We will proceed in 3 dimensions as we did in two dimensions.
We are Pure Awareness, with the ability to create through thought. Nothing exists. There is no space, no time, no somethingnesses. We think a thought as before, thus creating a POINT and a space between us and it. We take our point and move it about our self until it comes back to the same spot, thus creating a circle.

Figure 1

Then we get the bright idea to 'flip' the circle over ourselves. When we do this we are astonished to have created a uniformly symmetrical cocoon of energy that completely surrounds us. We call it a sphere. The sphere is the basic 3 dimensional shape in this universe (apologies to Buckminster Fuller!)
       
Figure 2
This excellent animation (1) was taken from  the U.S.  Naval Observatory (2).
Even though it is an animation of the phases of the moon, it serves to show how a circle rotated about it's center 180 degrees will form a sphere.
 

Now we can go to any POINT (call it A) on the surface of the sphere and repeat the process.
When we do this something very important happens: the 2 spheres intersect each other. like so:

Figure 3
We have created 2 intersecting points on the outside of the original sphere, at the outer edges of the sphere with center at A Lets call these POINTS B and C.  If this were a 3_D pic you could see more clearly the intersections of the 2 spheres. It forms a football shape called a vesica. Something interesting happens inside this vesica: the mingling of the energies of both spheres creates an excited state of energy. The convex shape of the vesica has an effect similar to that of a magnifying glass placed close to a source of light: it focuses it, makes it more intense.
The POINTS B and C are very important: they make possible the containment of the energy within the field. Whereas a lens is just circular but still focuses light, the vesica with its terminating POINTS B and C  helps not only to focus the energy with the vesica to an excited state, resulting in the creation of a more powerful energy than in the original two spheres, but it also creates points of transmission and enhanced consciousness. Remember POINTS are conscious! The POINTS B and C are where the entire energy 'package' is intensely focused. The vesica then is an enhanced or excited energy state delegated by the 2 terminating POINTS. The vesica is, as it were, a field of consciousness!
These 2 POINTS are  like singularities. They have the property of sending or receiving a focused beam of energy.
Also, the 2 points act somewhat like the 2 ends of a standing wave: If you take a wire and tie it down at both ends, then give it a rap, it will begin to vibrate and look like this:
[pic of standing wave 1 octave].
We will see later on how the intersection of spheres forms vesicas of smaller and larger dimensions, resulting in an almost infinite variety of vibrations of energy.

In other words,  the energy within the vesica is focused by the 2 POINTS B and C,  resulting in the creation of a more powerful energy than the energy contained within the 2 spheres originally. This is an example of synergy, where the interaction of 2 or more elements results in something which is greater than the sum of its parts. The vesica is a transmitter of energy, but it can also be a receiver of energy. It's no coincidence that the eye socket of the human being is shaped like a vesica.
 

OK.
Going to B and in turn C, creating spheres as we go along, we find D and E.


Figure 4
If we continue around the the original circle and draw spheres at D and E, we find one more point F.
Then drawing a circle at F we complete the circuit:

Figure 5

Just as we found in 2 dimensions, we have created a hexagon!  It is even cooler in 3 dimensions because we can see that the intersections of the spheres form egg-shaped enclosures called vesica's.
We can now go out to one of the 6 spheres we created on top of the original sphere at A, and do the same thing. In fact, we can continue this process indefinitely. Apparently  the ancients knew about this pattern because it appears (from the pink circle inward) in temples from Egypt to Ireland (3).

Figure 6 -- the intersecting sphere pattern
Notice that there are 2 sizes of vesicas in the pattern above:  the smaller (e.g. from O to F)
and the larger (e.g. from O to C to A to B). The ratio between the long axis (CB)
and the short axis (OA) is  \/¯3.  The ratio between the long and short axis of the smaller vesica's is
2 +  \/¯3.   (See  Maths Page ).

