Riemann for Anti-Dummies: Part 64 : Hypergeometric Harmonics

Hypergeometric Harmonics

by Bruce Director

In the same year that Riemann published his {Theory of Abelian Functions}, he also produced a companion piece of equal significance titled {Contributions to the Theory of Functions representable by Gauss’s Series F(a,b,c,x). The content of these works was the polished product of material Riemann developed in a series of lectures delivered at Goettingen University during the 1855-56 interval and through earlier discussions with Gauss and Dirichlet. When taken together with his 1854 Habilitation dissertation, these works developed the crucial discoveries that established the epistemological basis for the future development of science. Like all such fundamental discoveries, it has a history, that begins with the Pythagorean development of the Egyptian science of “sphaerics” , weaves its way through the Renaissance achievements of Cusa, Leonardo and Pacioli, and emerges in the work of Kepler, Fermat and Leibniz. Also, as with all creative discoveries, Riemann’s method is antagonistic to the sophistries of the Eleatics, Aristotle, the empiricism of Sarpi, Descartes and Newton, and the formalism of Kaestner’s and Gauss’s adversaries, Euler, Lagrange and D’Alembert. As such, the study of Riemann’s treatment of what was otherwise known as the “hypergeometric function”, provides us an opportunity to investigate a subject of universal import, whose significance extends far beyond its specific expression in the domain of physical science.
Therefore, to understand Riemann’s discovery, it is necessary to lay an adequate foundation by situating these concepts with respect to the history of ideas. This can be done pedagogically by pivoting our discussion on Kepler’s amplification of the Pythagorean/Platonic concept of harmonics into modern astro and micro physical investigations. But before turning to Kepler, it would be useful to summarily identify some specifics of the matter as it would have presented itself to Gauss and Riemann, because the general level of scientific illiteracy today is so high that only a few are able to recognize what would have been standard knowledge among Gauss’s and Riemann’s qualified contemporaries.

Calculating is Not Knowing

The series that Riemann referred to as “Gauss’s series”, is formally represented as an infinite series of the following form:

1+ ((ab)*x)/(1*c) + (a(a+1)b(b+1)*x^2)/(1*2*c(c+1))) + (a(a+1)(a+2)b(b+1)(b+2)*x^3)/(1*2*3*c(c+1)(c+2)….

