In nearly all descriptions of the principle conformal mapping one sees displayed a simple globe whose south pole is situated at the origin of a Cartesian x,y two dimensional plain and rays from the surface projected through the north pole. This represents a transformation from the flat plain to the positively curved sphere.
However, the great Carl Friedrich Gauss developed a different type of mapping of irregular extended curvature onto a unit sphere: the "Gauss map." Here the normal from the curved surface that is given by the differential curvature (but not its magnitude) is mapped by a parallel projection onto the unit sphere. It is the tendency for change at that "point" that is being mapped which is something altogether different than a mere point in the above mapping.
The great student of Gauss, Bernhard Riemann developed this principle yet further with his surface function. It is a surface that combines different sheets of a multi-valued function through singular branch points of that function. Here, our attention is turned from curvature per se to how a function that can take on many values. Viewed from the standpoint of an n dimensional manifold, a branch point might provide a pathway to a new degree of functional freedom to an n+1 dimensional manifold as Riemann detailed in his doctoral habilitation dissertation.
The conception of a singularity where a function branches becomes especially important in physical terms beyond the mathematical formalism. This can be shown in systems that undergo a critical condition and emerge into another regime or phase change such as in plasma physics. It can also be thought of as supplying a means of representing a multi-valued function of proteins in biophysics. There are a whole host of examples where proteins may have more than one role in disparate biological functions and sometimes wholly contrary ones.
Indeed, the most important case of this principle comes into play for the human economy. If we view an invention as a kind of creative singularity for the functional "space" of the physical economy. That is, due to the development of a new degree of freedom the economy "goes over" into a new functional set of relations via the branchpoint of that invention.
No matter, which type of extended space we have to deal with the principle of Leibniz' monadology comes into play. For as Leibniz clearly proved, the mind, or soul if you will, is not an extended substance. That is, it has no parts. Now this is what Riemann is referring to as its "compactness" in his statement "The mind is a compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop." Likewise, all true infinitesimals or monads have no parts. Leibniz called the non-mechanical force arising therefrom "vis viva." If one looks at the human economy as an extended functional system of physical relations one can perform a type of mapping of its distinct elements that operate together in much the same way as a Gauss map. The normal to the "curvature" is the inter-relationship of types of physical goods produced and operatives for the entirety of the reproductive cycle of the economy. In this way, we have a mapping characterized by distinct categories that were developed by Lyndon LaRouche's correction of Marxian economics.
This type of mapping of physical inputs and outputs can be carried out as energy throughput for the entire planet's biosphere for instance. Which brings us to back to the issue of "vis viva" and types of lower and higher monads. Firstly, it is clear that Leibniz qualified the human mind as the highest type of monad. Also that it, the substance of mind or soul is imperishable. How is this so? To show this, we must investigate the concept of pre-established harmony of the monads. In all "spheres" of reality as defined by Vernadsky, i.e. Lithosphere, Biosphere, and Noosphere there is a type of connatus that bounds it, hence its compactness. (Here one should adduce an Astrosphere to Vernadsky's as mankind enters the age of space colonization.) Now Leibniz proposed that not all monads have a kind of window upon the world. Some, however operate as a window for the higher monads. Such are the cilia signaling cells and the place cells, for instance. I would contend that this idea represents a type of mapping of the particular frame of reference of the monad to the universal. (It is of no small import that Einstein's principle of relativity is just such a kind of distinctly Riemannian mapping.) It is here that the harmonics comes into play and it is also in that sense that this enduring non perishable lawful relationship governing all reality makes this the "best of all possible worlds." (So much for ill begotten theories of stochastic and lawless chaos of the rule of irrationality.)
To conclude this survey, next take the physics of pressure waves that Riemann presented in his 1857 paper "On the propagation of planar airwaves of finite amplitude." In this paper Riemann in essence has predicted so called sonar shockwaves. For in any medium there is a governing and limiting characteristic rate of transmission. At the point of going beyond that, a singularity occurs where the curvature goes from a positive spherical (or technically for the case of spacetime hyperspherical) to a negative Mach cone type.
Now this is representative of the manner in which a nonlinear progression is pre-established in the ongoing composition of external and internal reality is distinctly cognized by human noetic or creative mentation in the arts and science. The searching out of such anomalous singularities is the true method for the production of necessary human progress. And here I will end this brief overview to remind my audience that whether these principles come to the fore in human relations will determine either the pathway to a new Renaissance or a new Dark Age.
Since I wrote this post there has been dramatic progress in research on the nature of grid cells. I am happy to report that my intuition that toroidal action is indicative of a principle of metastability across many facets of physical reality has been confirmed yet again. It turns out that the topology of grid cells in the brain take the form of a torus. This confirms to me that in the domain of animal life, there exists a wonderfully built in capacity for a sort of problem solving skill that relies upon a surface that has both positive and negative curvature embedded within it. Recalling that Riemann pioneered non Euclidean curved surfaces and the propagation of shockwaves (that result in negative curved horn surface in a series of Mach cones) as a student of Gauss, it is only lawful that this particular geometry would be in the words of their great predecessor Leibniz be pre-established. See this wonderful linked video.
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