Riemann For Anti-Dummies Part 48
Riemann For Anti-Dummies Part 48
In December 1822, C.F. Gauss submitted a paper to the Royal Society of Science in Copenhagen titled, “General Solution of the Problem: To Map a Part of a Given Surface on another Given Surface so that the Image and the Original are Similar in their Smallest Parts”.
Notably, the paper contained the motto: “Ab his via sternitur ad maiora” (“These results pave the way to bigger things”).
They did. Nearly 30 years later (1851), B. Riemann submitted, to Gauss, his doctoral dissertation on functions of a complex variable, which, along with his 1857 works on Abelian functions and the Hypergeometric series, developed the further implications of the method Gauss had initiated. The deeper epistemological implications of these results, however, were only brought to light in our present time, through Lyndon LaRouche’s discoveries in the science of physical economy, as in his most recent work, “On Visualizing the Complex Domain”. Therein is established the highest vantage point from which to re-live the discoveries of Gauss and Riemann.
Initially, the Royal Society had posed a more limited subject for the so-called “Copenhagen Prize Essay” than the one ultimately addressed by Gauss. The original question was directed toward solving some particular problems involved in the production of geographical maps. At the suggestion of his collaborator, the astronomer H.C. Schumacher, Gauss proposed the more general question to the Royal Society. After several years without anyone providing a serious solution to the question he posed, Gauss submitted his own solution, and, of course, was awarded the prize.
Obviously, Gauss was less interested in winning the prize than taking the opportunity to present the more general results he had been developing his whole life, beginning with his earliest work on the division of the circle and the fundamental theorem of algebra. The Royal Society’s challenge afforded Gauss the opportunity to demonstrate the extension of Leibniz’ calculus, under the concept of the complex domain that Gauss had developed in those earlier works. While this application provided the solution to the more limited practical problem of producing accurate maps, as Gauss indicated, it was really about something much more fundamental: specifically, the improvement in the capacity of the mind to grasp and communicate truths concerning the unsensed principles that govern the universe.
Mapping the Sensorium
The roots of Gauss’ method go deep into the history of Mankind’s efforts to increase its power in and over nature, beginning with the earliest attempts to map sense perceptual space- time by the development of calendars and geographical and astronomical maps. These maps expressed not merely the visible changes of the heavenly bodies. The unsensed principles were reflected as well, in the form of anomalies, paradoxes, and distortions. Thus, each map already implies another map that lies “behind’, so to speak, the visible map.
Members of the LaRouche Youth Movement currently involved in observing the motion of Mars, are confronting the types of paradoxes that arise from the development of such maps. Go out and look at Mars rising in the eastern sky. The arrangement of the visual image of Mars and the surrounding stars on the inside of the celestial sphere gives rise to a mental image, i.e., map. Over the course of the night, the motion of Mars and the stars, relative to the observer, changes, forming a succession of mental images, which gives rise to a map of the changes in the night’s succession of maps, or, in other words, a map of maps. From night to night, the image of Mars changes its relationship with respect to the images of the other stars. This change gives rise to a higher map, formed from each night’s map of maps. This map of maps of maps gives rise to an even higher type of map, a map that reflects the underlying principles governing the changes among the maps of the sense perceptual images. This higher map only becomes fully intelligible from the standpoint of Gauss’ and Riemann’s complex domain.
While these investigations are most ancient, the roots of our modern knowledge begin with Thales’ (624-547 B.C.) effort to map the celestial sphere onto a plane by the gnomonic, or central projection. Thales’ method was to define on the plane, the image of every point on the sphere, by drawing a line from the center of the sphere, through the surface, until that line intersected the image plane. (See Figure 1.) (The same result can be demonstrated physically by using a light source located at the center of a transparent hemisphere to cast shadows of figures drawn on the sphere onto a flat wall.)
