The climax of our studies is now within our reach. We are about to consider a truly marvelous structure in which the Cube, the interlacing Tetrahedra and the Octahedron are built upon a fivefold plan, and these within the Dodecahedron enclosed within the Icosahedron presents a geometrical figure of great beauty.
The Five Interpenetrating Cubes
Let us look once more at Plate 1. We see that the Cube is formed by 12 lines within the Dodecahedron. Across each pentagonal face of the Dodecahedron there is seen to be a line which is an edge of the Cube; twelve faces on the Dodecahedron, twelve lines forming the Cube.
In the course of our study, the Cube has been regarded as a symbol of Man as an intrinsic part of the Solar System. It is right, therefore, to consider the length of the edge of the Cub as the unit, with which the lengths of the edges of the other Polyhedra will be compared. From this we derived the numbers as found in Plate 4, wherein it is shown that an edge of the Dodecahedron is equal to .618, and that of the Icosahedron is 1.618. It will be also noted that the edge of the inner Icosahedron is .382, and placing these in the order of their lengths, we have four terms of the remarkable series of numbers associated with the Golden Section (See Appendix I).
Now, since all of the faces of the Dodecahedron are regular pentagons, it might immediately b asked why there should not be five lines crossing each of the faces, thus making twelve pentagrams or five-pointed stars. We may certainly do this, and w shall find that the resultant figure is that of five interpenetrating Cubes (See Plate VI).
We must take this study step by step, and we shall eventually draw all of the figures in the Greater Maze. Taking them first as simple figures, we shall draw each one in two views, and Fig. 13 shows the two views of a Cube that will be needed. Eventually the view to the left will be drawn twice, and that to the right, three times. It must be remembered that actually there is no difference in the Cubes themselves as they appear in the constructed figure; it is only necessary in the drawings to show them in two views.
The rectangular view to the right indicates that two of the faces are “edge on” as it were, and can be seen as straight lines only. Thus the overall appearance is that of a rectangle.
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Fig. 13 — Two Views of a Cube
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The next step toward an understanding of this very complex figure must be taken in learning about two irregular Tetrahedra based upon the Golden Section (one of them being, in fact, merely a reflection of the other). These will be quite different in appearance from the regular Tetrahedron. They are shown in Fig. 14. This irregular Tetrahedron is most interesting because its three edges are respectively: BD, .382; AD, .618; and CD, 1.00 — three terms in the Golden Section series. The reflected Tetrahedron indicates the same dimensions by the letters followed by (´). Thus we have a “left-hand” and a “right-hand” Golden Tetrahedron. Fig. 14 shows also how these Tetrahedra may be interpenetrated and resting upon a Square. Three sides of the Square are divided at the points of the Golden Section. (Note points A, A', B, B').
Figure 15 shows two right-hand Tetrahedra interlaced upon the Square. Figure 16 shows four Tetrahedra (two right and two left-hand) interpenetrated and resting upon the Square. As one might expect, the edges of these intersect one another at points of the Golden Section.
The construction of the five interpenetrating Cubs was accomplished by making a mold in the shape of the four interlacing Golden Tetrahedra; plaster of Paris was poured into this mold and six casts were made. A block of wood was carefully shaped into a Cube of the right size and these six plaster casts were carefully positioned and glued upon the six faces of the Cube. The result was the complex form of the five interpenetrating Cubes; when they were painted so that they could be distinguished one from another, the result was as shown in Plate VI.
When constructing these figures into the Greater Maze, where colored threads were used, all of the lines forming the five Cubes were made of red string, and they combined to form red stars within the faces of the Dodecahedron, which had been constructed of heavy wire and painted blue (See Frontispiece I).
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Fig. 14 — Left- and Right-hand “Golden” Tetrahedra (above).
The same interlaced upon a Square (below).
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Fig. 15 — Two left-hand “Golden” Tetrahedra interlaced upon a Square.
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However, as already explained, in the solid figure constructed of wood and plaster of Paris, the Cubes were colored white, red, blue, yellow and green, as it then becomes easy to distinguish them one from another. The resulting twelve five-pointed stars may now be examined (See once more Plate VI). It will be seen that now each of the lines forming a star is of a different color, so that in each star all five Cubes are represented. What an exercise in visualization it is whenever one sees a five-petaled flower, to construct in imagination a five-pointed star overlaid upon the flower, and to reflect that each line of the star is but an edge of a Cube. Let us picture what all five cubes would look like, delineating the existence of a Dodecahedron to contain them, itself within a surrounding Icosahedron! This gives us a clue to a sort of hidden beauty within the outward form of things. When we see that other designs in Nature are related to the Golden Section, we find that the physical universe is surrounded by a kind of mathematical albeit mystical universe, a concept which we can grasp more fully as our familiarity with these figures grows.
The Tetrahedra
Just as there is a pair of interlacing Tetrahedra within the cube in the Lesser Maze, these become five pairs of interlacing Tetrahedra in the Greater Maze.
In order to understand this construction, we take the case of a single Tetrahedron. This may be formed by drawing one diagonal across each face of the Cube, and as the Cube is
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Fig. 16 — Four “Golden” Tetrahedra interlaced upon a Square.
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drawn in two views, the resulting Tetrahedron will also appear in two views, as in Fig. 17.
