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Projective Fields in the Planetary Flux Model
This study is an attempt to unite three different fields under one technique. The first field is climatology. Examples will be given of recognized climatic patterns over North America. Second, this climate data will be placed in the context of ideas from the great astronomer, Johannes Kepler, regarding the concept of the music of the spheres. The third field of study makes use of projective geometric techniques to place data of motion in arc events taken from an ephemeris, into a context of Northern Hemisphere climate patterns. Motion in arc data placed in the context of projective geometric grid systems based on the harmonic theories of Kepler have proven very useful in modeling the onset and decay of unusual climatic scenarios. The combination of these three fields of study is the fundamental thesis of this paper.
To establish the context for this work the map of the northern hemisphere was divided using the longitudinal coordinates from the geodetic equivalency technique of Chris Mc Rae. The Geodetic World Map AFA 1988. This work was in itself an adaptation of the work of Sepharial in the 1920's. However, this placement was modified in this work according to the observed phenomena in the weather. As a result, there are no cusps, rulerships, ascendants, houses, midheavens or in effect any of the concepts so useful in standard astrology. This may seem to be heresy to a practicing astrologer but the system described in this paper is the result of a phenomenon driven protocol with very little reference to astrological conventions except for geodetic projection and the accurate ephemerical data concerning motion in arc reckoned in sidereal longitudes. I have also taken the liberty of displacing the McRae placement 23°to the west in order to allow for precession of the equinoxes.
Through time, certain indications given by Kepler have proved to be unerring guides in the process of observing the chaos of climate on a hemispherical scale. Coupled with precise data from solar and lunar eclipses the Keplerian harmonic indications have resulted in a technique which is complex yet very workable. This technique is in line with certain ideas from chaos theory and systems thinking. This paper is not about these researches however. Its aim is more general and introductory. The attempt is to present some indicators which Kepler has given in Book V of the Harmonica Mundi regarding the significance of particular angular aspects using a fundamental concept in the ancient world, the harmony of the spheres. The roots of this can be traced through Pythagoras in his studies of the notes and string length relationships in the monochord. In Kepler's work the application of the Pythagorean string length to angular aspects between planets was the esoteric basis for his practical applications of planetary motion, especially the studies of the weather. In these studies Kepler was seeking to find a correspondence between the harmonics of a fretted string and particular angular aspects between planets. The accompanying chart is a composite from the Harmonica Mundi.
From this we can see that a whole scale could be sounded from the motion in arc of planets taken from an aspectarian. In Kepler's time the inheritance of Greek thought allowed that the circumference of the earth could be compared to the length of a string on a monochord. If the string were fretted at half it's length then the octave above the fundamental earth tone would be sounding. This would happen during an opposition. During a conjunction the fundamental earth tone would be sounding. During a square the fourth above the fundamental would sound into the cosmos. This seems a bit quaint to hear of these correspondences. However during the Age of Enlightenment French geometers found a type of geometry that can support these views. The basis for this new synthetic geometry was the linear perspective found by artists in the Renaissance. Linear perspective has eventually evolved into modern projective geometry.
In synthetic or projective geometry there are no actual measurements such as form the fundamentals of most systems of geometry. Instead of measuring with compass and straight edge, the projective geometer projects lines into space in musical sequences or protocols, what is known to mathematicians as a process of iteration. Instead of measuring the lines and the 2-D forms they produce, the projective geometer watches for curves arising in time as the fundamental relationship is repeated or iterated. The repetitions of the forms in sequences are projections of the original formal relationships. The repetition transformed the single elements into field structures.
In this way projection can be used to model systems that are in constant metamorphosis and flux. The important element in projection is not the outcome or the final measurement but the process in which a new form or curve arises in time as a result of the interactive repetition of all of the variables in the whole system. Over time many mathematicians have refined the concepts of projection until it is now generally recognized that projective space/ time relationships are the archetypal geometric framework from which all other geometries arise. This makes Euclidian, 2D space a special case of projective time / space. Projection then is the archetypal geometrical method for deepening formal geometric motifs into field properties.
To further illustrate the idea of projection it would be useful to look at some basic geometric concepts in a simple way. In Euclidian geometry a point is considered to have the property of location. That is, it exists somewhere but that is all we can really say about it. Likewise in this way of thinking a line has the property of direction. That is it is like a string of points going somewhere. Finally a plane has the property of extension, that is it is going somewhere in a number of directions simultaneously. However, it is possible to see a plane in a number of different ways. For instance if we see a triangle drawn on a plane, we usually see it as a finished form which we can measure. This is the way the old geometries saw things. The forms were finished and measurable.
But even in the measurable space of Euclidian geometry some unusual things can appear that seem to defy common sense. For instance we can see the triangle as a field of points in which it is possible to find various positions or locations. There could then be different centers.
If we drew lines from the center of each side to each point (dotted lines), or if we drew lines from each point to the perpendicular on the opposite side (solid lines), there would be different centers according to how we measured. This illustrates a peculiar character of geometric forms in that the same form, such as a triangle, can be understood from many aspects each of which are each true from a certain point of view or perspective. In reality there are actually two more centers to a triangle besides these two.
