Tag: Ted Andromidas

by Ted Andromidas

Note: For this pedagogical discussion, you will need Appendices I and X to {The Science of Christian Economy}, {So You wish to Learn All About Economics,}, and the April 12, 2002 issue of {Executive Intelligence Review}.

“You have no idea how much poetry there is in a table of logarithms.” — Karl Friedrich Gauss to His Students

Developing a function for the distribution of the prime numbers has been one of the great challenges of mathematics. An exact solution to this problem, of how many numbers generated between 1 and any given number, N, are actually prime, has not yet been discovered, though there is a general notion of a succession of manifolds as determining to any solution.

One of the most stunning demonstrations of the generation of number by an orderable succession of multiply-connected manifolds, is Karl Friederich Gauss’ discovery of the “Prime Number Theorem.” The wonderfully paradoxical nature of Gauss’ approach, in contradistinction to that of Euler, is that we must move to geometries associated with the physics of higher-order forms of curvature, such as the non- constant curvature of catenary functions, and those forms of physical action associated with living processes, for a first approximation solution.

To understand the importance, and the elegance, of this discovery, we must first investigate a class of numbers called logarithms. Hopefully, it will all so demonstrate the inherent differences between a “constructive” approach to the questions of the generation of such numbers as logarithms as over, and against, the formalisms of the textbook. I have included as an addendum at the end of this discussion, a short rendering on the subject of logarithms, modeled on that of a typical textbook , so the reader might more appreciate the conceptual gulf separating the constructive approach from that of classroom formalisms.

“It is more or less known that the scientific work of Cusa, Pacioli, Leonardo, Kepler, Leibniz, Monge, Gauss, and Riemann, among others, is situated within the methods of what is called synthetic geometry, as opposed to the axiomatic- deductive methods commonly popular among professionals today. The method of Gauss and Riemann, in which elementary physical least action is represented by the conic form of self-similar- spiral action, is merely a further perfection of the synthetic method based upon circular least action, employed by Cusa, Leonardo, Keller, and so forth. [fn. 1]

It is in this domain, physical least action associated with the self-similar spiral characteristic of living processes, that we search for a solution to the ordering principle which, in fact, might generate the prime numbers. Gauss approach involves understanding the idea behind the notion of a logarithm.

Logarithms are numbers which are intimately involved in the algebraic representations of self-similar conic action. In previous discussions, we saw that number measures more than just position or quantity; number can also measure action. We discovered that numbers in one manifold measure distinctly different qualities, than numbers in another manifold, and that what and how you count can sometimes leave “footprints” of a succession of higher ordered manifolds.

All descriptions of logarithmic spiral action, and the rotational action associated with them, are of two types of projection:

1) The 3 dimensional spiral on the of cone; we understand that each increase in the radial length of the 2 dimensional, self-similar spiral on the plane, is a projection from the 3 dimensional manifold of the conic spiral. The projection of the line along the side of a cone, which intersects and divides the spiral is called “the ray” of the cone. [See {The Science of Christian Economy}, APPENDIX I]

2) The 3 dimensional helical spiral action from the cylinder; the rotation of the three dimensional manifold of the cylindrical spiral (helix) projects on to the two dimensional plane as a circle. Nonetheless, some action is taking place, and that action is represented, therefore, by a “circle of rotation”, as simple cyclical action, i.e. we “count” the cycles of each completed, or partially, completed cycle of rotation of the spiral.

Turn to the April 12, 2002 issue of EIR, page 16, (See figure), “The Principle of Squaring”; review the caption associated with that figure [“The general principle of ‘squaring’ can be carried out on a circle. z^2 is produced from z by doubling the angle x and squaring the distance from the center of the circle to z.”] and construct the relevant diagonal to a unit square. The side of the square is one, the diagonal that square equals the square root of two. Use that diagonal, the square root of two, as the side of a new square; the diagonal to that square, whose area 2, will also be a length equal to two. We are generating a series of diagonals, each, in this case, a distinct power of the square root of two. In this case, it is a spiral which increases from 1 to 16 after the first complete rotation; 16 to 64 after the second rotation, etc. As we will soon see, each of the successive diagonal beginning with the first square 1, is also part of a set of “roots” of 16.