It had been studied previously by Euler, who believed it to be a type of magic potion, which, if the right values for a, b, c and x were chosen, could be transformed into an infinite series that Euler could use to approximate the numerical values of algebraic, circular and logarithmic functions. For example, F(1, 1, 1, x) forms the geometric series 1+x+x^2+x^3….; t^nF(-n, d, d,-u/t) forms an infinite series that, algebraically, converges on (t+u)^n; xF(½, ½, 3/2, x^2) forms an infinite series that, algebraically, converges on the arcsine of x ; tF(1,1,2,-u/t) converges on the natural logarithm of (1+t). And there are many other cases of algebraic and transcendental relationships that can be given a formal expression by this series. Euler alleged that if a formal, deductive relationship could be presented between an infinite series and a function on which it apparently converged, the two should be considered equivalent expressions of the same idea.
But Euler was deliberately obfuscating the epistemological questions underlying infinite series that had been definitively settled by Nicholas of Cusa in his sermons on the quadrature of the circle. As is evident from Cusa’s constructions, the circle could never be defined as the limit of the infinite process of dividing the sides of a polygon, because no such repetitive process could ever produce a circle. On the other hand, a circle could produce a succession of polygons with an increasing number of sides. Consequently, Cusa insisted, the circle was generated by an entirely different power, one which Leibniz would later call “transcendental”, as distinguished from the lower, “algebraic” domain of the polygons. It was this transcendental nature of the circle, Cusa emphasized, from which the polygons obtained their characteristics, not the other way around.
The ontological implications of Cusa’s discovery re-emerged in connection with Kepler’s effort to determine the nature of the non-uniform physical action of the elliptical planetary orbits. As Kepler emphasized in his {New Astronomy}, the position and time of a planet could be calculated to any degree of accuracy he desired, but he could not give them an exact numerical value because their relationship depended on both the angle and its sine, whose proportions are incommensurable with each other, and as such, cannot be expressed by a finite arithmetical relationship. Kepler, recognizing that the presence of such an incommensurability signified the existence of an underlying, yet to be discovered principle, called for a new mathematics–which emerged as Leibniz’s calculus and the development of elliptical functions by Gauss, Abel, Jacobi and Riemann. Kepler’s call did not arise from a concern for a more accurate means to calculate the numerical values. He had already devised sufficient methods to achieve the highest degree of arithmetical accuracy possible. His concern was to express a more exact knowledge of the {principle} on which the planet’s motion was based, and the non-uniform changing effect of that principle on the planet.
Leibniz, through his investigation of the catenary and his discovery of natural logarithms, expressed Cusa’s discovery anew from the standpoint of his infinitesimal calculus. Exemplary is Leibniz’s demonstration that the Pythagorean relationships of the arithmetic, geometric and harmonic means had a fundamentally transcendental, not arithmetic origin. Exemplary is the case of the catenary which is expressed as the arithmetic mean between a pair of {transcendental} exponential-logarithmic functions. It is, of course, beautifully ironic to think of an “arithmetic” mean between two transcendental functions, since such an “arithmetic” relationship could never be given an arithmetic numerical value, hence Gauss’s later emphasis on “higher arithmetic”.
Another example is Leibniz’s discovery that the harmonic series 1-1/3+1/5-1/7+1/9… converges on the transcendental function Pi/4. This series is called harmonic because each term is the harmonic mean between its predecessor and its successor. This harmonic relationship is preserved when the negative values, 1/3, 1/7,1/11 or the positive values, 1/5, 1/9, 1/13… are taken separately from each other. The irony of Leibniz’s discovery is {not} that an infinite series of rational numbers produces a transcendental (as the sophistical followers of Euler write in modern textbooks) but that the harmonic relationship among these rational numbers is an artifact of the ultimately transcendental origin of the principle of the harmonic mean. The unified transcendental origin of the Pythagorean means (arithmetic, geometric and harmonic) was thoroughly established by Leibniz through his investigation of the principle of natural logarithms and hyperbolic functions, including his indication that these concepts must be extended into what Gauss later called the complex domain. (This historical fact puts the lie to the dogma of the official “state religion” of modern mathematics, that Euler discovered the principle of natural logarithms and extended the theory of logarithms into the complex domain.)
Despite Leibniz’s demonstrations (or perhaps it is better said to spite them), Euler and his cohorts, Lagrange and D’Alembert, as proponents of the empiricism of Sarpi, Descartes and Newton, were obsessed with the use of infinite series. But their fascination went further than simply looking for a means to calculate numerical values. It was a psycho-pathology. These empiricists insisted that since physical principles could not be observed directly through sense-perception, the functional relationship between a principle and its effect could only be known to the extent it could be given an arithmetical representation.
This method of Euler et al. had its origins in the same malignancy with which the Eleatics and Sophists attacked the method of the Pythagoreans. As Plato demonstrated in the {Meno} and {Theatetus} dialogues, to {know} a principle, such as the principle that has the power to double a square or cube, is to recognize the creative act through which that discovery is made, even if the result of the discovery can not be expressed in a finite arithmetical calculation. For example, the magnitude that doubles the square or cube can be precisely known by geometric construction, even if it cannot be given a finite expression in terms of the power that doubles a line. Thus, Theatetus could claim with confidence, and gain Socrates’ admiration for it, that he {knows} the entire species of square and cubic magnitudes without any reference to their calculation.
The sophists maliciously insisted that Theatetus could not claim to know these magnitudes unless he could give them a finite numerical expression. Yet it is just as absurd to demand that the principle that has the power to double the area of a square be expressed in terms of the principle that doubles a line, as it is to demand that the solutions to the current global economic crisis be expressed in terms acceptable to the reigning dogma of globalism which has brought about that collapse.
But Euler et al., in a direct attack on Cusa’s, Kepler’s and Leibniz’s demonstration of the physical primacy of transcendental functions, revived this ancient sophistry, and demanded that a method of arithmetical calculation must be given before transcendental functions could be admitted into science. With sophistical flourish, such ability to calculate was pronounced the only certain form of {knowledge}. For reasons of mental health, it must be recognized, that this empiricist trick was not designed for the pragmatic purpose of producing more accurate calculations, it was an evil-minded spear aimed at the creative process itself.
Gauss struck back at the sophistry of Euler and the gang, beginning as early as his devastating attacks on them in his 1799 treatise on the fundamental theorem of algebra. Though his own prodigious ability to calculate was legendary, Gauss recognized that knowledge concerned the capacity to discover, not calculate by such methods as infinite series. (Later Gauss was quoted as saying, “infinite series are like paper money. Sooner or later they have to be converted into gold.”)
Gauss insisted that any formal representation of a function must be preceded by a discovery of {the principle} that that function is intended to express. That principle is in turn expressed by a construction whose ironies direct the mind to recreate the original discovery of that principle. Such constructions are exemplified by Archytas’s construction for the doubling of the cube or Gauss’s own construction, in the 1799 treatise on the fundamental theorem of algebra, demonstrating the physical significance of complex numbers.
Gauss addressed these same fundamental questions with respect to infinite series in his 1812 {Investigations into the Infinite Series 1+abx/c….}, including establishing, for the first time, under what conditions could an infinite series be relied upon to converge on a given value. (Euler, for all his promotion of infinite series, never even attempted to prove whether the infinite series that he utilized with abandon actually converged on his assumed limit or diverged away from it. For example, it can be shown, algebraically, that the geometric series 1+x+x^2+x^3… converges on 1/(1-x). But this is only true if x is between -1 and 1. If x takes a value outside this interval, the alleged algebraic equality breaks down. Euler was famous for ignoring such fallacies in his thinking. Abel, reflecting on Euler’s sophistry and Cauchy’s revival of it, said, “divergent series are the work of the devil”.)
However, as Gauss stated in his summary announcement to his treatise, this published work contained only a small part of his investigation. The real subject of Gauss’s work on the hypergeometric series was far more general and profound. It was the fundamental principles underlying transcendental functions. Though the complete line of his thinking can only be gleaned from the fragments in his notebooks, he indicated its direction in the summary announcement to the published work:

