We now come to the greatest adventure of all, wherein we shall see the symbol of an endless complexity of inner worlds and planes of consciousness come to life. This will be the climax of our study of Mathematical Symbolism, one which extends even further our understanding of the Greater Maze, bringing us new concepts that reach into the deepest teachings about human consciousness.
Let us study in greater detail the Icosahedron lying at the center of either the Lesser or the Greater Maze. By the laws of its own structure, its twelve vertices touch the twelve lines of the Octahedron at the points of the Golden Section, as explained in the previous chapter.
Now we will remember that there are two points at which any line may be so divided. This implies the construction of a second Icosahedron within the Octahedron, and this possibility is shown by the lines in Fig. 21. The most interesting part of this study will be to show that this second Icosahedron has no physical reality even though it may be constructed as well as drawn, as will be shown. How can this be? Well, there are two good illustrations which will clarify the point. Every planet in the Solar System follows an elliptical pathway around the sun, even though in many instances the deviation from a true circle is slight. However, if we take the instance of the comets, we see orbits which are extremely elliptical, and we learn something of interest by studying them.
In the first place, a circle is defined as closed curve such that every point is equidistant from a fixed point called the center. All radii of the circle are equal. Thus there can be one and only one center.
By contrast, an ellipse is a closed curve such that the sum of the distances from any point to the two fixed points called foci, is constant. It might be said that in an ellipse where the foci are close together, the curve is approximately circular, whereas in the case of an ellipse in which the foci are widely separated, the ellipticity of the curve is more pronounced.
What this is leading to is that since the orbit of every planet and comet in our Solar system is elliptical, the center of the Sun occupies one of the foci of the ellipse. What is at the other focus? Apparently there is nothing there, although the location of that focal point may be determined in space. Thus, as a matter of pure science, a point which has no reality other than as a mathematical necessity may be plotted and shown on a chart.
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Fig. 21 — An “objective” Icosahedron above, and a “Subjective” Icosahedron below, drawn
as they would appear within an Octahedron.
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We are tempted to ask: Are there two foci because the orbit is elliptical, or is the orbit elliptical because there are two foci? The question might have no meaning except for the possibility that the several foci (one for each planetary orbit) are clustered about the center of the Sun, which is the focal point that is common to all of the orbits, and the though comes to mind that the Sun is a chain of Solar Globes, his children taking pattern after him. Could there be any relationship between these mathematical focal points and the Globes of the Sun Chain that might have a bearing upon the several planets?
The relationship between the real and the mathematical Icosahedron within the Octahedron may, again, be clarified by an illustration taken this time from the science of optics. It is well known that the properties of a lens-system in a movie projector, to take a convenient example, are such that the object (the film) throws an image of itself which may be focused upon the screen. This is known as the real image. There is also a virtual image, which cannot be thrown on the screen. So far as the audience in the theatre is concerned, this virtual image is non-existent, and yet the properties of the lenses are so well understood by the manufacturers that the virtual image can easily be drawn on paper, and its size and distance may be known with great accuracy. Thus, in a sense a thing may exist and yet not have any substantial reality so far as our sense are concerned.
Now in the construction of the Maze, only one of these Icosahedra has reality. This is the one which is formed by the internally intersecting lines from the Dodecahedron, as explained before. The position of the other Icosahedron within the Octahedron is such that none of its points coincides with the figures which properly belong to the Maze, and for that reason it may be said to exist as a mathematical concept only, having no reality beyond that.
Although we might call these two Icosahedra the “real” and the “virtual;” Icosahedra, it would be more accurate from the standpoint of our study in Symbolic Mathematics to use the terms “objective” and “subjective” respectively.
Now just as the objective Icosahedron is at the center of a “real” Maze, so we may rightly consider that the subjective Icosahedron is at the center of a subjective Maze; this could conceivably b drawn interpenetrating, as it must, the “real” Maze, were it not too confusing to do so.
The objective Icosahedron is common to all the five interpenetrating Octahedra in the Greater Maze; as explained in the previous chapter (see again Plate X), its vertices touch all five Octahedra at the 12 points where their edges intersect at their own points of the Golden Section.
Now it will become apparent that there will be a subjective Icosahedron within each of the interlacing Octahedra;; and although these Icosahedra have no reality so far as the objective Greater Maze is concerned, nevertheless they may all be constructed. Plate XI shows a drawing of the five subjective Icosahedra (colored), clustered about the objective Icosahedron (white).
