by Jonathan Tennenbaum

Printed in The American Almanac, July 21 and July 28, 1997

End of Page Paradox Site Map Overview Page


A pedagogical discussion--Part I


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The following two-part discussion is intended to prompt a richer reflection on what was presented earlier, concerning Analysis Situs, the paradox of ``incommensurability'' in Euclidean geometry, and Nicolaus of Cusa's discovery of a higher geometry based on ``circular action.'' At the same time, I will set the stage for a new series of pedagogical demonstrations, to be developed in coming weeks.

- * * * -

When you have encountered a new physical principle, you cannot just put it in your pocket and walk away. The new principle, if validated, implies a more or less revolutionary change in the entirety of existing knowledge. We have the task of integrating the new principle (``new dimensionality'') into a new, comprehensive hypothesis-system, incorporating the results of all pre-existing valid demonstrations of principle (i.e., the valid side of existing knowledge), as well as the new demonstration, as a new manifold of ``dimensionality N|+|1.'' What is the measure of the change in the per capita productive power of society, associated with the ``impulse ratio'' (N|+|1)/(N)? And how can we push the new manifold ``to its limits,'' uncovering new experimental anomalies which will provide us the stepping-stones on the way to future manifolds N|+|2, N|+|3|,|...?

From Cusa to Leonardo and Beyond

Would this sort of process be a fair way to characterize what happened during the 50-year period from Nicolaus of Cusa's ``De docta ignorantia,'' to the collaboration of Leonardo da Vinci and Luca Pacioli on the ``Divine Proportion''? Is it valid to conceptualize the scientific developments of the European Renaissance, from the Council of Florence through Leonardo and beyond, as a process of ``integrating'' Nicolaus of Cusa's crucial discovery, with the best previous accomplishments of Classical Greek, Arab, and other European civilization?

Before entertaining the possible merits of such a working hypothesis, we should first make sure to reject any temptation to impose ``linearized'' misinterpretations on what Nicolaus of Cusa actually discovered. Here, as always, there is no substitute for ``re-experiencing'' the {process} of discovery, which at the same time constitutes its real {content.}

Among the most ``tempting'' and commonplace misinterpretations, for present-day readers, is to substitute naive visual imagination's image of circular motion in empty space, in place of the radically different ontological conception of ``circular action,'' which Nicolaus actually adduced in his discovery. The promotion of this error by the Venetian agent Paolo Sarpi and his successors, as a willful fallacy, was key to the Enlightenment assault on the European Renaissance. Among other things, it provided the basis, via Galileo, Newton, D'Alembert, Lagrange, Euler, et al., for the elaboration of a so-called ``analytical mechanics'' as the model for an ``Establishment science,'' thoroughly ``sterilized'' against the seeds of discovery.

Circular Motion and Circular Action

Yes, there is a connection between the visible phenomena of rotation or circular motion, and Cusa's principle of circular action. But the connection is that of a shadow to the real object, whose existence it lawfully reflects.

Two brief quotes from Nicolaus of Cusa himself might be helpful in this context. Both are taken from his mathematical essays on the quadrature of the circle and related topics. The first emphasizes the Analysis Situs principle of ``relationship of species'' as crucial to his discovery:

``Since polygons are not magnitudes of the same species as the circle, it is still the case, even though we can always find a polygon which comes closer to the circle than any given polygon, that among things, which can be made smaller or greater, the absolutely largest can never be attained in existence or possibility. In fact, the area of the circle is the absolute maximum relative to the areas of the [inscribed] polygons, which are capable of being more or less and therefore cannot reach the circular area, just as no number can ever attain the encompassing power of the Unity, nor the Composite the power of the Simple.''

Full Scope of Circular Action

Another essay ends with a magnificent stretto, in which Nicolaus reveals the full scope of his conception of ``circular action,'' encompassing the relationship between hypothesis, higher hypothesis, the hypothesis of the higher hypothesis, and ``the Good'':

``We assert, therefore, that there exist beings of the nature of the circle, which could not be their own origin, since they are not like the absolutely greatest circle which alone is eternity. The other circles, which, indeed, seem not to have a beginning and an end, since they are conceived through abstraction from the visible circle, nevertheless, since they are not infinite Eternity itself, are circles whose being derives from the first, infinite and eternal circle. And these circles are, in a certain way, Eternity and complete Unity relative to the polygons inscribed in them. They possess a surface which incommensurably exceeds the surfaces of all the polygons, and they are the first images of the first, infinite circle, even though they cannot be compared with the latter on account of its infinity. And there are beings having an unending circular motion around the being of the Infinite Circle. These contain within themselves the power of all the other species, and from their enveloping power they develop, in imitation, all the other species; and, beholding everything within themselves, and beholding themselves as the image of the Infinite Circle, and through beholding this image--themselves--they raise themselves up to the eternal Truth or to the very Origin. These are the beings endowed with Reason, who comprehend everything by the power of their minds.''

