Riemann for Anti-Dummies: Part 41 : The Long Life of the Catenary

Riemann for Anti-Dummies Part 41


The following pedagogy is dedicated to the celebration of Lyn and Helga’s silver wedding anniversary on Dec. 29, which is an occasion for joy, not only for said happy couple, but for all people around the world to whom this marriage has contributed such happiness over the past quarter-century.

The Long Life of the Catenary: From Brunelleschi to LaRouche

The shift from a consumer society to a producer one is, fundamentally, a question of understanding power. For consumers, the world is a universe of magical powers, from which the apparent requirements of life are obtained through the intercession of high priests. Such people, when confronted with a turn of events, such as the present, in which the powers on which they’ve relied no longer deliver, become frightened. They demand action from their increasingly impotent priesthood, who, despite all boasts to the contrary, fail to produce results. They then take matters into their own hands, adopting ever increasing desperate rituals designed to appeal directly to the powers on which their hopes are pinned. The high priests, seeking to restore “consumer confidence” and regain their positions of lofty authority, suggest to the frightened populace, that new rituals be performed and new incantations be recited. Each desperate effort fails to bring about any respite from the crisis. Suspicions mount that the unseen potencies have either gone deaf or departed the scene. Dead or deaf, the thought that such powers might never have existed is their ultimate terror. Even if these forces are now believed gone, the idea of their previous existence not only persists, but continues to govern the thoughts and actions of the populari, feeding the hopeless pessimism that leads such unfortunate creatures into a contorted dance reminiscent of those depicted in a Breugel masterpiece.

There was a similar state of affairs in 14th century Florence, when in the aftermath of the Black Death in which nearly 80% of the Florentine population perished, a group of artists succeeded in launching the project to build a Dome with a diameter of 42 meters over the church of Santa Maria del Fiore. (See Figure 1.)

Figure 1

When the decision to undertake that project was made in 1367, the man who would ultimately execute it, Filippo Brunelleshci, was not even born, but the intention that would guide him was already embedded in the proposed size of the structure, and the requirements of its design. The Dome was to be equal in size to the Roman Pantheon, that temple to the mystical powers on whose behalf Roman popular opinion ruled. (See Figure 2a and Figure 2b.)

Figure 2a

Figure 2b

From the time of its construction in 128 A.D. under Emperor Hadrian, the Pantheon had remained the largest domed structure in the world. But, unlike the Pantheon, the Florentine Dome was to be beautiful from both the inside and out. It was to be a complete break with the pantheonic culture that, even though the Emperors had long since ceased to rule directly under its name, had continued to persist for more than 13 centuries and had brought about the very recent calamity from which Europe was still reeling.

The Dome was a project of bold optimism. It would not only span such a large structure, but by being self-supporting and free standing, it would demonstrate a principle that would transform the entire surrounding region, and through the minds of travelers and the imaginations of those with whom they would speak, the entire world.

The full implications of the principles necessary to construct the Dome were not known to the Dome’s original designers, but, to accomplish the feat, Brunelleschi would have to discover, apply, and communicate a form of the universal principle of least-action whose further elaboration would unfold over the ensuing 500 years. The crucial development occurred in 1988, when Lyndon LaRouche, faced with imminent unjust imprisonment, visited the Dome and recognized the implications of Brunelleschi’s discoveries for the subsequent breakthroughs of Leibniz, Gauss and Riemann, and for the future, development of a new physical science.(See 1989 issue of 21st Century with Dome on the cover.)

The Dome and Anti-Euclidean Geometry

Imagine yourself in 1420 looking at the octagonal drum of Santa Maria del Fiore without the Dome. What do you see? Empty space? If so, you would never envision, let alone build, the Dome. The construction of the Dome required a mastery of principles not visible to the eye. Not the invisible mystical powers of the Pantheon, but the universal physical principles, which, though unseen, are known clearly through the imagination. For the scientist, like the artist, there is no empty space, no empty canvass, no blank slate. There is a manifold of physical principles characterized by a set of relationships whose expression takes the ultimate form of the work of art. To visualize the unbuilt Dome, as the artist Brunelleschi would, imagine the physical principles, and the bricks and mortar take the required form.

This is the basis on which to begin to construct a physical geometry from the standpoint of Brunelleschi, Leibniz, Gauss, Riemann and LaRouche.

To grasp this, think of the difference between abstract geometrical shapes, and the physical geometry of bricks and mortar.

Start with a vertical column. An abstract geometrical line, according to Euclid, is that extension in empty space which has only length. No matter how long the line, its width, is always the same, namely, nil. However, when building a vertical column (line) of bricks, the higher the column, the greater the pressure on the lower bricks. To build the column higher requires strengthening the lower portions of the column, by increasing its width, or by some other means.