Figure 7  -- a vesica, the intersection of 2 spheres

Notice the sine waves that are interwoven throughout the structure. These sine waves, combined with the nodes at the ends, form standing waves. As you can see there are  wave patterns for both the smaller and larger arcs. Of course all of this is in 3 dimensions really, so the wave patterns go all over the energetic "surfaces" of each of the vesicas. These are the paths for energy interconnection. Remember we said earlier (2 Dimensional geometry basics) that the curved line represents something-that-can-be-perceived, since there is constant change of direction. Within this structure there are an infinite number of possible paths, which mean an infinite number of direction changes. This gives the structure its persistence and creates time, since the consciousness within it is constantly exploring itself.
Of course we have drawn all of the spheres with a radius of length OA. Now that we know how to do this, we can re-do the entire thing with spheres, say, of radius 1/2 OA. And 1/3 OA. And 2 * OA, etc. to infinity,. superimposing all of the new patterns of intersecting circles over each other. Imagine all of the intersections, all of the vesicas formed, all the excited and activated energies within each of the infinite number of smaller and larger vesicas that will form! The entire structure is totally dynamic and self-sustaining, because each of the POINTS which comprise all of the nodes are conscious, and the vesicas activate, transfer and reactivate the energy of consciousness within the entire structure. All of the energy is continually hitting nodes, so the POINTS  can dampen down the energy level or increase it. So the structure is self regulating.
The frequencies and amplitudes of all of the standing waves vary with the radius of each of the spheres. So there can be an infinite variety of wave frequencies and wavelengths within the structure.
This pattern of intersecting spheres may seem physically impossible until you remember that physics has shown there are no continuous surfaces. In the standard scientific model of the atom (which I believe is a metaphor) all atoms are 99.9% space: a nucleus surrounded by a probability cloud of electrons. The solar system is a good example of an atom: if you calculate the volume of space of the solar system and compare it with the volume of space occupied by the masses within it (sun and planets), the masses occupy a trivial portion of the overall volume. And then consider that these masses are not 'solid' but are themselves composed of molecules and atoms which are  themselves 99.9% space. So what appears as 'solid' matter to us isn't really solid!
If you remove every sphere with a center on the surface another sphere, you have a pattern of tangent, or 'closest packed' spheres. This pattern of tangent spheres is Buckminster Fuller's Isotropic Vector Matrix.
Buckminster Fuller said that there are no straight lines in nature and he was right. He claimed that all lines which appear straight are in fact  curved lines  with a very large radius (505.02, 555.21). Fuller claimed these lines were discontinuous because of quanta  energy packing, (thus explaining the wave/particle duality).  He further claimed that the basic unit, or quanta, of energy in the universe is a tetrahedron, and that all straight lines are actually tetrahedrons with one of the vertices pulled away from the other three. This may very well be, but if so, these tetrahedrons are curved, or spherical tetrahedrons.
Whether the curved lines in the intersecting sphere structure are discontinuous or not, they are still curved!

The pattern as it is drawn here only shows spheres in the xy plane, or across the screen as you are looking at it. In 3D there are spheres stacked above and below the drawing surface. In 3 dimensions the vesicas look like footballs going into and out of the drawing surface. Below is a crude representation of what one of the large vesicas might look like in 3D:

Figure 7B -- the blue 'football' (vesica) comes out of the page and down into the page in  3 dimensions.
 

Figure 7C: If you imagine the spheres coming out of the page and going into the page, you will have the pattern in 3D. Notice that the spheres in the xy plane (on the computer screen) are offset 60 degrees from each other, relative to the origin, O.
In 3 dimensions, of course, the pattern comes up out of the screen and down into the screen.
In 3D, the spheres are also constructed so that the angles between the origin O and the spheres on the positive and negative z axis (up and out, down and out  of the computer screen) are also 60 degrees.
Figure 7c shows Fuller's 3D Vector Equilibrium (Cubeoctahedron). The top of the cubeoctahedron is shown in bright yellow, the bottom half in lighter yellow.
You may be able to visualize this if  you imagine 6 spheres coming outward toward you, with centers at  F3, H3, J3, L3, K3, and I3, (sitting 'above' the pattern): 
Figure 7C -- the cubeoctahedron in 3D
 

Another interesting thing about the pattern is that in its 2 dimensional representation in Figure 6, it can be considered a diagram for a series of intersecting 3 dimensional torii (see also Figure 7D below). For example, in Figure 6,  the circle at A and the circle at F are tangent at O. But if one of the circles rotates 180 degrees around  a line JOI, it becomes the other circle and forms a horn torus.
Similarly, because the pattern is entirely symmetric, any 2 non-tangent circles may rotate to form a ring torus ( a doughnut, basically).
And any 2 intersecting circles (forming any of the large or small vesicas) may rotate around the axis formed by the endpoints of the vesicas to form a
spindle torus.
Because rotating torii form vortexes, this pattern can be seen to be able to funnel energy from just about anywhere to just about anywhere!

(5)
Figure 7D -- the three standard torii, showing cross-sections. Note that these cross-section circles exactly match the circles in the intersevcting circle/sphere pattern. The 3 dimensional vesicas can clearly be seen in the spindle torus.
 