Machine-Tool Design Prototype

By what mode of action do we expand the ``enveloping power'' of the human race, exercising increasing dominion over the Universe, and knowing Reason in the mirror of its own active participation in developing the Universe? What could be more fruitful, to deepen our understanding at this point, than to follow the track of Nicolaus's discovery into the busy workshops and ``design bureaus'' of Leonardo da Vinci and his Renaissance friends! Here is the prototype of the ``strategic machine-tool design sector,'' which has been key to the emergence and survival of the modern nation-state up to the present.

Much oligarchical effort has been expended, over the centuries, to mystify and conceal the ``machine-tool principle'' underlying Leonardo's work in all fields. For example, Leonardo is often portrayed as a ``speculative genius'' whose designs were wildly impractical in his day. As a matter of fact, much of Leonardo's time was spent in direct collaboration with machine-building workshops and factories, as well as with construction teams involved in infrastructure and other projects, developing solutions to problems as they came up. Thus many, if not most, of Leonardo's actual designs were implemented in his day.

Another malicious piece of gossip, spread by Joseph Needham among others, was that Leonardo made ``no fundamental breakthrough'' in the principles of machine-design. That assertion is commonly coupled with the assertion, that Leonardo was not a scientist, and that the real breakthrough, leading to the Industrial Revolution, came with the formal mathematical physics of Galileo, Newton, et al. For example, a book on Leonardo's engineering work, published by one L. Olschki in 1949, claims: ``The technical principles employed by Leonardo were hardly different from those handed down from antiquity and the Middle Ages.... He never attempted to frame new theoretical approaches or theories of mechanics.''

Leonardo's Breakthrough

Leaving aside such malicious nonsense, get out a good collection of Leonardo's sketches. Concentrate particularly on his designs for machines and mechanical devices of machines. Looking over those sketches, ask yourself: What was Leonardo's crucial breakthrough in these matters? What is stunning, revolutionary, about Leonardo's approach to the design of machines, and related matters, which went decisively beyond what had existed before? I am not talking about individual ``inventions,'' so often played up as isolated entities; I am asking for a ``One.''

Whoever tends to read Nicolaus of Cusa's principle of circular action as merely a form of ``motion,'' in the manner indicated above, will be plunged into a rather profound paradox at this point.

Looking at Leonardo's designs, what do you see except mere mechanical linkages--assemblies of gears, pulleys, and levers, which transmit motion from one place and direction to another, without ``adding'' any new motion? Didn't Archimedes already describe the basic mechanical principles involved, as typified by the action of the lever or pulley? Or is there something more than just ``mechanics'' in Leonardo's machine-designs, something absolutely banned from the textbooks of ``analytical mechanics,'' but which is a key to the unprecedented rate of increase in the productive powers of labor, unleashed by the Renaissance?

To be continued.

From Nicolaus of Cusa to Leonardo da Vinci:
The ``Divine Proportion'' as a Principle of Machine-Tool Design
Part II

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The first part of this two-part article appeared in the previous issue of New Federalist, dated July 21, on p. 9.

Lyndon LaRouche's discoveries in physical economy provide the key to unlocking the secrets of Leonardo da Vinci and the Italian Golden Renaissance, to a degree which would have been impossible at any earlier time, before LaRouche's work.

Observe that the leading features of Leonardo's designs for machine tools and other machines--most emphatically including the method of ``non-linear perspective'' employed in his drawings--all cohere with one central conception:

The emergence of {nation-state physical economy} as a {living process} based on development of the cognitive powers of individual members of society, imposes a unique ``curvature of space-time'' upon the Universe, such that each and every particular must be conceptualized and measured by reference to the ``horizon'' defined by that curvature.

That central conception subsumes the following features and consequences@s1:

1. A physical economy is a special type of living process, whose maintenance and growth depends on development of the cognitive powers of the individual members of society.