Each diagonal is 45 degrees of rotation from the previous diagonal; this should be obvious, since the diagonal divides the 90 degree right angle of the square in half. Therefore, each time we create a new diagonal and a new square, in turn generating another diagonal and another square, we generate a series diagonals, each 45 degrees apart. It should also be obvious that 45 degrees is equivalent to 1/8 of 360 degrees of rotation or 1/8 of a completed rotation of the spiral.

Let us now review a few fundamental elements of this action: we can now associate, in our spiral of squares, a distinct amount of rotation with a distinct diagonal value. In this case the diagonal values are powers of the square root of two or some geometric mean between these powers.

Table 1 
Rotation  Diagonal Value 
0        1 or ?2⁰ 
1/8      ?2 or ?2¹ 
2/8       2 or ?2² 
3/8      ?8 or ?2³ 
4/8       4 or ?2⁴ 
5/8     ?32 or ?2⁵ 
6/8       8 or ?2⁶ 
7/8    ?128 or ?2⁷ 
8/8      16 or ?2⁸ 
9/8    ?512 or ?2⁹ 
10/8      32 or ?2¹⁰

The diagonals of this “spiral of squares” function much like the rays [fn2] (or radii) of a logarithmic, self similar spiral. We can imagine an infinite number of self-similar spirals increasing from 1 to any number N, after one complete rotation. Each successive complete, whole rotation will then function as a power of N[table 2]:

Table 2 
Rotation   Power 
0       N⁰ or 1 
1       N¹ or N 
2       N² 
3       N³ 
4       N⁴

Since each rotation of the logarithmic spiral increases the length of the ray (or growth of the spiral) by some factor that we can identify as the “base” of the spiral. In other words the base of the spiral which increases from 1 to 2 in the first rotation ( and doubles each successive rotation), is identified as base 2; the base of the spiral which increases from 1 to 3, as base 3; from 1 to 4, as base 4;…. 1 to N, as base N, etc.. The spiral, base N, will after one complete rotation beginning with ray length 1, generate a ray whose length is N^1; after 2 rotations, the spiral will generate a ray whose length is N^2; after 3 complete rotations the ray length will equal N^3, etc.

To measure or count rotation, we now define a “unit circle of rotation”. We can map a point of intersection with a spiral, and a ray spiral whose length is equal to or greater than one, on to a point on a unit circle. In this way it seems that a point on our circle of rotation can map on to, potentially, an unlimited number of successive points of intersection of a spiral and any given ray. But, when we look at our circle of rotation, we are looking at the projection of a cylindrical spiral. We can therefore “count”, as cycles or partial cycles, the amount of rotation required to reach the point on the unit circle which a ray maps onto the unit circle and the spiral at the same time.

Look again at the musical spiral of the equal tempered scale. (see figure 1, page 50, {So You wish to Learn All About Economics}). Here, I am looking, not at successive ROTATIONS of the spiral, but DIVISIONS, in this case one rotation of the octave or base 2 spiral.

When I divide the rotation of the spiral by half (6/12ths), I get F# or the square root of 2.[see chart 2]. When I divide the rotation of the spiral by 3 (4/12ths) the first division is the G# or the cube root 2. So each successive rotation is a power of N, i.e. N^1, N^2, N^3, etc. Each successive DIVISION represents a root of N, i.e. ?N, 3?N, 4?N, 5?N, etc.

Chart 2
Division    Root of Two   Musical Note 
0           0            C 
1/12       12th          B 
2/12        6th          A# 
3/12        4th          A 
4/12        3rd          G# 
5/12     5/12th          G 
6/12     square root     F# 
etc. 
1         2            C

As we have now discovered, given any spiral base N, we can associate a distinct amount of rotation with a distinct power or root of N. Each successive complete rotation can be associated with a power of N; each division or partial rotation can be associated with some root of N, or a mean between N and another number. This distinct amount of rotation to a point on the “circle of rotation”, which can then be associated with a distinct rotation of a self-similar cylindrical spiral, is the logarithm of the number generated as a ray intersecting the spiral at a particular point.