“The logarithmic and circular functions, as the simplest kinds of transcendental functions, are those with which the analysts have been most occupied. They deserve this honor, both because of their constant interventions in almost all mathematical investigations, theoretical and practical, and because of the almost inexhaustible wealth of interesting truths which their theory expresses.”

Gauss also included in this domain of transcendentals his newly discovered elliptical functions, which, he emphasized, “must be considered as characteristic of an entirely different species.”
“Transcendental functions”, Gauss continued, “have their true source always, lying openly or concealed, in the infinite”. And as such these transcendentals have been approached through infinite series, of which the hypergeometric form was one of “far-reaching generality.”
But, Gauss insisted, this “far-reaching generality” of the hypergeometric series was not due to something special about the series, as Euler had maintained, but was due to a deeper connection among transcendental functions themselves. Rather than consider the series as the source of the transcendental functions, Gauss insisted they be derived, “from a more universal and applicable source and considered from a higher standpoint.” For Gauss, the properties of the series were an effect of a more general principle, which he identified as a new type of function, in which the differentiated hierarchy of species of transcendentals found their source.
Gauss sought to know this function, which has become known as the “hypergeometric {function}, and he delineated some of its essential characteristics. But it wasn’t until Riemann extended Gauss’s work on curvature, conformal mapping, and bi-quadratic residues, to develop his concept of an anti-Euclidean geometry and what have become known as “Riemann’s surfaces” , that a clear development of this higher domain of transcendental functions could be expressed. Armed with this physical-geometric approach, Riemann was able to elaborate a construction whose ironies recreated the discovery of the hypergeometric principle, as he emphasized, “virtually without calculation.”

