Introduction to a new pedagogical series

by Jonathan Tennenbaum

One of the reasons why you don't really understand the significance of Plato's five regular polyhedra, is because you have never questioned your own, completely unfounded assumption, that the sphere is a figure in 3-dimensional space.

We all remember the type of horror movie, where the Earth has been invaded by alien beings with the capability of taking over the minds and wills of human victims. The victims look the same as before, but their brains have been hollowed out or short-circuited by some sort of implanted devices, so that they effectively are no longer human any more.

However frightening the experience of such a horror movie, it hardly compares with the real horror story, of what standard school mathematics education has done to the minds of nearly everyone. As a result of what was done to you, the creative, cognitive processes of your mind are "turned off" most of the time, even when you are engaged in what you consider to be intellectual effort. Instead, a form of "artificial intelligence" operates, that was installed through school education and related, mostly early experiences. Under adversive conditions of intense cultural pessimism, even those who have known the joy of real thinking, will tend to revert back to those previously implanted, school-room (i.e. "career-oriented") habits of "artificial intelligence".

This "artificial intelligence" (otherwise known as Aristotelianism) excludes from consideration exactly that, which is the object of human cognition. The tactic is to divide the Universe into "sufficiently small" domains of experience, in which the cognitive considerations, thus ignored, are assumed "not to matter". Afterwards, the flattened, linearized pieces are fitted together again to construct a parody of human knowledge. The typical symptom of artificial intelligence is an obsessive fixation on the presumed existence of "objective, hard facts" --a fixation whose most revealing manifestation, perhaps, is the inability to conceptualize the fundamental significance of the sphere and the five Platonic solids.

A remedy is at hand, however. This being the best of all possible worlds -- and not the vicious, artificial world of a horror film-- the condition just described is more than reversible. By rooting out the problem at its deepest origins, we may be enabled, not only to restore our own creative powers to fullest blossom, but also to discover a means to render future generations forever immune to the disease of oligarchism.

That is the issue we are committed to fighting through, in the following series of pedagogical discussions on the sphere and the Platonic solids. Here, on this battlefield of choice, we are resolved to smoke out and defeat the internalized enemy of the human mind. In the process, all the most "advanced" topics we met with in our previous work -- including Gauss' biquadratic residues, modular functions and the Gauss-Riemann domain of multiply-connected action -- will reveal themselves as the most elementary sorts of notions, already implicit in the original discovery of the regular solids' uniqueness, more than 2500 years ago. All has been buried under the myth, that a non-existent "plane geometry" was the starting-point for Greek mathematics.

Start, therefore, with the following task: Given a clear night in which the stars are visible throughout the sky, how can we make a preliminary, but conclusive measurement of the curvature of the Universe?

Part II of a series

by Jonathan Tennenbaum

The spread of mythologies in the name of "history of science," began very early.

The Greek historian Herodotus reported, that geometry was invented in Egypt and transmitted from there to the Ionians. He also claimed, however, that geometry arose in connection with the practical problem, of measuring and reconstructing the division-boundaries of agricultural fields after each periodic flooding of the Nile (geo-metry = earth-measurement). If Herodotus intended the term, "geometry," to signify some specialized knowledge relevant to surveying, there may be an element of truth to the latter assertion; but if he meant the geometry of Thales, Anaximander, Pythagorus and Plato, then the account is certainly wrong and highly misleading. This story of geometry's alleged practical origin (whether Herodotus is to blame for it or not), found its way into the subsequent histories of science, up to this day. It reminds us of the theory of the "opposable thumb" and other absurdities of Friedrich Engels' "dialectical materialism." Contrary to this, the overwhelming evidence -- including that contained in Plato's Timaeus, in the Vedic and other ancient calendars, as well as the implied navigational skills of the "peoples of the sea" --, demonstrates that {all physical science originated in astronomy}. Astronomy, in turn, was cultivated in some form already tens, probably hundreds of thousands of years before the classically recorded Egyptian civilization, by maritime cultures spread across the globe. Geometry begins with nothing less, than Man's attempt to measure the Universe as a whole.

This should indicate that the practice of basing school mathematics education on so-called "plane and solid geometry" -- a practice that has dominated European education, despite the Renaissance, for over two millennia -- is profoundly in error. Henceforth, the teaching of geometry should begin with the {failure} of plane and solid geometry, to account for the most elementary features of visual astronomy. That failure has a precise, knowable structure; to characterize that singularity, is to carry out the first scientific measurement of the Universe.

Bearing in mind that we are dealing with matters of fundamental importance, we need not apologize for the elementary nature of the following account. It should help refresh the mind on familiar matters, while opening some new flanks at the same time.

Constructing a Star Chart

Imagine you are a prehistoric astronomer, attempting to produce a star chart on a clay tablet or papyrus sheet. You require that the chart should accurately represent the shapes of the familiar constellations of stars, and also the mutual orientations of the various constellations relative to each other, so that the chart can be used for navigational purposes.

As far as individual constellations are concerned, you find no difficulty drawing any one of them separately. You just naively transfer the image of what you see, {as if unchanged}, to the tablet. No problem? But, as you begin to map {larger} portions of the sky, adding more and more constellations to the chart, difficulties arise. The constellations don't fit together. You begin again, with another constellation as starting-point. Once again, things don't fit. Why? Although in each case you can specify the point at which the mapping process begins to break down, the underlying cause clearly lies {outside} the specifics of each attempt.

This problem embraces paradoxes, of the sort any curious child will have observed. I stand up and look straight ahead at some point on the horizon. Now I look to the right of that, and more to the right, and so on, until, by continuing my action of "looking to the right," I turn all the way around and come back to the original point...from the {left}! Or instead, if I start by looking straight ahead as before, and now look {up}, and keep turning my head in that "upward" direction further and further, I end up bending backward until I am moving my head {downward} toward the ground and seeing everything upside down!

(Let no one laugh off these simple paradoxes of linearity, who is not prepared, for example, to explain to any child or adult, how it can happen that the Earth can be in two different days, depending on the position on the Earth's surface, at one and the same moment in time.)

These sorts of paradoxes give rise to unavoidable, interwoven {periodicities} in our attempt to construct a star chart -- as for example when I attempt to represent the observer's looking "to the right" and "upward" by motion "across" and "up" on the chart.

(At a more apparently "advanced" level, the same problems plague the cartesian-like coordinate systems still used by astronomers to record the positions of the stars. To describe one such system in a perfunctory manner: Given any star, let "y" be its angular "height" above the horizon (i.e., the magnitude of angle from the position of the star "downward" to the point "directly below it" on the horizon), and "x" the angle along the horizon from that point to some chosen fixed point on the horizon. We might thus represent the position of any star by a point in the cartesian plane, whose rectilinear coordinates are proportional to x and y, repectively. The resulting mapping, however, grossly distorts the shapes and angular relationships of the constellations, especially those in the vicinity of the overhead or zenith-point, where the mapping "explodes.")

This mere descriptive approach, however falls short of identifying the underlying cause of the problem. In particular, it does not answer a crucial question which ought to pose itself to us: Does the difficulty arise only when we want to map {large portions} of the sky; or is it already present, albeit so far unnoticed, in the attempt to represent any {arbitrarily small} portion of the sky?

The Spherical Bounding of the Universe

To progress further, we need to examine the internal characteristics of that action by which we, as ancient astronomers and navigators, are attempting to measure the Universe. The ancient astronomer makes a series of {star sightings}, measuring, in effect, the {rotation} from one direction in the sky to another. Imagine that a movable "pointing-rod" of fixed length is fixed at one end to a universal joint at our point of observation. Observe that the tip of that rod moves on a {spherical surface} whose center is the fixed pivot point, and whose radius is the rod's length. Imagine we were to construct a transparent spherical shell of that dimension around the center, and mark the shell at each position where the end of the rod points to a star. The result would be a spherical star-chart, whose markings would coincide {exactly} to the observed star positions when viewed from the center of the sphere (and only then).

We have demonstrated a {spherical bounding} of our action to measure the Universe! The sphere is not an object in the sky, but a determinate feature of our act of measurement: a representation of its underlying {ordering-principle}. Does that make it arbitrary or "purely subjective"? By no means! This phase of astronomy is a necessary step in the self-development of the Universe, and thus an imbedded characteristic of the Universe itself.

It now appears, that the ancient astronomer's problem of drawing a star chart on a clay tablet or papyrus is equivalent to the problem of mapping the inner surface of a sphere onto a plane surface. (Note: "inner surface of a sphere" signifies -- paradoxically enough -- a {completely different} geometrical ordering-principle, than the "outer surface." "Inner surface" signifies the ordering of the surface with respect to the spherical center only.)

There exist innumerable possible methods to attempt such a projection, each of which fails in a different way. The simplest is the method of central projection onto a plane outside the sphere, defined as follows: For any locus on the inner spherical surface -- corresponding to a pointing-direction from the center -- prolong that direction outward until it intersects the plane. Readers should thoroughly investigate this species of projection with the help of a transparent plastic sphere and a suitable light source, noting several important characteristics.

For example: the action of simple rotation (e.g. of the pointing-rod) generates a {great circle} on the inner surface of the sphere; the projected image of a great circle, so constructed, produces the effect of a {straight line} on the plane surface. Encouraged by that result, examine the effect of the projection on various arrays of great circles. At the same time, observe that the projection maps only a {half} of the spherical surface, a hemisphere, onto the plane. The boundary of that hemisphere -- a great circle whose location we can determine by cutting the sphere by a plane surface parallel to the projection-plane -- defines a {singularity}: the mapping "blows up" when we approach that boundary circle. In the vicinity of the boundary, the projection introduces wild distortions relative to the relationships on the inner spherical surface. The least distortion apparent occurs farthest away from the boundary, in the "polar region" of the hemisphere.

The "catastrophic" distortions near the boundary, and the circumstance, that only half of the sphere is mapped (or actually much less, if we want to avoid the worst distortions), suggests to our ancient astronomer the following tactic: Instead of trying to map the entire spherical surface (or night sky) at once, divide the surface into regular, congruent regions, and construct the "truest possible" mapping for each one. The combination of such sectoral charts would hopefully fit together to replace a single one. Note, that a complete set of central projections, of the sort we now envisage, corresponds to a {regular array of great circles} on the sphere, each constituting the singular boundary of the corresponding mapping.

Out of the corner of our mind's eye we might already have anticipated a new source of failure: The attempt to "fit" the mappings together at the edges of the chosen regions, will result in {discontinuities}!

We have entered into the domain governed by the five regular solids. We propose to explore that domain, from a new standpoint, in next week's pedagogical discussion. To finish this one, consider the following:

We saw, that in order to reduce the effect of distortion in each spherical mapping to a minimum, the portion of the spherical surface mapped, should be made as small as possible. But, how finely can the surface of the sphere be subdivided?

The characteristic of linear, planar, solid or cartesian geometry in general -- a characteristics which distinguishes such hypothetical, "virtual" geometries from the real Universe -- is the purported possibility of unlimited, self-similar subdivision or "tiling" of space. Take a square in the plane, for example; by connecting the midpoints of the opposite sides, we can divide the square into four congruent subsquares, and so on ad infinitum. An analogous construction applies to any triangle. Similarly, a cube in so-called "solid geometry" can be divided into 8 (or any cubed number) of congruent cubes.

What about the inner surface of the sphere? Take the division of the spherical surface into six congruent, curvilinear-square regions -- i.e. a regular spherical cube. What happens when we try to subdivide those regions into smaller, congruent curvilinear squares? What happens for the division of the spherical surface, defined by the regular octahedron, and the other regular solids? What is the {common source} of the barrier to further subdivision?

by Jonathan Tennenbaum

Part III: Anti-Deductive Ordering Principles

How does the One subsume the Many? The key to the Enlightenment's "coup d'etat" against the Renaissance, was to remove the Platonic conception of higher hypothesis/change from its newly reestablished, leading role in scientific work, and replace it by the principle of {logical-deductive consistency}. Britain's Hollywood-style promotion of Newton and his famous Law of Universal Gravitation -- a "discovery" actually lifted out of the pages of Kepler's "New Astronomy" -- marked a late turning-point in this neo-Aristotelian coup. Generations of gullible minds were seduced by the promise of a "world formula": a single mathematical law, or set of laws, from which the entirety of physical phenomena could supposedly be derived by logical deduction and calculation. The British-Venetian propaganda machine succeeded in installing this cultish idea, which Leibniz had denounced and torn to shreds in his correspondence with Clarke and elsewhere, as the academically-accepted "norm" and "ideal" of the natural sciences to this very day.