2. The action of human Reason upon the Universe, occurs {solely} through the instrumentality of living processes. That is, through the activity of sovereign human individuals, working in and through society, upon the expanding domain of Man-altered Nature which constitutes the ``substrate'' of physical economy as a living process.@s2

Reason's Dominion Over the Universe

3. Hence, Leonardo da Vinci's conception of non-linear-perspective curvature is based on a relationship of Nicolaus of Cusa's ``species'': Living processes exercise increasing dominion over inorganic processes, and human Reason exercises increasing dominion over the entire Universe via its dominion over living processes (i.e., human individuals, the physical economy, and an expanding biosphere).

4. In particular, the required notion of ``technology,'' appropriate to the maintenance and development of physical economy, {cannot} be derived from inorganic physics. No mere physical laws, of the sort suitable to ``inorganic physics,'' could ever account for the impact of a new machine or other invention on increasing the productive powers of labor.@s3 Although it is possible to design a machine on the basis of a simple hypothesis, we cannot measure its economic {effect} that way. The survival of human society, therefore, depends on shifting attention from the mere ``engineering approach'' of simple hypothesis, to encompass the ``horizon'' defined by higher hypotheses. Leonardo's drawings have the included purpose, to communicate exactly that conception.

5. For these and related reasons, Leonardo's studies of anatomy, and his collaboration with Luca Pacioli on the ``Divine Proportion,'' were decisive inputs to his approach to machine-tool design. Leonardo sought to apply to the design of machines, a reflection on the principles and means by which living organisms exercise dominion over the inorganic domain.

6. When a living organism incorporates non-living material into its active domain, it {imposes} its own characteristic {ordering} upon that material. (One day soon, the environmentalists might turn against plants and trees, denouncing them for imposing their ``authoritarian values'' upon poor, defenseless dirt!)

Harmonic Proportions

7. Leonardo, Pacioli, and others demonstrated how the peculiar space-time ordering of living processes finds its lawful {visible} expression in self-similar elaborations of the harmonic {proportions} derived from the division of the circle and sphere. The latter all belong to the dominion of the circle's ``Golden Section.''

8. This sort of approach points to a principle of {harmonic composition of motion} for the evolution of machine-tool designs integrating an increasing number and density of degrees of freedom. The harmonic principle of the ``Golden Mean'' will be reflected, not necessarily in the individual machine per se, but rather in the context of the evolutionary series of species of technology. The latter constitutes, on the one side, a central functional feature of the growth of physical economy as a living process, while at the same time embodying an ordering of mutually inconsistent theorem-lattices of increasing ``power'' under the principle of ``higher hypothesis.''

9. In the continuation of this process, with the increase in energy-flux density and precision of machine-tool design, discoveries in microphysics oblige us to replace the concept of ``motion'' by a generalized notion of ``harmonically ordered physical action.'' The approach of Leonardo (and later Kepler) received preliminary, but brilliant confirmation in the domain of atomic and nuclear physics.

Beauty of Leonardo's Drawings

10. Hence, the stunning beauty of Leonardo's drawings! He communicates not merely a set of ``specifications'' for a machine, but a {conception}--a conception of that invention as seen in the perspective defined by the creative principle of the Universe as a whole. By this method of ``non-linear perspective,'' Leonardo is able to communicate the creative process itself, and not merely a particular product. Thus, Leonardo's designs and machines are vehicles for the communication of higher ideas, for the generation of higher qualities of labor power. Like great Classical music, they embody Reason's ironic reflection on the principle of life.


1. Besides Lyndon LaRouche's writings, I would especially recommend juxtaposing to our discussion, the relevant articles by Dino de Paoli on Leonardo and related matters.

2. Have you stopped to consider the significance of the fact, that we need a brain in order to think? Actually, we need more than that: To develop, individual creative reason must continually expand and intensify its ``active domain.'' By the term ``active domain,'' I mean, roughly, the region of the Universe which is directly subject to the deliberate actions of a given individual. The growth of the active domains of members of society, is obviously correlated to increase in per capita and per hectare consumption of energy and other components of the market baskets, as it is to increase of the productive powers of labor. To the extent that the creative contributions of individuals are communicated and realized by society, their active domains may encompass the entire physical economy, and more. Would it be justified to conside, that growth of the ``active domains,'' in some respects represents an enlargement of the physiological processes of the brain, as an instrument for the development and realization of valid ideas?

3. Some might reject such a categorical proposition as preposterous. Don't we know countless examples of inventions, whose labor-saving effects can easily be explained by any physics student? For example:

What could be more obvious, than the increase in productivity, caused by the above-mentioned inventions? Yet, such a casual affirmation overlooks at least one decisive point: What about the direct and indirect {costs} (in real terms) of developing, producing, and maintaining a given machine or technical improvement? How can we be {sure,} in any given case, that that additional cost will not actually exceed the saving in labor, or other benefits provided?

