by Bruce Director

As the pedagogical series on spherical geometry has indicated, a profound discovery arises, when you attempt to map spherical action on to a flat plane. Any such effort, immediately presents to the mind, the existence of two distinct types of action. Basic investigations of the physical universe, astronomy and geodesy, immediately confront us with the need to discover the conceptions that underlay this discontinuity.

Already we have presented several examples of this, which you can work through quickly in your mind before proceeding. Think of the various examples that demonstrated that spherical nature of the manifold of measurement of space. Think of the conception of the Platonic Solids from the standpoint of Kepler's re-discovery of the Pythagorean concept of congruence (harmonia). Think how we demonstrated that solids arise as the characteristic perfect congruences on a surface of constant positive curvature, as distinct from the perfect congruences that arise on a surface of zero curvature. And also, think of the pentagramma mirificum, and emergences of two distinct periodicities that arise from carrying out the same action, on surfaces of two different curvatures. (All the above examples were elaborated in pedagogical discussions published over the first three months of 1999.)

Now let's delve into this area once again. First, from the standpoint of mapping the stars, as represented on a surface of constant positive curvature, onto a surface of zero curvature, a most ancient investigation.

In our observation of the heavens, the stars are projected onto a spherical surface, as a function of our measuring their changing positions, as a change in the angle between the line of sight, the horizon and some arbitrary direction perpendicular to the horizon, such as north, or even "straight ahead." In this way, the changes in position of the stars, and their relationship to each other, are represented as arcs of circles and the angles between such arcs.

However, as we've seen before, when we try to project this spherical projection of the stars, onto a flat surface, discontinuities aries. Furthermore, the nature of these discontinuities changes depending on how we effect that projection. In other words, not all projections from a sphere onto a plane are the same.

You can carry out a simple demonstration of this, by drawing a series of great circle arcs, intersecting at different angles, on a clear plastic hemisphere. (For purposes of this description, call the circular edge of the hemisphere the equator, and the pole of this equator the north pole.) Hold a flashlight or candle at the position equivalent to the south pole of the sphere so that the great circle arcs cast shadows onto a marker board. Trace the shadows. Now, move the flashlight toward the center of the sphere, stopping at various intervals, and tracing the shadows of the arcs at each interval. Make one of those intervals the center of the sphere. Trace the shadows.

You will notice a change in the curvature of the shadows, as the point of projection changes from the south pole to the center of the sphere. At the south pole of the sphere, the shadows are arcs of circles. As the flashlight moves toward the center, the shadows straighten out, until at the center, the shadows are straight lines.

Now make a more precise demonstration. Draw on the hemisphere, an equilateral spherical triangle, such as the face of the octahedron, that has three 90 degree angles. Perform the above projections. When the flashlight is at the south pole, trace the shadows. Now move the flashlight to the center of the sphere, and trace the shadows.

The tracings of the shadows from the south pole projection are circular arcs. Measure the angle between the lines tangent to each arc at the each vertex. Now measure the angles between the sides of the straight line shadows projected from the center.

These are two specific projections, the first called the stereographic, the second called central projection, that transforms the great circle arcs on the sphere, to the plane. As you can see, each transformation is different. In the central projection, the spherical equilateral triangle with three 90 degree angles is transformed into a flat equilateral triangle with three 60 degree angles. In the stereographic projection, the spherical triangle is transformed into three circular arcs that intersect each other in 90 degrees. So the angular relationship between the vertices of the triangle is invariant under the stereographic projection.

With a little bit of thought, you should be able to figure out why that is the case. Think of the point of projection as the apex of a cone of light. The projection on the flat surface is formed by the intersection of a line that starts at the point of projection, and continues through a point on the sphere, and then intersects the marker board. If the point of projection is at the center of the sphere, than the lines connecting the point of projection to points on a great circle, will all be in the same plane. Consequently, the projection of these great circle arcs will be a straight line. In this way, the center of the sphere can be thought of as the unique singularity from which great circles can be projected into straight lines!