Physical Harmonics

The best vantage point from which to approach Gauss’s and Riemann’s work in this regard, is from the standpoint of Kepler’s development of the physical harmonics of the solar system.
Throughout his work, Kepler sought to discover the underlying principles of astrophysics that expressed themselves, visibly, as harmonic relationships among the planetary orbits. The first such relationship that Kepler discovered was the relationship between the five Platonic solids and the size of the orbits of the visible planets. But if the orbits were determined only in this way, they would be circular. But Kepler knew the orbits were really eccentric, not the perfect circles which Aristotle had insisted were the only form of physical motion. Once Kepler liberated science from Aristotle’s chains of perfect circle to the more perfect freedom of eccentric orbits, the question he confronted was, “What was the principle that determined these eccentricities?”
To answer this question he turned to the Pythagorean concept of harmonics. As he emphasized in his {Harmonies of the World}, the concept signified by the Greek word {harmonia}, or its Latin equivalent, {congruencia}, concerns the effect of unseen principles on the interaction among things in the sensible world. As he expressed this idea with respect to the five regular solids, polygons form different harmonic relationships depending on whether they are situated on a plane or sphere. For example, four squares fit together perfectly (are harmonic) in a plane, but only three squares fit together perfectly (are harmonic) on a spherical surface. Similarly, three pentagons are not harmonic on a plane, but are harmonic on a spherical surface. It is not a characteristic of the squares or the pentagons which determines this harmonic relationship, but the characteristics of the surface in which they exist. Thus, the uniqueness of the five regular {Platonic} solids reflects a harmonic characteristic of a spherical surface, not a characteristic of the squares, triangles or pentagons. Similarly, the “constructable” divisions of a circle reflect the harmonic characteristics of the circle, not the polygons that are constructed. In his essay on the snowflake, Kepler extended his investigation of harmonics into the micro-physical domain. (Gauss would later give a more general treatment of Kepler’s investigations into the regular solids and the divisions of the circle in his 1828{General Investigations into Curved Surfaces} and his 1801 {Disquistiones Arithmeticae} respectively.)
To emphasize the point, harmonics are not a characteristic of the things that fit together, but a characteristic of the underlying manifold in which they exist. As Kepler put it in his {Harmonies of the World}:

“Therefore, what is true in general of order and of relation is to be presumed by far the most strongly of harmony, which is based on proportion, and on the counting of parts which are equal in quantity. That is to say, for some sensible harmony to exist, and for its essence to be possible, there must be in addition to two sensible terms a soul as well which compares them. For if that is taken away, there will indeed be two terms which are sensible things, but they will not be a single harmony, which is a thing of reason.”

Kepler utilized this method of harmonics to discover the principle that governed the eccentric motions of the planets. He found that relationship between the minimum and maximum speeds of neighboring planets, which, he showed, corresponds to the same harmonic proportions that humans require to create bel canto polyphonic musical compositions. This “global” harmonic relationship is expressed within the individual planetary orbit as Kepler’s famous principle of equal areas in equal times.
That is, the planets fit together the way they do, not because of anything inherent in the planetary bodies, or even their orbits, but in the harmonic characteristics of the physical principles that generated the solar system itself.