Extend this thought to an area. From the standpoint of empty Euclidean space, an area is that which has length and width. A bounded area is enclosed by a line either straight or curved. A physical area, however, is enclosed by a physical structure, the shape of which is determined by physical principles. One approach to enclosing a physical area, would be to build two vertical columns and span those columns with a flat roof. But, this is a relatively weak structure, as the roof is only strong near where it is supported by the columns. The farther apart the columns are, the weaker is the roof. (See Figure 3.)

Figure 3

A far more stable structure for vaulting a vertical area is an arch. The circle is at first thought the simplest type of arch, because the circular boundary encloses the largest area by the least perimeter. If the arch is designed so that all the bricks point to the center of the circle, the arch will be relatively stable upon completion. (See Figure 4.) But, while under construction, the arch cannot support itself, requiring the use of a temporary scaffolding to support the arch under construction. Thus, the arch is self-supporting as a whole, but not in its parts.

Figure 4

The circular arch poses another problem. Even though it encloses the greatest area with the least perimeter, its height is a function of its width and the line of pressure does not conform to the circular curve. (See Figure 5.)

Figure 5

To enclose a taller area requires a wider arch which in turn decreases the overall stability of the structure because the downward pressure from the upper bricks pushes the sides of the arch outward. Thus, even though, from the standpoint of abstract geometry, the circle is isoperimetric, from the standpoint of physical geometry, some other shape provides the greatest stability for the tallest area. The shape that achieves this is a pointed arch in which the two arcs that make up the arch are circular arcs with different centers. (See Figure 6.) The pointed arch, thus, not only encloses a taller area, but is more stable because the curvature of the arch conforms more closely to the physical line of stresses in the structure. In other words, the shape of the arch is determined not by abstract geometrical characteristics but by physical ones.

Figure 6

Now, to the problem facing Brunelleschi enclosing a volume. Geometrically, a volume is enclosed by a surface, which is produced when a curve is moved. For example, from the famous construction of Archytas, when a circle is moved along a line, a cylinder is produced; when rotated around a point, a torus is produced; and when rotated around a line, a sphere is produced. And so, a dome can be produced by rotating an arch, either circular or pointed around an axis. (See Figure 7.)

Figure 7

But, a physical surface, such as a dome, is not simply a rotated arch, because a new set of stresses occurs in the dome that does not occur in the arch. In addition to the stresses along the arch, (from top to bottom, i.e., longitudinal), there are stresses around the dome (circumferential or hoop).

So, the problem Brunelleschi faced in building the Dome of the required size was to design a structure whose shape would balance these stresses without requiring external buttressing which would diminish the Dome’s beauty and undermine its effectiveness for changing society by changing the minds of the population.

Additionally, a dome, like an arch, generally requires a supporting scaffold, or centering, to hold it up until its completion. Here, Brunelleschi faced his most formidable obstacle. The Dome over Santa Maria del Fiore was so big that there was not enough wood available to build such a large scaffolding. Consequently, Brunelleschi took the bold step of building the Dome without centering, requiring him to construct a dome that was self-supporting in its whole and its parts. Such a shape could not be determined by the methods associated with Euclidean geometry. The shape Brunelleschi required was determined only by physical principles.

To do this, Brunelleschi decided to construct two domes, one inside the other, with a stairway between them. Both would conform to the pointed arch form called for in the original design. However, according to the architect Bartoli, (see 1989 21st Century) the curve of the inner dome was a based on a circle whose diameter was three fourths the inside diameter of the octagonal drum (pointed fourth) while the outer dome’s curvature was four-fifths the outer diameter (a pointed fifth) (See Figure 8.)

Figure 8

Since the use of centering had to be avoided, Brunelleschi had to control the curvature of both domes very carefully as the were being constructed. This entailed controlling three different curvatures: the longitudinal curvature; the circumferential curvature; and the curvature inward towards the center of the Dome. If all three curvatures could be controlled during all phases of construction, the Dome would not only be stable upon completion, but each stage would be stable enough to be a platform from which the next stage would be built. Thus, the Dome had to conform to a shape that would be self-supporting in its whole and its parts.

Brunelleschi had to solve a multitude of problems, each requiring revolutionary new ideas to accomplish the task. But, the discovery most central to his success, the one most relevant for the future development of the anti-Euclidean physical geometry of Kepler, Fermat, Leibniz, Gauss and Riemann, is the one identified by LaRouche. Brunelleschi used a hanging chain to guide the development of the curvature of the dome at each stage of construction. Thus, the overall shape of the Dome was determined, not by a curvature defined by abstract mathematics, but by a physically defined principle. Just as a hanging chain is self-supporting in its whole and its parts, the Dome, whose curvature is guided by the curvature of the hanging chain, is, likewise, self-supporting surface, in its whole and its parts.