From still another perspective we could also look at a triangle as a process or protocol of the different steps we used while constructing the form. For instance, consider that line ab would be drawn first then line bc then line ca. Only then could we draw line bd. This would be a process or procedural protocol of the drawing of the triangle and a different perspective on what constitutes the entity, triangle.
From still another perspective we could also look at a triangle as a perspective field or grid in which there could be a representation of 3D space in a 2D format. This can be seen in the third image. In this way of seeing geometrically the space in which the world appears becomes approachable to the cognitive imaginative powers of the soul. To illustrate this subtle shift of consciousness from 11th century orthographic space into 15th century geometric space of projection we can place point A of the triangle on a line h. The line h is now the horizon line and the point A will be renamed point V or vanishing point.
Line a v of our triangle will now be on one side of the street and line bv will be on the other. The bottom line of the triangle will be designated as line w for the world. Our flat triangle has now become a diagram for space and time in the world of appearances as we draw parallel horizontal lines in the perspective grid to measure the space moving out towards the horizon. We could also add line vc as an element of the perspective field and it would be included in the activity of the whole field through its relationship to the vanishing point.
From this we can see that a vanishing point shifts the 2- D character of the triangle in a significant way. This shift is the sense of the perspective of a line moving on a 2-D plane in a certain direction from, say, point V to point C. As part of a perspective field the line reveals a different aspect of its directionality to the observer. There is made available a soul experience that the line between V and E is not finished in 2-D but it is going on into the infinitely distant. Everything between line w and line h appears to vanish magically into orderly nothingness. What is going on? This is a far-reaching question.
In the complete system of perspective the points on line w must disappear again into the focal point, point F. This is seen in figure 4. In the language of projection, point F is a reciprocal of point V. This figure illustrates a construction known in projection as a perspectivity. In a perspectivity the lines which are laid down from line w into point F are lines which are counter to the "space of time " which is created in the illusion of 3d space in the perspective field "above" line w. The world which disappears "back" into the vanishing point on the horizon along the recessionals of the perspectivity also disappears "forwards" into the eye of the observer along lines which run counter to the formation of the recessional lines. In the language of projective geometry this perspectivity is an example of projective space and the counterspace lines to point F are essential because they include the observer and make the system complete from a 3D perspective.
However, since this way of experiencing space was developed there have been significant advances in geometry that extend the perspectivity of space into a projectivity of space. Projectivity enhances the experience in the soul so that the counterspace lines that meet in the eye of the observer continue through the eye and into an inner projective or synthetic space in which many phenomena find a basis. The essential element of projective space is the concept that there is a reciprocal one to one correspondence from points in one perspectivity, to points in another perspectivity. This is where it becomes interesting to consider the effects of planetary influences upon weather or even social phenomena since these phenomena appear in a context of reciprocal forces. Reciprocation is the characteristic condition of the laws of counterspace. To understand this it is useful to construct a projectivity.
In figure 5 the upper perspectivity is gathered together into a projector in the point F that we have said would be like the eye of the observer. The lines from this perspectivity are continued or projected onto another line ( line i) in a one to one relationship. That is every point on line w is projected through point F onto line i (inner space). This line could represent the retina of your eye. Point A on line w projects to point A' on line i. From line i then other perspectivities can be made.
In the projection field of figure 5, the projector is a point, point F . It is like the lens of a projector faithfully projecting every detail of a slide from line w (slide) through the projector onto line i (screen). It can be seen however that the projection is not without its mysteries. Even though the points are faithfully projected they are in a reverse or reciprocal order "counter" to the protocol of the points on the original line. This counter reciprocal placement is another symptom that the laws of projective geometry are in operation. The geometry of projection with its emphasis on protocol or process is a different geometry from the measured linear forms of Euclid. Projection can model systems that are reversing.
In order to see these applications in the realm of climate systems the idea of a projector needs to be developed with a line as the projector instead of a point. In such a construction there would be the interaction of two perspectivities placed in such a way that the perspective lines from a point on one line would form a perspective field on another line while the perspective lines from a point on the second line would simultaneously be projected onto the field of the first line. The construction would look like this.
In figure 6 the two points i and w are giving rise to mutually interacting perspectivities to form a perspective projection field in which lines arise as the projectors such as line a. In the following pages the unique properties of this kind of field will be applied to the interaction of celestial motion in arc events. These events in time will be projected onto a chart of the northern hemisphere in order to create a dynamic and flexible system of projection lines used to model changes in the atmosphere in very precise timeframes. For the past two decades, this technique has been integrated into daily weather observations of jet stream variabilities from 500mb weather faxes. This has further been integrated with numerous historical studies of eclipse positions. Using Pythagorean harmonic principles and the astronomical harmonic indications from book V of the Harmonica Mundi by Johannes Kepler a workable system for modeling long-term climatic scenarios has been developed.