Observe, for example, the vastly greater complexity and intensity of motion of Leonardo's machines, compared to the rudimentary gadgets of pre-Renaissance Europe. Even the simple act of introducing ball-bearings and related devices into machine design, adds new degrees of freedom to the system as a whole, raising the demands on the {quality of labor} required for the manufacture and maintenance of the machine. Actually, the purpose of the machines themselves, as means for urban-centered development of the nation-state, is not to ``economize labor'' per se, but to rather to {uplift its cognitive quality}. (Thus, while industrialization subsumes as a necessary aspect the reduction and final elimination of manual labor, a healthy industrial society actually increases the ``work load'' which must and can be borne by the average member of society.)

Reflecting upon such matters, we realize that the increase in the productive powers of labor, associated with the introduction of a new machine into the productive process, can hardly be determined from a mere analysis of the machine itself. It requires that we carry out a measurement of the entire economic process within which a proposed new machine design is to be ``inserted.'' Since the insertion of a new technology changes the characteristics of the economic process, that measurement must take into account, not only the present, but also its projected development in the future. In the last analysis, there is no adequate answer which does not center on the rate of improvement of the cognitive powers of labor, associated with any given ``pathway'' of economic development. Herein lies the cause of the essential ``incommensurability'' of real economic growth, relative to any linear sort of engineering or ``systems analysis'' standards of measurement. The significance of the Golden Section (``Divine Proportion'') comes once more to the fore.

