Measurement and Divisibility

By Bruce Director

In 1818, Karl F. Gauss accepted the assignment to conduct a geodesic survey of a large part of the Kingdom of Hannover, or, in other words, to measure a section of the surface of the Earth. The project involved many difficulties, and requires, first, that one reflect on the general concept of measurement.

Gauss' friend and collaborator, the astronomer Bessel, thought a man with Gauss' mathematical ability, should not be involved in such a practical project, to which Gauss replied:

``All the measurements in the world are not worth {one} theorem by which the science of eternal truths is genuinely advanced. However, you are not to judge on the absolute, but rather on the relative value. Such a value is without doubt possessed by the measurements by which my triangle system is to be connected with that of Krayenhoff, and thereby with the French and the English. However low you estimate this work, in my eyes it is higher than those occupations which are interrupted by it. ... you will agree with me, that, when one does without all real help in numerous petty affairs, the feeling of losing one's time can only be removed when one is conscious of pursuing a {great important} purpose...

``What do I have for such work, on which I myself, could place a higher value, except {fleeting hours of leisure?...''

How can you measure the surface of the Earth? Don't even think about using a yardstick. First think what it means to measure. You cannot measure one thing by another, unless you first can determine, if the two things are commensurable. If you worked through the last several weeks' pedagogical discussions, you know it is not always self-evident, whether two magnitudes are commensurable with each other.

To get a sense of this, look at a similar problem, investigated by Euclid, Archimedes, Cusa and Kepler, about which much commentary has already been written: Measuring the circumference of the circle.

One can measure a circle by another circle, or a part of a circle, but not by a line, or any other curve. A whole circle can measure another whole circle, only with respect to size, i.e., one circle is either greater or less than the circle by which it is measured. But, to measure along the circumference of the circle, the circle must be divided. The circumference can then be measured by the divided parts.

The first and most obvious division, is by half. This creates two semi-circles and a straight line diameter. Archimedes thought, that, by dividing the diameter into small parts, one could measure the circumference of the circle, but, Cusa proved, [and if you worked through last week's pedagogical, you would have proved to yourself], that the diameter and circle are incommensurable. One cannot measure the other. So in order to measure the circle, we must divide the circumference itself into smaller parts.

Well, if we continue folding the circle in half and in half again, we will divide the circumference into smaller and smaller parts. The number of parts, will be powers of 2. (That is, 2, 4, 8, 16, ....) But other types of divisions must be discovered, if we want to measure a part of the circumference which is not a power of two.

If we unfold the circle, after folding it into quarters, we will have constructed, two diameters, which meet at the center of the circle. Now fold the circle, so a point on the circumference touches the center. This will form a new line, shorter than the diameter, which intersects the circumference in two points. Once this fold is made, it is easy to find two other folds which will also meet at the center, forming two more lines, which will make a triangle. (It is easier for you to discover this by experiment, than for me to describe it without the use of diagrams.) This divides the circle in three parts.

By a more complicated process, the circumference of the circle can be divided into five parts, the description of which, would require a digression here, but will be discussed in future briefings.

It was long assumed, and Kepler proved, that it were impossible to divide the circle into seven parts. Until Gauss, it was believed, that this was the ultimate boundary of the divisibility of the circumference of the circle. Gauss discovered the divisibility of the circle into 17 parts, and other divisions also. But for purposes of today's discussion, what is important, is, that the process of division has a boundary. Not all divisions are possible, and since division is necessary for measurement, to measure requires one to discover, and if possible, overcome these boundaries.

To conduct his geodesic survey, Gauss had to determine how to divide the surface of the Earth, which presented many similar problems, albeit more complex, to our above example. For example, instead of measuring a curve, Gauss had to measure an area. This area, was on a curved surface, which in first approximation is a sphere, but is actually closer to an ellipsoid. How are these surfaces divided? How are these divisions, once discovered, measured on the surface of the earth itself? These and other problems, will be discussed in future pedagogicals.