JET STREAM HARMONICS
One of the greatest mysteries of climate studies is why the jet stream winds that guide storms across the earth veer and twist into their loops. Many physical facts are known about the jets but very little is known about why they change when they do and why they take a particular path at one time and another path at another time.
Sometimes the jet forms a flat curve across the continent or ocean. This pattern brings settled weather and a pronounced westerly flow to winds. Meteorologists call this pattern a zonal flow pattern. It parallels the zones of latitude. At other times the jet forms sinuous loops like a wildly oscillating, high- pressure fire hose. This pattern brings cold into southern latitudes and warm into cold latitudes. The weather patterns from this situation are often turbulent and unpredictable. This is known as meridional flow since the jet is paralleling the meridians of longitude. These facts do not adequately describe the almost mystical chaotic flow patterns that swirl over our heads in amazing and mysterious rhythms. Research has been done to construct motion in arc projections in which two perspectivities are woven into each other to create a grid upon which the motions of the planets could be projected upon the earth. This study is an introduction to these ideas.
The techniques of projective geometry involve in essence the construction of a grid or field of lines or points in which the relationships from one to another is determined by a protocol which is repeated over and over. In essence any point that gives rise to a line that connects the original point to another point is following a direction through space.
In figure 8, A is the original point and B is the other point which makes up the directionality of the line AB. Suppose that other points C, D and E are also in the field of points A and B (fig 9). Then these other points would have a part in the protocol of a perspective field. That is, we would always go from A to B then A to C, A to D etc.
In figure 9 these other points are on a common line, line f. This line is linked to point A by means of the perspective field. In figure 9, point A is also placed on a line, line h and that line is parallel to line f. As we have seen before, this creates a perspectivity in the language of projective geometry.
In fig 10 two perspectivities are woven into each other to create a projective field. In figure 10 line a is marked off in increments of similar units. Line b is marked off in units of the same length. Projecting the first point on line a to every point on line b, and then projecting the first point on line b to every point on line a results in a field of lines with crossing points spaced rhythmically throughout. By connecting the central crossing points of the two sets of lines, line c arises, which is parallel to both line a and line b. However, there are other projection lines that also arise in the projection field where the two sets of lines cross. These lines are curved and gradually approach being parallel to line c as the points on line c go to infinity on the right. These curves are a family of harmonically related curves that arise in the field between line a and line b. Two lines projecting onto each other have resulted in a family of harmonic curves even when the two lines are parallel to each other and the increments on each line are equal. The curves from these points in this protocol must arise as particular curves that are images of the harmonic relationships in the projection space that the two lines have in common.
In figure 11, the points on line b are spaced at 2 times the intervals of the points in line a. The result is a family of projection curves that are a picture of the ratio 1: 2. Once again the curves are harmonic to both line a and line b. Each curve is a projection of the ratio between these two lines. These simple projections are useful in understanding the ways in which geometric projection can be used to model climatic scenarios. In the drawings, each line represents a discrete motion in the projection field at a discrete time. That is, line a was drawn before line b etc. Taken together the construction represent a whole time / space event which is governed by the protocol for establishing the relationships between the two lines, and the sequence in which the lines were laid down. The line itself is a space element and the particular placement is a spatial event.
The two lines a and b can also be drawn in any relationship besides parallel. In figure 12 line a and line b are coming from a common point. Once again the family of curved lines arises in the field between the two lines even though they come from the same point.
In figure 13 points a and b are placed on a common line on the circumference of a circle. The harmonic curves that arise in the projection space are very subtle. Yet they are curves nonetheless. Placing the two projecting points on a circle opens the door to the work with combining projective geometry with climatology. Another type of projective transformation known as a translation, creates a projective space by moving only one projected line of a fixed length around the circumference of a circle.
In figure 14 a series of 90 degree lines are projected onto a map of the Northern Hemisphere in the following protocol. Each 90 degree line is translated or progressed along the equator by 5 °of arc. In this figure a single curve, instead of a family of curves, arises in the projection field near 30° N latitude. The curve in this case of projection, is always a circle. For this example the protocol is a 90 degree line progressing 5 degrees each time a new line is drawn. This protocol generates the translated projection curve. The degree angle of the line and the progression angle of arc will change in the following examples but the circular projection technique will be similar in each case study. In other words, the field will change but the process will not. From this 90° translation we can see that an angle of 90° of arc will translate into a curve at 30° latitude. This area climatologically is a sensitive area where the subtropical surface high-pressure areas form. These subtropical highs steer the jet stream and strongly influence climatic changes.
In figure 15 each successive line translates eastward 108° from a projection point that is progressed 5° of arc east from the proceeding point. It can be seen that when drawn one after another these 108° lines create a translation curve at about 40° north latitude. The space between the 90° translation and the 108° translation is a climatologically significant area in the Northern Hemisphere. In this next section we will look at the whole ensemble of translation curves that were given in figure 1, projecting them as the musical equivalents of aspect angles between planets, from the work of Kepler.