There is a rather complicated relationship among these five subjective Icosahedra themselves which is most interesting, however, it is different from their relationship to the “real” Icosahedron. And yet, even though of a different mathematical character, their relation to the “real” Icosahedron is a constant. Each one of these subjective Icosahedra might be though of as a center with its own subjective Greater Maze built upon it, so that clustered abut the objective Greater Maze there will be five subjective ones. Imagine the complexity of this figure! However, there is yet more to come.
Although we have been speaking of five subjective Icosahedra, this is only from the standpoint of the objective or “real” one; but it is conceivable that any one of these could be an objective Icosahedron in its own right and from its own viewpoint. As such, it would have five more Icosahedra clustered about itself, which to it are subjective. From the standpoint of our original “real” one, these would be subjective in the second degree. Then, around each one of these would be five more, subjective in the third degree; and so on ad infinitum.
The picture is developed thus: Around the original Icosahedron there are clustered five subjective Icosahedra. Around these, in their totality (five to each), there would be 25 subjective Icosahedra of the second degree. Around all these, there would be 125 Icosahedra subjective to the third degree, and then 625 more, subjective to the fourth degree, and so on, each degree bringing in a higher power of the number 5 and a greater degree of subjectivity.
We should not think of these as increasing in size just because they are conceived to be in differing degrees of subjectivity. Actually, they are all of the same size as the original Icosahedron. Their relationship gives more meaning to the phrase, so often found in theosophical literature: “in coadunation, but not in consubstantiality.”
This mathematical concept, quite in line with the laws of geometrical structure, gives us some clue as to the real nature of our Solar System. It is, indeed, the “Egg of Brahmâ,” filled with invisible worlds whose number it is impossible to estimate, existing on all conceivable levels of consciousness.
Within the diagram of the Greater Maze are to be found innumerable suggestive relationships. For instance, the objective Icosahedron with its five subjective Icosahedra “clustered” about it, suggest the relationship of the earth Chain to the Sacred Planets. Consider the Earth Chain as being the “real” planet — from our standpoint only — and clustered about it are the planetary chains of Mercury, Venus, Mars, Jupiter and Saturn, five in number.1 These Sacred Planets bear a certain subjective relationship one to another that they do not bear to the Earth; yet in a different capacity, their subjective relationship to the Earth is constant for all of them. This is because they all contributed toward the building of the various Globes of the Earth Chain. Obviously, the physical Globes of the chains are not here referred to.
Another aspect of the study which probes deeply in to the mysteries of human consciousness suggests that the original “real” Icosahedron may represent the Monadic Essence within Man. Paradoxically, although this is the most subjective aspect of human consciousness, from our personal standpoint, it is actually the most real part of us — objective on its own plane. Thus reference to the objective Icosahedron is quite apt in this point of symbolism The other Icosahedra clustered but it may well represent the manner in which the various Monads in Man are “clustered” about the Monadic Essence.
A deep meditation upon this theme leads us into still further interpretations of the symbolism involved, wherein we are led into further understanding of the mystic processes of Initiation. We learn that, just as we are trying to grasp the meaning of the subjective Icosahedra (representing the origins of so many subjective Mazes), so in Initiation, it is not the Spiritual Monad which is to be brought forth, but it is the latent Manasuptra within the disciple himself, which must be brought into fruition. That is the real secret that we are trying to discover: It is the hidden glory of the Initiant himself that must be brought forth, not that portion which has already achieved a high plane of spirituality.
This brings our present study to a close, but with no feeling that the final word has been said. If I were to elaborate all the points of significance that these geometrical figures hold for me, it might be interesting to some but less than satisfactory to others. By far the greatest value to the student will be found in the exercise of his own intuitions.
This study calls for more than the brain-mind approach. It demands and therefore calls for the transcendental faculties of the spiritual intuition, which all possess and which must be brought into play by the earnest student. The rewards of this study are great indeed, and perhaps the first thing that the student may learn is that there is no stopping point at which, having reached, he will say: “ I know all that there is to be known about it.” The wonder of this study is that it will lead him ever on to new and richer understandings and experience.
1 A hint as to the existence of others on a more subtil level of consciousness is to be found in the fact that the Sun and the Moon stand, as symbols only, for two more, which are said to "secret" planets.