Platonic Models of Trigrammaton: The Octahedron

The 27 Trigrams of Liber XXVII can be put into 4 categories, based on their structure. The number of Tao or null lines in each Trigram determines what category it belongs to. There is only one Trigram that has three Tao lines; this is the Zero, which is in a category by itself. The remaining 26 Trigrams occur in three sets of 6, 12, & 8, and this fact allows the modeling of this group of 26 components on the multi-dimensional grid of the octahedron.

The regular octahedron is one of three regular polyhedra that are composed solely of triangular sides. As with all polyhedra, the three major visual aspects that it displays are Corners, Edges and Sides, or Vertexes, Vectors, and Faces. It so happens that in the octahedron these aspects occur in three sets: 6 vertexes, 12 vectors, and 8 faces. This allows a one-to-one correspondence between the 26 ‘topological aspects’ and the 26 non-zero Trigrams.

The Trigrams naturally occur as 13 sets of polarities, and this fact will be used to determine how the correspondence is achieved between the 26 aspects of the octahedron and the 26 Trigrams. On the octahedron, all vectors are physically opposite other vectors, when a line connecting them goes through the center of the figure. The same is true of the vertexes and the faces. So these physically opposite aspects will be attributed to Trigrams that are also polar opposites.

The six initial Trigrams of the system are attributed to the six vertexes of the octahedron, such that each Trigram-vertex is opposite its own polar Trigram. Thus the Trigrams for 1 and 2 are opposite one another, as are the Trigrams for 3 and 6, and those for 9 and 18. Once this has been done, all the remaining attributions to the vectors and faces are achieved by utilizing the numerical characteristics of the vertexes.

Each vector is attributed to a Trigram whose value is the sum of the two vertexes that form the ends of that vector. For example, the ‘4’ vector connects the two vertexes whose values are 1 and 3. Opposite to this, the ‘8’ vector, connects the vertexes whose values are 2 and 6. So by adding the vertexes of any vector, you get the value for that vector.

Similarly, each face is attributed to a Trigram whose value is the sum of the three vertexes that form the corners of that face. For example, the ‘22’ Trigram is attributed to the face whose corners have the values of 1, 3, and 18. The face opposite to this is attributed to the ‘17’ Trigram, because the corners of that face are the vertexes whose values are 2, 6, and 9. This simple method allows for numerical balance in every direction, throughout the matrix of the octahedron. The sublime beauty of this balance is seen in the fact that in every case, polar opposite Trigrams are attributed to opposite aspects of the octahedron matrix.

A curious side-effect of this matrix is that if one considers the vertexes valued at 9 and 18 to be the neutral poles of spin of the octahedron, the four Trigrams with the values of 4, 5, 7, & 8 form the Equator of the octahedron. Since the four corners of this Equator are attributed to Trigrams with the values of 1, 2, 3, & 6, the entire Equator represents the sequence of numbers from 1 to 8, all of which can also be represented as Bigrams.

We now have a systemic matrix that involves an Equator whose four corners are the four Elements, and whose edges are four Zodiac signs, one of each Elemental type.

Platonic Models of Trigrammaton: The Cube

The method of attributing Trigrams to the octahedron can also be used with the cube. Both the cube and the octahedron have 26 cases of the three primary topological aspects; the octahedron has 12 vectors, 8 faces, and 6 vertexes, while the cube has 12 vectors, 6 faces, and 8 vertexes. The cube is therefore called the “dual” of the octahedron, since one figure can be changed into the other by reversing their vertexes and faces. In terms of the Trigrams, this means that they can be modeled equally well on the cube, with the same mathematical results and the same polar Trigrams opposite one another.

As the figure shows, the six vertexes of the octahedron have become the six faces of the cube, and these faces, or planes, remain the primary set of opposites that determine the whole figure. The vector where two faces meet equals the sum of those two faces, while the vertex where three faces meet is the sum of those three faces. As with the octahedron, the figure can be oriented so that the 9 & 18 Trigrams are the upper and lower faces, while the remaining faces are the four Elemental Trigrams, divided by the first four Zodiac Trigrams to form the same Equator found in the octahedron, consisting of all the numbers from 1 to 8.

This cubical array can be used as a template for the layout of a magical temple, with the East wall being Air, the West wall being Earth, the North wall being Water, and the South wall being Fire. The floor and ceiling would then be the poles of the system. This layout provides for the same Elemental directions as those based on the sequence of the Classic Trigrams, but on the cube the I Ching Trigrams are the corners of the room, not the perimeter.

The cubical matrix can also be elaborated by the creation of a three-frequency cube, i.e., a cube made up of 27 smaller cubes. Each of these smaller cubes can be attributed to one of the 27 Trigrams, with the Zero Trigram occupying the center of volume of the figure, the sole inner cube, invisible on the surface of the larger cube. The eight corner cubes are thus the Classic Trigrams. The six cubes on the center of each side are the Elemental Trigrams, and the twelve remaining cubes on the edges are the Zodiac Trigrams. These all retain their relative positions, as in the skeleton cube, with opposite cubes being attributed to opposite Trigrams. Of course this is difficult to illustrate in two dimensions, but the essential characteristic is that the path that joins any two opposite Trigrams travels through the central Zero cube. (Note that the three-frequency cube is the smallest subdivision of a cube such that it has a nuclear cube).

