry to which Fraser (and later Baker) had introduced him.

But there is a further story that is only incompletely lodged in this geography of the psychological. This other narrative entails an account of how the logic of drawing was "suppressed"; how thinking through drawing diagrams went from being celebrated across Europe in the mid-nineteenth century to being marginalized at the beginning of the twentieth. To complete this broader narrative properly would take us into the shifting fortunes of geometry in France and Germany, and into fundamental changes in pedagogy at Cambridge.(n47) I have only begun to sketch here the shifting role of persuasive visibilities in physics and their function in shaping an epistemological interior life for Dirac.

The Suppression of Geometry

To the mathematical generation that came of age after 1900 in England, geometry was no longer a science with claims to being descriptive of the world. Instead geometry, once the sun in the scientific sky, was being eclipsed by the formalized, devisualized system of logical relations exemplified on the Continent by mathematicians associated with David Hilbert and by physicists linked to Heisenberg. In Cambridge, it was Hardy who epitomized this new world of rigor--expressing the new mathematics in the formal relations of number theory not in a descriptive, physicalized, and drawn geometry. By the early 1920s, drawn diagrams felt ever more like a disappearing trace, a vestige of a system of inquiry, pedagogy, and values that was fast fading from the Cambridge scene. For the historian of mathematics Herbert Mehrtens, the geometrical-intuitive mathematicians in many ways stood for a Gegen-Moderne, an antimodernism fighting to bind mathematics to the physical world and beyond--to psychology, pedagogy, and progressive technology. The moderns, he argues, wanted to bound and restrict mathematics, guarding their authority through a professional autonomy; mathematics, they argued, was not "about" anything exterior to its own formal structure.(n48)

Dirac stood with one foot in the Cambridge of the older sort (through his association with Baker) and the other in the "new" Continent-leaning Cambridge (through his alliance with Heisenberg, Hilbert, and Hardy). It was a choice between Victorian geometrical tea parties and a post-Victorian modernism. Even as Dirac gave his own tea party talk in 1924, Baker's projective geometry was on the wane. Dirac had moved into the wing of Cambridge mathematics that had already lost the war to set the exam standards for the next generation of students and the mathematical standards for the next generation of researchers. Drawing diagrams gave Dirac an older safe point from which to venture into the new and, as he repeatedly emphasized, more fearsome unknown.

Heisenberg's paper of 1925 was antivisual without being, for that, formally and rigorously mathematical. It was physical and yet completely unvisual. Here was a final step away from the legacy of the Ecole polytechnique's physicalized geometry, away from Felix Klein's tactile mathematical models that formed part of his Erlanger program, away from the British Victorian effort to make descriptive geometry into the centerpiece of skilled reason binding head and hand. And yet, as Dirac launched a long and extraordinarily successful career expressed entirely in the language of algebra, there was another Dirac, privately sketching, figuring, reasoning with diagrams, translating the results back into algebra, and all but burying the scaffolding around an interior furnished with formerly public effects.

My inclination, then, is to use the biographical-psychological story not as an end in itself, but rather as a registration of Dirac's arc from Bristol to Cambridge, to an identification with Bohr's and Heisenberg's Continental physics. In that trajectory, Dirac was sequentially immersed in a series of territories in which particular strategies of demonstration were valued. Bristol University was a step away from the technical drawing of Merchant Venturers', the whole electrical engineering curriculum with its codified, abstracted, applied physics removed drawing to a form of depiction less tied to quasi-mimetic technical renderings and linked instead to more functional, topological circuit diagrams. Bristol's applied mathematics again took Dirac further away from engineering, as did Heisenberg's matrix mechanics.

Technical drawings idealize by removing nonfunctional textures; circuit drawings drop any pretense of mimetic depiction--they are topological insofar as they represent relationships and use icons to refer to component parts. Actual spatial positions and distances do not matter. Projective geometry is also topological in this sense--the distances are eliminated from consideration and only intersections and their relative locations count. Projective geometry began in the domain of the physical, crept somewhat away in higher dimensions and its representation of non-Euclidean geometries. But Dirac kept bringing projective geometry back to the world, using it to track each new topic in mathematical physics across a long career.

