SECTION TWO

Chapter III

“GOD GEOMETRIZES”

“1.  The eternal parent wrapped in her ever invisible robes had slumbered once again for seven eternities.

2.  Time was not, for it lay asleep in the infinite bosom of duration.”

—Book of Dzyan, Stanza I, verses 1-2.

“6.  From the first-born the thread between the Silent Watcher and his Shadow becomes more strong and radiant with every change.  The morning sunlight has changed into noonday glory…

This is thy present wheel, said the Flame to the Spark.  Thou art myself, my image, and my shadow.  I have clothed myself in thee, and thou art my Vâhana to the Day, ‘Be with us,’ when thou shalt re-become myself and others, thyself and me.  Then the builders, having donned their first clothing, descend on radiant earth and reign over men — who are themselves…”

—Book of Dzyan, Stanza VII, verses 6-7.

We shall now return to the moving circle, as seen in Fig. 4, remembering that the horizontal diameter has become a vector and is cut at the point K by the circumference of the circle.  One half of the line-segment AK was found to step exactly ten times around the circle, forming a regular decagon.  As a symbol, this was interpreted to represent the natural coming into being of the ten planes of consciousness of the Universe.

Now, Plate 3 shows that we have joined by straight lines the alternate angles of the decagon, and by doing so, we have drawn a plane projection of the very interesting three-dimensional figure known as an Icosahedron.  This is a regular polyhedron, having 20 equilateral triangular faces, 12 vertices and 30 edges.  It is the first of the five regular polyhedra with which we are to become familiar.

All these figures will be illustrated as they occur throughout our studies, but it will be best at this time to define the terms that will be used.

The names of the five polyhedra are from the Greek words that describe them.  Thus, the word polyhedron (pl. polyhedra) means many faces.  To qualify as a regular polyhedron, the geometrical solid must have an even number of faces.  All of the faces must be regular polygons (from the Greek, meaning many sides).  A regular polygon has all of its sides and angles equal.  Furthermore, the faces must all be congruent, that is, in any one polyhedron they must all have the same shape and size.  All the vertices of the polyhedron are similar polyhedral angles.  This will give the figure perfect symmetry.

An Icosahedron is thus a regular polyhedron, having 20 faces, as stated, all of them congruent equilateral triangles.  They meet in 12 vertices (polyhedral angles), and the figure is bounded by 30 lines, or edges.

There are only five regular polyhedra existing in nature.  See Plate 4, also Plates 1 and 2.  These are:

These figures will be discussed one by one as they appear.

An Icosahedron might now be thought of as consisting of 12 points equally spaced on the surface of a sphere, which points have been connected by straight lines, each such point then radiating five lines.  Thus, the 30 edges of the Icosahedron are really chords (the shortest one possible) in the sphere.  Looking now once more at its projection in the circle, there will be an interesting point to mention in respect to the center of the circle.  The center is really a projection of the north and south poles of the sphere, coinciding in one point.  There is a very interesting bit of symbolism here, related to the relationship between the numbers 10 and 12, before mentioned.  The ten points on the circumference were joined to the center of the circle and were considered to be radii; but it now appears that they are ten of the edges of the Icosahedron, five of them meeting at the north pole, and the other five meeting at the south pole of the sphere.  Thus there are really 12 points in all.  This is a matter worth pondering; it will be an important clue in understanding some aspects of the teachings about Hierarchies.

As a matter of fact, an Icosahedron may also be generated by a moving sphere, but no attempt has been made to elaborate this as it would be appreciated only by students who are familiar with Solid Geometry, and it might tend to confuse the issue if it were presented here.

We are going to take this study another step now and join internally by straight lines the vertices of the Icosahedron, and we shall find that if all possible chords are drawn within the body of the Icosahedron (omitting, however, the diameters of the sphere in order that the figures may be seen more readily), these internal lines meet in groups of 3, in 20 points.  Thus a new figure has been formed within the Icosahedron (Plate 5).  This is the Dodecahedron, above described as a regular polyhedron consisting of 12 pentagonal faces, 20 vertices, and 30 edges.

These two figures are complementary.  Both are constructed of 30 edges, but whereas the Icosahedron has 20 faces and 12 vertices, the Dodecahedron has 12 faces and 20 vertices.  The sum of all the plane angles in the Icosahedron is 3,600 degrees, and the sum of all the plane angles in the Dodecahedron is 6,480 degrees.

Now we have the remarkable fact that the moving circle has generated an Icosahedron enclosing a Dodecahedron.  In connection with the symbolism of this, remember that Brahmâ-Prakriti is dual.  Brahmâ may be said to be the “Silent Watcher of the Hierarchy,” and Prakriti, manifested Nature, the corporealized vestments, if you prefer, of the Solar Divinity called Brahmâ.