The Hypergeometric Domain

The truthfulness and superiority of Kepler’s harmonics over the empiricist methods of Descartes, Newton, Euler et al. was demonstrated anew by Gauss when he successfully determinated the orbit of Ceres after all those authorities who worshiped at the altar of infinite calculation had failed. (Euler famously lost his sight in one eye trying to calculate the orbit of a comet. Gauss commented, “I too would have gone blind had I calculated the way Euler did.”).
This success was followed up by a myriad of discoveries in the domain of astrophysics, geodesy, geomagnetism and electrodynamics. In previous installments of this series we have discussed the harmonic relationship among the least-action pathways of a potential field with respect to the investigation of Gauss and Dirichlet and Riemann’s generalization of this relationship as, “Dirichlet’s Principle”. (See Riemann For Anti-Dummies #’s 53 and 58.) It is relevant for this discussion, however, to add another example of Gauss’s treatment of Kepler’s harmonics.
One of Gauss’s most famous achievements in this respect was his determination of the harmonic relationship governing changes in the orbits of Ceres, Pallas, and the other large asteroids. While the orbits of these asteroids all conformed to Kepler’s harmonic relationship, Gauss investigated the tiny variations in the eccentricity of these planetary orbits which were due to the interaction of these asteroids with the larger planets, such as Jupiter. Such variations existed for the major planets, and were known to Kepler, but their magnitudes were so small that it was difficult to successfully study them. However, with the asteroids, these non-uniform changes in the non-uniform elliptical orbits were large enough for Gauss not only to measure, but to form an hypothesis concerning an additional harmonic relationship. To determine this relationship, Gauss imagined the mass of the outer planet, such as Jupiter, to be distributed in an infinitely thin ring along its orbit proportional to Kepler’s equal area principle. He then thought of the effect of this ring on the motion of the asteroid’s orbit around the Sun. Thus, he thought of the asteroid as moving in a potential field that was bounded on the inside by the Sun and on the outside by the imagined ring. From this construction Gauss was able to construct an elliptical function that precisely reflected the dynamic harmonics of the non-uniform effect of Jupiter’s orbit on the asteroid, even though this relationship is a highly non-linear transcendental relationship. Gauss further showed that this elliptical function was associated with his discovery of the arithmetic-geometric mean, which also led him into a study of the hypergeometric function. (For a summary account this work of Gauss see the pedagogical “Dance with the Planets” by Bruce Director at www.wlym.com)
Noting this and the many other appearances in physics and astronomy of such hypergeometric transcendentals, Gauss and Riemann insisted that it were necessary to develop a general understanding of the characteristics of this hypergeometric domain.
Since the clearest understanding of these characteristics is only obtained from the vantage point of Riemann’s employment of his surfaces, we interpolate, as Riemann did in his original work, a short, animated review of the Riemann surface. This review may seem a bit arduous when reading through the text, but let the animations do their work. Focus on the ironies among them and these difficulties will be greatly reduced.
Riemann’s development of his surfaces proceeds directly from Gauss’s work on conformal mapping, curvature and potential. In his Copenhagen Prize Essay, Gauss had shown that the conformal mapping of any surface onto another was the effect of the harmonic relationship between its curves of maximum and minimum curvature, and that such a harmonic relationship could be expressed by functions of a complex variable. (This is the origin of what have fraudulently been called the Cauchy-Riemann equations, which should be renamed, for historical accuracy, the Gauss-Riemann equations.)
Riemann took this to its next step. Instead of defining a complex function by an equation and then investigating the implied geometrical characteristics, Riemann showed that the geometrical characteristics of conformal mapping defined the function independent of whatever equation some formalist might wish to use to describe it.
It is important to emphasize that to understand Riemann’s geometric constructions (as is also the case in constructing pedagogical animations of economic processes), one must purge the mind of the formalist remnants of Cartesian graphs. Riemann mappings are not graphs that compare one linear parameter to another. Riemann mappings express a dynamic harmonic among parameters that are themselves multiply connected manifolds.
Pedagogically, this becomes most clear through an animated series of examples of Riemann’s surfaces. Animated figure 1 shows the geometry of the complex cubic function. On the left panel, an harmonic function is depicted extending over 1/3 of a circular disk.

Using Gauss’s representation, each point on the disk can be represented by a complex number denoted by the length of a line drawn from that point to the center of the disk, and the angle that line makes with a given direction. (That direction is marked by the black arrow in the animation.) Under the complex cubic transformation, each point on the disk on the left is mapped to a point on the disk on the right by tripling the angle it makes with the black line, and cubing the length of the line that connects it to the center. This produces a conformal map of the 1/3 disk on the left to the complete disk on the right. The conformality of the mapping is expressed by the invariance of the orthogonality of the arcs and radial lines on both sides. Further, as Gauss and Riemann emphasized, the conformality of the mapping reflects the preservation of a harmonic relationship between the minimum and maximum curvatures as, for example, in the case of a electro-magnetic or gravitational potential.
This geometrical construction brings to the surface two ambiguities of cubic functions that only appear when the function is extended to the complex domain. The first, obvious ambiguity is that since the 1/3 disk on the left maps to a complete disk on the right, what happens if the function is extended to the other 2/3 of the disk on the left? In animated figure 2, it is shown that when this is done, the other two thirds are apparently mapped onto the same disk as the first 1/3.

Inversely, every point of the disk on the right maps to three distinctly different points on the disk on the left.
The other ambiguity that emerges is: what is the effect of the function outside the boundary of the disk? Under Euler’s knuckle-headed approach of infinite series, the only way to know is to extend the disk a little bit, and see what happens, and then extend it some more, and some more ……
These ambiguities, however, are only an effect of the assumptions embedded in the fantasy-world of Cartesian geometry. To eliminate them, Riemann rejected Cartesianism and adopted a more advanced form of Pythagorean sphaerics as it had been pioneered, first by Cusa, and then by Gauss in his use of spherical mappings of curved surfaces. On the spherical form of Riemann’s surfaces, what had appeared to be unbounded, is mapped onto a self-bounded sphere. This removes the false ambiguities inherent in a Cartesian “fish-bowl” so that the discontinuities that appear on the spherical surface reflect matters of principle.
An example of this can be seen in the following series of animations. In animated figure 3, the cubic function illustrated in figures 1 and 2 is mapped, stereographically, onto the sphere. (Animated figure 4 shows the conformality of this mapping.)