A pedagogical discussion,
by Jonathan Tennenbaum -- Part I

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Last week's inquiry concerning Leonardo da Vinci's principles of 
machine-tool design, brought us face-to-face with an old friend: 
the significance of the so-called Golden Section. Although this 
topic has been discussed many times in our organization, I think 
there still exists a residue of mystification, remaining to be 
cleared away. Often enough in the past, mere mention of the 
Golden Section was liable to evoke fits of embarrassed 
hand-waving and numerological free-association from supposed 
experts, while the issues which really bothered people and have 
to be worked through in a rigorous way, were not adequately 
Lyn, of course, has dealt with the Golden Section repeatedly and 
from the highest standpoint. For those who are resolved to break 
through on this issue, I would particularly recommend rereading 
Lyn's essay "On the Subject of Metaphor" (Fidelio, Fall 1992) and 
his book-length study, "Cold Fusion: Challenge to U.S. Science 
Policy" (Schiller Institute, August 1992), particularly section 
II entitled "Six of the Crucial Discoveries in Modern Science". 
The following pedagogical discussions are intended to provide 
some useful geometrical "homework" on these matters, while adding 
a fresh view of the subject, thereby assisting the reader in 
"triangulating" the essential points to be mastered. 
On the most elementary geometrical level, we have the problem, 
that only a very few people (with the exception of Chuck Stevens 
and a few others, perhaps), have actually worked through the 
geometrical constructions which characterize the relationship 
of the regular solids to their inscribed and circumscribed 
spheres. Yet, the essential discoveries of Leonardo and Kepler 
were all referenced to Euclid's original treatment of exactly 
that issue, as later reworked by Leonardo and Pacioli in the 
book, "Divine Proportion" and "read" from the standpoint of 
Nicolaus of Cusa's concept of an "evolutionary" ordering of 
"species" (analysis situs). 
To present the "Golden Section" merely as a ratio derived from 
the regular pentagon, were a wild fallacy of composition. Even on 
elementary geometrical grounds, the only admissible approach to 
the "Golden Mean" is one which defines it as the unifying 
characteristic of the way in which the higher species (sphere) 
bounds the lower species (regular solids or spherical 
The really crucial problem, however, lies in the way people 
"read" (or misread!) the ontological significance of such 
elementary geometrical topics. 
Having witnessed many a member's more or less frustrated attempts 
to master the Golden Section, I am reminded of an often-cited 
anecdote from Russia: One evening, a man lost a ring in a dark 
corner of a park. Instead of looking for it there, the man spent 
hours carefully searching under a nearby street-lamp. When a 
passerby asked the man, why he kept looking in the wrong place, 
the man replied: "I am looking here, because here is plenty of 
People, who (consciously or unconsciously) are wont to stay away 
from  the "dark, uncomfortable" area of rigorous creative 
thinking, won't find an "answer" for the Golden Section, no 
matter how exhaustively they search for it. There does not, nor 
could there ever exist an explanation of the sort which would be 
acceptable to the aristotelean "norms" of contemporary classroom 
education. The Golden Section, as Leonardo and Kepler understood 
it, and as Lyn develops it further, is an idea. It does not 
exist in the "objective" world of geometrical forms per se. Nor 
does it arise from any amount of empirical evidence taken by 
itself. In his piece on Cold Fusion, Lyn demonstrates, in 
rigorous, step-by-step fashion, the inseverable relationship 
between Leonardo and Kepler's "reading" of the Golden Section, 
and Plato's notion of "hypothesizing the higher hypothesis", as 
that connection becomes uniquely intelligible from the standpoint 
of physical economy. Lyn adds an admonishment which the present 
author found most helpful: 
"Look at the Golden Section from the standpoint of what Plato and 
Cusa knew before Leonardo and Kepler. Do not attempt to read it 
as if Leonardo and Kepler were such fools as not to have studied 
intently the work of Plato, Archimedes and Cusa..." 
A few lines further, Lyn adds: 
"Until we have grasped so the fact that true science is 
subjective in this way, that its validity is located 
essentially in that anthropocentric subjectivity, we do not 
have the means to read intelligibly the crucial argument of any 
among the founders of modern science." 
The following discussion should cast some further light on the 
cited point, which is crucial to any adequate understanding of 
the physical significance> of the Golden Section. 
I would not be surprised, if there were a considerable number of 
persons, who routinely skip over Lyn's written discussions of 
certain scientific topics, rationalizing that practice to 
themselves by the argument, that Lyn intends these as merely 
"optional" illustrations of general points. The assumption is, 
that the same concepts could be communicated just as well without 
recourse to such difficult and "specialized" topics. In an 
extreme case, we might encounter the following train of thought: 
"Oh, yeah, Lyn is really just talking about the higher 
hypothesis, or negentropy or something like that. I already have 
an idea what those are. So why make such a big deal out of 
circular action, the Golden Mean, Riemann, Cantor and so forth, 
which just confuse me and cause me mental suffering?" 
Apart from exhibiting the typical intellectual sloppiness and 
lazyness of our "baby-boomer" generation, the quoted folly might 
usefully provoke us to consider quite the opposite thesis, 
