Focus your attention for a moment on the elements contained within the yellow triangle below. You can see the rectangle, the pentagon and the hexagon. That's 3, 4, 5 and 6 sided figures. We are reminded of a right triangle with short sides three and four units long, the hypotenuse of which is 5, and the area of which is 6.
The top corners of the pentagon, which appears artificial in relationship to the triangles and heaxgons at this point, are generated by extending the sides of the hexagon (below).
The overlap of the hexagon and pentagon above, generates a rhombus just like the overlapping circles of a vesica (below), which celebrates two other whole ratio triangles. The vesica begins with two circles drawn such that not only are they equal in diameter, but the center of each is on the circumference of the other, and it developes into the flower of life that we saw earlier.
The beauty of the vesica is that at this degree of overlap, all the lines depicted in red below are equal to the radius of these two circles. These five lines generate two equilateral triangles the sides of which have a 1:1:1 ratio. The twin circles symbolize equality but this triangle symbolizes identity. Like the 3-4-5 triangle, the equilateral triangle is easy to remember because all the angles are 60 degrees.
The chord of the circle that connects the intersection points (in blue above) splits two sixty degree corners in half, generating another whole ratio triangle, the 30-60-90 triangle. As you can see from the red lines, the short side of this triangle is half the hypotenuse, a 1:2 ratio. [A triangle with a 26+ degree acute angle has the short sides in a 1:2 ratio. This is the angle of the passages in the Great Pyramid.]
The overlap in this image of the cube produces the same 30-60-90 triangles as the vesica, but uses an overlapping pentagon and a hexagon. Note that the number 5 and the pentagon (or gram) signify the microcosm and the number 6 and the hexagon (or gram) represent the macrocosm.
This Escher print contains the same primal visual elements, the circle, the triangle, the rectangle, the pentagon and the hexagon; and illustrates the microcosm and the macrocosm. If you count the two halfs of the reptile stuck in the mat as one, you get seven of them; like 3,4,5,6,7.
Looking at this figure called the Tree of Life, we see the same elements; the triangle at the top, the rectangle around the central sphere, the pentagon below the triangle, and the hexagon of which the rectangle is a component. Note that this figure differs from the Kircher version of the tree, in that the top of this image is shaped like the letter "A", while the Kircher tree is H-shaped. Also, this tree features a pentagram, while Kircher's tree is composed only of heaxagons.
The 19 circle Cube accomodates both versions of the Tree of Life, but the 13 circle one does not. While the Norse Tree of Life, which features Nine Worlds, can be derived from the 13 circles by simply dropping the four on the corners of the rectangle, but keeping the two at the top and bottom, it was not so easy to make the 10 sphere tree fit.
The solution involved, splitting Daath (the black circle in the left hand figure) into spheres 2 and 3, and de-emphasing that sphere. Traditionally, Daath is not indicated on the tree, in order to preserve the notion of 10 spheres only. The Kircher tree (on the left) uses the centers of two of the circles in the 19 circle cube, while the other places these two on the triangle as indicated above.
And only one form maps to a dodecahedron. Note that the corners of the pentagon (2 and 3) can be found by projecting the hexagon. The line from 3 to 4 runs straight across the position occupied by Daath, and is the same between 2 and 5.
Not only does the tree map to that, but so does the 13 circle cube, once again illustrating the notion of 12 around 1.
Recall that the TOL depicts the descent of the divine spark (soul) from the Throne of God (at 1) to the earth plane (at 10), via the stars (2) and the seven planetary spheres [3-9, this includes the sun (6) and moon (9)]. This arrangement puts the sun at the center of the Tree of Life, as it should be, with the outer planets (Saturn 3, Jupiter 4, and Mars 5) above or beyond the sun, and the inner planets (Venus 7 and Mercury 8, the moon and the earth) below the sun, indicating that whoever fashioned the tree knew a good bit about celestial mechanics.
Looking at the cube, we see that the sun at 6 occupies the "front" corner of that as well as on the dodecahedron, while the other 12 circles map to one of the 12 pentagonal faces of the dodecahedron. This means that the latter figure can be derived from the former by making 12 cuts.
The six outer circles correlate to six faces, three of which face toward us, like 10, and three of which face a way, like 1. Circles at three corners of the hexagon in the tree/cube (Daath, 7 and 8) correlate to faces on the "front side", while the other 3 (4,5 and 9; the ones located where three pentagonal faces come together), correlate to the three remaining faces on the "back side".
While the A-shape tree resolves itself into Metatron's Cube and the dodecahedron, the H-shaped Kircher tree resolves itself into vesicas, and a double cube.