by Jonathan Tennenbaum

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 addressed.

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 divisions).

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 light!"

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, namely:

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.

By Jonathan Tennenbaum -

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 {essential}.

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 potential.

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 follows:

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 derived.

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 paper.

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 growth?

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(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]