Plato's Lambda Part 6

Correspondence between the Tetrahedral Lambda and the inner form of ten overlapping Trees of Life

Figure 9. 350 corners are intrinsic to the 70 polygons enfolded in 10 Trees of Life.

The following considerations indicate that it is not coincidence that the sum of the 20 numbers of the Tetrahedral Lambda is equal to the number of corners intrinsic to the 70 polygons (Fig. 9) enfolded in 10 overlapping Trees of Life:

the first two types of polygons — the triangle and square have five green corners, so that the 20 triangles & squares enfolded in 10 Trees have 50 green corners. This is the sum of the integers at the centres of the four faces of the Tetrahedral Lambda. The sum of the integers at the centres of the first & fourth faces is 30 and the sum of the second & third faces is 20. This 30:20 division corresponds to the 30 corners of the triangles and the 20 corners of the squares;

the next four types of polygons — the pentagon, hexagon, octagon & decagon have 20 blue corners, so that the 40 such polygons in 10 Trees have 200 blue corners. This is the sum of the 12 integers on edges of the tetrahedron between its vertices;

the last polygon — the dodecagon — has 10 red corners, so that the 10 dodecagons in 10 Trees have 100 red corners. This is the sum of the four integers at the vertices of the tetrahedron.

Eight other combinations of polygons are possible, although they are less natural because their numbers of corners do not increase with the order of sequence: 50→200→100. Which combination is the correct one is irrelevant to what is being demonstrated here, namely, that chance cannot explain why the same number 350 appears in the Tetrahedral Lambda and in the inner form of 10 Trees of Life because, if it did, it would then have to be also coincidental that subsets of types of polygons exist that generate the three types of numbers making up their sum 350 — a circumstance that is highly improbable.

Ten overlapping Trees of Life result from representing each Sephirah by another Tree of Life. The inner form of this holistic object is parameterized by the number 350 as the number of corners of the 70 polygons that belong solely to them. The actual number of corners of the 70 polygons is 351. But, as the top corner of the hexagon enfolded in the tenth Tree coincides with the lowest corner of the hexagon enfolded in the eleventh Tree, the number of corners intrinsic to the 70 polygons is 350.The implication of the correlation between the properties of this object and the features of the Tetrahedral Lambda is that the latter is an archetypal array of numbers that is the arithmetic representation of holistic systems. Its first face has long been known to generate the tone ratios of the notes of the Pythagorean musical scale as the ratios of its numbers, so it should not be too surprising that the Tetrahedral Lambda and its geometrical counterpart should refer to 10 octaves of this scale, as was pointed out in the section The seven musical scales (see here).