Not so if the point of projection is other than the center of the sphere. However, if the point of projection is the south pole, the angles between the projected arcs, are the same as the angles between the spherical arcs. This property has come to be called, "conformal".

Because of this angle preserving characteristic, this projection is particularly useful for mapping stars. The written discovery of the stereographic projection is attributed to the Greek astronomer Hipparchus, but its actual origins are most likely quite older. Under this projection, the entirety of the celestial sphere can be mapped onto a flat surface.

To do this, think of a sphere with a plane representing the horizon, going through the center of the sphere. (You can represent a cross section of this on a flat piece of paper as a circle with two perpendicular diameters. Call the endpoints of one of the diameters the north and south pole. Let the other diameter represent the horizon.) Now, draw a line that connects every point of the "northern" hemisphere with the south pole. Those lines will intersect the horizon and those intersections will form a stereographic projection. The north pole will project onto the center of the sphere. All the points of the northern hemisphere will project onto the inside of a circle formed by the intersection of the sphere with the plane, and all the points of the southern hemisphere will project to points outside that circle. Where will the south pole projet to? What other discontinuities or distortions emerge under this transformation?

YOU HAVE TO CARRY OUT THIS CONSTRUCTION IF THE ABOVE DESCRIPTION IS TO MAKE ANY SENSE TO YOU.

Over the last two millennia, the stereographic projection has been used to map the celestial sphere onto a plane and is the basis of the construction of the astrolabe, one of the earliest astronomical measuring instruments. (Rick Sanders has produced an interesting unpublished paper on the astrolabe available to those who are interested from RSS.)

The stereographic projection, therefore, represents a unique way of projecting one surface onto another, such that a certain characteristic, is invariant under the transformation. But, this projection is specific to the mapping of a sphere onto a plane. Can we find, for example in the case of a geodetic survey, where we are mapping the geoid, onto an ellipsoid, onto a sphere, onto a flat plane, a way to perform such a series of transformation, in which a certain characteristic, remains invariant under repeated arbitrary projections?

This formed the subject of Gauss' famous 1822 paper for which he won the Copenhagen prize. The paper was titled, "General Solutions of he Problem to so Represent the Parts of One Given Surface upon another Given Surface that the Representation shall be Similar, in its Smallest Parts, to the Surface Represented." In this investigation, Gauss delved even further into the nature of non-linear curvature in the infinitesimally small.

Last week we undertook a preliminary investigation into the projection of a sphere onto a plane. Now the fun starts.

If you carried out the constructions, you would have re- discovered, in a formal sense, certain principles whose ancient discovery was crucial for the development of human civilization. That discovery can be thought of in two aspects; 1) that elementary form of action in the physical universe is curved, and 2) that curved action is of a different "transcendental cardinality" than linear action. The nature of that difference is revealed in the investigation, not simply of each type of action, but by investigating transformations between each type, i.e., the "in betweenness." In that sense, the study of these projections has a significance for both the development of the higher cognitive powers of the mind, and the capacity of those powers to bring the physical universe increasingly under its dominion.

In general, there is no transformation of a sphere onto a plane that does not result in distortions and discontinuities, and it is by those distortions and discontinuities that the difference in "transcendental cardinalities" becomes apparent. But, there are a myriad of such transformations, each of which produces different characteristic distortions and discontinuities. (Last week, we investigated, preliminarily, two such transformations, the gnomic and the stereographic projection, but there are many others.) In order to more fully grasp the nature of the difference in "transcendental cardinalities" between the sphere and the plane, we cannot focus simply on specific types of transformations. We must investigate the general nature of transformations and not just between two specific types of surfaces, such as a sphere and a plane, but between any series of arbitrarily curved surfaces. That is, we must jump from investigating a particular projection, to the investigation of the general principle of projection itself. That puts us in the domain the hypergeometric. This is the domain unique to the contributions of Gauss and the subsequent discoveries of Riemann.

