Text: CONNECTIVITY AND CURVATURE Posted By: Rosalinda Date: Saturday, 26 June 2004, 10:49 p.m. In Response To: POWER FROM THE STANDPOINT OF THE COMPLEX DOMAIN (Rosalinda) But, there is another significant characteristic of these higher transcendental functions which Riemann emphasized, but which only comes to light when Gauss's general principles of curvature are taken into account. This can be introduced pedagogically by taking note of the change in the characteristic curvature of the surface associated with different transcendental functions. For example, a sphere, which is simply-connected, is everywhere positively curved, but a torus, which is doubly-connected, is positively curved only on the "outside", but negatively curved on the "inside". (Ironically, and interestingly, this combination of positive and negative curvature gives the torus a total curvature of zero!) Thus, a higher transcendental power is associated not only with a change in connectivity, corresponding to a change in the density of singularities, but also with a change in the characteristic curvature. Thus, a change in the power of a transcendental function , which occurs through the revolutionary discovery of an existing, but previously undiscovered universal principle, changes the characteristic curvature of the manifold of physical action. To illustrate this, we must again turn back to the work of Gauss. In his {General Investigations of Curved Surfaces}, Gauss showed that on a positively curved surface the sum of the angles of a triangle is always greater than two right angles (180 degrees), whereas on a surface that is negatively curved, the sum of the angles of a triangle is always less than two right angles. Inversely, the characteristic curvature of a surface can be determined by the characteristics of the triangles that exist on it. Furthermore, this characteristic curvature of a surface determines what Kepler called the types of congruences (harmonics) possible on that surface. For example, on a surface of zero curvature, six equilateral triangles can form a perfect congruence, because these triangles will all have angles of 60 degrees, and six such angles form one complete rotation. On the other hand, on a sphere, since any equilateral triangle will always have angles that are greater than 60 degrees, three, four or five triangles, but never six, will form a perfect congruence. Thus, from Gauss's standpoint, the uniqueness of the five regular solidS can be demonstrated to be a consequence of the characteristic curvature of spherical action. But something very different happens on surfaces of negative curvature. Since here the angles of an equilateral triangle are always less than 60 degrees, perfect congruences can be formed by any number of triangles greater than six. The problem Gauss understood, was that while surfaces of positive curvature could be represented as objects in visible space, such as a sphere, negative curvature acted on the visible domain from outside. Consequently, no negatively curved surface could be faithfully represented directly as a visible object! Gauss discovered, however, that the relationships of negatively curved surfaces could be represented visibly, but only as projections in the complex domain. Although Gauss never published his results, his notebooks document the direction of his thinking. Figure 8 shows one of Gauss's drawings depicting the projection of a congruence formed by eight triangles, each with three 45 degree angles. Such triangles could only exist outside the visible domain, on a negatively curved surface. To understand this projection, think of it as an analogy to the stereographic projection of the sphere onto the plane. In that case, the circles of longitude are projected onto radial lines, and the circles of latitude are projected onto concentric circles. (See Figure 9.) The circles of longitude are orthogonal to all circles of latitude, as are the radial lines to the concentric circles in the plane. But, whereas the circles of longitude all converge on the north pole, the radial lines spread out, approaching Cusa's infinite circle. Note, that these radial lines will, therefore, be orthogonal to the "infinite". Spherical triangles on the sphere are projected onto the plane as triangles whose sides are circular arcs, and whose angles are the same as on the sphere. (See Figure 10.) But, though the angles are preserved by the stereographic projection, distance is not. Consequently, as the distances measured approach the north pole of the sphere, the distances in the image on the plane increase exponentially. Now look at Gauss's projection of a negatively curved surface. Instead of an infinitely extended plane, the negatively curved surface projects onto a bounded disc. Here the sides of the triangles are formed by circular arcs, which, like the radial lines of the stereographic projection, are orthogonal to the boundary of the surface. Also, as in the stereographic projection, angles are preserved, but distances are not. But unlike in the projection of a sphere, where the distances become exponentially large as the boundary ("infinite") is approached, the distances in the projected image of a negatively curved surface, become exponentially shorter. (See Figure 11.) With this work of Gauss in mind, we can now begin to illustrate the relationship Riemann showed, between the increasing density of singularities associated with higher transcendental functions, and a change in the characteristic curvature of the manifold. This can be illustrated pedagogically by comparing the difference between the elliptical transcendental and the hyper-elliptical. As developed earlier, the elliptical transcendental, which generates four singularities, is expressed as a Riemann function on a torus, on which there are two distinctly different types of curves curves that go around the torus and the curves that go "through the hole". (See Figure 12.) This doubly-connected action maps into a network of rectangles. (See Figure 13 & Riemann for Anti-Dummies 56). As we just discovered through Gauss, such a congruence of rectangles can only be formed on a surface of zero or positive curvature. But the next highest transcendental, the hyper-elliptical, generates six singularities, and as Riemann showed, must be expressed on a triply-connected surface, such as a torus with two holes. On such a surface there are four distinct closed curves, instead of the two for the torus. (See Figure 14.) A mapping of these four pathways yields an octagonal congruence. (See Figure 15.) As Gauss showed, such a congruence can only exist on a surface of negative curvature, and so its appearance in the case of the hyper-elliptical transcendental is the image of a physical action, characterized by negative curvature, acting from outside the visible domain. (See Figure 16). Thus, as we now think of the hierarchy of the so-called "inexpressibles", from the algebraic, to the circular transcendentals, to the elliptical transcendentals, to the hyper-elliptic and higher, we can understand a successive transformation in curvature from zero (rectilinear/algebraic), positive (spherical/exponential), to positive/negative (elliptical/toroidal), to negative (hyper-elliptical/Abelian). Riemann emphasized that it is the relationship among these three characteristic curvatures, positive, zero and negative, that characterizes physical action. We cannot think of physical action as being characterized by any one type of curvature, but must consider the change in curvature that corresponds to the "power" governing the action. In the Habilitation lecture, Riemann posed a pedagogical construction of three such surfaces, represented by a sphere, cylinder, and the inside of a torus, all intersecting at one circle. (See Figure 17.) The circle is the unique pathway that at all times exists on all three types of curvature at once. Think of this circle as a new type of "infinitesimal", a moment of change from one manifold to another of greater transcendental "power". This relationship between curvature and the higher transcendentals is of extreme importance for the future development of modern physical science. As Riemann stated in his Habilitation lecture, the characteristics of physical action change when extended from the observable range, into the astronomically large, such as the Crab Nebula and the microscopically small, such as the sub-atomic domain. Such changes correspond to an increasing density of universal principles, i.e., singularities, which in turn is reflected as changes in the characteristic curvature, and connectivity, of the manifold of physical action. As science extends its investigations into these domains, an ever increasing number of universal physical principles will be discovered and incorporated into our knowledge of the universe. Such increases are associated with transcendental functions of increasingly higher power, of the type suggested by Riemann a type whose power is akin to that which connects us, through the mind of Keats, to those ancient people depicted on that Grecian urn. http://www.rumormillnews.com/cgi-bin/forum.cgi?read=51183

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