But, while contemplating the above, it is not unhelpful to reflect on the following statement of Gauss, excerpted from his ``Astronomical Inaugural Lecture'' in which Gauss argues against the idea of sperating so-called practical, from so-called theoretical science:

``To judge in this way demonstrates not only how poor we are, but also how small, narrow, and indolent our minds are; it shows a disposition always to calculate the payoff before the work, a cold heart and a lack of feeling for everything that is great and honors man. One can unfortunately not deny that such a mode of thinking is not uncommon in our age, and I am convinced that this is closely connected with the catastrophes which have befallen many countries in recent times; do not mistake me, I don not talk of the general lack of concern for science, but of the source from which all this has come, of the tendency to everywhere look out for one's advantage and to relate everything to one's physical well-being, of the indifference towards great ideas, of the aversion to any effort which derives from pure enthusiasm: I believe that such attitudes, if they prevail, can be decisive in catastrophes of the kind we have experienced.''

Measurement and Divisibility Part II

Last week, we investigated the measurement of the circumfrence of the circle. What was required, was to divide the circumference into commensurable parts. It was demonstrated, that division by 2, and powers of 2, was possible by repeated folding and division by 3 was possible, by folding in a different way. Division by 5 was stated as possible, and left to the reader to accomplish, and division by 7 was stated to be impossible, and the reader was refered to Kepler's proof (Harmony of the World, Book 1). To the eye, the circumference of the circle appears smooth, and everywhere the same, yet when one tries to divide the circle, one discovers boundaries, with each new {type} of division. Thus, the numbers 2, 3, 5, and 7 each signify a {type} of divisibility with respect to the circumference of the circle.

The word {type} here is used in the sense of Cantor and LaRouche. Each {type} of division, is seperated from the other, by a discontinuity. One cannot divide the circle into 3 parts, from the method of division by 2 or powers of 2. One can combine division by 2 and 3 to divide the circle into 6 parts, but a new {type} of division is required for 5 parts.

Let's experiment with other types of divisions, with respect to other types of curves and surfaces.

Once the circle is divided, polygons can be formed by connecting the points on the circumference, with each other, and triangles can be formed, by connecting the vertices of the polygon, to the center of the circle. It is easily demonstrated, that these triangles are all equal. Thus, the relationship of all parts of the circumference to the center are the same.

Now look at an ellipse. The ellipse differs from the circle, in that all parts of the circumference of the ellipse have a relationship to two points, (called foci) not one, as in the case of the circle. Specifically, the distance from one focus, to the circumference of the ellipse, plus the distance from the circumference to the other focus is always the same. In the case where these two foci come together, and become one, the ellipse becomes a circle.

Look further at the ellipse. One can fold the ellipse in half in only two ways (which for convenience we can call horizontal and vertical), whereas, the circle can be folded in half in an infinite number of ways. When the ellipse is folded in half, one of the lines generated, will be longer than the other, the intersection of these two lines, (called axes) will be called the center of the ellipse. Two circles can be drawn, using this center, related to this ellipse. One will have the smaller line as its diameter, and the other will have the longer line as its diameter. The former will be smaller than the ellipse, the latter will be larger.

Now divide the larger circle into any possible number of parts, and form the triangles associated with the polygon which is formed by the division. The sides of the triangles, which correspond to radii of the circle, will intersect the circumference of the ellipse, dividing the circumference of the ellipse. Now connect the points of intersection with the circumference of the ellipse, to one another, forming triangles in the ellipse. It is easily seen, that unlike the circle, these triangles are not equal, consequently, the divisions of the circumference of the ellipse, formed by these divisions of the circle, are not equal. Hence, the ellipse, cannot be divided, or measured, in the same way as the circle. A new discontinuity has been reached.

This new discontinuity arises from the difference in the characteristic curvature, between the circle and the ellipse. The curvature of the circle is constant, while the curvature of the ellipse is always changing.

This problem, of measuring the circumference of the ellipse, a crucial problem for physics and astronomy, was investigated by Kepler, and further developed by Gauss, by applying his hypothesis of the complex domain. These issues will be investigated in future pedagogical discussions. But for now, take one more step. Now think of a sphere. By what method, can one divide the sphere in half, and what will this tell us about the underlying hypothesis concerning the divisions of the circle and the ellipse?

