by Bruce Director

Let us turn our investigations to the domain of manifolds of a Gauss-Riemann hypergeometrical form. There is no need, as too often happens, for your mind to glaze over as you read the above mentioned words. Lyn has given us ample guidance for this effort, most recently in his memo on non-linear organizing methods.

Over the next few weeks, let us set a course, by way of several preliminary exercises that will shift our investigations from manifolds of constant curvature, that we've been looking at for the last couple of months, to investigations of manifolds of non-constant curvature.

A WARNING: these exercises should not be taken as some type of definition of the concepts involved, any more than bel canto vocalization should be taken as a substitute for singing classical compositions. However, without the former, the latter is unattainable.

As a first start, conduct the following experiment, that was alluded to in the previous pedagogical discussion on the pentagramma mirificum:

Think of a surface of zero curvature, represented as a flat piece of paper. This manifold is characterized by the assumption of infinite extension in two directions. The intersection of these two infinitely extended directions produces a singularity: a right angle, to which all geodetic action is referred.

Now, draw a right triangle, labelling the vertices BAC, with the right angle at A. Extend the hypothenuse BC to some arbitrary point D. (BCD will all lie on the same line.) At D, draw a line perpendicular to line BCD, and extend line AC until it intersects the perpendicular from D. Label that point of intersection E. (You will now have produced two right triangles, with a common vertex at C. The extension of leg AC of the first triangle, will form the hypothenuse CE of the second triangle CDE. Continue this action by extending line ACE to some point F. At F, produce a perpendicular line, and extend leg DE of triangle CDE until it meets the new perpendicular at some point G.

Now you will have three right triangles, BAC, CDE, and EFG forming a kind of chain. Continue to produce this chain of right triangles, by extending the hypothenuse EF of triangle EFG to some arbitrary point H. Draw a perpendicular to H and extend leg FG until the two meet at some point I. Now the chain has four triangles in it.

Keep adding to the chain of triangles in the same manner. You will notice that after every three triangles, the chain "turns" a corner. After the chain has eight triangles, if the appropriate lengths were chosen, the triangle will close. The closed chain of triangles, will resemble two intersecting rectangles. (We leave it the reader to discover what the appropriate lengths are for the chain to exactly close. As you will discover, the fundamental point is not lost, even if arbitrary lengths are used. In that case, the orientation of the 9th triagle will be identical to the 1st.)

Now produce the same action on a sphere, i.e. a surface of constant positive curvature. Begin with a right spherical triangle BAC. Extend its hypothenuse to some point D. At D, draw an orthogonal great circle arc. Extend the side AC until it intersects the orthogonal arc you just drew from D. Continue producing this chain of spherical triangles. You will discover, that the chain of right triangles on the sphere, closes after five "links" have been produced. In other words, the pentagramma mirificum!

(If each hypothenuse is extended to an arc length of 90 degrees, the chain will perfectly close after 5 links. If an arbitrary arc length is used, as in the plane, the chain will not perfectly close, but the orientation of the 6th and 1st triangle will be the same. On a sphere, the lengths need not be arbitrary, as a 90 degree arc length is determined by the characteristic curvature of the sphere. On a plane, no such ability to determine length exists.

Now, think about the results of this experiment. The same action was performed on a manifold of zero-constant curvature and a manifold of constant positive curvature. The same action on, two different manifolds, produces two distinctly different periodicities. What in the naive imagination's conception of the plane and sphere, accounts for two completely different periodicities arising from exactly the same process?

Now try a second experiment:

Stand in a room fairly close to two walls. Mark a dot on the ceiling directly above your head. Point to that dot and rotate your arm down 90 degrees so that you're pointing to a place on the wall directly in front of you. Mark a dot on that wall. Point to that dot, and rotate your arm 90 degrees horizontally to a point on the wall directly to your right (or left). Mark a dot on that wall at that point.