Today's pedagogical discussion seeks to start down the road to the re-discovery of Gauss' and Riemann's contributions. There is nothing contained below that is beyond the scope of most of the readers, but, be prepared to concentrate on the train of thought. You will find in it an illustration, typical of Gauss, of taking a previously discovered principle of classical Greek science, and approaching it from a new higher standpoint, which establishes that classical principle, as a special case of a more general concept. It is congruent with Beethoven's re-thinking of the significance of the Lydian interval, in his late quartets, to establish a new conceptualization of the domain of J.S. Bach's well-tempered system of bel canto polyphony.

From last week's discussion, you should have already demonstrated to yourself, some of the characteristics of the gnomonic and stereographic projection of the sphere onto the plane. Specifically, the gnomonic, (projection from the center of the sphere), transforms great circle arcs on the sphere, into straight lines on the plane. Obviously, since the sum of the angles of all plane triangles is 180 degrees, and the sum of the angles of triangles on the sphere are always greater than 180 degrees, angular relationships are changed under the gnomonic projection. On the other hand, last week's constructions provided the basis to demonstrate, at least initially, that under the stereographic projection, i.e., where the point of projection is a pole of the sphere instead of the center, the angular relationships are unchanged when projected from the sphere onto the plane. This characteristic is obviously crucial for geodesy and astronomy, as the relationships between stars projected onto the celestial sphere and positions on the surface of the Earth, as these relationships are measured as only as angular relationships. If a representation of these spherical relationships onto a flat surface is to be of any use, the angular relationships must be invariant under the projection.

When thinking of possible projections from the sphere onto a plane, the gnomonic projection seems to suggest itself most easily. For example, in the case of the celestial sphere, the point of projection is the observer, who projects the celestial sphere the stars along the lines of sight from the observer through the stars, to a plane. This projection was apparently discovered by Thales, but it is quite possible that it was known much earlier. However, because it distorts angles, it has obvious failings for a useful map of the stars or the Earth.

The stereographic projection is much less obvious. Here, the point of projection, a pole, is no where in the manifold of the observer. But, when the projection plane is the plane of the observer, (as in last week's example), the point of the observer is the only point that is unchanged under projection! This and the property that angular relationships are not changed under the projection, make the stereographic projection suitable for astronomical uses, such as a star chart, or astrolabe.

The experiment in last week's discussion, for pedagogical purposes, indicated by demonstration, but did not prove, that angular relationships are invariant under the stereographic projection, a characteristic called "conformal." One can, as Hipparchus did, prove by principles of Euclidean geometry, that this is the case.

(Such a proof is not very complicated. It relies on properties of similar triangles. But, to describe it in this cumbersome format would be, for the moment, distracting. So, we leave it to the reader to carry out.)

Gauss' standpoint was to go beyond the principles of Euclidean geometry, by inverting the question. Instead of starting with stereographic projection and asking, "Is it conformal?" Gauss asked. "What is the nature of the being conformal, and under what projections does it exist?" The former sets out to discover the existence of a general principle in a specific case. The latter question seeks the nature of the general principle, under which the special cases are ordered.

Gauss' approach is best grasped pedagogically by a demonstration. Take the clear plastic hemisphere you used last week, preferably with the 270 degree equilateral spherical triangle still draw on it. Cut out four circles out of cardboard, of different sizes. For my experiment, I made a circles with diameters, 3 1/2, 1 1/2, 1, 1/2. (For the circle of 1/2 inch diameter I used a thumb tack.) With tape, attach these circles to the sphere, all at the same "latitude", so that they are approximately tangent to the sphere at their centers.

Now, project this arrangement onto a plane. This is most easily done, by holding the hemisphere so that the plane of the equator is parallel to a wall or the ceiling, and use a flashlight to project the spherical images onto the wall or ceiling.