Do you think you have been immune to this operation? How many times a day, in organizing, do you try to explain to contacts the relationship between two events X and Y, by attempting to prove to them, in a deductive fashion, one of the following three propositions:

X implies Y;

Y implies X;

some Z exists, that implies both X and Y.

What if the most crucial events occuring in the Uuiverse -- including those most intimately related to each other -- cannot be reduced to deductive consistency? In other words: what if the actual ordering of cause and effect in the Universe is anti-deductive?

Take, for example, the question:

"Why does the well-tempered system permit only a discrete set of musical tones (12 in number) within each octave? Why does it reject an unbroken continuum of pitches, including the pitches {in-between} the 12 pitch-levels of the well-tempered scale?"

Consider, as a response, the proposition:

The necessity of a specific, discrete series of musical pitches for well-tempered polyphony, flows from the same underlying cause, which determines the {impossibility} of a singularity-free mapping of the celestial sphere onto a plane surface.

Does that mean to say, that we can {deduce} the well-tempered system of music from the geometrical properties of the sphere? Wouldn't we thereby be falling into a species of irrational, cultish belief: "sphere-worship," or (in an earlier phase of our discussions on these issues), "spiral-worship," or "the cult of the Golden Mean"?

Reflecting on the difficulty experienced by many in grasping the significance of the regular solids, my attention was called to a crucial step in those solids' derivation, which few people have even noticed, and even fewer have thought through in a rigorous way. The point in question touches upon the much-misunderstood concept of the "celestial sphere." Omitting or glossing over the relevant step, opens the door to serious confusions and misinterpretations of a sort which appear to be rampant among us, and can derail the whole effort. It is therefore urgent to clarify this matter now, before proceeding further along our orbit. The habit of focussing unblocked attention on just such matters, as the professionally-educated trend to dismiss as "too trivial to be worth thinking about," which most often yields flashes of insight into the most advanced issues in science.

These are some of the reasons, why I deliberately began last week, not with the sphere per se, but with astronomy and the problems posed by the attempted construction of a flat star map. Note: I made no assumptions about the shape of the heavens or anything like that, but set out instead to {measure} -- first in rough way, by attempting to draw the sky directly onto a flat surface, and then using a pivoted pointing-rod as an instrument. That {action} of attempted measurement, called forth an ordering principle, the which (for reasons indicated last week) I qualified as "spherical."

We must, however, not gloss over a very crucial point here: The ordering principle in question is {not directly visible to the eye}; the immediate result of my measurement effort was a pattern of distortions and {discontinuities} -- singularities of "failed" mappings!

So, forget the sphere as a visible form. Get it out of your head entirely. It tends to drag your thinking into a downward, aristotelian direction. Don't say "sphere," until you have generated the concept.

(An aside: Remember, Baby Boomers tend to throw words at things, as a substitute for working problems through. This is called "verbal skills": the magic powers by which Baby-Boomers were typically raised and taught to manipulate their liberal parents, and to succeed in school, university and career ... reinforced, naturally, by an occasional temper-tantrum. For this and related reasons, it is {mandatory} that the reader actually carry out the experiments indicated in last week's, this and the following pedagogical discussions. The worst mistake of all, is to think you don't need to actually carry out a construction or related experiment, because you presume you already know what the result will be, or can discern it from the text. That is pure information theory, pure post-industrial ideology. DO the experiments described. Don't just do them in your head, don't just try to imagine them, don't watch somebody else do them, don't read a description of them... DO THEM! Otherwise, you may read the words and make interpretations, but you won't know what I am really talking about. Afterwards you may devise more elegant and powerful ways to evoke the relevant concepts, for which I and others will be most grateful.)

Make sure you have really worked through the main experiment from last week -- the attempt to represent the visible arrangement of stars in the sky, on a flat surface. If a full, clear sky is not available, you can do a roughly equivalent experiment in the middle of a room. Take a large piece of paper, and try to draw your whole surroundings, as they appear to you, on that flat surface.

Leaving aside for a moment the spherical projections described last time, let's examine the problem anew from the standpoint of multiply-connected circular action. Taking the cue from Leonardo da Vinci and later Johannes Kepler in his "Snowflake" paper, examine how the most elementary, multiply-connected features of circular action are expressed in the harmonic motions of the human body.

Standing straight from the vantage-point of your drawing experiment, point your arm straight ahead. Now, rotate the arm to point to the right (or left). That defines a first interval of rotation. From that position pointing right, now rotate into the straight upward direction. That defines a second interval of rotation. Finally, rotate down from the upward direction to the straight-ahead direction. With these three rotations you have generated a triangle: not a visible triangle, but a {triangle of rotational action}.

Now compare the two intervals: {forward==>up}, and {right==>up}. Observe, that if and rotate our whole body around to face to the right, then the interval {forward==>up} now becomes the interval {right==>up}. Observe also, that if we bend forward at the waist, so the trunk of our body is pointing straight forward, then the motion of our arm, which produced the interval {right==>up}, now does {right==>forward}. In this way, by {rotating the rotations}, each of the three rotations forming our "triangle" can be rotated into any of the others. We have an equilateral rotational triangle! The rotations which carry each of those three rotations into any of the others, constitute the {angles} of the triangle. (Note, that the rotations in question are all of the type described in ordinary geometry as "right angles." Anticipate the paradox: a triangle whose angles are all right angles!)

Explore, in the same way, the interrelations of the total of 12 mutually-similar rotational intervals and 8 equilateral rotational triangles arising from what Kepler identified as the astronomically-derived, "three distinctions" embedded in the construction of any animal: forward-backward, up-down, left-right. Don't confuse them with coordinate axes in cartesian space; we make no assumption of scalar, linear extension here, but only angular, rotational action, implicit in our astronomical measurements.

Indeed: What is the crucial distinction of the manifold of rotational action, we have begun to explore, as compared to a flat, cartesian manifold?

Note the following: In a flat plane, for example, the linear displacement "to the left-and-right" vis-a-vis motion "up-and-down" are apparently {independent} degrees of action. If we, for example, move one unit distance to the right in a plane, and then one unit upward, the result will be the {same}, as if we would first go up, then to the right. Compare the composition of motions, that constituted our "rotational triangle." Are rotation "up," and "to the right," for example, strictly {independent} dimensions of action? Or is not the very existence of the equilateral right-angled triangle, just generated, characteristic of the multiple-connectedness of the rotational manifold?

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Note: Our ongoing pedagogical exploration should provide guide-posts for sorting out the real history of ancient geometry, and demolishing encrusted mythologies. The following note from my own, preliminary readings, will hopefully encourage comments and contributions by others.

An 1870 German treatise on the development of geometry before Euclid, refers to an ancient Egyptian treatise on geometrical constructions, the so-called Rhind papyrus from 1100-1000 B.C. According to the German author, that papyrus documents familiarity with the regular solids, as well as the elements of spherical (i.e. rotational) geometry. The papyrus contains a note to the effect, that it is a copy of a treatise dating from much earlier, probably to 3400-3200 B.C. Much later, around 600 B.C., the Ionian Thales devoted much of a lengthy visit to Egypt, to studying the methods and results of Egyptian astronomy. Back in Miletus, Thales and his school, including most notably Anaximander, reworked the Egyptian results and launched a revolution in Ionian-centered scientific development. The next phase appears to eminate from the philosophical-political movement of Pythagorus, who (among other things) is credited by later Greek writers with discoveries concerning the construction of what were referred to as the "cosmic figures" (kosmica skema). Of course, the Greek sense of "cosmic" has nothing to do with the present-day connotation of the mystical or other-worldly. Quite the opposite: the Greek expression connotes "ordering" in the sense of "ordering of the Universe" or "the Universe as ordering principle." This is the platonic conception Wilhelm von Humboldt's brother Alexander intended as the title of his many-volume summary of the natural science of his day: "Kosmos."

It would also be worthwhile to investigate the obvious astronomical origin of the Chinese "Book of Changes," which (among other things) contains unmistakable references to the characteristic, octahedral singularity of visual astronomy, explored above in a preliminary fashion. The Chinese and Egyptian developments are evidently coherent.

by Jonathan Tennenbaum

When we attempt to relive Kepler's discovery of the efficient ordering-principle of the solar system and its crucial empirical feature -- the exploded planet between Mars and Jupiter -- a chief obstacle we encounter is our own, deeply-ingrained assumptions concerning the nature of space. However much some people might scream and hurl epithets at Newton, when you scratch the surface, you often find their idea of space essentially coincides with Newton's; indeed, it seems virtually impossible to them to imagine anything essentially different, than an infinitely-extended, featureless void in which straight-line motion (or something equivalent to it) is the elementary form of action. This typically goes together with an awful sense of smallness, the existentialist's squatting in the middle of an endless parking lot.

Fortunately, remedies are at hand. The Universe is much, much smaller than you think. What at first glance might appear to be "merely subjective" paradoxes of visual geometry, can help us free ourselves from the prison of Cartesian space, and provide a preliminary insight into a notion of anti-Newtonian curvature of the Universe, in which no isolated events are possible. The following, experimental exploration paves the way.

For this purpose, let's go back once again to our starting-point -- mapping the stars -- from a somewhat different angle. Rather than attempting to draw the heavens (or other features of your surroundings) onto a piece of paper, proceed as follows. Take a flat surface of transparent material (such as plexiglass) and fasten it somehow in a fixed position in front of you, so that you can see a chosen constellation of stars (or arrangement of objects in a room) through that transparent window. Using a marker pen, and being careful not to change your eye's position in space (better use one eye and keep the other closed) mark onto the window the positions of the stars or other objects that you see. By construction, the positions and configuration of the resulting marks will coincide {exactly} with those of the corresponding objects -- at least, as seen from the chosen vantage-point of your eye.

What is the problem with this procedure? For one thing, we evidently cannot map more than the one-half of the visible world which lies in front of us through the window, at one time. We might of course set up the window on the opposite side of our vantage point, turn around at map the other "half-world". But the two maps do not fit together smoothly. Separating the two half-worlds is a singularity, where our ray-of-sight becomes more and more skew to the surface, and finally does not touch it at all.

This is not the only difficulty. Taking the window down from its fixed position and examining directly various parts of the image drawn on it, we that they generally {do not} match what we see at all, but become more and more distorted as we move outward from the center (i.e. the region of the window directly in front of the vantage-point). Distorted how? Take a constellation of stars, for example. As the ancients did, we assist our memory by imagining the stars of the constellation joined by imaginary lines in the sky; the resulting {shape}, as reflected in specific angles between those lines, helps us to recognize the constellation. Now look at the image of constellations which are far from the center of our projection. If we measure the {angles} made by the corresponding lines on the surface of the window, we find they are generally very different from the angles we see in the sky. Try it!

The paradoxical nature of the difficulty involved, will become clearer, it you do the following very simple additional experiment: Stand in the middle of an approximately cubical room, facing one of the walls. That wall, bounded by the edges where it meets the ceiling, floor and adjacent two walls, is clearly {square} or at anyway {rectangular} in shape; and that is exactly what you see when you look toward the middle of the wall. In particular, the angles at the four corners are obviously right angles, right? But now look directly into one of those corners. You see three lines representing the edges of the cube coming together... at equal angles of each 120 degrees! What happened to the right angle at the corner of the wall? It now appears as a 120 degree angle! How is that possible? How can an angle change just because I look at it differently? Note, that the change is not explainable as the result of a shift in the position of the vantage-point in space. That point remains the same; all that changes is the {direction} we are looking in.

You can use the device of our transparent window to verify this bizarre phenomenon. If I set up the surface parallel to the wall, and trace out the outlines of the boundaries of the wall as they appear from my vantage-point, I get a square. The images of the boundaries are straight line segments intersecting at right angles. If I now look straight at the upper right-hand corner, and hold the transparent surface at a perpendicular to my line-of-sight to the corner, what I mark on the surface is three line segments intersecting symmetrically at a common point. The 90 degre angle has now become 120 degrees!