As presented in previous pedagogicals, the manifold of action, that generated the positions of these three dots, is characteristic of a surface of constant positive curvature, i.e. a sphere. The three dots are vertices of a spherical equilateral triangle.

Now, take some string and masking tape and connect the dots to one another with the string. Since the strings form the shapes of catenaries, those same dots are now the vertices of a negatively curved triangle.

Finally, in your mind, connect the dots with straight lines, and those same dots represent vertices of a Euclidean triangle.

From this construction, the same three positions lie on three different surfaces.

But, there is also another type of "surface" represented in this experiment. A hypergeometric manifold characterized by the change in curvature from negative, to zero, to positive curvature.

This is not simply a trivial class room experiment. In our previous discussions, we generated the concept of a sphere, as a manifold of measurement of astronomical observations. Instead of being in a room, the three dots can be thought of as stars, whose positions on the celestial sphere are 90 degrees apart.

But, couldn't the relationship of these three stars, also be conceived to lie on a surface of constant negative curvature? In 1819, Gauss' collaborator Gerling forwarded to Gauss the work of a friend of his named Schweikart, a professor of law whose avocation was mathematics and astronomy. Schweikart had developed a conception, that he called, "Astralgeometrie", that conceived of the spatial relationship among astronomical phenomena as a negatively curved manifold. Gauss replied, that Schweikart's ideas gave him, "uncommonly great pleasure" to read and agreed with almost all of it. In his reply, Gauss added a few additional ideas to Schweikart's hypothesis.

It should come as no surprise, that Gauss would receive Shweikart's work so warmly. Three years earlier, Gauss had expressed an even more advanced notion, in his April 1816 letter Gerling, that we have cited several times before, most recently two weeks ago:

"It is easy to prove, that if Euclid's geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid's Geometry were not true, because then we would have, a priori, a universal measurement, for example, one could use for a unit of space (Raumeinheit), the side of an equilateral triangle, whose angle is 59 degrees, 59 minutes, 59.99999... seconds."

I'm sure you found Gauss' choice of a triangle whose angle is 59 degrees, 59 minutes, 59.99999... seconds curious. But, think about it in the context of the above reference to a hypergeometric manifold characterized by a change from negative to zero, to positive curvature. The surface of zero curvature, is nothing more than a singularity, in that hypergeometric manifold. The sum of the angles of a triangle in a manifold of negative curvature will be less than 180 degrees. The 60 degree equilateral triangle is the maximum. On a surface of positive curvature, the sum of the angles of a triangle is always greater than 180 degrees. The 60 degree equilateral triangle in this manifold, is the absolute minimum.

The triangle Gauss proposes for an absolute length, does not exist in a manifold of negative curvature, nor in a manifold of positive curvature. And, on a surface of zero curvature, it can no longer define an absolute length. On the other hand, in a hypergeometric manifold, that characterizes the change from negative, to zero, to positive curvature, such a triangle represents, a unique singularity, a maximum and a minimum, existing in the infinitessimally small interval, in between two mutually distinct curvatures.

Enjoy the exercises. We'll be back next week.

by Bruce Director

A common mistake can occur, when replicating Gauss' method for determining the Keplerian orbit of a heavenly body from a small number of observations within a small interval of the orbit, that has wider general implications. The error often takes the form, of asking the rhetorical question, "What did Gauss do, exactly?" and, answering that question, with a rhetorical step-by-step summary of a procedure for calculating the desired orbit. In fact, Gauss himself never published, or wrote down any such procedure. Gauss determined the orbit of Ceres in the summer of 1801, and communicated only the result of that determination, so that astronomers watching the sky could re-discover the previously observed asteroid. It wasn't until 8 years later that Gauss, after repeated requests, published his "Summary Overview," and a year after that, his "Theory of the Motion of the Heavenly Bodies Moving About the Sun In Conic Sections."