That the universe is constructed in such a way, that it were 
impossible to master the notion of the higher hypothesis, 
except through the included means of certain, 
uniquely-defined series of geometrical discoveries! 
By "geometrical discoveries", I do not mean to imply that the 
discoveries emerge from the domain of mathematics per se. Rather, 
it is the "provocation" provided by otherwise unresolvable 
physical anomalies, which causes us to evolve new species of 
geometry, in such a way as to permit us to "integrate" those 
anomalies as new "dimensionalities" in a revised, unified 
conception of multiply-connected physical action. 
It is quite remarkable, that up to the present time those 
fundamental geometrical discoveries, so defined, all have the 
direct or indirect effect of redefining -- or "unfolding" as it 
were --, the significance of the circle (circular action) and the 
sphere within geometries of ever higher "order". So it is with 
the pythagorean discovery on incommensurability, the proof of 
"transcendence" of circular action as conceived by Nicolaus of 
Cusa, the reworking of the universal significance of the Golden 
Section by Leonardo da Vinci and Kepler, and C.F. Gauss' 
introduction of the complex domain's "anti-euclidean geometry". 
Through all these transformations and revolutions, the circle and 
sphere remain "the same" as visible forms; the crucial thing that 
changes, is how we "read" them. 
Might not the key, the physical significance of the Golden Mean 
lie in that, profoundly subjective condition of human 
existence? Could it be, that the circle, sphere and Golden 
Section-derived harmonics are embedded in the structure of the 
Universe itself, in the form of "transfinites" undergoing a 
continuous process of conceptual "redefinition" through validated 
experimental discoveries, as necessary characteristics of any 
pathway for the survival and development of the human race? 
With these issues of method in mind, turn now to some geometrical 
experiments! If carried out thoughtfully, the insights from those 
experiments can help to sharpen our appreciation of the same 
points, when we return to them later. 
As Kepler himself emphasized, his own "reading" of the 
significance of the sphere and regular solids, was based on 
Nicolaus of Cusa's development of the concept of lawful ordering 
of axiomatically-separated "species" (analysis situs). The sphere 
bounds, as a higher species, the regular solids with all their 
mutual relationships and quasi-regular "offspring". The "Golden 
Mean" should signify, first of all, a unifying characteristic of 
the relationship between the lower (regular solid) and higher 
(spherical) species. 
That simple remark already points to something wrong with the 
commonplace approach, in which the solids are constructed as 
isolated entities -- typically out of sticks fastened at the 
ends, by gluing together regular polygonal faces, or similar 
means) --, without reference to inscribed and circumscribed 
spheres. Constructions with circular "hoops" have the same 
drawback, except insofar as we observe the way the curvature of 
those hoops and their mutual relations are determined by an 
invisible spherical bounding. 
Why is it not permissible, to first construct a regular solid, 
and then create the inscribed and circumscribed spheres, as it 
were, by "spinning" the solid? Because, I say, that would 
misrepresent the true ordering of the species. The very existence 
of the regular solid, and virtually each step of any proposed 
construction, presupposes and embodies circular action applied to 
results of circular action. What, after all, is an angle? Just 
look at the constellation of angular displacements which 
accompanies the "birth" of each and every singularity (vertices, 
edges and faces) of a solid! 
Accordingly, I propose that the following task be explored: 
To inscribe into any given sphere, by construction, each of the 
five regular solids. Similarly, to circumscribe a given sphere by 
each of the five regular solids. (In the first case the solid is 
to be so constructed in the sphere's interior, that its vertices 
touch the inner surface of the sphere; in the second case, the 
solid is to be constructed around the sphere, in such a way that 
the midpoints of its vertices touch the sphere's outer surface.) 
Try to do the constructions in two ways: (1) by the means of 
classical euclidean geometry "in three dimensions"; (2) by 
rotations of the sphere. In the latter case, the task is to 
construct the great-circle division of the spherical surface, 
corresponding to each of the regular solids. (For the purposes of 
this exploration, we may consider that a given rotation 
generates, as singularities, the corresponding poles and 
equatorial great circle on the sphere, as well as the arcs traced 
by already-generated singularities.) 
Observe the relationship between the two modes of construction. 
The point of this exercise, is not necessarily to complete all 
the proposed constructions immediately. Indeed, people will 
observe, that the case of the duodecahedron and icosahedron gives 
rise to rather extraordinary difficulties! Those difficulties are 
very much connected with the unique role of the Golden Section. 
The important thing is to explore the terrain, and to pick up and 
conceptualize the paradoxes which arise in any given approach. 
Readers will probably find that the inscription of an octahedron 
presents itself as the simplest case, and also opens a pathway of 
approach for the cube and tetrahedron (in that order). Note, that 
the constructions involved (at least, the most direct ones) all 
share certain common features. Observe also, that analogous 
methods fail for the duodecahedron and icosahedron, although 
the latter provide a pathway for constructing the first three. 