Do you think the cubical shape of the room itself is the cause of this problem? I say no. For, imagine we would install a large, transparent sphere (for example) around our position at the mid-point of the cubical room. Taking the common midpoint of the cube and sphere as our vantage-point, we could project the image of the 12 edges of the cube onto the spherical surface. If done with great care, in fact, an observer situated at our vantage point, and looking only at the pattern of lines traced on the spherical surface, would have the impression of standing in the middle of a cube! Again, looking first toward the middle of what appears to be one of the walls, and then looking toward one of the apparent corners, the observer would experience {exactly} the same change of corner angle, from 90 to 120 degrees, as before.

(Some industrious persons should prepare "pedagogical museum" demonstrations for each local, along the lines just sketched. The key item to be procured, is a set of large (preferably at least 20 cm-diameter), transparent plastic hemispheres, which fit together to form a full sphere. Use water-soluble markers to trace the spherical equivalent of the cube, and later the octahedron and other regular solids in succession, on the surface of the sphere. Now have people look with one eye into any of the hemispheres, from a location close to the midpoint of the corresponding sphere. Note, that the great circles, corresponding to the edges of the solids, at appear as straight lines when viewed from the spherical center. In the case of the octahedron, when we look toward the middle of any face, we see what appears to be an rectilinear equilateral triangle, whose angles are 60 degrees. But when we look at any vertex, we see edges intersecting at right angles! Demonstrate an equivalent phenomenon, by placing a small, bright light source (e.g. the bulb of a small halogen lamp) at the center of the sphere, and examining the projected images of the curvilinear solids onto flat screens (e.g. heavy white cardboard) placed in different positions outside the sphere.) With a bit more care, we can demonstrate the same phenomenon of shifting angles in the observation of any constellation of stars that sweeps across a sizeable section of the sky. As we shift the center-point of our vision from one star to another, the apparent, overall shape of the constellation {changes}. This can be verified using projection on a flat transparent window.

Evidently, the cause of these phenomena is not located "out there" somewhere, not in some specific feature of the objects we are observing, but rather in the "infinitesimally small" of visual space itself.

Investigate this further with the help of the following experimental device. Take a small ball of polystyrene or a clump of putty to represent any given observation-point. (It is best to mount the ball of clump at the top of a slender rod or stick, whose lower end is fixed to a flat base). Taking something like slender bamboo skewers (thin shashlik sticks or equivalent), we can represent any given {direction} from the given observation-point, by sticking the tapered end of a skewer, pointed in the given direction, into the center of the ball or clump. So, for example, let two such sticks, stuck into the ball, point in the directions of any two stars. Note the {angle} formed by the two directions. What is the value or measure of that angle? It would appear to be nothing but the magnitude of the {rotation} necessary to rotate the one direction into the other. Note we have made an implicit assumption or hypothesis here: the notion, that for any two directions taken from our vantage-point, it is actually possible to transform the one into the other by a simple rotation.

Now consider {three} directions, represented by three sticks pointing out from the common center. These might, for example, represent the directions of three stars, as seen from the given vantage-point. What are the {true angles} formed by that triangular constellation of stars? I don't mean here the angles we might imagine are formed "out there" between the stars themselves, as objects supposedly existing somewhere in some sort of space, hundreds or thousands or millions of light-years away; let's avoid making any assumptions about that. Rather, I mean the angles formed directly inside a hypothetical "monad" located at the given vantage-point.

Looking at the configuration of the three sticks, perhaps you might suddenly realize something you never noticed before: Any two pairs of directions define an {angle} -- a unique rotation carrying one to the other. That defines 3 angles of rotation. But this is not all! Name the three given directions A, B, C. Compare the two rotations from A to B and A to C, respectively, and recall our earlier discussion of the notion of "rotation of rotation": From the standpoint of the direction A, the two rotations, A->B and A->C are characterized, beside a definite magnitude or angle of rotation, by two different {directions of rotation}. Between those two directions of rotation there is an {angle}, namely the angle of a rotation carrying the one direction of rotation to the other. Thus, any three directions determine not 3, but a total of {6} angles of rotation!

(Some may be accustomed to a different approach to the same relationships in terms of standard solid geometry, as follows: two directions from the common center define a common {plane}. Two such planes, defined for example by pairs A, B and B,C, intersect to form a "plane angle". The three "additional" angles are the angles formed by the pairwise intersections of the three pairs of planes through A,B and A,C and B,C. Fundamentally, however, the planes in question represent nothing but directions of constant rotational action, and the concept of "plane angle" is just a disguise for multiply-connected rotational action. We require no assumption of self-evident linear extension, of the sort which pervades so-called "standard classroom mathematics".)

Now examine the relationship between the array of six angles, just defined, and the {changing} shapes which an observer, located at the given vantage-point, will observe when looking in different directions at one and the same constellation of stars. Just look at the effect of projection of the three directions onto a variable plane.

To close this week's work, try a final experiment: What is the effect of two rotations, carried out in succession? Hold a book in front of you, for example. First rotate it 90 degrees in the clockwise direction. After that, rotate it 90 degrees around the horizontal axis (the upward part rotating downward away from you). Note the resulting orientation of the book. Now, do the rotation on the vertical axis first, and then the clockwise rotation. Why is the result different? Compare this with the case of combining relatively linear displacements, as when we slide the book on a table a certain distance, parallel to itself, in each of two different directions. Is the multiply-connectedness of the rotational manifold, just demonstrated, responsible for the paradoxes of vision?

by Jonathan Tennenbaum

We now enter a crucial phase of our journey, which begins by discovering the axiomatic implications of what at first appears as a mere optical illusion, takes us to Kepler's discussion of "the curved and the straight" in his Mysterium Cosmographicum, and on from there to a fresh view of the regular solids.

First, an experiment.

Take a large transparent hemisphere, held or fixed with its border-circle in the vertical plane, so you can look into the inside with your eye in the spherical center and the pole of the hemisphere straight in front of you.

Trace a great circle (or actually half-circle) approximately at mid-height on the hemisphere, so that it has the appearance of a horizontal line when viewed from the center. Now trace a "vertical" great circle, which cuts the horizontal one at right angles at a point X. Viewed from the hemisphere's center, this second line appears as a straight line running perpendicular to the first one. Note, that relative to the horizontal line (circle) running right-left, the up-down line is perfectly {symmetrical}: it does not "lean" in either direction. Now chose a point Y about 30 degrees to the right (or left) on the horizontal circle from the position of X, and draw a third great circle through Y, at right angles to the original circle. What do you see when you look from the sphere's center? You see a straight, horizontal line with perpendiculars drawn to it at two points X and Y, don't you? And as perpendiculars intersecting the common base-line, they must be parallel to each other, must they not? Indeed, focussing attention on a point mid-way between X and Y, the perpendiculars appear as perfectly parallel, vertical straight lines coming off the horizontal.

But as you look upward from the horizontal line, you notice that the "parallels" come closer and close together, as if leaning toward each other! How is that possible, if they remain straight? You recheck the angles at the horizontal line. No question, they are right angles, which means {complete symmetry}: the perpendiculars cannot lean in either direction, left or right. And yet, you just found them converging toward each other! Did they somehow get bent? You follow each of the perpendiculars carefully, and find no divergence anywhere from what appears to be {perfect straightness}! How could it happen, that perfectly straight lines, making right angles to a common line, stop being parallel?

Compare this paradox with that of our earlier investigation of the great-circle triangle with 90 degree angles, traced on the surface of a sphere. Looking from the center the sides appear as perfectly straight lines, forming an equilateral triangle. Looking at the angles, you see that each one is a right angle. Try to draw what you have seen on a piece of paper. Impossible? Why? A triangle is a triangle isn't it? If so, why can't you draw it?

Evidently, something anomalous is going on here, which is much simpler and more fundamental than most ordinary sorts of optical illusions. I suggest, that the problem is not located in your visual apparatus per se, but in your mode of {interpretation} of visual perceptions, so that you experience the paradox as "unheimlich", as bursting forth inside your own mind. Let's try to trap this critter for closer examination:

1. An arbitrary configuration of great circles, when viewed from the center of the sphere, appears to the viewer as a configuration of {perfectly straight lines}.

2. No {single} view of that configuration contains {anything} which were incompatible with the assumption, that what we are looking at, is an array of straight lines drawn in a plane.

3. A difficulty arises only, when we compare {more than one view} of the same apparent configuration. Indeed, when we try to {correlate} our various perceptions of that array, as we look from the center in various directions, we encounter phenomena (i.e. the equilateral triangle with right angles, or the converging perpendiculars) which are absolutely incompatible with the assumption just articulated.

4. Why does this surprise and baffle us? Evidently, one and the same perception, and one and the same array of predicates (the straight-line images) can be interpreted in more than one way, from the standpoint of more than one set of assumptions concerning the geometry in which those predicates are embedded. It would seem as if the very appearance of an array of straight lines tends to evoke, in our minds, the assumption of a linear, plane geometry. Whereas, that same appearance is not only consistent with a curved geometry, but the {changing} array of appearances, arising when we change our direction of viewing, is compatible {only} with a non-zero curvature of a certain type.

5. That, then, is where the implicit flaw is located; not so much, I submit, in the formal assumption of a plane geometry per se, but rather the deep-seated tendency to regard the characteristics of a geometry as something emanating from, or self-evidently determined by, the predicates (appearances, objects, isolated "facts") in and of themselves. Whereas, what {distinguishes} the curved from the flat geometry, in this case, is not the predicates per se, but the characteristic of {change} in the adducible relationships within the array of predicates, or more precisely, in our cognition of that change and its implications.

Kepler's Argument

From this standpoint, let us turn to the kernel of Kepler's argument in his Mysterium Cosmographicum, the section entitled "The sketch of my main proof". I hope the pedagogical devices of this and the preceeding pedagogicals will throw some new light on Kepler's notion of "the curved versus the straight" (or, for reference to our present discussion, "curved versus linear, or flat"), not as an "objective" contraposition of types of forms in space, but rather in terms of the {mental processes} we have just begun to explore, and particular the process of {shift of basic assumptions}, from one type of geometry to another. Much more could be said about this, and we shall come back to it again, but let us go ahead and read what Kepler writes:

"God wanted, that Quantity should be created before all other things, in order that a comparison of {curved} and {straight} might occur. Exactly for this reason I find the Cusaner (Nicolaus of Cues) and others possessed of divine greatness, namely because they attached such high importance to the behavior of the straight and the curved toward each other, and dared to attribute the curved to God, the straight to the created things...What the Cusaner ascribed to the circle, and others to the space enclosed by the sphere, I attribute only to the surface of the sphere alone. I am firmly convinced, that no curved thing is more noble and more perfect than the spherical surface. For, the (solid) ball is more than the spherical surface, and is mixed with the linear, by means of which alone the interior is filled. A circle arises only in a plane, i.e. only when the ball or sphere is cut by a plane....

"But why did God chose the difference between curved and straight, and the noble nature of the curved, when he wanted to form the world? Why, indeed? Only because the most perfect architect must necessarily construct a work of the highest beauty.... In order that the world might be the best and most beautiful, in order that it might be able to receive this idea, the All-wise Creator produced Magnitude and brought forth the Quantities, whose entire nature is comprehended, in a sense, in the differentiation of the two concepts, the straight and the curved... It is probable, that God from the first moment selected the curved and the straight, in order to engrave in the world the divine nature of the Creator; to make possible the existence of these two concepts, quantity was created; and in order that quantity might be conceived, He created before everything else (spatial) body.

"As we before chose the sphere, because it is the most perfect quantity, so we now make a {single} jump to the bodies, which are the most perfect among the straight quantities and consist of three dimensions."

In this light, let's review the ground we have traversed, once more. We had two geometries, a linear geometry represented on a plane surface, and a spherical geometry. The two geometries are fundamentally incompatible, hence Kepler's expression: a "jump". We cannot construct a single, consistent, "literal" representation of the curved surface of a sphere (i.e. the manifold of rotations), within the bounds of linear plane geometry. Might there exist some {other} form of representation? We already have the answer on the tip of our tongues: Given the impossibility of a consistent representation of spherical geometry within the linear, plane domain, the spherical domain's {existence} could have no other lawful manifestation within that linear domain, than through the generation of characteristic patterns of {anomalies}! What, then, is the {minimum} set of anomalies, sufficient to characterize what Kepler describes as the "single jump" from the "straight" to the spherical geometry?