Both these works, refrain completely from presenting any step-by-step procedure -- because no such procedure existed. Instead, Gauss presented, first in summary form, than in a more expansive way, the totality of interconnected principles that underlay the motion of bodies in the solar system. These principles are not a collection of independent functions that are mutually interdependent. Rather, that mutual connectedness is itself a function, a representation of a higher principle that governs planetary motion.

To illustrate this point, think of Kepler's principles of planetary motion, maliciously mis-characterized as Kepler's three laws. The elliptical nature of the orbit, the constant of proportionality for each orbit (the "equal area" principle), and the constant of proportionality between the periodic times and the semi-major axis of the elliptical orbits, were each demonstrated by Kepler as a valid principle governing planetary motion. But (as those who've worked through the Fidelio article will recognize), all three principles are inseparably linked in each small interval of every planetary orbit. It is the functional relationship among these principles, the "hypergeomtric" relationship, that is the essence of Kepler's discovery.

It is the "disassembly" of this hypergeometric relationship, into separate independent functions, that has been the hysterical obsession of the oligarchy and its lackeys, from Newton, to Euler, to today's academics.

Leibniz, in a letter to Huygens exposed this hoax from the get go:

"For although Newton is satisfactory when one considers only a single planet or satellite, nevertheless, he cannot account for why all the planets of the same system move over approximately the same path, and why they move in the same direction...."

Or, from another angle: Nearly 20 years after his discovery of the orbit of Ceres, Gauss took on the task of measuring the Kingdom of Hannover, by means of a geodetic triangulation. In the course of this investigation, which had many practical implications, Gauss demonstrated a similar "hypergeometric" relationship. Each triangle he measured was "infinitesimally" small with respect to the entire Earth's surface, and the deviation of those triangles from flat ones was also small. As the network of triangles was extended, however, the small deviation in each individual triangle, became an increasingly significant factor in the measurement of the larger area covered by the connected network of these triangles. Not only did the area measured deviate from flat, but it also deviated from a spherical surface, and more closely resembled an ellipsoidal surface. Furthermore, Gauss discovered an "infinitesimally" small deviation from the astronomical observation of his position on the Earth's surface, and the position determined by his triangulation. This led Gauss to the discovery of the deviation of the Earth's surface, from one of regular non-constant curvature, such as an ellipsoid, to a surface of irregular, non- constant curvature, that today is called the Geoid.

This defines a functional relationship of the measurement of the relatively "infinitesimally" small triangles, and the multiple surfaces on which these measurements were performed. That is, each triangle measured, had to be thought of simultaneously as being on a surface of zero-curvature (flat), constant curvature (spherical), regular non-constant curvature (ellipsoidal), and irregular non-constant curvature (the Geoid). The characteristics of each triangle changes from surface to surface. But, in the real world, these surfaces are not independent surfaces, simply overlaid on top of each other. There is a functional relationship among them. Gauss' genius was to recognize, not only the interaction between the characteristic of curvature of the surface, and the characteristic of the triangles measured, but also the functional relationship that transformed one surface into another.

Or, from an even different angle: In 1832, after nearing the completion of his geodetic survey, Gauss published the results of the work he had been doing along the way. In his second treatise on bi-quadratic residues, Gauss extended the concept of prime numbers into the complex domain, transforming Eratosthenes' Sieve. Gauss showed that the characteristics of prime numbers, were also a function of the nature of the surface, such that, for example, 5 is transformed from a prime to a composite number. The number 5 exists in both domains, but it's nature changes, as the domain changes. The number 5 is not two separate independent numbers. Again there is a functional relationship between these two domains, the transformation, that provokes our minds to a higher mode of cognition.

The above three examples, presented in summary form, have been elaborated in previous pedagogical discussions, and will be further elaborated in future ones. The intent in presenting this summary juxtaposition, is to provoke some thought on the functional relationship among these three. They are not three independent concepts. There is a connection, whose active contemplation, gives rise to a conception of functional relationship, that governs the generation of each concept.

As Lyndon LaRouche has wisely advised us, "If you want to know anything, you have to know everything."