Seen from above, the levels of the cube look like this:

Top Level: 17 11 14

15 9 12

16 10 13

Middle Level: 8 2 5

6 0 3

7 1 4

Bottom Level: 26 20 23

24 18 21

25 19 22

Another cube image can be found on the C.F. Russell website. His work touched upon areas directly related to Trigramaton,and his planetary cube is essentially the same layout I have described above, but concerned only with the eight corners:

http://www.cfrussell.homestead.com/files/cubed2.htm

Platonic Models of Trigrammaton: The Tetrahedron

The cube and the octahedron are two of the five Platonic solids, along with the icosahedron, dodecahedron, and the tetrahedron. Like the former pair, the icosahedron is the “dual” of the dodecahedron; one has 30 vectors, 12 vertexes, and 20 faces, while the other has 30 vectors, 20 vertexes, and 12 faces.

The tetrahedron is in the curious position of being its own “dual”, for it has 6 vectors, 4 vertexes, and 4 faces. When the number of the vertexes and faces reverse, the same total, and thus the same figure, is created. But one would be mistaken to think that these are one and the same tetrahedron. They are in fact a pair of distinctly positive and negative tetrahedra. This fact can be illustrated by attributing the Trigrams to the tetrahedron, as was done with the octahedron and the cube.

As in the previous examples, the six Elemental Trigrams remain the additive units. Thus the Elemental Trigrams are attributed to the six vectors of the tetrahedron, with the opposite Trigrams attributed to opposite vectors. Each vertex is the meeting point of three vectors, and the sum of these vectors is the value of that vertex. Similarly, each face is bounded by three vectors, and the sum of these three vectors is the value of that face.

This is where the positive and negative tetrahedra appear, for there are two, and only two, distinct layouts of the six vectors such that the opposite Trigrams remain opposite. One layout results in the even numbers 14, 16, 22, & 26 being the vertexes while the odd numbers 13, 17, 23, & 25 are the faces. The other layout results in the odd numbers being the vertexes and the even numbers being the faces. The reason that there are two possible layouts is because, unlike the other Platonic solids, each vertex of a tetrahedron is opposite a face, and not another vertex. This allows the tetrahedron to be its own dual, with the values of the four faces and the four vertexes changing places.

These tetrahedra then show the integration of the 6 Elemental Trigrams with the 8 Classic Trigrams to make up the 14 cases of the tetrahedron’s primary topological aspects. Lest we forget the Zodiac Trigrams, these make up the 12 cases of the fourth topological aspect: the vertex-angles.

Each face of the regular tetrahedron is an equilateral triangle: 3 angles times 4 faces equals 12 angles in all. Each of these angles is one of the Zodiac Trigrams, having the value of the two vectors that meet to form that particular angle. Thus the tetrahedron provides the primary model for the interaction of all the Trigrams. Each Element is a single line, a vector. Where two intersect is an angle, a Zodiac Trigram. Where 3 vectors intersect or interlock is a vertex or a face, an I Ching Trigram.

In the Positive Tetrahedron, Trigrams 1, 3, & 9 form the edges of one triangular face, whose value is the 13 Trigram and whose angles are 4, 10, & 12; the three Fire signs. The Negative Tetrahedron has Trigrams 2, 6, & 18 forming the edges of one face, whose value is 26 and whose angles are 8, 20, & 24, the three Water signs of the Zodiac.

Vectors Face Angles of Face Opposite Vertex

Pos. 1, 3, 9 13 4, 10, 12 26

2, 6, 9 17 8, 11, 15 22

2, 3, 18 23 5, 20, 21 16

1, 6, 18 25 7, 19, 24 14

------------------------------------------------------------------------------------------

Neg. 2, 3, 9 14 5, 11, 12 25

1, 6, 9 16 7, 10, 15 23

1, 3, 18 22 4, 19, 21 17

2, 6, 18 26 8, 20, 24 13

It should be noted that the arrangement of the Trigrams can also be accomplished symmetrically on the two remaining Platonic solids, the icosahedron and dodecahedron. In those cases, however, the number of topological aspects is so large that some of the numbers have to be repeated in order to flesh out the matrix.

The trigrams can also be assigned to the cuboctahedron, the figure that Buckminster Fuller called the ‘Vector Equilibrium.’ Although not a Platonic solid, the cuboctahedron is the keystone of all of his Synergetic Geometry.

Archimedean Model of Trigrammaton: The Small Rhombicuboctahedron

Another geometric figure that can display the interactions of the Trigrams is one of the Archimedean solids, the Small Rhombicuboctahedron. This is a 26-sided polyhedron onto which the 26 Trigrams can be attributed. 8 Sides are triangular, and these are the eight I Ching Trigrams, the other 18 sides are square. To see a 3D representation of this figure, follow this link:

http://mathworld.wolfram.com/SmallRhombicuboctahedron.html

The diagrams that follow were inspired by the work of Fr. Zalanes, and revised according to the mathematics of the Trigrams.

The first figure is the rhombicuboctahedron laid out in two dimensions, with the numbers from 1 to 26 arrayed on it. One aspect of immediate note is that all numbers that reduce to the same single digit are in the same row together; e.g. 10, 1 and 19 all reduce to the numeral 1, and are all in the top row.

When folded back into a 3-D shape, numbers that are Antigrams will appear on opposite sides of the polyhedron.

The next figure is the same layout, but using Trigrams instead of decimal numerals.

Antigrams are on squares or triangles of the same color.

The next image shows the letters of the English Alphabet instead of their Trigrams.

(The letter I is placed on 26 to complete the set, but its gematria value is actually 0).

The next image shows the letters of the Futhark Runes. By tradition these 24 letters come in three sets of eight, and so are attributed to the numbers 1 – 8, 10 – 17, 19 – 26. Thus the top row is the first letter of each of the three ‘aett’ of the runic system, and each of the three ‘aett’ has its own column. A lovely symmetry.