Aha! We already encountered a relevant sort of anomalies, which arose in the attempt to {correlate} different views of one and the same configuration of great circles on the spherical surface, as seen by looking in different directions from the center of the sphere.

Juxtaposing an Array of Projections

Investigate this phenomenon, by placing a light source at the center of the transparent (hemi)sphere, and projecting images of various great circles traced on the spherical surface, onto a large, flat screen which we have mounted in any chosen, variable position relative to the sphere. We see that the images projected on the screen are indeed straight lines. Indeed, if we keep the screen in its given position, and replace the light source again by our eye, then the positions of the straight lines on the screen will be seen to coincide {exactly} with the appearances of the lines (great circles) on the sphere, as seen from that center.

Note, however, that the actual array of lines projected on the screen, including the magnitude of the angles between those lines, change greatly when you move the screen from position to position around the sphere. What is the significance of those changes? Evidently, the various projections correspond to what we referred to above as "different views of one and the same array of great circles" when viewed in {different directions} from the center.

The relationship becomes clear, when we determine the line running from the center of the sphere to the nearest point P on the plane of the screen; in other words, the perpendicular from the sphere's center to the plane of the screen. Let Q mark the position where that line passes through the spherical surface. If we trace the projected images on the screen, and hold the screen in front of us so we are looking at P, then what we see is a "photograph" of how the sphere appears to us, when we look from the center into the direction of the point Q.

For any given position of Q, the corresponding positions of the screen are defined by the perpendicular planes to the corresponding direction. Clearly, moving the screen closer or further away from the screen, while keeping it perpendicular to that line, only blows up or contracts the dimensions of the image, while keeping the angles the same. If we choose to regard such changes as non-essential -- which indeed seems justified in view of our search for a "minimum" representation of the anomalies -- it makes sense to choose only one plane for each Q. Which one? The only unique choice, at this present stage, is to slide the screen up to the sphere until it touches it, at Q; in other words, project onto the {tangent plane} to the sphere at Q.

If, now, the anomalies which we observed earlier, are connected with the discrepancies or changes between such projections, when made in different directions, then two tasks confront us: First, to determine a minimal set or sets of projections, needed to display the type of anomaly in question; and second, to characterize the anomaly itself.

Bearing in mind what was said in the next-to-last paragraph, each projection involves the choice of a point Q on the sphere, such that the line from the sphere's center through Q defines the direction of the projection; the screen being located at the tangent to the sphere at Q, perpendicular to that line. Accordingly, choosing an array of projections of the indicated type, amounts to choosing a {set of points "Q" arranged on the spherical surface}. Each projection is equivalent to a "snapshot" of the spherical surface, taken from the sphere's center with our "camera" pointed at the corresponding point Q. The interesting phenomena will obviously be located in the regions where any two projections, say corresponding to points Q and Q' from the set, intersect or overlap with each other.

With a bit of thought, we conclude that the character of the transformation or change between two such projections, depends only on the change of relative directions, i.e. of the relative positions Q and Q' on the sphere; or in still other words, on the {arc} between them. As a result, to obtain the simplest, minimum characterization of the anomalies we are looking for, we must choose the array of points Q in such a way, that the arcs between adjacent points of the set, are all {equal}. In other words, they must form a {regular} array.

Isn't it now obvious, where our journey is taking us? Look at the array of tangent planes (our "projection screens") corresponding to the various points of the regular array of points Q. They form a kind of "envelop" within which the sphere is inscribed. Observe the edges formed where adjacent planes intersect, and cut off the portions of those planes which protrude on the outside of the intersections, in the obvious manner. What do you get?

Finally, do the following experiment. Trace a single great circle on your transparent sphere, and install a small, but bright light source in the middle of the sphere. Next, using some appropriate translucent material (i.e. plastic sheet), build a regular solid around your transparent sphere. The points of tangency with the sphere, defines the regular array of points "Q" in our previous discussion. Now observe the image of the great circle, projected onto the faces of your regular solid. It appears as a closed chain of straight line segments, whose "links" are at the edges of the solid. Observe the {discontinuous change of angle of inclination}, when the image crosses an edge, from one face of the solid to an adjacent one. Does this not remind you of the refraction of light? Finally study how te image behaves, when the sphere is rotated inside the solid.

By Jonathan Tennenbaum

Recapitulation

Pursuing Kepler's juxtaposition of the "curved and the straight" in terms of the attempted mapping of a spherical surface onto a plane, I last week suggested the following:

Given the manifest impossibility of a simple, consistent representation of spherical geometry within the linear, plane domain, the spherical domain's {existence} could have no other lawful manifestation within that linear domain, than through the generation of characteristic patterns of {anomalies}! What, then, is the {minimum} set of anomalies, sufficient to characterize what Kepler describes as the "single jump" from the "straight" to the spherical geometry?

I sought to answer that question, by studying the anomalies, which occur when I project the sphere from the center onto a tangent plane. The most obvious anomaly is the apparent {incompatibility} between any two such projections: they don't fit together, at least not in any way that can be accounted for by "plane geometry."

I ended up with the idea, that a minimum representation of spherical curvature would be achieved by an array of projections, whose "incompatibilities" all have the same form. This led to the requirement, that the {directions} or midpoints of the projections, should form a {regular array} on the surface of the sphere. I argued, that this amounts to {projecting the sphere onto the faces of a regular solid}.

Some readers surely recognized at least one major inadequacy in my approach, namely, that I posed the choice of a regular array as a {requirement}, but I didn't account for {where} such regular arrays come from and {why} they must exist.

The effort to fill this lacuna will takes us into new territory, a territory inhabited and ruled by a wonderous creature, called the "pentagramma mirificum."

Now, the territory in question has a fearful reputation: Many of those who venture in, either never return again, or if they do, they tend to emerge in a scrambled-up state, suffering from dizziness and giddyness and unable to report what they saw in a coherent way. To avoid falling victim ourselves, it is necessary to proceed step-by-step, and above all to fix our conceptual bearings from the start.

The Singularities of Rotation

For most of us today, the idea of simple rotation of a body around a well-defined axis seems self-evident. That idea is deeply embedded in human culture, it would seem, since no later than the proverbial invention of the wheel. Have you ever stopped to think, that an {hypothesis} is required? Indeed, if we put aside astronomy, and observe the motions of "natural" Earth-bound objects, then, apart from man-made objects, constructed or selected on the basis of that very hypothesis, we hardly find any case of rotation around a well-defined, precise axis. Pick up an irregular rock, throw it into the air or try to spin it on a hard surface. You never get a simple rotation, but rather a very complicated, wobbly motion. Imagine someone challenges you to go into a forest, without any modern tools or other products of our technical culture, and construct a wheel from the natural materials you find there. How would you do it? How would you, for example, starting from "nothing", build a rotating table for producing pottery, an elementary form of machine-tool? If you make the attempt, you might develop a healthy respect for the level of technology embodied in the simplest household artifacts of ancient cultures.

So the idea of simple rotation as a fundamental principle of physical action, does not arise from mere sense-perception of objects around us. Nor does the notion of {axis of rotation} as a subsumed singularity of such action arise so. Might it not be the case, that the notions of rotational action and axis of rotation, at least insofar as they became a concious principle of ancient machine-tool design, developed from astronomy? Remember how, millenia later, Gauss and Wilhelm Weber initiated a revolution in electrodynamics, by carrying over principles of astronomical measurement, into the microscopic domain. But let's be careful not to jump over crucial steps here. Bringing machine-tool principles from heaven down to the Earth, is no self-evident linear process.

Observe the heavens on a clear night. Do you see the rotational motion of the stars? Not directly, anyway. But suppose, we as very prehistoric astronomers, have once established, with the help of our memory, the existence of the daily cycle of star motion and finally the circumstance, that the individual stars move in what appears to be a system of concentric circles in the sky. (Having filled in, in our minds, those motions unseen during the interruption of daylight and the periodic disappearance of stars below the horizon.) Now someone will probably jump up and say: What's the big deal? You already have it: rotation!

But wait a moment. {Where} is the {axis} of that rotation? Through what points in space does it pass? Does it go through your body? Does it pass through somebody a mile away, who also observes the circular motions in the heavens? Or does the rotation have any axis at all in the sense of a line going through There is still a big topological distinction between the cyclical motion we adduce in the heavens, and that of the wheel we are about to invent.

To make my point a bit clearer, take out the measuring apparatus introduced earlier, consisting of a thin pointer rod or stick whose end pivots around a fixed locus, the latter corresponding to the view-point of the observer. (Many variants of this sort of instrument are possible; what is important is the functional result, namely to determine and record the {directions} in which stars are sighted, when seen from the given locus). Now examine the {manifold} of positions of the pointer rod, as it follows a given star in the course of an evening. Supplement those positions according to the presumed motion of the star when it is not directly visible. Do the same thing for a variety of stars. What do you find?

In the course of a daily cycle, the pointer rod describes the surface of a {cone}! [Show this with the bamboo skewers (shashlik sticks) stuck into a small ball of putty or styrofoam, or equivalent means.] Different stars determine different cones. Some are narrow, some wide, in correspondence to the apparent size of the circular path of the star in the heavens. But the entire array of cones is ordered in a very beautiful way, as a {nested} series. We find there are stars which barely change position in the course of the night (and day); such a star generates a very thin cone. A star a bit further away from that region of the heavens generates a larger cone, which contains the first one, and so on. The cones open out more and more, until they become virtually flat (for stars near the so-called celestial equator, which I shall define rigorously in a moment); after that they begin to close away from the direction of original narrow cones, becoming narrower again.

Now pay attention to the really interesting features of this family of cones, its {singularities}! On the one side, we have the narrow cones, which, as they become narrower and narrower in a nested manner, appear to converge toward a certain definite direction in the heavens, common to all. How shall we characterize that singular direction? It is the direction of {least motion}.

On the other hand, as we examine stars located progressively further and further away from the locus of the heavens corresponding to least motion (known as the "celestial pole"), the cones open outward, and we encounter another singularity: an ambiguity separating two subfamilies of cones, the ones opening toward the pole, and the ones opening in the opposite direction. That ambiguity corresponds to a hypothetical, "perfectly flattened" cone, which makes what we today call a {right angle} to the direction of least motion. The corresponding "ring" around the heavens, correponding to all possible directions of the pointing rod moving in the flattened cone, is known as the celestial equator. Stars in this region have the {greatest motion}, compared to everywhere else. At the same time, we conceive the existence of a {second} pole of {least motion} opposite to the other pole, although the earth under our feet blocks it from view.

Now, how do the characteristics of motion, which we have adduced from our observation of the stellar motions as a whole, project from the astronomical scale, down to our own, earthbound scale?

Take a putty or styrofoam ball, and 4-5 shashlik sticks. Insert one stick into the ball so that it points in the direction of the celestial pole, and insert the remaining sticks so that they point in the direction of as many stars, including one on or near the celestial equator. With time, of course, those stars will change position. Is there a single continuous motion of the ball, such that each of the pointers remains pointed at its assigned star? Now we have it: the {rotation of a solid body around a (relatively) fixed axis} is the form of action, on our earthly scale, which corresponds to the adduced motion of the heavens. The axis is the direction of the stick which points to the celestial pole. This also suggests a possible astronomical determination of a "primordial straight line": motion along the constant direction of a pole. (There is more to be said on this, but not now).

Now imagine our pointing device placed in the middle of a transparent sphere. If we mark the locations on the surface of the sphere, corresponding to the positions of various stars as seen from the center, then the two celestial poles correspond to two points on the sphere, located opposite each other from the center, and the celestial equator corresponds to a great circle, located on the plane through the center perpendicular to the direction from the center to the poles (that plane is the "flattened cone" we spoke about earlier). We can represent the relationship of the pointing device to the heavens, by the relationship of the center of the sphere to the spherical surface. The daily motion of the stars corresponds to a rotation of the entire sphere, around the axis through the center and the two poles. In that rotation process of the sphere, the poles constitute the {regions of least motion}, the equator the {region of maximum motion}. That, in turn gives us the principle of the wheel, which combines maximum motion on the perimeter with minumum motion of the axel.

