Riemann for Anti-Dummies Part 57

Pythagoras as Riemann Knew Him

by Bruce Director

There is a widely circulated report that when Pythagoras discovered the

incommensurability of the side of a square to its diagonal, he sought to conceal its discovery on

pain of death to whomever would disclose it. But such an account is of dubious veracity, as it

attributes to Pythagoras an attitude more appropriate to his enemies than to his collaborators. For it was the Eleatics, Sophists and Aristotle, who insisted that what was inexpressible could not be known; and it was Aristotle's Satanic disciples, as Bertrand Russell would come to exemplify,

who demanded physical death for those who posed the potential for discovering new ideas; and it

was Aristotle's method itself, when practiced as directed, that caused so much mental disease from his day to ours. For Aristotle: control what can be expressed, and you control what can be known.

On the other hand, those who considered themselves Pythagoreans realized that the

inexpressible was the frontier, not the barrier, of human thought. As Plato expressed it in the

{Laws}, those who don't know the significance of the incommensurability of the line with the

square, and the square with the cube, were closer to "guzzling swine" than human beings. The

issue for the Pythagoreans was not that the inexpressible could not be known, but simply that it

could not be expressed, in terms consistent with an {a priori} set of axioms, postulates and

definitions, as Aristotle insisted. Thus, for the Pythagoreans, the discovery of something

inexpressible was not a cause for alarm, but a joyful occasion to demonstrate, that man was not

constrained by mere Aristotelean logic, but was, unlike a swine, free and unbounded.

Therefore, as Plato insisted, it is of great benefit, and to be highly recommended, that

political leaders discover for themselves the significance of incommensurability, in the terms that

that discovery was known to Pythagoras and Plato. However, the true profundity of that

discovery becomes much more fully illuminated when viewed from the standpoint of its more

advanced development--the complex domain of Gauss and Riemann as that concept is expressed

by Gauss's 1799 {New Proof of the Fundamental Theorem of Algebra}, and Riemann's crucial

1854 Habilitation lecture, and his 1855-57 lectures and writings on elliptical, Abelian and

hypergeometric functions. These breakthroughs show that the principles discovered by the

Pythagoreans were simply the first of an extended, and virtually unbounded, succession of

transcendental functions, that express the increasing power of the human mind to discover, and

communicate, ideas concerning universal physical principles.

Knowing Is Not Calculating

Much to the disdain of the Leibniz-hating followers of Euler, Kant, Lagrange and

Cauchy, Riemann insisted that physical principles could be known, and given a mathematical

expression, "virtually without calculation." In taking this approach, Riemann was directly in the

Pythagorean tradition of Plato, Cusa, Kepler and Leibniz, who all recognized, that to know a

physical principle, meant to have an {idea} concerning that principle's generative power, the

which could never be discovered, nor expressed, by merely calculating that power's visible

effects. As Gauss noted in comparing Euler's attempt to determine the orbit of a comet by

calculation (an effort that left poor E. blind in one eye), with his own uniquely successful

determination of the orbit of Ceres, "I too would have gone blind had I calculated like Euler!"

Gauss's comment was consistent with, and inspired by, Kepler's earlier attack on the

Aristotelean Petrus Ramus's diabolical demand that the tenth book of Euclid, (which concerns

the incommensurables) be banned. Ramus insisted, as did Aristotle, that since only ratios of

whole numbers were susceptible to finite calculation, no physical action was knowable, that

could not be calculated thus. (Ironically, Gauss's, {Disquisitiones Arithmeticae}, {Treatises on

Biquadratic Residues I & II} and the subsequent work of Lejeune Dirichlet and Riemann on the

subject of prime numbers show, that even the principles governing whole numbers cannot be

expressed by the linear arithmetic advocated by Aristotle and Ramus.)

In his {Hamonices Mundi}, Kepler demonstrated that the physical principles that govern

planetary motion cannot be expressed by the ratios of whole numbers, but only by those

magnitudes which the Aristoteleans considered "inexpressible", specifically the magnitudes

associated with the regular divisions of the circle, the five regular spherical solids, and the

harmonic relations of the musical tones.