When you hold the flashlight so that the bulb is at the center of the hemisphere, the shadows of the spherical triangle will, as we saw last week, be straight lines. The shadows of the tangent disks, will be ellipses. When you pull the flashlight back to the position of where the south pole of the sphere would be, you will see that the shadows of the spherical triangle will be circular arcs, intersecting at 90 degree angles, and the shadows of the tangent disks will be almost circular.

The change in the projection of the tangent disks, from ellipses in the gnomonic projection, to circles in the stereographic, is a reflection of a crucial element of Gauss' discovery.

Gauss' first step, was to abandon the idea of the sphere and plane being objects embedded in three dimensional Euclidean space, and instead, he thought of each as a two dimensional surface of different curvatures. On any two dimensional surface, the angular relationship of 90 degrees is a singularity, consistent with Cusa's notion of maximum and minimum. That is, geodetic arcs, or lines that intersect at 90 degrees are at the maximum point of divergence. Or, in other words, any two such arcs, or lines, define two divergent directions. Any other angle, at which geodetic arcs lines intersect, is merely a combination of these two directions. (Gauss goes to great lengths to point out that these two directions are arbitrary, but once one is chosen, the other is determined.)

Now look back to the difference in the transformation of the tangent disks in the two projections. In the gnomonic projection, the change of those disks from circles to ellipses, is a reflection that the gnomonic projection changes one direction in a different way than the other. The transformation of those disks into circles in the stereographic projection, is a reflection of how this projection changes both directions exactly the same.

But, there is another principle at work here that you can discover with some careful observation. If you look closely at the tangent disks, you should notice that in the gnomonic projection, the shadows of the disks become more elliptical, the smaller the disk. And, in the stereographic projection, the shadows of the disks become more circular the smaller they are.

Remember these disks are not on the sphere, but tangent to it. Therefore, the smaller the disk, the closer to the surface of the sphere it is. As the disks become infinitesimally small, the characteristic change in curvature, becomes even more pronounced. In other words, the characteristic curvature of these projections, or any other for that matter, is reflected in every infinitesimally small area of both surfaces. And, the smaller the area, the more true is the reflection! Just the opposite of linearity in the small.

Do this experiment and play with this idea a while. You are getting close to a very fundamental principle discovered by Gauss and Riemann, which we'll take up in the final installment of this series next week.

I hope you had fun conducting the experiment described at the end of the last pedagogical discussion. This week, we will conclude this preliminary phase of pedagogical discussions on the early development of the Gauss-Riemann theory of manifolds, with a discussion of the general principles of Gauss' theory of conformal mapping. In future weeks, we can extend these investigations, using this preliminary work as a starting point.

It is important to remember the context in which these investigations of Gauss and Riemann occurred. The thread begins with Cusa's {Learned Ignorance}, and his insistence that action in the physical universe was elementarily non-uniform. The discoveries of Kepler on planetary orbits, and Leibniz and Huygens on dynamics, and light, confirmed and validated what Cusa had anticipated. In each case, the general nature of the non- uniformity of physical action, was discovered by the manifestation of that characteristic in an infinitesimally small interval of action.

Gauss' geodesy is a good case in point. Between 1821 and 1827 Gauss supervised and conducted a geodetic triangulation of most of the Kingdom of Hannover. That undertaking confronted him with a myriad of scientific problems, that sparked a series of fundamental discoveries about the nature of man and the physical universe.

A short review is necessary, from the standpoint of the last several month's pedagogical discussions on spherical action. Think back to the question of the measurement of the positions of the stars with respect to a position on the Earth. Those positions will change over the course of the night, the course of the year, and the course of the longer equinoctial cycle. The geometrical form of the manifold of such changes, is the inside of sphere. The daily, yearly and equinoctial changes of the stars' position trace curves on the inside of the sphere. Those curves can be thought of as functions of the Earth's motion.