The Singularities of Multiple, Self-reflexive Rotation

Now obviously, the preceeding astronomical derivation of rotation and its singularities, does not exhaust the Universe. Although the Sun does have a daily rising and setting, if we plot its position on the transparent sphere, now made to rotate so that it follows the motion of the "fixed stars", the relative locus of the Sun {changes}. In fact, it traces out what looks like a circle on the sphere (corresponding to the so-called ecliptic in the heavens). That circle intersects the equator (the circle corresponding to the celestial equator) in two points, apparently exactly opposite to each other (these are the Spring and Fall equinox points, the two days when night and day have equal length.) In fact, if we apply rotation to the sphere, taking as the axis of that rotation the direction through the center and those two points, we find that we can rotate the equator {exactly} onto the ecliptic. At the same time, the points corresponding to the celestial poles are carried into new positions, which have the same relationship to the ecliptic as the original poles had to the original equator. Evidently those new positions constitute the poles of a {third rotation}; a rotation whose equator is the ecliptic. You will probably see these relationships a bit more clearly, when we generate them in a slightly different way, in a moment.

Still another degree of rotation reveals itself, when we move our observation-point to a different location on the Earth's surface. As we go toward the north, the axis of rotation of our sphere must be {rotated} into an increasing angle relative to our apparent horizon increases. This fourth degree of rotation Thales and probably Eqyptians and others long before, interpreted correctly, as a manifestation of the curvature of the Earth.

These circumstances, among others, suggest, that action in our Universe involves nothing less than a combination of many degrees of rotational action. What is the interrelation or interconnection among those various rotations? Our comparison of the celestial equator with the ecliptic suggests the idea: rotational action applied to rotational action.

To explore this notion further, as follows, using the surface of a medium-sized plastic or styrofoam ball, and marking singularities with a non-permanent marker). Start with a first rotation R, which generates as singularities two poles N, S and an equator E (mark them). Now imagine any arbitrary second rotation R'. The second rotation generates a second pair of poles N', S' and a second equatorial (or "great", i.e. maximum) circle E'. The two equators intersect in a pair of points X, Y lying on opposite sides of the spherical center. Examining the two great circles and their corresponding pairs of poles marked on the sphere, it is manifest how to rotate one onto the other: the required rotational axis, is the axis passing through the intersection-points X,Y of the two equators. In other words, X and Y are the {poles} of the rotational action which carries E to E'. That same action carries the poles of R into the poles of R'. Note, that the equator corresponding to that third rotation, passes through {all four poles} of the original two rotations.

If you look carefully now, you will find at least two fish flapping around in your net.

The first fish is the remarkable suggestion: all rotations can be generated from any single one, by rotating it in the manner just described! Actually, that is not exactly true; we only demonstrated we could rotate the chief {singularities} of the two rotations into coincidence with each other, but didn't take into account the different modes and quantities of rotation -- fast or slow, continuing (indefinite) rotation or terminated (definite) rotation, and in the latter case through what magnitude of angle, etc. So, the more accurate conclusion so far would be: any rotation can be obtained from any given one, by applying a definite rotation to that one, and in addition possibly changing its mode and quantity.

The second fish, caught in the corner of our eye, so-to-speak, is the suggestion of a self-reflexive sort of "connectivity" among rotations. Let's try to catch this one. To do that, avoid the element of arbitrariness in the angle between E and E' in the previous discussion, by considering the effect of a {continuing} rotation of E, i.e. one that does not stop at E'. Call that continuing rotation R1. The poles of R1 are still the points we called X and Y. As I noted before, the equator E1 of R1 contains the poles N, S of the original rotation R; in fact, each of those poles traces out E1 in the course of rotation by R1.

Interesting. That means E1 is, in a sense, covered {twice} in the course of a single cycle of R1. Look at the process a bit more closely. As we proceed to apply the rotation R1, the equator E rotates into a {variable} circle E' which intersects E in X and Y, making ever larger angles to E. Suddenly, however, we run into a singularity: when the angle is what we now call 180 degrees, E' coincides with E, although with an {opposite direction} of rotation! At that same moment, the poles N and S have {reversed position}. As we continue to rotate further, E' separates again from E, only to coincide with it again when the total angle of rotation is 360 degrees, i.e. a full cycles. To sum up the result: rotation applied to rotation, results in the division of a full cycle into {half-cycles}, divided by a singularity corresponding to "reversal of direction."

Choose a victim, and ask him to prove to you, why "1/2" and "-1" are equivalent as geometrical numbers!

So far, we have rotations R and R1, poles N,S and X,Y and equators E and E1 respectively. Observe, that E and E1 intersect at two additional points, Z and W, which lie opposite each other across the spherical center, and divide both equators in half. At the same time, notice that {E is carried into E1} by a rotation whose poles at Z and W. Call the corresponding {continuous} rotation R2, and its equator E2. Note that all four points N, S, X and Y lie on E2, and they divide a full cycle of rotation according to R2, into {four} congruent sections.

If we start with E and begin to apply R2, we again get a variable circle E'', intersecting E at the points Z and W. After a rotation of 90 degrees, E'' coincides with E1. Continuing past that, we get to the singularity at 180 degrees, when E'' coincides with E, except for a reversal of direction. Next, at 270 degrees, E'' coincides with E1 but with reversed direction, before finally coming back to E. What shall we call the rotation from E to E1, the which, when repeated, takes us to the reversed-direction version of E? Call it "i", otherwise famous as the first imaginary or complex number.

Now go back to your victim, and demand that he immediately explain to you the equivalence of "1/4" and "i". Also the equivalence of both of these with "3", since we required a uniquely-determined series of {three} rotations to divide a full rotation into four congruent subcycles and to generate "i".

With the addition of the rotation R2, its poles W, Z and its equator E", a new phenomenon occurs: closure! The attempt to continue the process of generating new rotations and poles from the configuration just created, in the manner pursued so far, yields nothing new. If, for example, we take the intersection of E1 and E2, we get the poles N and S of the original rotation; and that one-fourth of a full cycle of that rotation carries E1 into E2. Thus, the construction process has an intrinsic {periodicity}, returning to the starting-point after three steps.

Any {two} of the rotations R, R1, R2 are carried into each other by the third, through the same quarter-cycle of rotation. The equator of each of the 3 rotations contains the poles of the other two, which in turn divide that equator into 4 congruent segments. The combination of all 3 equators E, E1, E2 divides the surface of the sphere into 8 congruent regions, bounded by 12 arcs and 6 vertices (the poles). Each of the regions is bounded by an equilateral curvilinear triangle whose angles are all 90 degrees.

Note: There is nothing {arbitrary} about that configuration. If you begin with {any} continuous rotation (as R), and rotate it around {any} axis that lies on R's equator (as R1), then you end up with the {same} -- i.e. precisely congruent -- configuration of three rotations R, R1, R2 and the same array of singularities (poles, equators, division of the spherical surface).

The reader has surely recognized the curvilinear octahedron, discussed in Part 3 of this series, and may also be familiar with the way the octahedron produces -- with hardly any outside assistance! -- the cube, and the cube the tetrahedron. But here we seem to encounter a natural boundary. To proceed further we must add a singularity. That will bring us face to face with the legendary pentagramma mirificum.

By Jonathan Tennenbaum

"It is as if one were travelling, alternately, in two worlds. In one world, there is action-at-a-distance along straight-line pathways, a linear, empiricist or Cartesian world. In the adjoining world, a circular action is produced by {rotation}, not by action-at-a-distance along straight line pathways... These two worlds are two Types, of which the rotation-world is the superior, the bounding, the limiting, the determining, the higher one." (Lyndon LaRouche, in "Cold Fusion: Challenge to U.S. Science Policy", Chapter III)

We have now come to the construction, that Kepler's enthusiastic contemporary Napier dubbed, "the wonder of the pentagram." My description in words will be a bit awkward, probably unavoidably so, nor could static diagrams by themselves convey the required sense of self-reflexive, multiply-connected rotation. There is no substitute for the reader's active exploration and replication of the following constructions.

The locus of the pentagrammum is the rotational manifold, that arose as a product of our attempt to map the heavens (see Parts 3 and 4 of this series). We have represented such rotations in two ways: {first} in terms of changes of direction, as when we observe the sky from a single viewpoint, i.e. the celestial sphere as seen "from the inside"; and {second}, in terms of the rotations of a spherical surface as seen "from the outside." I shall start with the second representation, which is easy to experiment with, using plastic spheres and erasable markers of various colors to mark the singularities (great circles, poles etc.)

Start with an arbitrary rotation of the sphere. Call the equator for that rotation E1, its pole R. Choose any position P on the great circle E1. Think of P and R at first as reference-positions, relative to which we now juxtapose a third, arbitrary (variable) position. Let that third location, Q, be given anywhere on the sphere outside the equator E1. (To avoid certain difficulties, which I shall discuss below in part, it were best to begin with the case, were Q does not lie too far away from either P or R, i.e. forming an arc of less than 90 degrees from either of those two locations.)

Now, by "unfolding" what is implied in the relationship between the arbitrary locus Q and the two loci P and R, we obtain the following, seemingly endless {chain} of singularities:

First, there will be a {unique} great circle passing through P and Q, corresponding to the least rotation which carries P to Q (*1). Construct that great circle. Call the pole of that rotation S, and call the circle itself (i.e. the equator of the rotation) E2.

Next, there will be a unique great circle E3 passing through Q and R, corresponding to the least rotation carrying Q to R. Call the pole of that rotation T.

Again, there will be a unique great circle E4 passing through R and S. Call its pole U.

Still once more, there will be a unique great circle E5 passing through S and T. Call its pole V.

And so forth. At first glance, this chain of relationships might seem to go on ad infinitum: a "bad infinity."

But do the experiment. You will find, to your initial surprise, probably, that the process {closes} by itself, after generating the {fifth} point! Indeed, the pole U appears to {coincide} with the starting-point of the chain, P, while V coincides with Q and so forth. The whole process repeats, generating {exactly} the same sequence of five great circles and poles once again! The points P, Q, R, S, T form the vertices of a (generally) non-regular, 5-sided spherical polygon.

This periodicity was first studied in detail, as far as we know, by Johannes Kepler's contemporary Napier. What strikes us as so extraordinary (mirificum!), is the circumstance, that the character of periodicity does not depend on the choices of the initial points P, R and Q. More precisely: all that is assumed in the construction, is three arbitrary points P, Q and R, subject only to the condition, that P lies on the equator of a rotation whose pole is R. (As the reader can easily ascertain, the latter condition signifies that P and R are separated by an arc of 90 degrees as seen from the center of the sphere).

Evidently, the self-closure of the chain into the form of a non-regular spherical pentagon, reflects a {universal} characteristic of spherical geometry, having no obvious equivalent in simple plane geometry. That characteristic determines the outcome of the construction, as it were, "from outside"; standing above and beyond the seemingly arbitrary choice of starting-points for the construction.

But let's try to see more clearly, {why} the pentagrammum {must} close. For this purpose, let's review the chain of relationships once again, this time from "inside" the spherical geometry. We shall find that the pentagrammum is already implicit in the simplest astronomical observations.

Under a clear night sky, stand facing due north, looking toward the corresponding northermost point on the horizon. Take that point as your "P". At the same time, note that the zenith point (directly overhead), corresponds to the pole of the horizon. In other words, if we point our arm toward the horizon and rotate our arm left-right so that it follows the horizon, then the axis of that rotation will be vertical, and the poles are the zenith and the point opposite to the zenith, "directly down" under our feet. Call the zenith-point "R".

Now choose a star anywhere in the sky. Designate its position "Q". Observe the relationships between Q, the horizon-point P and the zenith-point R. Note two imaginary arcs formed in the sky: from P to Q, and from Q to R. These are the first two sides of our pentagram. Note also the arc from the zenith R down to P, which makes right angles to the horizon -- a celestial right triangle! That same arc will be a {diagonal} of the pentagram.

Now trace out the arc PQ, by pointing first at P, and then applying the relevant rotation to your arm until you are pointing at Q. Point with your other arm in the direction of the axis of that rotation, i.e. toward its pole. Call that pole "S".