This projection immediately presents us with a crucial paradox. Certain relationships among the images on the sphere are different than the relationships among their images on the plane. For example, the image of a spherical triangle whose vertices are three stars on the celestial sphere, is a rectilinear triangle on the plane. Consequently, the angular relationship among the three stars cannot be preserved in the image, for the sum of the angles of a spherical triangle is always greater than 180 degrees, while the sum of the angles of a plane triangle is always equal to 180 degrees.
However, the measurable relationships among the images of the stars on the celestial sphere are determined by angular measurements, which are not accurately represented by Thales’ gnomonic projection. The first solution to this problem is attributed to Hipparchus (160-125 B.C.), to whom is attributed the discovery of the stereographic projection. In this projection, the sphere is mapped onto the plane from one of its poles. (See Figure 2.) (This can be demonstrated physically by moving the light source in the previous experiment from the center of the sphere to its pole. Notice the resulting change in the relationship among the shadows.)
In this stereographic projection, the lines of longitude of the sphere are transformed into radial lines on the plane. The latitude lines on the sphere are transformed into concentric circles. If we think of the point touching the plane as the “south” pole, and the point of projection as the “north” pole, the circles of latitude in the “southern” hemisphere all map to circles inside the circle which is the image of the equator. On the other hand, the circles of latitude in the “northern” hemisphere all map outside this circle. The radial lines on the plane make the same angles with each other, as the longitude lines of which they are images. However, the radii of the concentric circles that are images of the circles of latitude, increase exponentially, the farther the latitude circles they represent get from the “south” pole and the closer they get to the “north” pole.
In the case of the stereographic projection, the angles among the images are preserved.
Another projection that preserves angles was developed by Gerhard Kremer (1512- 1594), otherwise known as Mercator. On the Mercator projection, the image of the equator is a straight line, and the images of the circles of longitude are perpendicular lines spaced equally along it. The images of the circles of latitude are straight lines parallel to the equator, but the distance between them increases. (See Figure 3.) This is because on a sphere the longitude lines get closer together as they approach the poles. Consequently, the ratio of the distance along the surface of a sphere for a given angle of latitude, to the distance along the surface for the same angle of longitude, changes from the equator to the poles. (See Figure 4.) This change is reflected in the Mercator projection by increasing the distance between the lines of latitude so that the proportion between the lengths of latitude and longitude is the same as on the sphere.
The Mercator projection, although entirely different than the stereographic, also preserves angles. It also has the characteristic that the so-called rhumb line, or path that makes the same angle with all lines of longitude, is a straight line. This exemplifies the types of paradoxes that emerge, for the straight-line is the shortest path on the flat plane of the projection. But on the sphere, the shortest path is a great circle, and the rhumb line is a spiral path called a loxodrome, which is longer than the great circle arc. (See Figure 5.)
The Mapping of Principles
Thus, at first glance, the development of these angle-preserving projections (a characteristic that Gauss would later call, “conformal”), has a very important significance for the representation of the images of the Sensorium. Nevertheless, Gauss had something much more significant in mind. The distortions and paradoxes that result from these projections are not only due to the visual representation, but reflected something “behind” the visible. By developing a general means for transforming one surface into another conformally, Gauss paved the way for Riemann’s more general investigations into the nature of these transformations themselves, and their relationship to the underlying principles behind the Sensorium.
To do this, Gauss rejected the reactionary, a priori Euclideanism of Kant. For him a “surface” is not an object embedded in empty Euclidean space that is infinitely extended in three directions. Rather, a “surface” is generated by some physical action. For example, we don’t measure the celestial sphere by two angles. The celestial sphere is a physically determined idea, generated by the physical action of rotation with respect to the direction of the pull of gravity, and the direction around the horizon from some physically determined direction, such as the position of the rising (or setting) Sun.
In Riemann’s terms, these two acts of rotation are the physical modes of determination of the celestial sphere. Were these modes of determination created by some other physical action, they would produce a different surface. This is the basis for a physically determined geometry. The transformation of one surface into another, Gauss demonstrated, is accomplished by finding a function that transforms one pair of modes of determination into another.
