by Ted Andromidas

I began my investigation of the implications of the use a minimal surface by Brunelleschi, not merely as a theoretical or experimental investigation of physical principle, but as a "machine tool" breakthrough in constructing the cupola of Santa Maria de la Fiore, by investigating the historic scientific foundations upon which this breakthrough depended. I began, therefore, looking at the Classical Hellenic scientific tradition.

First let us re-acquaint ourselves with the physical principle used by Brunelleschi in the Dome's construction. Why do we call a soap film bound by one or more wire hoops or boundaries a minimal surface? With amazing elegance and simplicity soap film solves an historic mathematical problem, namely, the soap film finds the least surface area amongst all imaginable surfaces spanned by the wire. For example, a "trivial" minimal surface which connects the interior of a circular hoop is a flat circular plain.

In a minimal surface the surface tension stabilizes the whole surface because the tension is in equilibrium at each point on the soap film. In other words, the tension at each point on the surface is equal to the tension at any other point on the surface. Just as the hanging chain or cable equally distributes the weight across its entire length, so the minimal surface also distriubtes the tension equally across its entire surface.

To see this for yourself, take a simple wide rubber band and begin stretching it. As you apply greater tension across the rubber band, you will notice that the middle of the rubber band is narrower, thinner, and almost translucent. The tension across the surface, at that point, is greatest. In fact, you know that the band will snap at that point if you continue to pull it apart. The stretched rubber band is not a minimal surface!

What a wonderful paradox! The surface which creates the minimal of all possible areas within any given set of boundaries also creates equal and minimal tension across the surface.

As we discovered last time, the current history of science indicates that the first non-trivial examples of minimal surfaces were the catenoid and helicoid found by J.B. Meusnier in 1776. Yet, as LaRouche discovered, Brunelleschi uses a minimal surface as a principle of physics in the construction of the Dome.

At this point I thought: Does this principle of least action, though not "proven' mathematically go back to the classical Hellenic period? If the Archimedian screw has been described as a kind of helicoid, was, perhaps, the common bolt thread the first minimal surface studied? In reading a small article on the history of the bolt, I learned that the first comprehensive studies and development the screw or bolt thread are attributed to Archytas of Tarentum, the last and greatest of the Pythagoreans.

I went looking for Archytas.

A close friend and collaborator of Plato, it is if Plato had Archytas in mind when he says that "...those cities rejoice, whose kings philosophize and whose philosophers reign." Archytas himself was so loved and respected in his native city that, though there was a one year "term limit" for anyone to act as chief executive of the city of Tarentum, the citizens suspended these rules and elected him to hold that position for seven consecutive years. We get a sense of his collaboration with Plato in the "Seventh Letter".

Here, Plato discusses his various attempts, at the behest of his student and friend Dion, to teach the just anointed ruler of Syracuse, Dionysus the Second how to become a "philosopher king". Plato says: "Dion persuaded Dionysios to send for me; he [Dion, ed.] also wrote himself entreating me to come by all manner of means and with the utmost possible speed, before certain other persons coming in contact with Dionysios should turn him aside into some way of life other than the best. What he said...was as follows: 'What opportunities,' he said, 'shall we wait for, greater than those now offered to us by Providence?'" Archytas certainly helped Plato in this endeavor: "...it seems, Archytas came to the court of Dionysios. Before my departure I had brought him[Archytas, ed.] and his Tarentine circle into friendly relations with Dionysios."

Plato makes clear his regard for Archytas when he says again in the "Seventh Letter", that when Dionysus invited Plato to Syracuse a second time, he sent the invitation with one of the students of "...Archytas, and of whom he supposed that I had a higher opinion than of any of the Sicilian Greeks-and, with him, other men of repute in Sicily."

Finally, when it becomes clear to all that not only is Dionysus deaf to Plato's teaching, but, infact, the tyrant is determined to kill him, Plato turns to Archytas for help: "I sent to Archytas and my other friends in Taras, telling them the plight I was in. Finding some excuse for an embassy from their city, they sent a thirty-oared galley with Lamiscos, one of themselves, who came and entreated Dionysios about me, saying that I wanted to go, and that he should on no account stand in my way."

Most of what we know about Archytas and his thoughts comes from either references from the writings Plato, Eudoxos, Plotinus, Eratostenes and others, and a handful of fragments his own writings. Nonetheless Archytas' contributions seem to have been substantial and essential, to classical Hellenic science. In the following fragment Archytas writes of the science of mathematics: "Mathematicians seem to me to have excellent discernment...for inasmuch as they can discern excellently about the physics of the universe, they are also likely to have excellent perspective on the particulars that are. Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music. These seem to be sister sciences, for they concern themselves with the first two related forms of being [number and magnitude]."