For example, take our spiral of the squares; that spiral is base 16. The logarithm of 16 is one, written as Logv16(16) = 1[footnote 3]. Using our table 1, we can create a short “Table of Logarithms” for base 16. Turn once again to the April 12, 2002 issue of EIR, pages 16 and 17; as Bruce indicates, if I double the rotation, I square the length. Let us try various operations with the table of logarithms below. Table of Logarithms, Base 16 Logarithms unit value of diagonal or “ray” 0 1 or ?2^0 1/8 ?2 or ?2^1 2/8 2 or ?2^2 3/8 ?8 or ?2^3 4/8 4 or ?2^4 5/8 ?32 or ?2^5 6/8 8 or ?2^6 7/8 ?128 or ?2^7 8/8 16 or ?2^8 9/8 ?512 or ?2^9 10/8 32 or ?2^10

Add the logarithm of 2 to the logarithm of 4, base 16. What is the result? (2/8 + 4/8 = 6/8 or the logarithm of 8, base 16.) If I add the logarithm of 2, base 16 to the logarithm of 4, base 16, the two ADDED rotations give my the logarithm of 8, base 16, which is the product of 2 x 4.

Now subtract the logarithm of 4, base 16, i.e. 4/8 from the logarithm of 8, base 16, i.e. 6/8 and the remainder will be the logarithm of 2, base 16 or 2/8. Now take any of the logarithms from our table, base 16; add or subtract the logarithms of any number of numbers and see if they correlate with the division or multiplication of those same numbers. In other words: adding or subtracting the logarithms of numbers (i.e. the amount of rotation) correlates with multiplication or division of those numbers,

When I am looking at the number we call a logarithm, I am actually looking at the measure of two distinct forms of action in the complex domain of triply extended magnitudes, i.e. the cyclical nature of helical action, with the continuous manifold of the logarithmic spiral. Which is precisely why Gauss understood “…how much poetry there is in a table of logarithms.” We will look at this relationship in another way next time when we investigate why: “It’s Really Primarily Work.”

Footnotes

1) NON-LINEAR ELECTROMAGNETIC EFFECTS WEAPONS: IN THE CONTEXT OF SCIENCE & ECONOMY speech by Lyndon H. LaRouche, Jr. Milan, Dec. 1, 1987

2) The ray of a cone is a line perpendicular to the axis of the cone, intersecting the spiral arm [It can also be constructed as a straight line from the apex of the cone to an intersection with the spiral. Both project onto the plane as the same length. When we project from the 3 dimensional cone to the two dimensions of the plane we assume that the incidental angle of the cone is 45 ray of the cone and the axis are of equal length.

3) LogvN(N) = 1 is the equivalent of saying “the logarithm (Log) in base N (vN) of N (N) equals 1. In the above case we’re saying the Logarithm of 2 in base 2 is 1

ADDENDUM I: “What is a logarithm?” according to the book.

“… a logarithm is number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number.”

Presuming we understand the concept of “the power to which a number is raised”, then a definition for “exponent” and a “base” might be necessary at this time. An exponent “…is a symbol written above and to the right of a mathematical expression to indicate the operation of raising to a power. In other words, in the simple function of 2^2 = 4, ^2 is the exponent, in the function 2^3 = 8, ^3 is the exponent, etc. The definition of a “base” is a little more complicated.

When we write our numbers we use the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since we use these 10 digits and each digit in the number stands for that digit times a power of 10, this is called “base ten”. For example, 6325 means:

6 thousands + 3 hundreds + 2 tens + 5 ones.

Each place in the number represents a power of ten:

(6 x 10^3) + (3 x 10^2) + (2 x 10^1) + (5 x 10^0), or 6325

We could also use base 2, 3, 5, or any other that would seem most appropriate to our requirements.