It might be helpful, in grasping these relationships, to tilt yourself in such a way, that the arc PQ appears as your new "horizon", and your new "above" (zenith) is S. Similarly for the arc QR and its pole T, the arc RS and its pole U, etc.

In this way, we trace the pentagram as an imaginary "constellation" in the sky, unfolded from the relationship of any given star Q to the reference-points P and R. Note, that if we change the position of Q, the shape of the pentagram will also change.

Now, what makes the chain of arcs and poles close after exactly 5 steps? Perhaps the reason will emerge, if we draw up a list of the chain of relationships in the construction:

1. P is a point on the horizon. 2. Q is any arbitrary point off the horizon. 3. R is the pole of the horizon (i.e. the zenith). 4. S is the pole of the rotation P->Q. 5. T is the pole of the rotation Q->R. 6. U is the pole of the rotation R->S. 7. V is the pole of the rotation S->T. 8. W is the pole of the rotation T->U. 9. X is he pole of the rotation U->V. etc. etc.

From the list itself, we don't see any reason why U should coincide with P, V with Q and so on. Have we failed to take account of something? Recall last week's discussion of the multiple-connectedness of spherical rotation. Aha! We didn't pay attention yet to the various {angles} in the pentagramma. For example: the very first angle in the construction, which is the angle formed at P between the horizon and the arc PQ. This is the angle an observer would have to tilt himself by, in order to make the great circle containing PQ into his new "horizon." Or in other words, in the language of our earlier constructions on the sphere: it is the (lesser) angle formed between the great circles E1 and E2, which in turn is the amount by which we would have to rotate the sphere itself, to carry E1 into E2. Evidently, the point P, which is the intersection of E1 and E2, represents one of the {poles} of that rotation.

Now what happens to the pole of E1 (i.e. R), when we carry out that rotation of E1 into E2? Evidently, E1's pole is carried to E2's pole, i.e. R moves to S. Our conclusion: the rotation from E1 to E2 -- a rotation whose pole is P -- {coincides} with the rotation R->S. The latter rotation, however, appears as the 6th step in our list above, where the point "U" is defined as its pole.

So P and U are poles of one and the same rotation! Now we begin see why the chain of relationships closes.

"The Theory of Ambiguity"

But here a difficulty arises: The circumstance, that P and U are poles of the same rotation, does not necessarily mean they {coincide}. They might instead be {antipodes}. Indeed, a rotation always has {two} poles, at diametrically opposite positions on the sphere.

By failing to consider the ambiguity in our expressions, such as "S is the pole of the rotation P->Q" or "S is the pole of the great circle E2", we left open two possible choices. If at each step of our construction, we permit either of the two choices, we evidently end up with many more possible constellations. The chain is no longer uniquely defined, and in some cases will not close after 5 steps. The chain only becomes well-defined, if we introduce some "external" criterion for choosing between the two poles at each step: as, for example, by requiring that P, Q. R, S, T etc all lie on the same hemisphere. Alternatively, we could require that all the rotations (arcs) are all less than 90 degrees -- which can always be accomplished by the proper choice of poles --, or that at each step the pole chosen should be "upward" with respect to an observer who has tilted himself through the angle between the successive great circles (or more precisely, the lesser of the two pairs of complementary angles, the one less than 90 degrees) to get from one great circle as his "horizon" to the next one. In the course of the construction, that "upward" direction pivots around in a closed cycle, pointing always toward the "interior" of the pentagram. Examining the entire configuration of five great circles, we see that not one, but {two} identical pentagons are formed on the sphere. Their vertices -- 10 in all -- are antipodes of each other.

At first glance, the ambiguities might seem a bothersome complication. Yet, as Gauss and Riemann developed the point in great richness: it is the ambiguities which determine, to a great degree, the whole character of a process. We began to study a similar, related case in Gauss' approach to the Pothenot problem. Evariste Galois, a disciple of Gauss, referred to this elementary part of analysis situs as "the theory of ambiguity" (*2).

If you think the problem of ambiguity can be avoided, just try to define the vertices of the pentagram as a {continuous function} of the variable Q. Watch what happens when Q approaches, and crosses, the great circle through P and R, or when Q runs around the back and underside of the sphere as seen from P and R. The ambiguities associated with the double nature of the poles, come out as discontinuities in any attempt to impose a single, simple continuous function on the pentagram relationships.

Higher Self-similarity

Now behold the array of self-reflexive relationships subsumed by the pentagram, which the reader should be able to verify without much trouble:

The {diagonals} PR, QS, RT, SP and TQ are all equal in magnitude, corresponding to quarter-circle arcs (90-degree arcs).

Each vertex P, Q, R, S, T is a pole of the rotation defined by the opposite side (arc) of the pentagram (i.e. P is the pole of the arc SR etc.). The exterior angle formed at each vertex by the great circle-prolongations of the adjacent sides, is equal to the angle spanned by the arc on the opposing side as seen from the center of the sphere.

Of the total of 20 intersection-points of the five great circles, 10 are vertices of the two, antipodal pentagons formed by those circles (namely the intersection-points of E1-E2, E2-E3, E3-E4, E4-E5 and E5-E1). At the other 10 intersection-points of the great circles (those of E1-E3, E2-E4, E3-E5 and E4-E1), {right angles} are formed.

Most important is the self-reflexive characteristic, that the pentagram can be "regrown" from any three consecutive vertices. In other words: if for example I take R, S and T as starting-points, instead of P, Q and R, and construct a pentagram from {them} in the same way as before, then I end up with {exactly the same figure}.

Thus, although Napier's pentagram -- unlike a regular pentagon -- can take on a continuum of different visible shapes, including very irregular ones, the periodic character of the construction-process points to a higher form of symmetry and self-similarity. Instead of the five equal angles and sides of the visible regular pentagon, the pentagram embodies five equal {transformations}. The reader who has worked through the above constructions, should already have a sense of this (*3).

Needless to say, the existence of such a five-fold, self-similar periodicity embedded in the rotational manifold, points toward the existence of the duodecahedron whose sides are regular pentagons. Indeed, as we shall see next week, the pentagramma mirificum is the crucial singularity leading us beyond the domain of the spherical octahedron and its "children" -- the spherical cube and tetrahedron, as well as the corresponding straight-line polyhedra which they bound -- to the duodecahedron/icosahedron and the notion of a unique, universal characteristic of the "rotational world", subsuming, and reflected in, the whole simultaneous array of the five regular solids.

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(1) By "least rotation" I have in mind the following. Given two loci X, Y on the sphere, there are {many} rotations of the sphere which carry X into Y. The total angle through which the whole sphere must be rotated, in order to carry X to Y, will differ depending on which axis we choose. For example, if we choose the axis passing through the midpoint between X and Y, then a rotation of 180 degrees is required to carry X to Y. If, on the other hand, we take the rotation which carries X to Y along the arc of the great circle through those two points, and whose axis passes through the corresponding poles of the sphere, then in general a much smaller angle will be required. In fact, the latter choice of axis provides the least rotation carrying X to Y.

(2) LaPlace and Cauchy bear direct, personal responsibility for Galois' early death at the age of 21, as they do for the tragic, early death of another brilliant Gauss disciple, the Norwegian Niels Abel.

(3) One way to express that periodicity, albeit in a somewhat formal way, is as follows: Given any three loci A,B,C on the sphere, such that A and C are separated by an arc of 90 degrees (i.e. A,B,C satisfy the requirements to be consecutive vertices on a Napier pentagram), construct a locus D as the nearest pole of the rotation A->B, and constitute the {new} triple of loci "B,C,D". Now conceptualize a transformation T, which carries the triple "A,B,C" to "B,C,D" (so defined), as a kind of geometrical function. T has the effect, when applied to three consecutive vertices of a Napier pentagram, of "shifting ahead by one" in the order of vertices. Thus, T("P,R,Q") = "R,Q,S", T("Q,R,S") = "R,S,T" and so on. Evidently, the effect of applying T {five times}, is to come back to the original triple. Although T is not at all like a simple rotation, T's self-similar periodicity makes it the higher analog of rotation by 360/5 = 72 degrees, which is the characteristic transformation of a regular pentagon.

by Bruce Director

Before starting this pedagogical discussion, make sure you've worked through the report by Jonathan from two weeks ago (99056jbt001). In that discussion, you will have constructed the pentagramma mirificum, and begun to discover why Napier and Gauss referred to it as "mirificum", i.e. miraculous. This week, we'll take a further look.

First, from the construction itself a very surprising property emerges. Each side of the spherical pentagon, is the equator of the opposite spherical vertex, and, that vertex is the pole of that equator! Make sure you have grasped that property before proceeding.

In recent discussions, both written and oral, Lyn has emphasized the importance of knowing the difference in cardinality between a spherical surface and a flat one. These words will be only that, mere utterances, unless you work through a crucial paradox that brings this concept fully alive into your mind. For that, we have Gauss' fragmentary investigations of the pentagramma mirificum, into which we will take a preliminary look today.

To begin, think first of the property mentioned above. Each side of the spherical pentagon, is perpendicular (when extended) to the other sides of the pentagon, that are not adjacent to it.

Compare that to a pentagon drawn on a flat piece of paper. Extend the sides of that pentagon. The non-adjacent sides will intersect at points outside the pentagon, forming a pentagram. (Like on the sphere, the pentagon and pentagram are not regular ones). The angles at which these non-adjacent sides intersect cannot all be perpendicular, but in the pentagram we constructed on the sphere, they all were.

Try a little experiment. On a flat piece of paper, draw a line labeled a. Now draw another line perpendicular to a called b. Now draw a third line perpendicular to b called c. Now, draw a fourth line perpendicular to c called d. Can you draw a fifth line perpendicular to d that is also perpendicular to a? But, when we constructed the pentagramma mirificum on the sphere, this is precisely what we did. In fact, on the sphere, our construction "automatically" closed after five perpendicular transformations. On the plane, the construction closes after only four.

Okay, this doesn't surprise you. Of course, you say, the plane and sphere are of two different curvatures and so, as Lyn says, the geometry of each will be different in every small interval. So, it is to be expected that things that occur on the spherical surface, do not occur on the flat one. These are mere words, unless you can form in your mind a concept of the difference in cardinality between the two surfaces. I am NOT speaking of WHAT is different between the two surfaces, but the nature of the difference. (Think of the Socratic concept of change, embodied in Kepler's concept of congruence, later adopted by Gauss in his geometrical study, Disquistiones Arithmeticae.)

The nature of that difference, is precisely the direction of Gauss' fragmentary investigation into the pentagramma mirificum, and can be discovered by looking at fragment 2. (Before proceeding you should review part 6 of pedagogical discussion on spherical geometry by Jonathan Tennenbaum filed in the Alpha computer as 99036bmd001).

Go back to the spherical pentagon and take another look at the "self-polar" property. On the spherical pentagon, a "line" (great circle) connecting any vertex to its opposite side, will intersect that side in a right angle, since each vertex is a pole and the opposite side is its equator. If you were to connect each vertex to the opposite side, the five "lines" (great circles) might or might not all intersect each other in the same point. Taking the inversion, you could pick a point inside the spherical pentagon, and be able to draw perpendiculars from each vertex to its opposite side, so that they all intersect at the chosen point.

Now, perform a variation on the experiment discussed in part 6. Draw the pentagramma mirificum on a clear plastic hemisphere, and project that pentagramma on a flat surface by placing a light source at the center of the hemisphere. The flat surface should touch the hemisphere at one point. The spherical pentagramma will project, on the flat surface, a straight line pentagram. Keeping the hemisphere still, move the flat surface. First pivot it around the same point. Then move it from point to point. (To be most effective, make your flat surface out of stiff plexiglass covered with tracing paper. Trace the projected pentragam on the tracing paper. Use a different piece of paper each time you change the projection by moving or pivoting the plexiglass. That way you can draw a series of snapshots of the different projections, corresponding to the different angles and places the flat surface makes with the hemisphere. When tracing the projections, be sure to mark the point at which the flat surface touches the hemisphere.)