When Dirac moved to Cambridge to begin studying physics, he took with him this projective geometry and used it to think. But that thinking had now to be conducted only on the inside of a subject newly self-conscious of its separation from the scientific world. Dirac's maturity was characterized again by flight, this time to Heisenberg's algebra, an antivisual calculus that at once broke with the visual tradition in physics and with the legacy of an older school of visualizable, intuition-grounded descriptive geometry With an austere algebra and Heisenberg's quantum physics, Dirac stabilized his thought through instability: working through a now infolded projective geometry joined by carefully hidden passageways to the public sphere of symbols without pictures.

Freud often argued that what cannot be expressed in private is manifested in public. In a sense I am suggesting the contrary here: at the turn of the century in Britain, projective geometry was shifting away from the status of a state-endorsed liberal epistemology that joined university to factory and toward a form of knowledge that was distinctly second class. Physicalized geometry--geometry grounded in spatial intuitions, visualizations, diagrammatics--collapsed under the language of an autonomous science. In a sense Dirac's suppressed drawings were the hidden remnants of an infolded Victorian world. Public geometry became private reason.

CONSTRUCTIONS USING CONICS

Source: Mathematical Intelligencer, Summer2000, Vol. 22 Issue 3, p60, 13p, 14 graphs, 2bw Author(s): Bainville, Eric; Geneves, Bernard

The classical problems of constructibility using ruler and compass (duplication of the cube, trisection of an angle, quadrature of the circle, construction of the regular polygons) have been solved through the works of Rend Descartes (1637), Karl Friedrich Gauss (1796), Pierre Laurent Wantzel (1837), and Ferdinand Lindemann (1882) (see [3, 8]).

In a recent paper, Videla [11] characterizes the points constructible by ruler, compass, and a "conic drawing tool." In this article, we present constructions using these three tools. The effective realization of these constructions is possible using the Cabri-Geometry software, which integrates the conics as base objects.

To begin, we will recall the definitions of "constructible" using different tools. Then we give some theorems characterizing constructible objects. Next, we discuss construction of regular polygons. We show some known constructions of the polygons with 5, 7, 9, 13, and 17 sides, and some new constructions of the polygons with 19, 37, 73, and 97 sides. We close with some remarks on the automated construction of regular polygons.

Constructibility

Definitions

Given two distinct points a and b, they define a unique line (containing a and b) and a unique circle (centered in a and containing b). No line or circle can be defined by two identical points.

DEFINITION 1 (RC-constructible point). RC stands for "ruler & compass." Let A +/- R[sup 2] be a set of points. Let RC(A) be the smallest set containing A such that the intersection points of two primitives (line and circle) defined from points of RC(A) are in RC(A). Trivial intersections (when the number of intersection points is infinite) are considered empty. RC(A) is called the set of points RC-constructible from A.

A complex number x + iy is RC-constructible if the corresponding point (x, y) is RC-constructible from the set {(0,0), (1,0)}.

DEFINITION 2 RC-constructible number), RC = RC({(0,0), (1,0)}) denotes the set of RC-constructible numbers.

We can now add conics as primitives. Five points are usually on a unique algebraic curve of degree 2 (conic). When more than one conic passes through five given points, we will say that these points define no conic. The definitions of RC-constructibility can then be extended as follows.

DEFINITION 3 (C[sub 2]-constructible point). Let A Subset R[sup 2] be a set of points. Let C[sub 2](A) be the smallest set containing A such that the intersection points of two primitives (line, circle, conic) defined from points of C[sub 2](A) are in C[sub 2](A). Trivial intersections are considered empty. C[sub 2](A) is called the set of points C[sub 2]-constructible from A.