This is exactly what takes place in a stereographic projection. The two modes of determination, represented as circles of latitude and longitude on the sphere, are transformed into concentric circles and radial lines, respectively, on the plane. In the Mercator projection, the same two modes of determination of the sphere, are transformed differently, that is, into straight lines. Nevertheless, both projections are conformal. Thus, the characteristic of conformality reflects a more general principle, not specific to a particular projection.
Gauss recognized that for a projection to be conformal, it must transform one surface into another equally in all directions. This is expressed geometrically by the fact that the images of circles are also circles. This can be illustrated physically in two ways. Think of an image on a stretchable, say, rubber, surface. If the surface is stretched proportionally in all directions, then the shape of the image will be the same, only larger. If it is stretched by a different amount in different directions, the image will be distorted. The former, represents a conformal transformation, the latter, non-conformal.
Another physical example has been pointed out in previous pedagogicals. Take a clear plastic hemisphere and tape to it circles of differing sizes, around what would represent a circle of latitude on a sphere. Shine a light from the center of the hemisphere. Look at the shadows. The circles are transformed into ellipses. Now, move the light to the pole. The shadows become more circular, with the shadows of the smaller circles being the most circular. This illustrates the difference between the gnomonic projection and the stereographic. In the former, the circles are stretched differently in different directions, thus producing ellipses. As such, this projection is non-conformal. In the latter, the circles are stretched into circles, i.e., they are conformal.
This characteristic of circular action, Gauss had already developed as the principle of complex numbers, as early as 1796, in his discovery of the division of the circle, and his 1797 fundamental theorem of algebra (published in 1799). That is, a complex number, Gauss showed, was not arbitrarily defined as the solution to an algebraic equation. Rather, the complex number expressed that rotational action, which transcended, and thus determined, all possible algebraic magnitudes. The transformation of one complex number into another, therefore, was the transformation of one rotational action into another rotational action, exactly the condition necessary for the mapping to be conformal.
For this reason, Gauss considered not the visible surface, but its representation in the complex domain. Each point on the surface corresponded to a complex number, which in turn was determined by the physical modes of determination of the surface. To transform one surface onto another, required transforming the modes of determination of one surface into the modes of determination of the other, which in turn transformed each complex number of the first surface into a definite complex number of the second. This is what Riemann would later call, “a function of a complex variable.”
This was illustrated above by the examples of the stereographic and Mercator projections. In the former case, the circles of latitude and longitude that were equally spaced around the sphere were transformed into exponentially spaced concentric circles and radial lines on the plane. Gauss notes that this corresponds geometrically to the transformation of the complex exponential. (See Figure 6.) The Mercator projection corresponds to a transformation of the stereographic projection. (See Figure 7.) To map a sphere onto a plane, Gauss applied these complex transformations, to the modes of determination of a sphere, that is, the two modes of rotation.
In his paper, Gauss demonstrated why these types of projections would be conformal. This can be illustrated geometrically, by looking at the behavior of a small square undergoing the desired transformation. As the accompanying animation illustrates, in order for the diagonals of the square to remain perpendicular, the sides of the square must change accordingly. Gauss expressed this geometric condition by a formula in the language of Leibniz’ calculus, which was restated by Riemann in his doctoral dissertation. It is one of the continuing frauds of modern mathematics, that this formula has become known as the “Cauchy-Riemann” formula, despite the fact that Augustin Louis Cauchy added nothing to its development. For the sake of historical accuracy, and mental health, this relationship should really be known as the Gauss-Riemann relationship.
Still Lurking Behind the Scenes
Still Lurking Behind the Scenes
Within Gauss’ discovery, something is lurking behind the scenes, a spirit from the domain of unseen principles. In both the above examples, the projection becomes increasingly distorted– as the projection approaches the north pole in the case of the stereographic, and as the projection approaches both poles in the case of the Mercator. At the poles, the projection “blows up” completely and ceases to exist. Is this just a failing in the projection, or, is this an indication of some yet unknown, hidden principle?
In this question lay the “bigger things” for which Gauss paved the way.