THE NEXT DIMENSION OF   page6 As a result of suggestions from various sources I ended up looking at the geometry of this formation with the 3rd dimension added. This means looking at it as a 3-dimensional shape seen from above. The circles become spheres and the equilateral triangle becomes its 3D counterpart, the tetrahedron (a three sided pyramid, where all of the sides, including the bottom, are identical in shape). The images below show the 3D shape I modelled from two angles, the first from top.  The diagram looks complicated, but don't worry. The explanation makes it understandable (I hope!). If you are not interestic in mathematics, you can see the ANIMATED PRESENTATION page8  <--side view from this side Of course since we only see the geometry from above, we can't directly see the vertical positons of the elements. At first I thought perhaps the tetrahedron fits inside the large sphere (E). Unfortunately it didn't. However, then I noticed that the smallest sphere in the middle (A) happens to fit perfectly between the 3D center of the tetrahedron (X) and the bottom of it (a). Based on this discovery I built a 3D model so that the three concentric spheres A,B and E (visible in the formation) are also concentric in 3D and the tetrahedron is placed according to sphere A. As you can see, then the bottom corners of the tetrahedron touch the inside surface of sphere E. Since this solves the "mystery of the too large circle" (see the bottom of page5), I thought I must be on the right tracks!
 Now it happens to be that the distance between the tetrahedron center (X) and its bottom is one quarter of its height (which I didn't know before, so I'm learning a lot about tetrahedron geometry here myself!). In other words, the height (h) is four times the diameter of sphere A (h=4xA as seen in table 1 on the upper right corner of the diagram) . The height of the flat equilateral triangle (a, see the small image in the lower right corner) is three times the diameter of sphere B (a=3xB, table 1). So, now we have a connection from both of these spheres to the tetrahedron. Surprisingly it happens to be that the size of B can also be concluded straight from A through a rectangle drawn around it. This means the diameter of B is A multiplied with the square root of 2! The rest of the geometry unfolds in a very straight forward manner. It was natural to assume the three middle-sized spheres (C) would be positioned symmetrically on the corners of the tetrahedron, so that they touch sphere B. Sphere D is a reference for the center of these. The following astounding discoveries can be made (most of them match the discoveries on page 5): The diameter of C is twice the diameter of B. The diameter of D is 3 times the diameter of B. So the sphere C center's (Cx) distance from the spherical center (below X) is 1.5 times the diameter of B. The diameter of D is exactly the same as the triangle's height (a), which is 3 times B. This height is also 1.5 times the diameter of C, which means that C fits perfectly inside the triangle as seen in the small top view. If you don't believe, you can measure all these for yourself! In table 1 the important angles are presented. They are 19.5 degrees and 35.25 degrees. In the lower left corner of the tetrahedron you can see how alpha plus two times beta makes 90 degrees. If you are familiar with Richard Hoaglands studies of the Mars Cydonia geometries, this 19.5 angle might ring a bell! It appears to be an important angle in sacred geometry. In table 2 various relations between the spheres and the tetrahedron are listed. In table 3 you can see how the sizes of each sphere relate to sphere A. Here you see how the 3D shape seen from top matches the formation. See the animated presentation Return to the 2D analysis   