Let us look at base 2, the mathematics of the computer. There are 2 digits in base 2, 0 and 1; as with base ten, each digit represents a power of the base number, in this case 2. For example the number 1101, base 2, is: (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0) or 13, base 10.

Base 10 is called “the common base” and was most widely used in developing the Logarithmic tables. Let us take an example: the logarithm of 100 in base 10, which is 2. To say it in another way, in base 10, 10 ^2 (^ denotes exponent or power) = 100, and the exponent, in this case, is 2. We will note this relationship in the following way: v denotes the subscript followed by the base number, such that, in mathematical shorthand, the logarithm of 100 in base 10 will be written Logv10 (100) = 2.

The logarithm of 10 base 10 or Logv10(10) = 1, Logv10(100) = 2, Log v10(1000) = 3, etc. Therefore, if I add:

Logv10(10) + Logv10(100) = 3

I get a logarithm of 1000 in base 10, which is also the exponent of 10^3, or 1000.

If I subtract:

Logv10(10,000) – Logv10(100) = 2

I get 2, which is the logarithm of 100 base 10, which is also the exponent of 10^2, or 100.

In other words, adding logarithm of any number, N, to the logarithm of any other number of that base number system, N1, generates the logarithm of the product of those numbers:

Log(N) + Log(N1) = Log(N x N1)

Subtracting logarithm of N from N1 generates the logarithm which is the quotient of those numbers:

Log(N1) – Log(N) = Log(N1/ N)

Consequently, tedious calculations, such as multiplication and division, especially of large numbers, can be replaced by the simpler processes of adding or subtracting the corresponding logarithms. Before the age of computers and rapid calculating machines, books of the tables of logarithms of numbers were for engineers or astronomers or anyone else who needed to calculate large numbers.

I think the preceding discussion has been a relatively accurate one page “textbook” introduction to logarithms and their use. If it seems somewhat confusing, one solution is that described by a typical professor of mathematics identified as “Dr. Ken”, who, using the Pavlov/Thorndike approach to arithmetical learning, suggests that:

“The way you think about it is this: the log to the base x of y is the number you can raise x to get y. The log is the exponent. That’s how I remembered logs the first time I saw them. I just kept repeating ‘the log is the exponent, the log is the exponent, the log is the exponent, the log is the exponent,…’ “

A singular problem arises when we use the Pavlov/Thorndike approach, replacing the name of one number with that of another, “x is y” or “the log is the exponent”, and then simply memorizing it. If we don’t know the characteristic of action generating the exponent, then what the heck is the logarithm anyway; if this simple equivalency were all there was to the matter, then we have no concept of the characteristic action corresponds to this class of numbers.

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.

How Archimedes Screwed the Oligarchy, Part 2

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.

by Ted Andromidas

When I presented the draft of our last discussion on prime numbers and the notion of indicative quota, to one of my closer collaborators, I was filled with a sense of satisfaction, and wonder, at having gotten a glimpse at what I thought was the idea of number, the generation and distribution of the prime numbers and their connection to the notion of indicative quota. She read it, looked up and said, not the “Ah-ha” for which I had so patiently waited, but “Yeah, so what?”

I was stunned! I sputtered: “What do you mean ‘SO WHAT?’? Are you confused.”

“No, not really.” she said. “I was confused for a moment when you tried to convince me that Eratosthenes and Earthshines was a clever play on words, until I realized that you screwed up the spell checker; AND, despite the fact that you then tried to convince me that there was some pedagogical significance to putting the footnotes out of order, rather than just sloppy editing, I do see how Eratosthenes’ method works; it’s just: What’s the significance of this to indicative quota? And, as a matter of fact, what’s the significance of this problem of prime numbers at all?”

I walked away baffled and disoriented. I thought it was all so clear.