For reasons which we plan to make more evident as we go along, the Icosahedron is best taken to represent Brahmâ, and the Dodecahedron, Prakriti.  We shall come to see that the Dodecahedron is more directly involved in the other regular polyhedra, whereas the Icosahedron is indirectly involved, and yet it may be said to give the “breath of life” to the other figures.

Thus far we have been looking at the Icosahedron as though held with one of the vertices in our line of vision, and the Dodecahedron positioned with one of the faces directly in our line of vision.  It is important to view these figures also in the reverse manner, that is to say, with the Icosahedron positioned with a face directly in our line of vision; the Dodecahedron within it will then have one of its vertices in our line of vision; the Dodecahedron within it will then have one of its vertices in our line of sight.  In many respects this will be a more favorable view because the outline of the Icosahedron will have changed from the ten-sided to the six-sided view, and a new symbolism will then become apparent.  Plate 6 will make this clear.  Here we have the Tetraktys, and the seven dots at the center, which we hold to represent the seven manifested planes, will now form the basis for a drawing of the Icosahedron surrounding the Dodecahedron.  The three dots at the angles of the Tetraktys still remain the symbol of the unmanifested planes.  Here the meaning of the Tetraktys is greatly expanded.  We see it as a symbol of the Solar System, at least as far as we have developed the theme up to this point.  To the best of our ability we should attempt to see these figures as solids rather than as flat pictures.

Just as the Dodecahedron was formed by joining internally the vertices of the Icosahedron, so it will now be possible to make another Icosahedron within the Dodecahedron, and then another dodecahedron may be formed from the smaller Icosahedron; and thus we find that we may make an endless series of Icosahedron-Dodecahedron, never reaching the center.  This is shown in Fig. 6.  Conversely, the edges of the Icosahedron may be extended until they meet in 20 points in space, and these points may then be joined, making a new Dodecahedron; and the edges of this figure may also be extended until they in turn meet in 12 points, which, again, may be joined, making a new Icosahedron, and so on ad infinitum.  Thus, the Icosahedron enclosing a

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Insert Figure 6 here.
Fig. 6 — An Infinite Series
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Dodecahedron, which was originally formed by the moving circle, is found to be but one term in an infinite series of these regular polyhedra, extending outwards and inwards, ad infinitum.  What a magnificent symbol this is of the concept that this Universe is but one term in an infinite series of universes, extending in either direction, inwards and outwards, through all conceivable levels of consciousness.

This remarkable geometrical process throws a brilliant light on another Theosophical concept: the relationship between the Microcosm, the “little universe” which is Man, and the Macrocosm, the “greater universe” about him.  “Nothing is great, nothing is small, in the Divine economy.”

We are now ready for what is perhaps the most beautiful and significant aspect of this whole study.  We are going to see a symbol made before our eyes: the manner in which the spiritual-divine fire of Brahmâ blends with Prakriti to give birth to the spark of Divine Consciousness which is the core of the core of Man himself.

Heretofore we have joined internally the points of the Icosahedron to form a Dodecahedron; and the points of the Dodecahedron were joined internally to form an Icosahedron.  Now, we are going to join each point of the Icosahedron (of which there are 12) with certain of the points of the Dodecahedron.  The 12 lines so formed will converge in 6 points within the Dodecahedron (See Plate 7).  These points are actually the vertices of a new figure, an Octahedron. The Octahedron, as before stated, is a regular polyhedron having 8 triangular faces, 6 vertices, and 12 edges.

It will be observed that only 12 of the vertices of the Dodecahedron were employed in constructing the Octahedron.  We shall soon see that the other points of the Dodecahedron will be accounted for; but we must take the study one step at a time.

The symbolism shown in the Octahedron formed by the union of the Icosahedron and the Dodecahedron probes into some of the most recondite of the teachings about Man's consciousness.  Man is verily a child of the Gods and Divine Nature!

We must realize that when we speak of Brahmâ as the Hierarch of our Solar System, this Divine Consciousness is in reality a very complex entity.  We can approach an understanding of its nature if we imagine it to be a great light emanating certain rays, or portions of itself.  We speak of these as the Logoi of Brahmâ, and we number them as being twelve, in keeping with the more advanced teachings.  The twelve points of the Icosahedron may properly be taken to represent these twelve Logoi, radiating their own twelve energies to form the highly complex entity that we call Man.  This process is represented in the joining of the points of the Icosahedron with those of the Dodecahedron, forming the Octahedron, which is therefore quite aptly taken to represent the Monadic Essence of Man.  The symbolism of the Octahedron will become even more important as we proceed.

Seeing the octahedron as the main actor in the cosmic drama about to be unfolded, we are going to consider how we can portray the process of manifestation of the Monadic Essence, which it represents.  The Octahedron as formed represents the condition of the Monad before its own manifestation takes place; and we are to understand that from the Monad itself are produced the various sheaths of consciousness in which it is to clothe itself.  We learn that these sheaths are of themselves bi-polar; that is to say, they have their own energic and their own material or corpuscular aspects.  We can employ the terms more widely used and say that each such sheath of consciousness is at once Spirit and Matter.