Now, think of those same observations as taken from another position on the Earth's surface. A new set of curves will be generated that are a function the same motion of the Earth. But, the nature and position of those curves will be different than the curves traced by the observations from the first position.

These two sets of curves, give rise to a new function, that transforms the first set of curves into the second. That function reflects the effect of the curvature of the surface of the Earth. This function can not be visualized in the same way, as a set of curves, as in the case of the first two functions. This new type of function, a function of functions, is congruent with what Gauss and Riemann would refer to as a complex function.

In this example, a complex function is discovered that maps spherical functions into other spherical functions, which is another way of thinking about the concept of projection. The previous two discussions in this series, looked into types of complex functions that project spherical functions onto a surface of zero curvature (a plane), such as the gnomonic projection and the stereographic projection. These two complex functions transform the same curves from the sphere onto the plane, but in different ways.

The stereographic projection had the unique characteristic that the angles between great circle arcs on the sphere are not changed when projected onto the plane. This characteristic Gauss called conformal.

In his announcement to the first treatise on Hider Geodesy, Gauss points out that the curves conform in the infinitesimally small. However, in the large, the projection of the great circle arcs are magnified, the degree of magnification changes, depending on their position with respect to the point from which the projection is made. The experiment projecting circles tangent to the sphere, suggested in the last pedagogical, illustrated this point, at least intuitively.

In other words, if you think of the stereographic projection from Gauss' standpoint, it is a special case of a complex function. A complex function that transforms curves on a sphere to curves on the plane, according to a law, that conforms in the infinitesimally small.

In the course of his geodesic investigations, Gauss was confronted with the requirement of discovering other complex functions that transformed functions on one surface to another. Rather than tackle each case separately, Gauss went into the matter more deeply, discovering the general principles on which these complex functions rested. This was the subject of his 1822 paper referred to in previous weeks, "General Solution of the Problem to so Represent the Parts of One Given Surface upon another Given Surface that the Representation shall be Similar, in its Smallest Parts, to the Surface Represented". These investigations formed the foundation for Riemann's theory of complex functions.

In his paper Gauss gives an example of such a problem from Higher Geodesy. In his geodetic survey, Gauss measured the area of a portion of the Earth's surface, by laying out a series of triangles whose vertices were mutually visible. By measuring the angles between the lines of sight between these vertices, the area of the triangle could be computed. As this network of triangles was extended over the Kingdom of Hannover, the entire area of the entire region could be computed by adding up the areas of the smaller triangles in the network.

As discussed in previous weeks, the area of these triangles is a function of the shape of the surface on which they lie. If a spherical shape of the Earth is assumed, then the size of the triangle is a function of the sum of the angles comprising it multiplied by the diameter of the Earth.

Look back on our first example above. Between two positions on the surface of the Earth, a complex function characterizes the difference between the observed positions of the stars at those two positions. (For purposes of this example, consider the two positions as lying on the same meridian. Then the measurement of that complex function can be expressed as simply the difference in the angle of observation of the pole star between the two positions.) Based on an assumption about the size and shape of the Earth, the distance between the two positions along the surface of the Earth can be calculated.

The distance between those two positions can also be calculated by a geodetic triangulation carried out over the area of the Earth's surface between the two positions. That distance, when compared with the enables us to test the original assumption of a spherical shape for the Earth. That type of measurement determined the shape of the Earth to be closer to an rather than a perfect sphere.

This confronted geodesist with the requirement of projecting those ellipsoidal triangles onto a sphere, conformally. Gauss was the first one to be able to solve this, by applying his general method of conformal projection. The method employed is analogous to Kepler's measurement of planetary motion in an elliptical orbit, by the eccentric and mean anomalies, but with the use of complex functions, of the type described above.

In future weeks we will develop pedagogical exercises from Gauss' examples, and then go on to a more thorough examination of Riemann's revolutionary extension of Gauss' discovery.