If someone were to say that an “indicative” quota, i.e. a systematic approach to raising money, is just like any other idea of quota, that it’s just a number reached by adding the money raised in any given week, and that any change in the that quota is merely an process of adding or subtracting from that number, could we not characterize that as an “axiom of the system of quota”? Then couldn’t…

“You say to somebody, ‘Here is the axiomatic problem.’ Everybody in mathematics who has a terminal degree–which is what happens to you before they put you in a body-bag–knows the hereditary principle. Even Bertrand Russell knows the hereditary principle–or knew it, wherever he is today. Everybody knows that if you construct a logical system– and mathematics as usually defined is nothing but a logical latticework– everyone knows that if you start out with a system based only on axioms and postulates, and you develop only deductive theorems based on these axioms and postulates, that the entire latticework, which can never be closed, consists of nothing but echoes of the axiomatic assumptions with which you started. Therefore, if one of the axioms is false, the entirety of that field of knowledge collapses.

“An example: If you say that the only thing that exists in arithmetic is the integers, as counting numbers–that everything else is synthetic–therefore, so the argument goes, all mathematics must be derived from the counting numbers as the axiomatic foundation. So you start with an axiomatic counting system, 1 + 1, you construct that, and from that elaborated basis you must develop all mathematics. This is essentially what Russell and Whitehead demanded: radical nominalism. Therefore, as the case of prime numbers implicitly proves–the Euler-Riemann theorem, the work of Gauss on prime number sequences, the ingenious foresight of Fermat on this question, the work of Pascal on the question of differential number series–the entire history of mathematics, centering around this fantastic little problem of prime numbers…”  —Lyndon LaRouche, Schiller Institute Conference, 1984

If we look at the process of counting as iteration, as a function of a one dimensional manifold, we are confronted with “…this fantastic little problem of prime numbers…”: we can not determine the distribution or “density” of prime numbers between 1 and any given number N by any other means than that of Eratosthenes? We can not determine what the next prime number, or for that matter, any future prime number in a counting series, will be before it is actually generated by counting?

Begin counting 1, (2), (3), 4, (5), 6, (7)…;(all prime numbers are bracketed in parentheses) at first it seems that all the prime numbers are also all the odd numbers; that is by just adding 2 + one, then add 2 + 3, i.e. f(p)= (1+2x). We see, to generate the primes, but as we continue to count, (2), (3), 4, (5), 6, (7), 9, 10, (11), (13), 14, 15, 16, (17), (19), 20, 21, 22, (23) 2…,the “pattern” or function seems to change. For a while it seems to be f(p)= (6x +/- 1) till we reach (23), then it changes again. We seem unable to discover a successor function for, not just all the prime numbers, but any particular continuous series of prime numbers.

There have been innumerable functions, theorems, hypotheses, corollaries and conjectures written on this problem: The Prime Number theorem, the Reimann Hypotheses, the Twin Prime conjecture, the Goldbach Conjecture, and the Opperman Conjecture just to name a few.

Let us look at a conjecture referenced several times by LaRouche, that of Pierre de Fermat. Fermat conjectured that every number of the form (2^{2^n +1}) is prime. So we call these the Fermat numbers, and when a number of this form is prime, we call it a “Fermat prime”; the only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537.

In 1732 Euler discovered 641 divides F5 and F(n) has been extended to 31, (i.e. 2^{2^31 + 1}) and no other primes have been generated by this function. It is, therefore, likely, yet not proven, that there are only a finite number of Fermat primes.

You might remember that the Fermat primes were the subject of a recent “Reimann for Anti-Dummies”. Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g., by compass and straightedge) if and only if n is a power of two times a product of distinct Fermat primes. (Hopefully we will look at this problem from the vantage point of Riemann’s and Dirichlet’s correction of Euler, if I can figure it out by then.)

Anyway, for now, let’s just continue to investigate the phenomenon of counting and the primes.

The Spiral of Prime Numbers

As a prime number is generated it is implicitly the modulus of an ongoing cycle, which intersects all past and future cycles of previous prime numbers, transforming the entire number field past and future. Yet, we seem unable to account for the generation of that singular event in the number field, the generation of a prime number, till, in fact, it occurs.