This posed an ontological paradox for the Aristotelean. The principles governing

physical action were inexpressible in terms acceptable to the Aristotelean. Therefore, as

Aristotle's syllogism went, the physical universe was unknowable.

But for Kepler, the principle governing physical action could be {discovered}, by physical

hypothesis, and {known} as a simple, i.e. unified, idea ({Geistesmasse}) . The effect of that

principle could be expressed mathematically only by the appropriate, "inexpressible",

magnitudes. An inexpressible magnitude was thus known, not in itself, but as that which was

produced by the effect of a discovered physical principle.

{In other words, the principle is not known by a magnitude. Magnitude is known by the

principle whose effect it expresses.}

Here, Kepler took his approach directly from Nicholas of Cusa, who, citing the

Pythagoreans in {The Laymen on Mind}, insisted that such inexpressible magnitudes, such as

the proportion of the side of a square to its diagonal, or the relationships among the musical

tones, lead to an understanding of " a number that is simpler than our mind's reason can grasp":

"By comparison then, see how it is that the infinite oneness of the Exemplar can shine

forth only in a suitable proportion a proportion that is present in terms of number. For

the Eternal Mind acts as does a musician, who desires to make his conception, visible to

the senses. The musician takes a plurality of tones and brings them into a congruent

proportion of harmony, so that in that proportion the harmony shines forth pleasingly and

perfectly. For there the harmony is present as in its own place, and the shining forth of

the harmony is made to vary as a result of the varying of the harmony's congruent

proportion. And the harmony ceases when the aptitude-for-proportion ceases."

John Keats makes clear in his great poem, {Ode on a Grecian Urn}, that all human

knowledge is gained in this way. Looking at the urn, Keats sees the images of an ancient Greek

society-- images of real people who lived and died, with passions much like ours. Yet all the

questions he poses, which attempt to determine what the formalist would consider precise

knowledge of those people and their culture, go unanswered. However, what is completely

known, with absolute precision, is that {principle} of whose effect this urn is an image--the

eternal power of human thought:

When old age shall this generation waste,

Though shalt remain, in midst of other woe

Than ours, a friend to man, to whom thou say'st,

"Beauty is truth, truth Beauty" that is all

Ye know on earth, and all ye need to know.

Toward an Extended Class of Higher Transcendentals

To understand Riemann's essential discovery, we must take a quick look back, at the

early development of the knowledge of inexpressibles, from the higher standpoint of Riemann's

work.

Begin with the magnitude which doubles the line. It can double the line but not a square.

Yet, the magnitude that doubles the square is inexpressible, in terms of the magnitude that

doubles the line. Inexpressible, but known--as that magnitude, that expresses the effect, of the

physical principle, that has the {power}, (i.e., {dynamis}), to double a square. Thus, this simple,

yet inexpressible magnitude, is known.

The magnitude that doubles the square, however, cannot triple, nor quadruple, nor

quintuple, etc., a square. These magnitudes are associated with different physical actions.

Though each is distinct, they are nevertheless mutually related, and expressed by the general

relationship, which the Pythagoreans called one geometric mean between two extremes. Thus,

each particular square power is generated by a still higher species of power--the power that

generates all individual square powers.

This higher power can be given a clear mathematical expression as the geometrical

relationships among the sides of the connected right triangles formed by a certain motion in a

semi-circle. (See Figure 1.) While this construction expresses the effect of this power, as one

unified action, it is not the power itself. The power is in the {idea} of that which has the power

to generate all individual square powers. By giving the effect of this idea such an expression, our

{mind's} power to control, and act on this {physical} power, is increased.

But to know more of this idea, we must know not only what it can do, but what it cannot.

This square power, while unlimited with respect to squares, is impotent to double a cube. The

doubling, tripling, etc. of the volume of a cube, is the effect of a different species of power,

which the Pythagoreans understood could be expressed as two geometric means between two

extremes.

As Archytas's construction demonstrates, the generation of this cubic power, can be given

a mathematical expression by the proportions generated by a series of connected right triangles

formed by the relative motion of two orthogonal semi-circles. (See Figure 2.) The relationships

among the right triangles so produced, though changing, always express two geometric means

between two extremes.