More next week.

MEASUREMENT AND DIVISIBILITY PART III

Last week's discussion ended with the question: By what method can we divide a sphere in half? Let's compare this problem, with the problem of dividing the circle in half. This was accomplished by folding the circle on itself, and, we discovered certain boundary conditions, with respect to that process. How can we apply this method to the problem of dividing a sphere?

First think about what we did when we folded the circle. We weren't simply dividing the circle. We were applying a rotation to the circle, in a direction different then the rotation which generated the circle itself. That is, a circle of 2 dimensions, is rotated in 2 + 1 dimensions. Division in n dimensions, was effected by a transformation in n + 1 dimensions.

Now apply this to the sphere. Obviously the sphere can not be folded, but it can be spun. Or, in other words, if we consider the sphere, as a surface of 2 dimensions, we must take action in 2 + 1 dimensions, in order to divide it. So, if we pick a point on the surface of the sphere, and, spin the sphere around that point, every point on the sphere, except the one exactly opposite the initial point, will move. These two points can be connected by the equivalent of the diameter of the circle, which on the sphere is a great circle, that divides the sphere in half.

Now apply this principle, of measuring n dimensions, with respect to n+1 dimensions, to the initial discussion three weeks ago about Gauss' efforts at measuring the surface of the Earth. How do we locate our initial position? With respect to north and south, we can measure the angle at which we observe the North Star. The higher overhead the North Star is, the farther north our position on the Earth. To measure our position on the surface of the Earth, we must look up, to the stars. This measurement is, therefore, n+1 dimensions, with respect to the n dimensions of the surface of the Earth. Now for our position with respect to east west, we must refer to the rotation of the earth on its axis, which goes from east to west. We measure this, with respect not only to a change in position with respect to heavenly bodies, but with respect to a change in time. Another dimension, (n+1)+1.

Once this position is determined, we now measure other locations in a similar manner, and then measure the distance between those locations, using triangles. In order to meaningfully measure the surface of the Earth, these triangles must be large. Too large to measure with rulers, yardsticks, or chains. If we start with two relatively close points on the earth, and precisely mark off the distance between them, we can then measure the distance between these two points and a third point, by measuring the angles that form the triangle between these three points. This is done, by placing an object at each point, that can be seen, using a telescope, from the other points, and we measure the angle at which the telescope has to be turned, to see each point.

Gauss invented a device, called the heliotrope, that used a small mirror to reflect sunlight, that could be seen, by a telescope, from many miles away. If three such devices are positioned at three different points on the Earth's surface, a very large triangle can be formed, that can be measured precisely. In this way, the surface of the Earth, can be covered with a network of triangles, and measured.

But, when we look through these telescopes, to see each point, the light is refracted (bent), by the atmosphere, and the lens of the telescope. This makes what we see, different from the actual position of the point on the Earth. So this physical property, refraction of light, must be taken into account in our measurement--another dimension, [(n+1)+1]+1.

But since our measuring points are at different elevations, we use a level, which adjusts its position with respect to gravity. So we must measure variations of the gravitational field of the Earth, yet another dimension, {[(n+1)+1]+1}.

Likewise, when using a compass, which reflects changes in the magnetic field of the Earth, we must measure variations in the magnetic field of the Earth--yet another dimension, {[(n+1)+1]+1}+1. And so on, with each new physical principle discovered.

The inclusion of each new dimension is not a simple addition, but a transformation in the hypotheses underlying our conception of physical space-time. Just as the idea of dividing a circle, contained within it, an underlying assumption of a higher dimension, which wasn't apparent, until thought of in terms of dividing the sphere, each new dimension, corresponding to a physical principle, uncovers previously ``unseen'' assumptions, with respect to the hypothesis of lower dimensions.

But, these assumptions, expressed in the form of anomalies and paradoxes, won't be ``seen,'' unless you look for them, not in n dimensions, but in n+1 dimensions. You can't measure where you are, except with respect to the horizon, which cannot be ``seen'', except with respect to the higher dimensionality, which you are seeking to discover, but which you will not find, unless you have the passion to ``look'' for it.