So let us represent this in keeping with our geometrical study by making of each face of the Octahedron the base of a small triangular pyramid, or Tetrahedron.  It will be remembered that the Tetrahedron is one of the regular polyhedra consisting of four equilateral triangular faces and four vertices; it has six edges (Plates 1 and 2).

When eight such Tetrahedra are constructed on the faces of the octahedron, we find that they form two large interlacing Tetrahedra as in Plate 1.  This gives us the same symbol with which we are already familiar: the well-known interlacing triangles; but in this three-dimensional form it is far more graphic.  Thus the interlacing Tetrahedra represent Spirit and Matter, the bipolar sheaths of consciousness in which the Monad clothes itself.

It is remarkable geometrical fact that these Tetrahedra are of such a size that their vertices just touch the eight points of the Dodecahedron, which are not occupied by the golden lines emanating from the points of the Icosahedron.  This places Man within the Solar System, showing his inseparable link with his environment.  He is part and parcel of it.  He is the “Jewel in the Lotus.”

Now the Cube or Hexahedron, representing fully embodied Man, is not an arbitrary symbol, as will be made clear by the next stage in our study.  If we join by straight lines the vertices of the interlacing Tetrahedra, we shall find we have constructed a Cube, which encloses the interlacing Tetrahedra.  The edges of these Tetrahedra are seen to be the diagonals on the faces of the Cube, while the vertices of the Octahedron are coincident with the midpoints of the faces of the Cube.  Furthermore, the Cube with its 12 edges fits perfectly within the Dodecahedron with its 12 faces.  Turning to Plate 1 again, we see that each face of the Dodecahedron is crossed by a line; and upon closer observation it will be seen that these lines form the Cube. (See also Plate I)

Thus the Cube may be taken to represent fully manifested Man within the Solar System, or within Prakriti.  The implications of this study touch on some of the deepest mysteries of human consciousness.

A very interesting point with regard to the Cube is that the sum of all its plane angles is 2,160 degrees.  Students will be quick to recognize the importance of this number.  Twice 2,160 is 4,320, a key number in calculating the durations of the Root-Races, and in a wider field, the Rounds of the Earth Chain.  Furthermore, it takes 2,160 years for the Equinoctial Point to advance through one Sign of the Zodiac.  (See appendix II.)  This is only a hint of another aspect of the study, which will be taken up in due course of time.

It might be mentioned here that, having put all the regular polyhedra together, the Icosahedron surrounding the Dodecahedron, the two combining to form the octahedron, which in its own turn generates the interlacing Tetrahedra and the Cube, we have constructed what is known as the Lesser Maze.

This procedure is shown step by step in Plate I.  The Octahedron is first shown in yellow lines, the Tetrahedra in green lines, and the Cube in Red.  These are enclosed within the Dodecahedron shown in blue lines.  Plate II shows all of this enclosed within the Icosahedron.

If we now add all of the plane angles in the Lesser Maze we have the following:

Plate III shows the Lesser Maze as constructed of wires and threads.  The outermost figure in white threads is the Icosahedron; the Dodecahedron is constructed of wires painted blue; the Octahedron of yellow threads, surrounded by the interlacing Tetrahedra in green threads; and the Cube within the Dodecahedron is constructed in red threads.

Before closing this chapter, we should bring out the important concept of the Icosahedron being formed by joining with straight lines the 12 points equally spaced about the surface of a sphere.  In like manner, it must be said that the Dodecahedron is formed by joining with straight lines 20 points spaced symmetrically about the surface of a smaller concentric sphere, and that the octahedron is formed by joining 6 points on the surface of a concentric sphere, smaller yet.  It is to be noted, however, that the Cube and the Interlacing Tetrahedra share certain points with the Dodecahedron.  Because the points of these two last-named figures coincide with points on the Dodecahedron, it is obvious that their spheres will coincide, appearing as one sphere.  This means that in the Lesser Maze we have three concentric spheres, the largest enclosing the Icosahedron, the intermediate enclosing the Dodecahedron with the Cube and the Interlacing Tetrahedra, and the last, and smallest, enclosing the Octahedron.  This concept is going to be of great importance in the study to be taken up in a subsequent chapter.

Let us point out once more that there could be a second Icosahedron formed by joining internally the vertices of the Dodecahedron.  One of the marvels of the interrelationships of these figures is that this internal Icosahedron is so placed by nature, that its 12 vertices just touch the edges of the Octahedron at the points of the Golden Section of these edges!  A marvelous bit of symbolism, truly.  It points to the fact that it is in the inmost spark of Divinity that man carries within himself a link with other universes, other hierarchies that regard his as their own Silent Watcher.