Perhaps it is the way we count; let us “count” differently. Rather than imagining the number line as straight, one dimensional manifold, i.e. 1, 2, 3, 4, 5, 6, 7…etc., is it possible to count in two dimensions? As we’ve seen from the dialogues of Philosph and Cando, numbers in a 2 dimensional manifold are not necessarily what they are in one; but rather than looking at the characteristic differences between two dimensional and one dimensional measure, let us take a more simple construction, and see what happens. Let us generate the number field in 2 dimensions by using a simple a kind of “Archimedean spiral”.

In the center of a piece of note paper write the number 0. To the right of that write the number 1; above one write 2. Now count to the left 3, 4. Below 4 write 5(at the same level as 0); below 5 write 6. To the right of 6 write 7, 8, 9; above 9, at the same level as 1 write 10, and go up 11, 12, 13; now count left of 13, 14, 15, etc. As you count this way, you will generate a spiral of numbers.

Now, beginning with zero, start counting the numbers spiralling out from there.(see figure 1). Try it; it is really not difficult.

(figure 1)

4-(3)-2 | | (5) 0–1 |

6—(7)-8—9

What do you notice, almost immediately: a certain number of prime numbers are generated along various diagonals of the number field.(see footnote 1)

Now, if we begin counting with 5 as our first number in the center of our spiral, we notice that between 5 and its square, 25, all the numbers that lie on a diagonal connecting 5 and 25 are prime numbers. They are not all the prime numbers between 5 and 25, but they do define a successor function of prime numbers between 5 and 25.

Begin counting at 11 and we generate a diagonal of prime numbers along the axis between the prime number 101 through 11, to 121, the square of 11. If we start counting with our Archimedean spiral at 41, we discover the same generating characteristic: it is a line prime numbers which stretch along the 41 and 1681 diagonal. In fact, if we count in this manner from 41 to 10,000,000, half of the numbers on that diagonal will be prime.

When we count in the one dimensional manifold, we can, through the sieve of Eratosthenes, determine that the cyclical, modulo characteristics of the counting numbers, the integers is ordered through the two dimensions of circular action. Yet, there seems to be no “connectedness” at all to the ordering characteristics of the prime numbers, no “pattern” seems to emerge.

When we actually begin to count in a two dimensional manifold, a “connectedness” emerges almost immediately, numerical shadows on the wall of Plato’s cave. Why? Is there some ordering principle from a higher, perhaps 3 dimensional manifold, ordering the two dimensions of our spiral counting? It doesn’t provide us with an actual function for determining the distribution of the prime numbers, nor does it help us develop a successor function, but it does impel us on to a notion of a succession of ordering principles, as we will begin to see in our next discussion.

And finally, to my collaborator’s insistent, “So what? What’s the significance of the prime numbers, anyway”, Karl Friedrich Gauss would reply:

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimated men, i.e. for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator… The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

— Karl Friedrich Gauss, Disquisitiones Arithmeticae (translation: A. A. Clarke)

Footnote 1.

Here is a list of the first prime numbers to aid you in your investigations.   2 3 5 7 11 13 17 19  23 29 31 37 41 43 47 53 59  61 67 71 73 79 83 89 97 101  103 107 109 113 127 131 137 139 149  151 157 163 167 173 179 181 191 193  197 199 211 223 227 229 233 239 241  251 257 263 269 271 277 281 283 293  307 311 313 317 331 337 347 349 353  359 367 373 379 383 389 397 401 409  419 421 431 433 439 443 449 457 461  463 467 479 487 491 499 503 509 521  523 541 547 557 563 569 571 577 587  593 599 601 607 613 617 619 631 641  643 647 653 659 661 673 677 683 691  701 709 719 727 733 739 743 751 757  761 769 773 787 797 809 811 821 823  827 829 839 853 857 859 863 877 881  883 887 907 911 919 929 937 941 947  953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693