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        The task laid out in the first part of this discussion, 
opens up numerous avenues of fruitful exploration.  What we shall 
attempt to do now, is to go as directly as possible to something 
        For this purpose, focus on constructions on the sphere (1). 
As always, it is imperative that the reader work through the 
necessary constructions, including making drawings on spheres, 
making sketches as well as consulting the familiar models of the 
regular solids in their mutual relationships. 
        Start with a sphere, considered as a featureless space.  By 
rotating, generate a first great circle, G1.  That action is our 
first singularity.  Next, rotate G1 on itself, i.e. rotate the 
sphere on an axis through G1.  This second singularity generates 
a second great circle, G2, which intersects G1 at right angles. 
Then, rotate the sphere a third time, around the axis defined by 
the intersections of G1 and G2.  This third singularity generates 
a great circle G3, which is perpendicular to both G1 and G2.  The 
constellation of G1, G2 and G3, their points of intersection and 
the curvilinear triangular areas bounded by them on the spherical 
surface, constitutes a spherical octahedron. 
        By joining the intersection-points of G1, G2 and G3 in the 
obvious fashion by straight lines through the interior of the 
sphere, we obtain the "skeleton" (i.e., edges and vertices only) 
of a regular octahedral solid inscribed in the given sphere. 
Projected outward to the sphere's surface from the center of the 
sphere, the faces and edges of the octahedron project to the 
corresponding elements of the spherical octahedron. 
        (Observe the paradoxical feature of the equilateral 
curvilinear triangles forming the spherical octahedron, compared 
with the faces of the solid octahedron.  If you want to torment 
someone with a linear mind, ask them how it is possible to 
construct a triangle having three right angles!) 
        So far, we didn't need to invent much of anything; G1, G2 
and G3 seem nearly self-evident predicates of the sphere.  What 
is more, once the spherical octahedron has been formed in that 
way, the spherical cube and spherical tetrahedron practically 
fall into our lap! 
        We have only to generate the great circles which bisect the 
right angles formed by the each pair of the circles G1, G2 and G3 
already constructed.  (With a bit of thought, the reader will 
readily discover how to bisect angles in spherical geometry, for 
example by analogy to the familiar method of plane geometry.) 
The result is a "net" of great circles whose intersection 
generates the mid-points of the spherical octahedron's faces. 
The constellation of those mid-points defines a spherical cube. 
If we connect those points by straight lines inside the sphere, 
we obtain the "skeleton" of a solid cube inscribed in the sphere, 
in the same manner as the octahedron earlier.  Note, however, 
that the net of great circles just constructed, divides each 
"face" of the spherical cube along its "diagonals" into four 
isoceles right triangles; in the world of spherical geometry, the 
sides of any given square "face," when continued further, 
automatically form the diagonals of the four adjacent faces. 
Observe, that the smaller angles of the isoceles right spherical 
triangles are each 60 degrees (one-sixth of a complete circle) 
instead of the 45 degrees (one-eight of a circle), which we would 
get in plane geometry.  Note, again, the paradoxical 
relationships defined by the projection of the inscribed, solid 
cube onto its spherical "father." 
        As for the spherical tetrahedron, it is already "there" (in 
fact, two times).  It jumps into view, for example, when we 
"color" every other face of the spherical octahedron, in 
checker-board fashion.  The mid-points of the colored faces (i.e. 
four out of eight) form the vertices of the spherical 
tetrahedron, whose "faces" are equilateral spherical triangles, 
each of whose angles are 120 degrees.  Taking the non-colored 
faces instead gives us a "twin" tetrahedron.  We get inscribed 
solid tetrahedrons by the same proceedure as earlier for the 
octahedron and cube. 
        Easy going!  But what about the dodecahedron and 
icosahedron? Now the real fun starts. 
        Let us go for the spherical dodecahedron (the construction 
of the icosahedron is essentially equivalent).  Looking at the 
network of circles we have created on the sphere, there is no 
lack of angles to bisect and vertices to connect.  We think to 
ourselves: try to find pentagons, pentagons!  We construct more 
great circles.  The thing just becomes more complicated and 
confusing.  A "bad infinity"!  We realize that the regular solids 
embody a form of "closure," but beyond the 
octahedron-cube-tetrahedron, we never seem to get it. 
Frustration sets it, then rage.  Why can't I find the trick? 
Soon many of us are in a fit, just connecting things at random 
(the famous "connectoes"), and generating garbage.  Others have 
drawn back into the secrecy of their rooms, scribbling equations 
in the hope that the "secret" will emerge by some sort of magic. 
        Enough of this!  Only fools allow their approach to a 
problem to be defined in terms of the so-called "givens" (as most 
of us were drilled to do in school)!  Rather, successful survival 
depends on being able to think {backwards} from the point in the 
future where we know need to go, and to define our approach to 
the present {from that vantage-point}. 
        So, let us start afresh.  Juxtaposing the 
dodecahedon-icosahedron to the {species} cube, octahedron and 
tetrahedron defines a metaphor.  Looking to the "future," imagine 
spherically-bounded geometry, within which the existence and 
relationship of {both} is predetermined, as a unified concept. 
But, wait!  The dodecahedron, as the "maximum" of the polyhedra, 
generates the rest.  In other words, the "top-down" ordering is 
from sphere, to dodecahedron-icosahedron, to 
octahedron-cube-tetrahedron.  In construction, on the other hand, 
we appear to build "from the bottom up," even though the "top" is 
in a sense already "immanent" in the circular action which is the 
"minimum" of the construction process.  The solution?  You have 