Besides tutoring Eudoxos, some historians contend that Archytas also tutored Plato in mathematics at some point during the ten years that Plato spent in Sicily and Southern Italy.

Besides saving Plato's life, itself no mean contribution to the future of humanity, Archytas' is also known as the founder of scientific mechanics. Other numerous contributions were in the fields of music, astronomy, mathematics, and aerodynamics. He also provided the first solution the age-old problem of "doubling the cube", i.e. constructing the side of a cube that is double the volume of a given cube.

As I said, Archytas speaks to us only through fragments, yet his thoughts on human creativity and resonate with our own when he says in one fragment: "To become knowledgeable about things one does not know, on must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find. ...."

In astronomy Archytas first put forward the notion of an infinite and boundless universe when in another fragment he says: "...since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit." [This is not to be confused with the idea of simple extension of three linear extensions in space. Ed.]

As with all leading Pythagoreans, Archytas studied music. From these studies comes his discovery and development of the so-called "harmonic mean".

Archytas is also credited with having developed a geometrical method for the famous "doubling of the cube" using a cylinder, cone and torus. Though not attributed to him there, some historians insist that Archytas approach to this problem can be found in Book VIII of Euclid's "Elements" .

Since Archytas avowed that geometry was came from the study of physics, this particular solution to the "cube" problem could well have developed out of his work as an inventor and machine tool designer. As I said, Archytas is sometimes called the founder of mechanics.

As reported last week, General of the Revolution and student of Monge, Jean Baptiste Meusnier not only "discovered" the minimal surfaces of the helicoid and catenoid. But also designed and flew the proto-type of the first Dirigible.

In an historical parallel which is certainly not accidental, Archytas is credited with designing and flying the proto-type model of the first heavier than air aircraft.

According to Hero of Alexandria, Archytas designed and built an apparatus wherein a wooden bird was apparently suspended from the end of a pivoted bar, and the whole apparatus revolved by means of a jet of steam or compressed air.

Which takes us to the bolt or screw thread, in principle, the first use of a minimal surface. The which Archytas created and Archimedes then developed even further. Over the next week, why don't you investigate this problem for yourself.

Construct a cylinder and a helix on that cylinder. You can do this by either constructing a paper of cardboard rectangle with a diagonal, and bending the rectangle into a cylinder; or get an empty paper towel role, which has the helical structure built in. Using the helix as a guide and the cylinder as your unthreaded "bolt", with paper or any other "bendable" material, try to construct the "threads" of the "bolt" around your cylinder.

I urge you to take some time and try various ways of creating the appropriate shape of the surface that you will "bend " around the cylinder. I actually spent several hours drawing and cutting various shapes out of paper and then trying to fit them around a cylinder. So give it a try See what you get. Find out for yourself.

Next installment we will look at exactly what kind of surface we need to construct.

Once I determined to investigate the implications of LaRouche's 1987 discovery of the use of "minimal surface" or " least action' physical principles in the design and construction of Fillipo Brunelleschi's Dome of the Cathedral of Florence, I began to look at some of the history classical Hellenic and Hellenistic science.

Among the first "connecting references to minimal surfaces in Classical Hellenic and Hellenistic science was between the Archimedean Screw a water pumping device, though developed sometime in the 3rd century BC, still widely used today, and the helicoid surface as discovered by French Revolutionary General J.B. Meusnier.

Initial investigations of Archimdes' invention, led to several references comparing the minimal surface helicoid to his invention. Yet none of these references noted the obvious paradox that the former discovery of the helicoid is attributed to Meusnier, the student of Monge, 2000 year later.

This in turn led me back to the 4th century BC founder of mechanics and rescuer of Plato, Archytas of Tarentum, as a way of coming back to the Archimedean principle two centuries later. It is important to note that, in principle, the "machine tool physics" as developed by Archimedes rested upon an historical foundation of at least two centuries or more. This in turn, and in steps, I'm convince will lead back to the implications of Brunelleschi's Dome of Cathedral.

The problem of design faced by Archimedes would have been:

What kind of surface is the thread* of a bolt or screw?

How would I investigate and map such a surface? Put in another way: How would I "blueprint" the necessary specifics of a new machine tool product like the bolt thread?

Let me be clear. Despite what many historians assert, the engineering methods used by these early "machine tool" designers were not based on trial and error.