This construction expresses not only the effect of the cubic power, but also the connection

between the cubic and the square power, because here, the effect that generates the square

powers, is itself generated as an effect, of the motion that generates the cubic.

Even more importantly, the Archytas construction provides an insight, if seen from the

standpoint of Cusa, into that still higher power, from which the square and cubic powers are

themselves generated. While the specific magnitudes that correspond to the edges of squares

and cubes are generated in the above construction as specific relationships among the lines

forming the sides of right triangles, those relationships are determined not solely by lines, but by

the connected effect of circular and rectilinear action.

This can be seen clearly in the above cited figures. In figure 1, the relationships among

the sides of the triangle are formed as an effect of the connection between the uniform motion of

"P" along the circular arc which generates the non-uniform motion of "Q" along a line. But in

figure 2, "Q" now moves both along a straight-line, {and} around a circular arc, while the

motion of "P" is along both a circular arc {and} along the curve formed by the intersection of a

torus and cylinder.

Thus, it is a type of doubly-connected circular action that generates the rectilinear

relationships that determine the effective changes in squares and cubes. Cusa, in {On the

Quadrature of the Circle}, became the first to identify, and prove, that this circular action was

an effect of an entirely different species of power, than the cubic and square powers. Leibniz

identified this species of power as {transcendental}, as distinct from the lower species of powers

(such as the cubic and square), which he called {algebraic}.

Power From the Standpoint of the Complex Domain

The above review is pedagogically helpful as a starting point for approaching the work of

Gauss and Riemann. As these simple examples illustrate, physical processes are the effects of a

connected action of physical powers (principles). Each power is expressed by a distinct species

of magnitude. But, when a physical action is generated as the effect of a connected action among

a group of powers, it generates a manifold, the which expresses a new, and completely different,

characteristic species of magnitude. Riemann called such manifolds, "multiply-connected".

A strong word of caution is in order. As will become more clear as we work through

Riemann's ideas, by "multiply-connected", Riemann did not mean the Aristotelean idea of a set

of theorems connected to one another through a lattice of logical formalism. Rather, Riemann's

multiply-connected manifold is a unity of demonstrable physical principles, which, like Leibniz's

{monads}, are distinct, but connected, not directly to each other, as if point-wise, but only

through the higher organizing principle of the manifold itself.

A few physical examples, with which readers of this series will be familiar, will help

illustrate this point:

--As Kepler's principles of planetary motion illustrate, the planet's motion, at every

infinitesimal moment, is being determined by the connected action of all those principles that

govern action in the solar system. This action is expressed mathematically by the combined

effect of Kepler's treatment of the five regular solids, the principles of elliptical motion, and the

harmonic relationship among the musical pitches. As Gauss later showed through his

determination of the orbit of Ceres, and his later work on the secular perturbations of the planets

and asteroids, there are an even larger number of physical principles affecting the motion of the

planet at each moment, than those expressed by Kepler. Gauss showed that the manifold of

these connected principles can only be expressed in the complex domain. (See pedagogical

discussion {Dance With the Planets.})

--The case of the intersection of a beam of light with a boundary between two different

media, such as air and water, in which some of the beam is reflected and some of the beam is

refracted. On the macroscopic level, we can see that this action must be thought of as occurring

in a manifold that connects the two principles, reflection and refraction. But as we take this

investigation into the microscopic domain, many more principles, those governing action in the

atomic and sub-atomic domain, come into play, requiring a re-conceptualization of the manifold,

into one with the power to connect a greater number of principles.

--The catenary's expression of the universal principle of least-action as the arithmetic

mean between two, oppositely directed exponentials. Each exponential itself denotes a manifold

that transcends all algebraic powers. The catenary, therefore, must exist in a manifold that

connects two such transcendental manifolds. In this higher manifold, both exponentials are

acting, not only arithmetically, as indicated by their visible relationship, but also geometrically,

the latter acting in the direction perpendicular to the visible plane of the hanging chain. (See

Figure 3.) As Gauss showed, a manifold with the power to act on both exponentials

arithmetically and geometrically, must be expressed as a surface in the complex domain.