to look "from the top-down," in order to define the pathway to 
realize, "from the bottom up," what is already there in 
        Examine, accordingly, how the octahedron, cube and 
tetrahedron are contained in the dodecahedron as derived 
entities.  The simplest relationship is with the cube, and is 
most easily visualized, perhaps, in the solids.  The cube is 
"inscribed" in the dodecahedron, in such a way, that its vertices 
coincide with 8 of the dodecahedron's 20 vertices.  Observe the 
relationship of each square face of the cube passing into the 
interior of the dodecahedron, and the configuration of four 
pentagons which share vertices with that face of the cube.  Note 
the two vertices of the dodecahedron, which lie "on top of" the 
said face. 
        Aha!  How do get from the cube to the dodecahedron?  What 
singularity must be added.  How is the just-mentioned pair of 
vertices lawfully related to the cube?  Observe, that each of 
those vertices is joined, by pentagonal edges, to three other 
vertices.  The corresponding triangle is equilateral, and one of 
the sides coincides with an edge of the cube.  Suddenly, we see 
the pathway to construct the dodecahedron from the cube, as 
        Start with the spherical cube, and one of the "faces" of 
that cube.  For purposes of discussion, identify the vertices of 
the given face by A, B, C, and D, going around the face in 
clockwise order.  Now, rotate the cube around the axis defined by 
A and its antipode on the oppostive side of the sphere.  Under 
that rotation, the edge AB of the cube (i.e.  the great-circle 
segment AB) with its endpoint B, describes a circle.  Next, 
rotate the cube around B, letting the endpoint A trace a circle 
of equal radius to the first.  The intersection of those two 
circles, constructed on the surface of the sphere, defines two 
new points, of which one of them lies inside the curvilinear 
square ABCD, and the other outside.  Call the point inside the 
square "E." 
        Now repeat the same construction, but with C and D instead 
of A and B.  Let "F" denote the interior point, which results 
from the intersection of circles with "curvilinear radius" CD (= 
AB) around C and D respectively. 
        For reasons which {could only be made intelligible from the 
standpoint of the "finished" dodecahedron, with all its 
relationships}, the just-constructed points E and F, constitute 
precisely the "missing singularities" required to transform the 
cube into a spherical dodecahedron!  The great-circle segments 
EF, EC, ED, FA, FB are all edges of the spherical dodecahedron, 
forming sides of two adjacent curvilinear pentagons.  To 
construct the rest of the vertices and edges, we merely repeat 
the same proceedure with each of the remaining faces of the cube, 
in proper orientation (2).  The vertices of the dodecahedron 
consist of the 8 vertices of the cube, together with 2 additional 
vertices for each face; 8 + (2x6) = 20.  Having obtained the 
spherical dodecahedron in this way, the spherical icosahedron, as 
well as the corresponding inscribed solids, can easily be 
        Now stand back a moment and think over what we have done. 
Leave aside the details of the constructions, which admit of many 
variations and alternative pathways.  What is significant, is the 
following: Although we constructed all five spherical polyhedra 
by rotation of the sphere, the {species of idea}, which we needed 
in order to {devise} the construction, was {not} the same in each 
case.  The octahedron, cube and tetrahedron came out almost as a 
linear series, through a process of successive bisections. 
Relative to that sort of process, the dodecahedron and 
icosahedron are utterly inaccessible.  We were able to jump over 
that apparently "unbridgeable chasm" -- how?  By introducing a 
new principle into the process -- a principle which we adduced 
from the "end result," {before we had that result}!!  Cheating? 
No. Time-reversal. 
        Now think back to the starting-point of this whole 
discussion: the work of Leonardo da Vinci and Kepler on the 
Golden Section.  Keep in mind the essential subjectivity of 
science.  Is not "time-reversal" a determining characteristic of 
any negentropic process?  Including living processes?  And did we 
not just experience {in our own minds} the requirement of 
"time-reversal" as a {necessary} characteristics of any pathway 
to construction of the dodecahedron?  As opposed to the 
relatively linear ("inorganic") octahedron, cube, tetrahedron. 
Compare Leonardo's studies of the morphology of living processes, 
and Kepler's discussion of the monad principle in his Snowflake 
        Much more could be said here.  But let me end with a little 
paradox: If what we have said is not far from the mark, then 
where does the functions of {growth} come in, which we connect 
with the idea of self-similar spiral action?  Or, to put it 
another way: How could it be that the sphere, which in itself 
appears bounded and finite, could embody a principle of unlimited 
        (1) I will address the construction of inscribed and 
circumsscribed solids, "by the means of Euclidean geometry," in a 
future pedagogical discussion.  The task, in the Euclidean sense, 
is not so much to physically build the solids with the spheres; 
rather, given the radius of the sphere as "unit" the task is to 
determine the sides, angles and other parameters of the inscribed 
and circumscribed solids; not as algebraic values, but in terms 
of the geometrical constructions involved.  On a higher level, 
these constructions take us to the threshhold of Monge and 
Carnot's "descriptive geometry." 
        (2) For each face the construction can be done in two 
possible ways, depending on which set of opposite sides are used 
-- AB, CD or alternatively AD, BC, for the case just described. 
To complete the dodecahedron, the choice of pairs of sides must 
alternate, so that the constructions in adjacent faces are at 
right angles to each other, i.e., in such a way each edge of the 
cube is used exactly once.  The reasons for this will be obvious 
to those who work through the construction.  [jbt] 

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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.

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