Metatron's Cube and the Tree of Life
Actually, the first version of the ten sphere Tree of Life image was published before Kircher's, but it is not well know or displayed. The next image was the bookplate from Portae Lucis (Gates of light, 1516) which was a Latin translation of "Shaarey Orah" written by Rabbi Joseph Gikatalia (1248-1323).
You will notice that, like the Kircher tree, this image features a hexagon with the top corner missing at the center; but unlike that figure, this tree has a pentagon at the top, instead of another hexagon. This image is also derived from Metatron's Cube, but when Daath (at the center of the pentagon) was split, the resulting spheres (2 and 3 below) were only moved as far as the edges of the triangle.
[*Side Trip -
Enoch is said to have hidden a white cube, engraved with a triangle and the name of God. The name of God, JHVH (Jehovah), adds to 26 (J =10, H=5, and V = 6). Looking at the triangle and Tree of Life image at the center of Metatron's Cube, we can see that the total for the numbered spheres on the middle pillar of the tree is 26 (1+6+9+10).
This version of the Tree of Life, like Metatron's Cube, is built around a hexagon and an equilateral triangle. Sphere Six, at the center of the cube/tree, is located where the diagonals of the hexagon cross. The equilateral triangles are formed by connecting every other corner of the hexagon. The tree can be drawn in any of six directions on the cube.
For future reference you should note that the position of sphere 2 can be determined by "sighting" from 5 across Daath, and the same is true for 3 and 4; that is, the position of the top corners of the pentagon is determined by extending the sides of the hexagon at the center, and Daath lies where the diagonals of that pentgram cross.
Notice how, like the yin yang symbol which has white at the center of the black half and black in the center of the white half, in this tree form one corner of the pentagram falls at the center of the hexagon while one corner of the hexagon lies at the center of the pentagon. These symbolize the macro and micro-cosms.
To me it is a more elegant symbol than the Kircher tree. Consider how when the spheres 2 and 3 on the tree fall on the triangle, we see within that tree, the triangle, a rectangle, a pentagon and a hexagon; that's 3, 4 , 5 and 6, which reminds us of the triangle with short sides of 3 and 4, which has a diagonal of 5 and a total area of 6. We see all the same elements in this MC Escher print.
We see circles, a triangle, rectangles, pentagons, and hexagons; and if we count the lizard that is half in and half out of the map, we get 7 of those. I am counting the one on top of the dodecahedron as number 5. The juxtaposition of the macrocosm (hexagonal mat) and microcosm (dodecahedron composed of pentagonal faces) is the main theme though, just like in the Tree.
We add another layer of meaning when we see the hexagon as a cube, Plato's symbol for the element earth and the dodecahedron, as the aether. Remember that Plato said that God embroideried on the dodecahedron.
In the left panel of the image below we see how we can imagine either a cube or a dodecahedron in the Cube figure. Note how the corners of each of these two triangulate, and how three of the circles of the cube fall at the centers of the three faces at the corner. The other three of the inner ring of six circles correspond to the three faces on the back side of the dodecahedron.
On the other side, there is a corner just like the one facing us now, except that the three faces there form an upside down triangle. This means that Metatron's Cube of thirteen circles maps to the point and the 12 pentagonal faces of the dodecahedron; BUT six of the faces are on the back side, so it does not literally wrap around it.
The middle panel shows the hexagon that is generated by marking these points at the centers of these six faces (3 on the front and 3 on the back). As you can see, the resulting cube overlaps three pentagons. Notice how extending the hexagons points to the corners of the pentagons.
Compare the overlapping pentagons and hexagons in the cube/tree to the geodesic dome.
The last panel shows how the A-shaped Tree of Life maps to the dodecahedron. This means that it takes 5 of these tree figures to cover the dodecahedron.
Note that the triangle, pentagon and hexagon are essential to this whole complex of connected images, and that this does not work with the Kircher tree, since it does not feature a triangle or pentagon.
A Double Cube
The tree easily maps to both the cube and the dodecahedron, while mapping the cube to the dodecahedron is not so easy. Conceptually it's easy enough, because the thirteen circles correlate to the central corner and the 12 faces of that solid figure. But visually you have a problem in that you can only see six faces at a time, meaning that 6 circles represent visible faces and 6 represent "hidden" faces.
In the normal presentation of Metatron's Cube, three of the outer ring of six circles, and three of the inner ring of six, represent pentagonal faces on the back side of the dodecahedon.
Mapping to the back side of the dodecahedron requires the use of a "double cube" of sorts. This figure is something that you could use for a paper model to wrap a dedecahedron in, so that each of 12 circles would fall on each of 12 faces, with the central circle indicating the corner, both front and back.
Some of you will recognize the grid on the cube below as the same as that which was used by cathedral builders to place the elements in their elevation plans. The lines themselves are refered to as "regulating lines", and the method is known as "polygonal projection". The hexagon image produces square (90 degree) angles (blue), 60 degree angles (purple) and 30 degree angles (red) for what was called "ad triangulum". Ad quadratum uses diagonal squares, and features 45 degree angles.