In all of the above examples, the powers determining the physical action, are acting, from

outside the visible domain, but their effects are present everywhere. Therefore, as Riemann

made clear in his 1854 Habilitation lecture, to understand physical action, we must ban from

science all considerations of geometry formed from a set of {a priori} axioms, postulates and

definitions, and consider only {ideas} concerning physical manifolds, whose "modes of

determination" are physical principles. With axiomatic assumptions now eliminated from

geometry, the characteristic of action associated with Euclidean geometry, i.e., infinitely

extended linearity, in three directions, disappears as the phantasm it always was. Instead, the

characteristics of such a physical manifold are determined only by the physical principles which

form the "modes of determination" of the physical action under consideration.

In his work, Riemann established the elementary principles to construct an image that

faithfully reflects the means by which such physical "modes of determination" determine the

characteristic of action in such a multiply-connected manifold, by showing how the effect of

these principles determines the topology and characteristic curvature of the image. Most

importantly, what is gained by Riemann's method, is a means to determine and express the type

of change that occurs, by the discovery of a new physical principle.

Riemann based his discovery on the previous work of Gauss, most notably, Gauss's 1799

treatise on the fundamental theorem of algebra, and Gauss's work on the general characteristics

of curvature. Thus, it is most efficient pedagogically, to begin with a quick review of these

features of Gauss's work.

In rejecting the methods of Euler, Lagrange, and D'Alembert, Gauss showed that any

formalist treatment of algebraic expressions, according to the logical rules of algebra, lead to a

contradiction, (i.e. the square root of -1), within the domain of the formal system of algebra itself.

This was not the result, Gauss insisted, of some hidden flaw within the logical system. It was a

flaw of the system itself, arising from the fact that the algebra of Euler, Lagrange and D'Alembert

was merely a logical system. As Gauss emphasized, the system could not be reformed, it had to

be abandoned all together. In other words, Gauss did not come to save the system of algebra. He

came to free science from its mind-killing constraints.

As Gauss showed, the inherent flaw in the formalist's algebra, was the treatment of an

algebraic power by simple rules of arithmetic. Gauss, in referring back to the Pythagorean

principles of the doubling of the line, square and cube, insisted that the "power" in an algebraic

expression must be understood to reflect a physical principle. For example, an algebraic

expression of the second degree, must concern what Riemann would later call a "doubly-

extended" relationship such as areas; an expression of the third degree, must concern a "triply-

extended" relationship such as among volumes. A change from one power to another, therefore,

denoted a change in the number of principles under investigation, not the number of times one

number is multiplied by another. By constructing his surfaces as images that reflect this physical

idea of power, the addition of a new power is reflected in the image, as a change in what he

called the geometry of position, or topology, of the surface. (See Figure 4.) Thus, what is

counted in algebra is not numbers, but powers. For Gauss, it was mind deadening brainwashing

to consider an algebraic expression as a set of formal rules. Instead, he insisted, such expressions

are, at best, only a short-hand description of a physical action, whose real characteristics could

only be truthfully expressed through his geometric constructions.

Riemann insisted that only a method similar to Gauss's could be applied when

investigating the transcendental, elliptical and Abelian functions. As Leibniz had already

indicated, such functions, by their very nature, could never be expressed by any formal algebraic-

type means. For example, assigning a set of rules for calculating the expression "sine of x" does

not give us any knowledge of the transcendental relationship between circular and rectilinear

motion, let alone the profound connection that Leibniz discovered between circular, hyperbolic

and exponential functions. Yet, as Leibniz emphasized, following Kepler and Cusa, universal

{physical }action could only be expressed by such non-algebraic, "inexpressible" magnitudes.

Thus, for Riemann, to "know" a transcendental function, meant to know its geometrical

characteristics, because all attempts at formal expression, as typified by the work of Euler, Lagrange,

and the bigoted Cauchy, were always impotent. (See The Dramatic Power of Abelian

Functions, Riemann for Anti-Dummies Part 54.)