We are considering the edge of the Cube as representing 1 or unity, and the lengths of the edges of all of the other Polyhedra are related to it. Thus, the length of the edge of the Tetrahedron is equal to the square root of 2, or 1.414.
Plate VII shows how the five Tetrahedra will be placed in space, being constructed within the five Cubes. As in the case of the Cubes, they all fit within the Dodecahedron, but whereas each point of the Dodecahedron carries the vertices of two Cubes, each point of the Dodecahedron carries the vertex of only one Tetrahedron. The Dodecahedron has 20 points or vertices, and as there are four vertices on the Tetrahedron, five Tetrahedra provide all twenty points required to fill the Dodecahedron.
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Fig. 17 — Two Views of a Regular Tetrahedron.
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Fig. 18 — Two Regular Tetrahedra interlaced.
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Because a Cube contains a pair of interlacing Tetrahedra, Fig. 18 shows two views of these interlacing Tetrahedra, and Plate VIII shows the construction of the five pairs of interlacing Tetrahedra. This naturally fits within the Dodecahedron, as it should.
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Fig. 19 — Two Views of a Regular Octahedron.
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The Five Octahedra
Fig. 19 shows two views of an Octahedron as it would appear if drawn at the center of a pair of interlacing Tetrahedra. From all that has gone before, we should expect to find a cluster of five Octahedra in the Greater Maze, and we shall not be disappointed (See Plate IX).
Again, as one might expect, we find the edges of the combined Octahedra intersecting one another at points of the Golden Section. Another amazing fact is that it is just these points of the Golden Section that are also the vertices of the inner Icosahedron which would be the framework on which an entirely new, but smaller Greater Maze would be constructed. And this process could be carried on to infinity. Plate X shows five Octahedra clustered about a central Icosahedron. Note the intersection points of the edge of the Octahedra in relation to the vertices of the Icosahedron.
The overall appearance of the figure in Plate IX is that of many irregular quadrilateral pyramids. The bases of these pyramids are not rectilinear, nor are the faces equilateral triangles. All of the faces are congruent triangles however, and their angles are of special interest. The angle at the apex is 60° and the two angles at the bases are 48º and 72º. These numbers, 48, 60 and 72 are most interesting as they fit in so well with the study of the occult numbers: they are 4 x 12, 5 x 12, and 6 x 12, respectively.
In another respect, the five interpenetrating Octahedra present a most interesting figure, which is really a hybrid. If we were to join externally all the points, we would have the Icosidodecahedron, pictured in Fig. 20, which is a combination of an Icosahedron and a Dodecahedron. It consists of 20 equilateral triangles and 12 pentagons.
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Fig. 20 — An Icosi-dodecahedron
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Since our first acquaintance with the Octahedron came about as the result of joining internally the points of the Icosahedron with certain points of the Dodecahedron, it should follow that all five Octahedra might have been constructed in the original manner by drawing five lines from each point of the Icosahedron; such is actually the case. When we construct the Greater Maze, which contains the totality of these figures, that is just how the five Octahedra are formed. This fact gives the completed figure its marvelous beauty. Frontispiece II shows the Greater Maze as drawn, whereas Frontispiece I shows it constructed of wires and colored threads, totaling 270 lines.
In the Greater Maze these five Octahedra are worked out in yellow threads, but in the solid form, they are made in the five colors as were the Cubes and Tetrahedra.
It is another amazing fact that whereas there are now five lines radiating from each of the points of the Icosahedron (5 x 12 making 60 lines in all), we find that these lines combine in groups of three at the 20 points of the Dodecahedron (3 x 20 is once more 60 lines). Each Octahedron having 12 edges, all 60 lines are accounted for (12 x 5 is once more 60).
Now it is most remarkable that these golden lines emanating from the points of the Icosahedron penetrate the faces of the Dodecahedron at the very points of intersection of the lines of the interlacing Cubes. This means that if you look for the red five-pointed stars on the faces of the Dodecahedron, it is just where these red lines intersect (at the points of the Golden Section, of course) that the yellow or golden lines penetrate the faces of the Dodecahedron!
Before leaving this chapter let us emphasize the significant point that as the lines of the Octahedra intersect one another at their own points of the Golden Section, it is with precisely these points that the vertices of the internal Icosahedron coincide. (See once more Plate X). We showed in Fig. 6 that there is an endless series of alternating Icosahedra-Dodecahedra, suggesting strongly that this Universe is but one term in an infinite series of universes.
The other geometrical figures are constructed primarily upon the Dodecahedron, and yet the Octahedron, lying at the center of them all, is a marvelous connecting link between one such hierarchy and the next, in the sense that the smaller Icosahedron, constructed in the correct manner, by joining internally the vertices of the Dodecahedron, is by its own nature so positioned that its own vertices just touch the edges of the Octahedron where they intersect at their own points of the Golden Section!
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The full beauty of this Greater Maze becomes apparent when, after having given due attention to its structure and the truly remarkable mathematical and geometrical relationships within it, we turn our attention to the mysteries which are symbolized in this particular manner. Remembering that a genuine symbol carries within itself (albeit here in mathematical form) the principles of the thing that it symbolizes, we may now see the Greater Maze as the window that we may approach for a panoramic view of the secrets of cosmic life.