DEFINITION 4 (C[sub 2]-constructible number). C[sub 2] = C[sub 2]({(0,0), (1,0)}) denotes the set of C[sub 2]-constructible numbers.

In the definition of "conic-constructible" by Videla [11], conics are defined from a point F (focus), a line L (directrix), and a number e (eccentricity). The conic is then the set of points M of the plane satisfying dist(M, F) = e dist(M, L). These definitions are equivalent (five distinct points of the conic can be RC-constructed from these elements and, conversely, the elements of the conic can be RC-constructed from five points defining the conic uniquely).

Characterization

THEOREM 1 (Wantzel, 1832). RC is the smallest subfield of C stable under conjugation and square root.

THEOREM 2 (Videla, 1997). C[sub 2] is the smallest subfield of C stable under conjugation, square root, and cube root.

These theorems can be proven using the tools of Galois theory, as explained by Stewart in [10].

Constructions

Sum and product, bisection and trisection of angles Given two complex numbers x and y, RC-constructions of x + y, xy, 1/x, -x, and x are well known. Given a positive real r, the RC-construction of r[sup 1/2] is also well known, C[sub 2]-construction of r[sup 1/3] was first presented by Menaechmus (350 BC): given two numbers a and b, he showed how to construct two numbers x and y such that a/x = x/y = y/b using two parabolas (see [3, 11]). The trisection of an arbitrary angle using conics was first accomplished by Pappus (third century); see [11].

Roots of polynomials of degree 2 and 3 Given reals s and p, the two real zeros of P = X[sup 2] - sX + p, whose sum is s and product is p, are RC(s, p)-constructible. A simple construction uses a Carlyle circle, named after Thomas Carlyle though found earlier by Descartes (see [4]). For A(0, 1) and B(s, p), let c be the circle of diameter [AB] (see Fig. 1). c intersects the x axis if and only if P has one or two real zeros, and in that case, the abscissas of the intersection points are the zeros of P [4].

Given reals a, b, and c, and P = X[sup 3] + aX[sup 2] + bX + c. The real roots of P are constructible as the abscissas of the intersection points between two conics defined from points RC-constructible from {a, b, c}. Several convenient choices of the pair of conics can be made, using either a fixed hyperbola (the right hyperbola XY = 1) or a fixed parabola (Y = X[sup 2]):

• XY = 1 and cY[sup 2] + X + bY + a = 0. This parabola has Y = -b/2c as axis (dashed line) and passes through the points (-a + b - c, - 1), (-a - b - c, 1), and (-a, 0) (black dots). See Figure 2.

• XY = 1 and X[sup 2] + aX + cY + b = 0, a parabola with axis X = -a/2. See Figure 3.

• Y = X[sup 2] and XY + bX + aY + c = 0. This hyperbola has axes parallel to the x and y axes (dashed lines) and the point (-a, -b) as center. Its equation can be rewritten (X + a)(Y + b) = ab - c; that is, X'Y' = ab - c in a coordinate system with origin at the center. See Figure 4.

Descartes used such methods involving circles and parabolas to find the roots of third-degree polynomials. The methods presented here can easily be defined as macroconstructions and used as building blocks for complex figures. The constructions of the regular polygons with 73 and 97 sides presented at the end of this article would have been far more difficult to carry out without using these macroconstructions.

Regular Polygons

Let R[sub p] be the regular p-gon having the points (cos(2kpi/p), sin(2kpi/p)) as vertices, k = 0, l, ..., p - 1.

Gauss [6] has shown that the RC-constructible regular polygons have p = 2[sup n]p[sub 1]p[sub 2] ellipse p[sub k] sides, where n, k >/= 0 and p[sub i] are distinct prime numbers of the form 2[sup a] + 1 (numbers known as Fermat primes). Up to 300 sides, this corresponds to the 38 values 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, and 272 (prime numbers in boldface). This list is given by Gauss in [6] (item 366).