In Riemann's geometrical expressions, as in Gauss's, the change from one transcendental

power to another, is reflected as a change in the topology of the Riemann surface. For

example, the circular/hyperbolic transcendental, which is associated with the catenary, is simply

periodic, has two branch-points, and thus can be characterized by the topology of the sphere. (See Figure 5.) Whereas

the elliptical transcendental associated with the elliptical orbit of a planet, or the motion of a

pendulum, is doubly periodic, with four branch-points, and is characterized by the topology of the torus. (See Figure 6.

See Riemann for Anti-Dummies 49, 52, 54, and 56 ).

Just as in the case of Gauss's treatment of algebraic powers, each transcendental power is

distinct. Consequently, the transition from one transcendental to another, because it involves the addition of a new principle, is not continuous. Like

a discovery of a revolutionary new idea, the shift to a new transcendental, suddenly and

completely, transforms all pre-existing relationships, that had been considered, until then,

fundamental.

For example, think of how Riemann expressed the effect of a simply periodic

transcendental function, through the image of a stereographic projection of a sphere onto a plane.

In this image, the circles of latitude on the sphere are images of concentric circles in the plane,

and, as such, are orbicular. But, the circles of longitude are images of radial lines which

converge at the image of the "infinite", i.e., north pole. Consequently, motion along these

longitudinal circles can never be periodic, as a complete rotation must always "cross over the

infinite".

In this way, Riemann's image fixes in our mind the idea of a physical process in which

simple periodicity is a physical characteristic, not simply a mathematical formalism.

On the other hand, a doubly periodic action is a connected action with two distinct

periods. Such an action could never be represented on a sphere with an infinite boundary. As

Riemann showed in his treatment of the elliptical transcendentals, the type of surface on which

these elliptical transcendentals "live", must correspond topologically to a torus, whose "hole"

allows for these two distinct, but connected, periods. However, as Riemann emphasized, the

transformation from a sphere to a torus is discontinuous, because an entirely new possibility of

action is added. In this way Riemann showed, that the essential characteristics of a

transcendental function, {and} the characteristic of a change in transcendental power, could be

made intelligible, even though such characteristics were utterly "inexpressible" in formal

algebraic terms.

Riemann called the type of transformation just illustrated, a change in the "connectivity"

of the manifold. For Riemann, the sphere is "simply-connected", because it has no hole and

requires only one closed curve to cut it into two distinct parts. The torus, on the other hand, is a

surface that Riemann called "doubly-connected", because it has one hole and requires two closed

curves to cut the surface into two distinct parts. A "triply-connected" surface is one that has two

holes, etc. (See Figure 7.).

Riemann emphasized that connectivity is a characteristic, like the number of "humps" in

Gauss's surfaces, that is independent of all measure relations of that surface, or calculations

within the formal expression. For example, in the case of an algebraic expression, it doesn't

matter how wildly the coefficients of the expression vary, the physical characteristics of the

action that expression describes are determined solely by the number of principles involved, as

denoted by the expression's highest "power". This is what is reflected by the topology (number

of "humps) of the corresponding Gaussian surface. In the case of Riemann's investigation of the

higher transcendentals, the "power" of the transcendental is expressed by a similar type of

invariant characteristic, the surface's connectivity.

It is important to note here, but reserve for the future its more complete development, that

Riemann showed that this characteristic change in the topology of the image, is a function

{solely} of the "power" of the transcendental function, which, in turn, is determined by the

number of characteristic singularities generated by that transcendental function. Thus, the

"holes" in a Riemann surface do not signify "nothingness", or that something is missing or left

out. Rather the number of holes signifies the density of singularities associated with the power of the

transcendental function.

In this way, Riemann showed, in his lectures on Abelian and hypergeometric functions,

that Abel's "extended class" of transcendentals could be expressed by surfaces of increasing

degrees of connectivity, or what Riemann called "multiply-connected" surfaces. A change in the

number of singularities associated with a transcendental function, is expressed as a change in the

connectivity of the surface that expresses that function.

Connectivity and Curvature

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 solida 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.