Now what had appeared to be infinitely far way, in the Cartesian form of animations 1 and 2, is mapped to a single point, the north pole, which becomes a second pivot for the rotation ascribed by the cubic function.
Riemann resolved the indeterminancy of having one point map to three and three points mapping to one by creating a multi-layered sphere. In this way, those points that appeared to map onto the same place in the Cartesian form, were mapped onto different layers (branches) of the Riemann sphere. In our cubic example, the part of the sphere on the left which is colored red, maps to the red layer on the right, which covers an entire sphere. Similarly with the blue and yellow sections. The branches of the sphere on the right are connected to each other at the poles by what Riemann called branch points which he said should be thought of as, “helicoids with infinitely small pitch”.
The boundaries between the branches on the left map to a single curve on the right which, on both spheres, extends from one branch point to the other. This curve Riemann called a branch cut across which the layers of the sphere on the right could be connected to each other, in conformity with their images on the left.
In this way both spheres become continuous surfaces. But it is important {not} to think of this construction as two different surfaces. Construct in your mind as one idea, as Riemann instructed, one single surface comprised of the essential characteristics of both spheres. Such a surface cannot be visualized externally, but like all works of art, or true ideas of science, it can be sensed in the mind as a real object of thought. The visual representation, as depicted in these animations, must be considered as merely two different views of the same object, and the mapping to be a continuous transformation within that object. To facilitate the generation of this idea, use the principle of inversion, as in a musical composition. The mapping depicted in the animation is just as much a mapping of the right sphere onto the left as vice versa. Thus, to understand a complexfunction, one must also think, simultaneously, of its inverse. The unified action and its inverse are only separated in this visible form because of the inherent limitations of sense-perception. Nevertheless, the cognitive powers of the mind enable us to form the required, unified image.
The continuity of these surfaces is illustrated in animated figure 5.

As the black curve traces its pathway on the left sphere, its image follows a conformal path on the right. When the curve crosses the different partitions of the left sphere, its image crosses from one branch to another on the right. Again think of this also, inversely, as the curve on the right being “unwound” onto the surface on the left.
The different branches of the function appear in the left view as partitions of a single sphere, while in the right view they appear as different layers over an entire sphere. Any action in a part of the left sphere, maps to an action on the same branch of the right. If an action crosses from one branch to another on the left, its image crosses to the same branches on the right, and vice versa.
In this way, the Riemann spherical mapping can express a complex function as an advanced form of the Pythagorean-Keplerian harmonic division of a surface.
On the layered sphere every point will correspond to other points on different layers that are directly above or below it. These points will each be an image of distinct point on the sphere on the left. Gauss called such points {congruent, i.e., harmonic} to each other, relative to the modulus of the function. In our example of the cubic function, that modulus is 2Pi/3, because a rotation by that angle will exactly map each branch onto another. Further, as is the case for all algebraic functions, this modulus has a finite periodicity. In our example, one rotation of 2Pi/3 will map the red part to the blue and the blue to the yellow and the yellow to the red. A second rotation will map the red to the yellow, the blue to the red and the yellow to the blue, and so on. But, after three such rotations, the process only repeats itself.
Further, the divisions into three branches (for the case of the cubic function) is independent of the particular position of the branch-cuts. It is the nature of the cubic function to generate a harmonic division with three branches, in the same way that the uniqueness of the regular spherical solids is independent of the positions of the polygons on the spherical surface. The former is a function only of the harmonic characteristics of the sphere, not of position on the sphere. This is a characteristic of what Riemann adopted from Leibniz as “analysis situs”.
Riemann called this characteristic the {modulus of periodicity}, which implicitly defines a new type of function, which he and Gauss called, {modular functions}. The characteristics of this modular function are defined by the way the branch points, singularities and branches fit together harmonically. For Riemann, this characteristic of analysis situs {defines} the function, not any explicit algebraic formula.
To illustrate this look at animated figures 6, 7, 8, which show the change in the Riemann’s surface corresponding to a change in the algebraic formula from w=z^3 to w=1-z^3.