Now take the array of projected flat pentagrams drawn on the pieces of tracing paper, and draw lines from each vertex that intersect the opposite side at right angles. These lines will all intersect at one point, and that point will be the one at which the plexiglass touched the hemisphere! The self-polar property of the spherical pentagramma remains embedded, in the projected plane one. In fact, as Gauss notes in his fragmentary investigation, {every} plane pentagon is nothing more than the projection of a spherical one.

To help make this point sink in, take its inversion. Start with an arbitrarily drawn pentagon on a plane. Draw the perpendicular lines from each side to the opposite vertex. These lines will all intersect in one point. But, this pentagon was just drawn on a flat paper. No sphere was used in its construction, yet the spherical property of the pentagramma mirificum is still in there. Spherical action is present, even without the sphere.

(On this point, I refer the reader to the very important, but far too little read, Science Memo by Lyn on Cold Fusion, written in prison and released during the 1992 Presidential campaign.)

We leave you this week, by introducing for future contemplation, another piece of Gauss' fragmentary investigation. Go back to the spherical pentagramma drawn on the hemisphere with a flat surface touching the hemisphere at one point. The light source at the center of the hemisphere will project a cone, whose apex is the center of the hemisphere, and whose base will extend through the points of the spherical pentagon. The flat surface will cut that cone obliquely, defining an ellipse, that circumscribes the projected pentagon. What does this have to do with Kepler's determination of elliptical orbits, Gauss determination of the orbit of Ceres, and Gauss' later investigations into the perturbations of planetary orbits? In future weeks, we will delve into these questions.

An Exercise in The Division of the Sphere

by Bruce Director

To begin with, the hastily written end of last week's pedagogical might have caused some confusion for those who carried out the construction. And, as Kepler said, "A hasty dog produces blind pups." In the next to the last paragraph, the reader was asked to draw an arbitrary pentagon, such that the altitude lines all intersected in one point. The intersection at one point, of the altitude lines of the plane pentagon, will only occur on those pentagons which are central projections of a spherical one. An arbitrary plane pentagon, may not necessarily be such a projection. In those cases, the altitude lines will not necessarily meet at one point. However, those pentagons, can be transformed into projections of spherical ones. We will take that up at a later date.

That said, this week we will look at the difference in cardinality between a flat and spherical surface from another standpoint; the principle that Kepler and Gauss called congruence, or in the Greek harmonia. Unfortunately, due to the political mobilization of the past weeks, this week's discussion is also written hastily, so I beg your indulgence in advance for what may seem to be a rushed presentation. However, the issues are crucial, and I would not want to postpone your enjoyment of working through them, by delaying it's appearance in the briefing.

In the second book of the Harmonies of the World, Kepler re-introduces the Greek concept of harmonia, as equivalent to the Latin term congruentia, or in English, "to fit together." Kepler, demonstrates that on a surface of zero curvature, or a plane, certain polygons, i.e. squares, triangles, and hexagons, will fit together perfectly. He called this a perfect congruence. However, in three dimensions, (i.e. solid angles) the formation of perfect congruences is entirely different. Perfect congruences can be formed by three, four, or five triangles, three squares and three pentagons. In this way, the uniqueness of the five Platonic solids is demonstrated.

But, there is still an underlying assumption not completely revealed in the above demonstration. The difference in which perfect congruences can be formed, is a function of something not stated explicitly, an underlying curvature of space. On the other hand, if we look at this principle of congruence from the standpoint of the surface of the sphere, as the Greeks and Kepler undoubtedly did, we see that this difference in congruence between two and three dimensions, is a reflection of the difference in cardinality between a surface of zero curvature-- a plane, and a surface of constant curvature -- a sphere.

To create this concept in your mind, think of a crucial difference between a plane and sphere. On a plane, the sum of the angles of any triangle are equal to two right angles, or 180 degrees. On a plane, triangles can change their size and relative shape, but the sum of the angles are always 180 degrees. Additionally, there is no maximum triangle. A triangle can be as big as can be imagined.

Think of a triangle on a sphere. In the constructions in the earlier pedagogical discussions on this subject, we constructed triangles with three 90 degree angles, e.g. the triangle between the zenith, a point straight ahead on the horizon, and a point directly to the left or right on the horizon. The great circles which form the sides of this triangle intersect each other at 90 degree angles, and the area enclosed by them is 1/8 the entire area of the sphere. Now, in your mind, move one of the horizon points towards the other, keeping the zenith and second horizon point fixed. What happens to the angles of the triangle and the area enclosed? The great circles intersecting on the horizon will remain at 90 degrees each, but the angle between the great circles meeting at the zenith will decrease from 90 degrees to 0. Simultaneously, the area enclosed by the triangle will also decrease. When the two horizon points meet, the resulting triangle, will look the same as a great circle from the horizon to the zenith. This "triangle" will have two 90 degree angles at the horizon, and a 0 degree angle, for a total of 180 degrees. In other words, when the sum of the angles of a spherical triangle are 180 degrees, the triangle ceases to be!

Next, do the reverse. Rotate one of the horizon points away from the other, keeping the zenith and the second horizon point fixed. The great circles intersecting the horizon will remain at 90 degres, but the angle at the zenith will increase, and, so will the area enclosed by the triangle. When the two great circles intersecting the horizon come together, the resulting "triangle" will have a zenith angle of 360 degrees, and two base angles of 90 degrees, for a total of 540 degrees. The area enclosed by this "triangle" will be 1/2 the surface of the sphere. But this "triangle" will appear to be the same as the triangle of 180 degrees, but constructed in exactly the opposite manner.

From this we can begin to arrive at a concept of a maximum and minimum triangle on the sphere. To get this idea more firmly in the mind, think of an arbitrary triangle on the sphere. If we increase the lengths of the sides of this triangle, the area enclosed will increase, as well as the sum of the angles. But, as the triangle grows, the angles between the sides will get greater and greater, until the angles are all 180 degrees, for at total of 540 degrees. Like in our previous example, this maximum triangle, encloses an area equal to 1/2 the surface of the sphere. On the other hand, if we shrink this triangle, the angles will get smaller and smaller, and so will the area enclosed.

From this demonstration, you should now be able to grasp the concept, that, unlike on a surface of zero curvature, a triangle on the surface of a sphere, has a maximum and minimum area, and the sum of the angles has a maximum and minimum boundary. But, there is a crucial distinction between the nature of the minimum boundary and the maximum. The minimum sum of the angles is 180 degrees, but that is the sum of the angles of a plane triangle. Since the sphere is nowhere flat, even in the smallest interval, a 180 degree triangle does not exist on the sphere. On the other hand, the maximum boundary, is a great circle, which, when considered as the maximum triangle, contains three 180 degree angles and encloses an area equal to 1/2 the sphere. Consequently, the sum of the angles of a spherical triangle, is always greater than 180 degrees, but never greater than 540 degrees. And, the area of a spherical triangle is always greater than zero, but never greater than 1/2 the area of the sphere. Since the area of the triangle is proportional to the sum of the angles, and since a triangle whose angles equal 180 degrees has zero area, the area of a triangle is proportional to the amount by which the sum of its angles are greater than 180 degrees. This quantity is called "spherical excess."

With this principle established in our minds, lets look at the formation of perfect congruences on the surface of the sphere. As Kepler did, we want to find what spherical polygons will make such perfect congruences. However, since the angles of a spherical polygon change with size, we must consider both shape and size when forming perfect congruences.

We begin with discovering which perfect congruences can be formed with triangles. Because of the crucial difference in the nature between the minimum and maximum triangle, we must start with the maximum triangle, i.e. a triangle whose angles sum up to 540 degrees and that encloses 1/2 the area of the sphere. We can then shrink this triangle, until we find one whose size is such that it can make a perfect congruence with at least three other triangles. Because, unlike the plane, the sphere is bounded, this process has two boundary criteria. First, since the triangles must form a perfect congruence, that is "fit together," the angles of the triangles must add up to 360 degrees when they come together at a common vertex. And, since the sphere is bounded, these congruences must divide the total area of the sphere evenly.

Since the area of the sphere is 4 x Pi x the cube of the radius, the area of the maximum triangle, on a sphere whose radius is 1, will equal 2 Pi. This same area can be thought of as an angular change from the center of the sphere, or 360 degrees. The area of the entire sphere will thus be 4 Pi, or as measured from the center of the sphere, 720 degrees.

To make a perfect congruence with three spherical triangles, we shrink the maximum triangle until it has three angles of 120 degrees, or 1/3 of 360. The total sum of the angles of such triangles will be 3 x 120 or 360 degrees making a spherical excess of 180 degrees. Since the total area of the sphere is 720 degrees, 4 such triangles will fit exactly onto the sphere, forming a spherical tetrahedron.

If four spherical triangles are to be fitted together, we must continue to shrink the triangles until the internal angle are 1/4 360 degrees or 90 degrees. The angles of these triangles will have a sum of 270 degrees, or a spherical excess of 90 degrees, or 1/8th the entire surface of the sphere, forming a spherical octahedron.

For five spherical triangles to be fitted together, the internal angle must be 1/5 of 360 degrees or 72. The total sum for these triangles will be 216, making a spherical excess of 36 degrees, or 1/20 the total area of the sphere. This forms the spherical icosahedron.

If we make our triangle still smaller, so that six triangles fit together, the internal angles will be 60 degrees, for a sum of 180 degrees. But we already discovered that such a triangle can't exist on a sphere, and so we've reached the boundary of dividing the sphere into equal regions with triangles.

Now, try dividing the sphere with spherical squares. Like with the triangle, a great circle is the maximum square, comprised of 4 180 degree angles, for a total of 720 degrees. The sum of the angles of a square on a surface of zero curvature, is 360 degrees. So the maximum spherical excess of a spherical square is 720 degrees - 360 degrees = 360 degrees. If we make the square smaller so that 3 can be fitted together, the internal angles must be 120 degrees, with the angles of each square having a total sum of 480 degrees. This makes a spherical excess of 480 degrees - 360 degrees = 120 degrees, or 1/6 the total area of the sphere, forming the spherical hexahedron, or cube.

If we make the square smaller, so that 4 fit together, then the internal angles must be 90 degrees, for a total sum of 360 degrees. But this is equal to the maximum spherical excess, and so such a square cannot exist on the sphere.

We can similarly show that the spherical pentagon will divide the sphere into the spherical dodecahedron, and that is the limit of equal divisions of the sphere. We leave this demonstration to the reader.

From this standpoint the nature of the difference in cardinality between the sphere and the plane can be seen anew. When we begin with the maximum polygon, a great circle, we form the simplest perfect congruence, division in half. Then as we descend from the maximum polygon, there are certain sizes which form perfect congruences, or harmonies. The polygons in between the maximum and the harmonic ones, form imperfect congruences or even dissonances. The spherical divisions, corresponding to the five Platonic solids, are the only perfect congruences, or perfect harmonies of the sphere surface. Work through this construction yourself, so we can discuss it further in future weeks.

by Bruce Director

Next week we will bring to a conclusion, this series of pedagogicals on spherical action, that began in the Dec. 18, 1998 briefing, and continued through last week's discussion on the the spherical development of the five Platonic solids. You are strongly urged to review this series as a whole. In the meantime, this week we offer you some words of wisdom from our predecessors, Cusa, Kepler and Gauss.

Nicholas of Cusa

Leonard Ignorance Book 1, Chapter 23:

"... Hence, Parmenides, reflecting most subtly, said that God is He for whom to be anything which is is to be everything which is. Therefore, just as a sphere is the ultimate perfection of figures and is that than which there is no more perfect figure, so the Maximum is the most perfect perfection of all things. It is perfection to such an extent that in it everything imperfect is more perfect -- just as an infinite line is an infinite sphere, and in this sphere curvature is straightness, composition is simplicity, difference is identity, otherness is oneness, and so on. For how could there be any imperfection in that in which imperfection is infinite perfection, possibility is infinite actuality, and so on?

"Since the Maximum is like a maximum sphere, we now see clearly that it is the one most simple and most congruent measure of the whole universe and of all existing things in the universe, for in it the whole is not greater than the part, just as an infinite sphere is not greater than an infinite line. Therefore, God is the one most simple Essence (ratio) of the whole world, or universe. And just as after an infinite number of circular motions an infinite sphere arises, so God (like a maximum sphere) is the most simple m easure of all circular motions....