The beauty of the Dome demonstrates the truth of Brunelleschi’s discovery, but, it would take the discoveries of Kepler, Fermat, Leibniz, Gauss, Riemann and LaRouche to fully grasp the underlying principle.

The Development of the Physical Idea of Shape

The success of Brunelleschi’s Dome demonstrated that the architectural principles of physical geometry on which it was based were universal. This view was expressed by Johannes Kepler, who approximately 150 years later wrote concerning the construction of the solar system in his Mysterium Cosmographicum, “We perceive how God, like one of our own architects, approached the task of constructing the universe with order and pattern, and laid out the individual parts accordingly, as if it were not art which imitated Nature, but God himself had looked to the mode of building of Man who was to be.”

Kepler went on to develop, in that work and in his subsequent New Astronomy and Harmonies of the World, that the shape of the solar system, like the Dome, was determined not by considerations of abstract mathematics, (which would have indicated perfectly circular orbits) but by physically determined harmonic principles. Thus, the elliptical planetary orbits, like Brunelleschi’s Dome, were the size and shape that they had to be in order to express the harmonic relationships of those physical principles.

This physically determined idea of shape took another step in its development with Fermat’s determination that the shape of the pathway of light was determined by the principle of shortest-time:

“Our demonstration is based on the single postulate, that Nature operates by the most easy and convenient methods and pathways — as it is in this way that we think the postulate should be stated, and not, as usually is done, by saying that Nature always operates by the shortest lines … We do not look for the shortest spaces or lines, but rather those that can be traversed in the easiest way, most conveniently and in the shortest time.”

Leibniz, following up on the discoveries of Kepler and Fermat, generalized these discoveries as a universal principle of least-action:

“…the Architect of all things created light in such a way that this most beautiful result is born from its very nature. That is the reason why those who, like Descartes, reject the existence of Final Causes in Physics, commit a very big mistake, to say the least; because aside from revealing the wonders of divine wisdom, such final causes make us discover a very beautiful principle, along with the properties of such things whose intimate nature is not yet that clearly perceived by us, that we can have the power to explain them, and make use of their efficient causes, along with their artifacts, such as the Creator employed them in order to produce their results, and to determine their ends. It must be further understood from this that the meditations of the ancients on such matters are not to be taken lightly, as certain people think nowadays.”

Leibniz’ most far reaching discovery of this principle of least-action, made in collaboration with Johann Bernoulli, demonstrated that the catenary, the guiding principle of Brunelleschi’s Dome, embodied the most universal expression of least-action. As we’ve developed in other locations, the physical characteristic by which the hanging chain supports itself, expresses all the elementary transcendental relationships of geometry: the circular, hyperbolic, and the powers associated with lines, surfaces and volumes. (See Figure 9.)

Figure 9

Look back at our earlier comparison of the difference between abstract geometrical notions of line, area and volume, with the physical requirements of constructing a column, arch and dome. As is already implicit in the concept of powers developed by Pythagoras, Archytas, Plato, et al., even the purely geometrical concepts of line, area and volume are ultimately determined by the physical principles which Leibniz demonstrated are expressed by the catenary. The idea of lines, areas and volumes, separated from this idea of power as universal physical principles, is as chimerical as the mystical powers of the Roman Pantheon.

From Pathways to Surfaces

Brunelleschi’s Dome points the way to a still further development of the universal principle of least-action. Planetary orbits, the curvature of light, and catenaries are all pathways, i.e. curves. Brunelleschi’s Dome is a least-action surface.

The concepts to understand this distinction were developed by Gauss who, looking back as we’ve been doing, on the discoveries of Kepler, Fermat and Leibniz, developed the foundations of a physical theory of surfaces.

The context for Gauss’ discovery was his measurement of the surface of the Earth, which, because it is physically determined, must, in keeping with Leibniz’ principle, be a least-action surface. Over a more than 20 year period, Gauss made careful astronomical and geodetical measurements of the Earth. Abstract geometrical considerations would suggest that the Earth would be a perfect sphere, because the sphere enclosed the largest volume inside the smallest surface. But, because the Earth is a rotating body in the solar system, its physical shape is not spherical, but ellipsoidal.

As we have developed in earlier pedagogicals, Gauss’ measurements led him to discover that the physical shape of the Earth was not ellipsoidal, but something more irregular. He identified the, “geometrical shape of the Earth, as that shape which is everywhere perpendicular to the pull of gravity.” In other words, Gauss did not try to fit the Earth into a shape pulled from the text books of abstract mathematics, rather, he invented a new geometry that conformed to the physical characteristics of the rotating Earth.

Gauss reported the generalization of his discoveries in his 1822 Copenhagen Prize Essay, on conformal representation and his 1827, “General Investigations of Curved Surfaces”. Future pedagogicals will develop these concepts in greater detail, while here we focus on the general ideas most relevant to this discussion.