The change in the algebraic formula has the geometric effect of moving the roots of the function from one point (the south pole in the first case) to three different points (all located on the equator) in the second case. Though this change produces a slight variation in the effect on the sphere on the right, the analysis situs, i.e., the modulus of periodicity, the number of branch points and branch-cuts, does not change. The harmonic relationships expressed by the modularity of the function are essentially the same.
Animated figure 9 illustrates the Riemann sphere for a different algebraic function with an added singularity–a pole.

This results in an harmonic division that divides the left sphere into four quadrants. Each quadrant maps to an entire layer of the sphere on the right. But as the animation illustrates, because of the presence of the pole, the two quadrants in the “southern” hemisphere on the left are mapped from the “north” pole southward, on the right. This effect is reversed in the mappings of the left sphere’s “northern” quadrants.
The further effect of the analysis situs can be seen in animated figure 10.

A pathway around the south pole on the left maps to a pathway around the north pole on the right, and, because the pathway on the left crosses two quadrants, its image produces a doubling of the rotation in its image on the right. In animated figure 11, a pathway around a branch point where all four quadrants come together, still produces only a doubling of the rotation on the right.

This is because the pathway on the left only spends one-fourth of its rotation in each quadrant.
This harmonic relationship is associated with a change in the characteristic of the associated modular function. For the example just illustrated, that modular function must express two actions.: One that rotates the sphere by 180 degrees and the other that inverts the northern and southern hemispheres. As in our previous example a finite combination of these actions, along with the characteristics of the branch points, defines the totality of the action expressed by the function.
But these are only the Riemann surfaces for algebraic functions. The fun only really begins with the transcendental functions. In what follows we will be greatly aided by Riemann’s direction and investigate these functions from the standpoint of analysis situs freed from the illusions inherent in an algebraic formula.
The simplest transcendentals, as Cusa and Leibniz demonstrated, are the higher species from which all the lower algebraic functions are derived. This characteristic emerges clearly from the standpoint of Riemann’s surfaces as a fundamental shift in the nature of the associated modular function. As we just illustrated, the modular function associated with the algebraic species divides the entire self-bounded surface of the sphere into a finite number of parts, which by a finite number of rotations, or inversions, can completely describe the harmonic relationships of the function.
However, the transcendental functions, when expressed by the associated Riemann surfaces, are characterized by a harmonic relationship that divides the sphere into an infinite number of parts.
This is a manifestation of Gauss’s assertion that the transcendental has its true source in the infinite. But as the following examples will illustrate, that source cannot be depicted by the undifferentiated endless blah of Aristotle’s, Descartes’ or Euler’s infinite. Rather it must be thought of as the domain of increasing perfectability which Plato, Cusa and Leibniz ascribed to the actual universe. As Leibniz put it in his {Discourse on Metaphysics}:

“God has chosen the most perfect world, that is, the one which is at the same time simplest in hypotheses and the richest in phenomena, as might be a line in geometry whose construction is easy and whose properties and effects are extremely remarkable and widespread.”

Thus to understand this domain of the transcendental functions we must proceed with Riemann’s method, that is we must investigate those functions from the standpoint of analysis situs, {not} algebraic formulas.
Begin this stage with an investigation of the complex logarithmic-exponential which is the function that expresses the relationship between the arithmetic and geometric. As we illustrated in previous installments in this series, these simple transcendentals are simply-periodic. (See Riemann for Anti-Dummies Part 62.)
The Riemann surface for such a simply-periodic function is illustrated in animated figure 12.

The different colored segments of the left sphere are conformally mapped to a branch that covers an entire sphere on the right. And as is evident from the animation, the number of segments of the left sphere is infinite. Thus, the Riemann surface for the complex exponential divides the left sphere into an ifinite number of segments and produces a sphere on the right with an infinite number of layers.
Animated figure 13 shows this same mapping with the sphere on the left projected down onto a plane.