"... Therefore, all beings tend toward Him. And because they are finite and cannot participate equally in this End relatively to one another, some participate in it through the medium of others. Analogously, a line, through the medium of a triangle and of a circle, is transformed into a sphere; and a triangle is transformed into a sphere through the medium of a circle; and through itself a circle is transformed into a sphere."

Johannes Kepler

Epitome of Copernican Astronomy; Book 4

[written in Q and A format in original]

"What is the cause of the planetary intervals upon which the times of the periods follow?

"The archetypal cause of the intervals is the same as that of the number of the primary planets, being six.

"I implore you, you do not hope to be able to give the reasons for the number of the planets, do you?

"This worry has been resolved, with the help of God, not badly. Geometrical reasons are co-eternal with God -- and in them there is first the difference between the curved and the straight line. Above (in Book 1) it was said that the curved somehow bears a likeness to God; the straight line represents creatures. And first in the adornment of the world, the farthest region of the fixed stars has been made spherical, in that geometrical likeness of God, because as a corporeal God -- worshipped by the gentiles under the name of Jupiter -- it had to contain all the remaining things in itself. Accordingly, rectilinear magnitudes pertained to the inmost contents of the farthest sphere; and the first and the most beautiful magnitudes to the primary contents. But among rectilinear magnitudes the first, the most perfect, the most beautiful, and most simple are those which are called the five regular solids. More than 2,000 years ago, Pythagoreans said that these five were the figures of the world, as they believed that the four elements and the heavens -- the fifth essence -- were conformed to the archetype for these five figures.

"But the true reason for these figures including one another mutually is in order that these five figures may conform to the intervals of the spheres. Therefore, if there are five spherical intervals, it is necessary that there be six spheres; just as with four linear intervals, there must necessarily be five digits.

"Why do yo call them the most simple figures?

"Because each of them is bounded by planes of one species alone, viz., triangles of quadrilaterals or pentagons, and by solid angles of one species alone -- the three primary figures by the trilinear angle, the octahedron by the quadrilinear angle, and the icosahedron by the quinquelinear angle. The other figures vary either with respect to the angle or with respect to the plane....

"Why do you call these figures the most beautiful and the most perfect?

"Because they imitate the sphere -- which is an image of God -- as much as rectilinear figure possibly can, arranging all their angles in the same sphere. And they can all be inscribed in a sphere. And as the sphere is everywhere similar to itself, so in this case the planes of any one figure are all similar to one another, and can be inscribed in one and the same circle; and the angles are equal."

Carl F. Gauss

Letter to Gerling; April 11, 1816

"It is easy to prove, that if Euclid's geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid's Geometry were not true, because then we would have, a priori, a universal measurement, for example, one could use for a unit length, the side of a triangle, whose angle is 59 degrees, 59 minutes, 59.99999 seconds."

by Jonathan Tennenbaum

(In partial celebration of the second anniversary of the pedagogical discussions.)

Whoever has worked through the previous installments of this series in a thoughtful manner, should now have a fairly solid grasp of 1) how the rotational manifold and spherical curvature arise in the most elementary astronomical measurement of the Universe; 2) the characteristic sorts of anomalies, that result from any attempt to map a spherically curved surface onto a flat surface; 3) the origin of the regular solids in this context, as a single interconnected unity with the dodecahedron/pentagrammum as the centerpiece; and out of this, 4) why the five regular solids constitute the necessary and sufficient (least action) expression of the singularity, that separates spherical curvature, as a {type}, from flat, linear geometries typified by classroom plane geometry.

While much more could be said on these geometrical matters, and the pedagogy should be further refined, the preceeding installments provide at least a first approximation to what is needed. But one big issue has still been left hanging. Many readers, I am sure, have a nagging thought in the back of their minds, concerning the meaning of the whole exercise. To put it crudely, but otherwise accurately, I read the thought as follows:

"Your spherical geometry is lots of fun, and now I understand the regular solids much better. But I just gotta ask you: Do you really want me to believe, that the {solids} determine the planetary orbits? They are just abstract ideas, aren't they? How could they have any {physical} effects? I mean, don't get me wrong, I know Newton was a bad guy and all that, but ... what should I say?... I really can FEEL that gravitational force. It's really there. You know what I mean?"

Here we have a clear case, where no decisive progress can be made, until certain entrenched, false ideas and habits of thinking are fully demolished and the rubble cleared out of the way. People should study Lyn's most recent memo (reproduced in Friday's briefing), which deals with exactly this topic. In honor of the second anniversary of the pedagogical discussions, I would like to add a few additional observations.

Firstly, observe that the form of the indicated, nagging doubt corresponds {exactly} to the what many people react to, in Lyn's "triple curve" characterization of the curvature that is governing the collapse of the present global financial-economic system. They cannot accept the idea, that the reason for the collapse -- and the emergence of the Russia-India-China-Iran "survivor's club" -- lies entirely {outside} the domain of Newtonian-like mechanical causality. They see events as being caused by the interaction of a huge number of "forces": market forces, political forces, sociological forces, "lone assassins" etc. They reject the idea, that the entire manifold of current history might be shaped {as a whole} in such a way, that the possible courses of events at this juncture are restricted to a very few alternative pathways, and no others. Ignoring this higher bounding of history, they entertain all kinds of scenarios and "solutions," which do not exist in reality.

Just so, the Newtonian imagines arbitrary planetary orbits at arbitrary distances from the Sun, while the real solar system permits only a discrete array of harmonically-determined orbital bands. (Offending objects, it appears, are ejected from the system, or end up in the "garbage can" of the asteroid belt).

What is the problem? Project a curved surface on a flat surface, and observe the distortions produced. If you stubbornly insist on regarding the linearity of the flat surface as an inherent feature of reality, then you will be obliged to invent a complicated system of "forces" to explain the distortions in the image.

This is exactly what Sarpi, Galileo, Newton, Descartes etc. did.

Look at the Universe. Look at the impossibility of constructing a "flat" projection of the heavens, and look at the spherical geometry (often refered to as the celestial sphere) we demonstrated to underly all astronomical measurement. Look at the hierachy of {periodicities}, {cycles} which the ancients found to govern all apparent motions of the stars and planets. Look at the spherical (or spheroid) curvature of the Earth, measured by Erathosthenes, and the spheroidal curvatures of all other celestial bodies. Look at the harmonic system of the planetary orbits, whose unique coherence with the regular-solid spherical harmonics was demonstrated by Kepler. Look down toward the microscopic scales. Look at Kepler's founding of crystallography (in the snowflake paper), and Mendeleyev's ensuing discovery of the periodic system of the elements. Look at the Huygens-Fresnel demonstration of the spheriodal geometry underlying the process of light propagation, and its organization in cycles of wavelength and frequency -- work that demolished Newton's linear fallacy of "light corpuscles" travelling in straight lines. Look at the Ampre's preliminary discovery of the non-linear (angular) nature of electromagnetic action. Look at Wilhelm Weber's derivation, from his own experimental confirmation of Ampre's principle, of the necessary existence of an essential singularity of electromagnetic action at a "critical length" corresponding to subatomic scales. Look at the implicit (if somewhat flawed) extension of the Huygens-Fresnel-Gaus-Weber work to atomic physics, by Planck, De Broglie, Schrdinger and others. Finally, look at Dr. Robert Moon's preliminary demonstration of the Keplerian ordering of the subatomic domain. Compare this with the harmonic characteristics of living processes, and with the harmonic characteristics of human Reason, as reflected for example in the well-tempered system of bel canto polyphony. And so forth.

Review, thus, the panorama of the Universe, as the actual process of discovery has thus far revealed the Universe to be. Do you find, anywhere in this, any trace of a supposed primacy, or even mere existence of simple, straight-line action in the Universe? No, not the slightest trace! Rather, we discover everywhere reflections of a universal curvature, coherent (to a first approximation) with the characteristics of spherical bounding as understood by Nicolaus of Cusa and Kepler.

But now arbitrarily stipulate, that all events in the Universe are taking place in an "empty", featureless, euclidean three-dimensional space, extended indefinitely in all directions. Stipulate straight-line motion at constant velocity as the "natural" form of action inhering in that notion of space-time. Build that into your physics as a basic assumption. You have now transformed the entirety of the actual physical evidence into a gigantic anomaly!

Any motion, for example, which departs from constant, straight-line motion -- i.e. all real motions! -- is anomalous. So, postulate the existence of "forces" that are "bending" the motions into the observed curved trajectories. Elaborate that curve-fitting into a sophisticated mathematical structure. Congratulations! You have just received a Nobel Prize for virtual reality! The main anomaly left to be explained, is how Galileo, Sarpi, Newton etc. were able to get away with it.

"But can't you understand, I really FEEL those gravitational forces." We can hear Descartes swearing, pathetically: "I feel it, therefore it exists"! But sense perceptions are mere phenomena, they have no meaning in and of themselves. Some action, some change has occurred. So what?

Consider the following experiment: we suspend a magnet by its midpoint on a thread. A meter or so away, we set up a coil. When we pass an electric current through the oil, the magnet on the other side of our table rotates. What is the significance of that correlation of events? Does it mean that some physical entity (Leibniz called Newton's forces "occult qualities") emanates from the coil, reaches out through space across the table to the magnet, and turns it? Or were it not more reasonable, in place of such extravagant and arbitrary speculations, to report, that the magnet {responded} to a {change in the Universe}, which we generated with our actions, and that the Universe is manifestly bounded in such a way, that the correlation of events in the Universe takes a certain form, and not another. The phenomena remain the same, including the weight-lifter's conviction, that he is working against "gravity".

But the nagging starts in again and somebody asks, "Well, if you don't believe in forces, then please {explain} to me, {why} the planets go around like that, why the Earth is spherical, and so forth."

"You want an explanation? Forget it. That's the way it is, buddy. Our Universe is (approximately) spherically-bounded, and you're going to have to live with it!"

Sometimes, in science as in organizing, blunt answers are appropriate. Sometimes you make a bad mistake by trying to "explain" things. (Explain in terms of what?) Why? Because a certain mode of demanding explanations is really just a ruse for refusal to except reality. Because, what the person is really saying is, "I will refuse to except that X is happening, if the existence of X contradicts my deepest beliefs." What people commonly mean by "explanation," is to demonstrate the {deductive consistency} of an event, with their own underlying assumptions and beliefs. But, what if their beliefs are wrong? If the entire coherence of the evidence contradicts a firmly-held belief or habit of thought, then as scientists and truth-seekers, we must part with those beliefs and habits.

Riemann put forward exactly this, {opposite} sense of "explanation" in a posthumous fragment on scientific method:

"If an event occurs, which is necessary or probable according to the given system of concepts, then that system is thereby confirmed; and it is on the basis of this confirmation through experience, that we base our confidence in those concepts."But if something unexpected occurs, being impossible or improbable according to the given system of concepts, then the task arises, to enlarge the system, or, where necessary, to transform it, in such a way that the observed event ceases to be impossible or improbable according to the enlarged or improved system of concepts. The extension or improvement of the conceptual system constitutes the {`explanation'} of the unexpected event. Through this process, our understanding of Nature gradually becomes more comprehensive and more true, while at the same time reaching ever deeper beneath the surface of the phenomena."

Thus, "explanation" in Riemann's sense means a successful {change} in fundamental concepts and assumptions, which have been overthrown by the generation of an event in the Universe, which is incompatible with the previously prevailing beliefs and assumptions. The question, which Riemann does not fully answer, but LaRouche does, is the nature of the {bounding principle} of that process of change.

These remarks bears crucially on the deeper side of the fallacy of Newtonianism. The epistemological equivalent of straight-line action and Cartesian-Newtonian space-time, is deductive reasoning. What we encounter is a strong resistence to the notion of an efficient {bounding} of events, which does not have the form of logical-deductive implication. The existence and form of such bounding principles is an {experimental} question; they cannot be derived from mathematics. Their existence is demonstrated historically, however, by the manner in which the Universe reacts to creative human Reason, by such changes as lead to harmonically-ordered increases in the relative potential population density of the human species. Thus, Nicolaus of Cusa and Kepler understood the ontological significance of spherical (and higher) curvature, as a lawful expression of the principle of perfection of human Reason.