For Gauss, all surfaces had a characteristic curvature, which in turn determined certain least-action pathways, that he later called, “geodesics”. For example, in a plane, the geodesic is a straight-line, while on a sphere, the geodesic is a great circle. In these two cases the curvature is uniform and so the geodesic is the same every where on the surface. In contrast, an ellipsoid, for example, is a surface of non-uniform curvature. Consequently, the geodesic is different depending on its direction and position on the surface. (See Figure 10.)

Figure 10

To illustrate this, the reader is encouraged to do some physical experiments. Take a plane, sphere, spaghetti squash or other irregular shaped object. Mark two points at different places on the surface and stretch a thread between them so that the thread is taught. The thread will conform approximately to the geodesic between those two points. Notice that on the plane, the geodesic is always a straight line, while on the sphere it is always a great circle, while on the squash, the geodesic changes from place to place, and direction to direction.

There is a further distinction between the plane and the sphere or ellipsoid. On the plane there are an infinite number of pathways between any two points, but only one of these paths is a geodesic, i.e. least-action. This is also true on a sphere or ellipsoid, except, if the two points are at the poles. Then there are an infinite number of geodesics between these two points. Thus, the bounded nature of the sphere and ellipsoid, produce a singularity with respect to the nature of the geodesics. The significance of this distinction will become more clear as we develop more of Riemann’s geometry in future pedagogicals.

What Gauss investigated was the general principles by which the curvature of the surface determined the characteristic of the geodesic. Of immediate relevance for this discussion is Gauss’ determination of a means to measure the curvature of the surface at any point. It is sufficient for our purposes here to illustrate this by a physical demonstration. On the squash, draw a circle by tying a marker to one end of the thread and rotating it while holding the other end of the thread in a fixed position. The radii of this circle are all geodesics in different directions. Now examine the curvature of each geodesic, which will vary for each direction. However, one geodesic will be the least curved, while another will be the most curved. Now, try this on a different type of surface, such as a butternut squash shaped like a dumbbell. The part of the round ends of the butternut squash have the same characteristic as the spaghetti squash, in that the center of curvature is always inside the squash. But, in the middle of the squash something different happens. Here the center of curvature is either inside or outside the squash, depending on the direction of the geodesic. This characteristic Gauss called, “negative curvature” and is the characteristic of curvature expressed by a surface formed by a rotated catenary called a catenoid. (See Figure 11.)

Figure 11

Brunelleschi’s Dome expresses this characteristic of negative curvature.

Furthermore, Gauss proved that on any surface, no matter how irregularly it was curved, the geodesics of maximum and minimum curvature would always be at right angles to each other!

Thus, at any place on a surface, the curvature of the surface expresses a physical principle that in turn determines the geodesic, or least-action pathway within that surface.

From Surfaces to Manifolds

Working from Gauss’ discovery, Riemann generalized this concept still further to the idea of a geodesic within a manifold of universal physical principles. The manifolds cannot be directly visualized but the characteristics of that manifold can be directly determined by a change in geodesic.

For example, light under reflection and refraction follows a pathway within a surface, but each type of action expresses a different pathway because the physical manifold of refraction includes a principle, changing speed of light, that does not exist within the manifold of reflection. The addition of this new principle to the manifold of action, changes the geodesic.

Riemann developed the means to represent these higher manifolds by complex functions. For example, as was developed in the previous pedagogical, the conic section orbits and catenary are both least-action pathways with respect to the manifold of universal gravitation. In other words, each represents a changing geodesic with the manifold of universal gravitation. But, when the catenary is expressed as a function in the Gauss/Riemann complex domain, the conic section orbits are seen as a subsumed geodesic within the higher principle represented by the catenary. (See Riemann for Anti-Dummies Part 40.)

More general examples are illustrated in the accompanying animations. These illustrate how the same action, when carried out in different manifolds, is changed by the characteristics of the manifold. Think of the orthogonal nets in each figure as the minimum and maximum geodesics in each manifold. The loopy curve maintains the same angular orientation with respect to these geodesics in each case. But, because the geodesics change, from manifold to manifold, the action changes. Thus, a change in the principles that determine the manifold, change the geodesics, which in turn change all action within that manifold. Conversely, to effect a physical change in any action, one must act to change the characteristics of the manifold.






Now look at Brunelleschi’s Dome from this standpoint. The Dome is a surface whose geodesic, in principle, conforms to the catenary. As a least-action surface, it expresses a geodesic with respect to the principle of universal gravitation. With respect to the manifold of universal history, building the Dome was the geodesic from that dying culture of the Roman Empire to the Renaissance.

At our present place in the manifold of universal history, building LaRouche’s youth movement combat university on wheels and making LaRouche President of the United States, is for us, Brunelleschi’s Dome– the geodesic from this dark age to our Renaissance.