Let's look at the "physics" we began investigating last week; the physics out of which the Archimedean Screw must have developed. As I indicated earlier, this device, invented sometime in the 3rd century BC, is still in use today. It is an ideal, relatively inexpensive means for pumping large volumes of water or other fluid like material, i.e. sand, fine gravel, ore, etc. Therefore improvements in design and development have continued to the present day.

In the latest study of "Optimal Design Parameters for the Archimedean Screw," as printed in the Journal of Hydraulic Engineering, March 2000 edition, it has been determined that, given various critical parameters, the Archimedean Screw as designed by Archimedes and described by the Rome architect in Book VIII of the Architecture, is in fact, if not the optimal design...the best design! Given design parameters like angle of pitch of the tread surface to amount of thread rotation, or the width of the thread surface compared to the diameter of the overall structure, pumping screw as designed by Archimedes is 7% off from the optimal as determined by today's engineering capabilities.

In other words, the Journal of Hydraulic Engineering concluded, there is no cost effective way to improve upon the original 2000 year old design. Yet that same Journal's authors assert that the incredible success of this design is a result of mere experience with the technology over centuries. This is quite an arrogant assertion on the part of the Journal, as none of Archimedes thoughts on the invention of the screw are extant, owing in part to the Roman's burning of the library of Alexandria. The only course left to the modern investigator, therefore, is to replicate Archimedes thinking, which, in no way, can be considered trial and error.

Two centuries earlier, Archytas was inventing the bolt and screw, whose function can be studied as the intersection of several different, intersecting and interacting surfaces. Archytas is also credited with providing a solution to the age old problem of doubling the cube, using the intersection of those surfaces, i.e. the cone, cylinder and torus.

Archimedes developed a machine tool of such efficient design that, to date it is the best design for doing the job, moving large volumes of fluids. This design also requires the intersection several different surfaces. Archimedes is the first to scientifically investigate volumes of spheres, cylinders and cones, and their inter-relationships. He studied the relationship of weight to volume, using water, to develop the idea of specific gravity. He was not only a mathematician, he was a master inventor and hydraulic engineer.

With this all said; what is the relationship between the "thread" of the bolt and the "cylinder of the bolt? What kind of surface is that thread?

We can "develop" a cylinder by "bending" rectangular plane such that two parallel sides are joined to form the side of the cylinder, while the other two parallel sides from the base and top circles. A cone can be "developed" from a circular plane. Simply cut an arc out of the circle in a "slice of pie" shape. Bend that circular slice of pie arc such that two radii of the circle meet forming the side "ray" of the cone; the point where the two sides of the pie meet, the center of the complete circle of the circular arc is the apex of the cone; the semi-circle forms in the circular base of the cone.

In both cases there is no "ripping" of the surface to make it fit. You just bend it. As you know, we can not "develop" the sphere from a plane; it is not a developable surface. If you did some experimentation last week you might have discovered that the surface of the thread is also not developable.

It is the case though that the circular place surface and the helicoid share common features: 1) They are both minimal surfaces. They define the least area connecting a set of boundaries. The circle, for example is the maximum area for the minimal circumference. The helicoid is a surface which in connecting the boundary defined by a helix also describes the minimal area.

As we pointed out last week, the easiest way to construct a minimal surface is to dip a wire in the shape of the boundary with which you wish to construct the surface, i.e. circle, two circles, cube, pyramid, etc. The soap film will quite beautifully "describe" the shape of the minimal surface connecting those boundaries. A helical wire with a central axis will "describe" the surface called a "helicoid". 2. Both the circular plane and the helicoid are "ruled" surfaces. If you rotate a straight line such that one end is fixed at a point and the other end of the line rotates around tat point, the straight line become the radius of the circle which seeps out a circular plane surface.

Now look at the helix on your cylinder. The cylinder is bound by two circle whose radii are the radii of the helix as well. Now begin to wind one of those radii along the helix, keeping it perpendicular to the side of the cylinder. Think of a winding staircase inside a lighthouse or turret. Think of the edge of each step as the radius of the helix.

This process will describe the helicoid as "discovered" by Meusnier. Now this is fascinating. We've discovered the minimal least action surface of the helicoid as developed in the Archimedean screw in the 3rd century B.C. Now, while trying to convey the idea of constructing a helicoid, we discover that the spiral staircase, an ancient architectural and engineering feature, also describes the helicoid minimal surface. One of the best examples of this is Tycho Brahes observatory in Copenhagen Denmark.

It must be the case that for centuries, if not millennia, architects have been incorporating least action principles of minimal surfaces into their engineering techniques.

More next time.