Here one can see each layer of the sphere on the right is mapped onto a strip on the left. That strip is bounded in the vertical direction, but is infinite in the horizontal directon.
This harmonic division, generated by the complex exponential, defines an entirely different type of modular function. The modulus of periodicity is finite (i*2Pi) but the periodicity of the modulus is infinite!
But the logarithmic-exponential is only the first, and simplest species of transcendental. As Gauss emphasized, the elliptical transcendentals are an entirely different species. As we discussed in previous installments of this series, these transcendental functions arise when two transcendental actions are acting together to produce a single effect. The simplest examples are the case of the elliptical orbit, in which the double incommensurability of the arc with the sine and the arc with the angle combine to produce the unified non-uniform elliptical motion and the simple circular pendulum. As Riemann and Gauss emphasized the distinguishing characteristic of elliptical functions is their double-periodicity. (See Riemann for Anti-Dummies Parts 62 and 63.)
Here again Euler’s fraud is exposed. Both the simple and elliptical transcendentals can be given an infinite expression in terms of Euler’s formal construction of the hypergeometric series. But from the standpoint of the harmonic relationships that emerge with the relevant Riemann’s surfaces, it can be shown that the “infinite” from which these two transcendentals arise is not the same.
Animated figure 14 begins to illustrate this.

Since the elliptical function is doubly periodic, it appears on the Riemann surface as a mapping of a doubly bounded parallelogram on the left sphere to a layer covering the entire right sphere. To show that this mapping is conformal and harmonic, this animation shows the orthogonal green and white curves on the left being mapped to a set of orthogonal periodic curves on the right. In the subsequent animations, only one set of these curves is shown in order to make the animation more visually intelligible.
Animated figure 15 illustrates that there are an infinite number of parallelograms on the left sphere, each of which constitutes an entire branch of the function and thus maps to a layer on the right sphere that covers the entire sphere.

Animated figure 16 shows the same function with the branches of the sphere on the left projected onto a plane.

Thus the branches of the elliptical functions are doubly bounded parallelograms, producing an infinite harmonic division of the sphere. This harmonic division has a greater density of branches than the infinite strips of the simply periodic simple transcendentals.
What becomes even more evident from the standpoint of Riemann’s surfaces, and most relevant for understanding the more universal epistemological implications of Riemann’s ideas, is that the elliptical modular function expresses a greater density of singularities than the simple transcendentals.
This increased density of singularities is brought to light when we trace the pathways of the Riemann surface that go from zero to the infinite. This is illustrated in animated figure 17.

In the figure, the elliptical period parallelograms are outlined in white, and are drawn so that their corners correspond to the points which map to the south pole (zero) on the right sphere. Here we see on the left, orthogonal lines emanating from a corner where the parallelograms meet, and extending to a point near the center of the period parallelogram. The images of these lines, on the right sphere, go from the south pole to the north (from zero to the infinite) and there are four distinct directions. Animated figure 18 shows this same mapping with the left sphere projected onto a plane.

Animated figures 18a and 18b show these paths to the poles with the period parallelograms drawn so that the center of the parallelogram corresponds to zero and the corners correspond to the poles.

Figure 18c shows a surface representation where each pole has both a positive and negative direction.

Thus, in the elliptical transcendental, each branch of the function contains a pole, and since there are four paths to each pole, this pole has double the effect on the mapping of the simple poles of the sphere. Riemann denoted this as a pole of the second degree. What is most important to emphasize, is that the modular function associated with the elliptical transcendentals has a greater density of singularities than is possible for the simple transcendentals.
Because of this doubly periodic nature of these elliptical functions and the associated greater power of its poles, Riemann insisted the geometrical structure of these functions did not conform to the simple spherical action, and could only be competently mapped onto a torus. Animated figures 19 and 20 show that the intrinsic geometry of the torus is congruent with this double periodicity elliptical functions.

Riemann continued these investigations beyond the elliptical case and into the domain of the hyper-elliptical or, Abelian functions. When these functions are expressed by Riemann’s surfaces and their associated modular functions. It becomes clear that each higher species of transcendental function is characterized by a harmonic division with a greater density of singularities. We will have to leave to future installments a further investigation into the anyalsis situs of these higher transcendentals, but you can already glimpse its nature from figure 21 which shows a sketch by Gauss of the modular harmonics associated with the next highest transcendental.

Looking back from the standpoint of the richness of Gauss’s and Riemann’s ideas, it is a wonder why anyone would allow themselves to be enslaved by the sophistry of Euler’s infinitely boring calculations.