A ring torus is formed when a circle rotates around an axis. If the rotation is clockwise, energy will come into the torus at one end, forming a vortex, and exit through the other end. This will form a vortex with energy flowing in one direction only. But if we postulate another, counter-rotating circle superposed over the original, we get energy flowing in exacrty the opposite direction, meeting at the center and forming a stable mass. Perhaps this is how matter and energy is formed in the universe! When a torus rotates, then, energy is flowing in and out of the torus, rotating clockwise and anti clockwise, forming a spherical, toroidal field.
If one postulates the intersecting sphere pattern filling the entire universe, one can see that it is possible to form super-galaxies as well as the tiniest subatomic particles. In this formulation, all matter and energy is vibrational, and all matter and energy is surrounded by a spherical-toroidal field of energy formed by counter rotating energy formed from the pattern. So all things in the universe are alive, for they are all surrounded by a nurturing field of energy which creates the object, and which is, at the same time, "breathing" in and ejecting (refreshing itself) from the universal medium!
Here is a crude diagram:

We will show in the next diagram how the 5 basic regular polyhedra (including the tetrahedron) can be formed within the structure:

Because the pattern of intersecting spheres is inherently hexagonal (triangular) and based upon the \/¯3, it is not immediately obvious how we can get to structures like the pentagon, which involve the \/¯5. The \/¯2 (and thus the square and the cube) can be derived easily from this pattern, because it just involves the bisection of  2 of the central angles within the hexagon pattern, as we saw earlier:  (e.g., bisecting <COF and <BOD is trivial because of the nodes J and I, see Figure 6).

Figure 8  -- roots
The   \/¯2 is CD in red. It's just the side of the square CDHG. Notice that it easily fits into the pattern because the points G and D are simple bisections of the line CH which comes from the hexagon (in purple).
The \/¯3 is line CB, which is the long axis of the vesica COBA, which we demonstrated above. It also fits easily into the pattern because it's just 2 vertices (C and B)  of the hexagon.
But the \/¯5 (CL in orange) is different. The ending point for it is somewhere ugly.  In order to get it we need the double square CMDO and ODLH. CL is the hypotenuse of the triangle CHL. CH is 2, because the radius CO =OH = 1. and HL = OD = radius = 1, so we have a 1,2,\/¯5 triangle.

The golden ratio is itself built upon the \/¯5. Life is built upon the \/¯5! If you look into the structure of DNA you find the pentagonally-shaped deoxyribose-sugar component of the nucleotide, the basic building block of DNA:

(4)
Figure 9  - the nucleotide
As you can see from the above pic, the phosphate group (to the left)  is square, and the nitrogenous base is hexagonally shaped.
The nucleotide combines \/¯2 geometry (the square phosphate group), \/¯3 geometry (the hexagonally shaped nitrogenous base) and \/¯5 geometry (the pentagonally shaped deoxyribose-sugar), but the basis for its structure is the centrally located deoxyribose-sugar!

Life, at least human, plant, and animal life,  is an approximation! Just as the golden mean must be approximated by nature in the Fibonacci sequence, so to must DNA itself be a 'stretch' in the fabric of existence. It is the division in extreme and mean ratio which can never be precisely nailed down which gives life its quality of tension and dynamic growth.
You can see from the pattern of spheres how easy it is to get growth triangularly, cubicly, hexagonally -- snowflakes, minerals, crystals, etc. all grow effortlessly on this pattern.
(For a detailed description, with pics,  of crystal lattice structures, and thus life forms based upon \/¯2 and \/¯3 geometry, see the fantastic web site at  <http://www.kings.edu/~chemlab/vrml/index.html>.)

But pentagonally-based life as we know it must think real hard and 'reach' for an uattainable goal -- the perfect division into mean and extreme ratio in order to form the pentagon. Pentagonal designs do not 'fit in' precisely  to the structure of existence!
I believe that this is/was done on purpose. It is what gives life it's dynamic and yes, its quality of tension-- the constant search for perfection. That is what creativity is.
In other words, pentagonally-based life has an inherent element of tension built into it, unlike other life forms which effortlessly replicate along the easy-to-find nodes of the intersecting spheres.
The fact that tension generates energy is found in tensegrity. Tensegrity structures generate energy internally and are immediately responsive to changing conditions. (See  The Geometry of Harmony ).

Buckminster Fuller found that many polyhedra can be easily found within his Isotropic Vector Matrix, but the icosahedron and especially the dodecahedron caused difficulties. Fuller used the 'jitterbug' to describe how the cube-octahedron transformed into the icosahedron (which although basically triangular is pentagonal as well) but never could  incorporate  the dodecahedron, which is composed of 12 pentagons. This is because the IVM is incomplete in  the sense that the pentagon is a wild card and will not fit precisely into it.  When we get to the dodecahedron we will see how it is formed 'ugly' within the structure of intersecting spheres.

Why do we say it is impossible to perfectly construct pentagonally-based forms?
From A Geometric Construction of the Golden Mean Spiral we see how easy it is to divide any line segment in Phi ratio. Therefore, it should be possible to build perfectly harmonious life forms based upon the pentagon! (I am reprinting this sequence for clarity):

Take any line AB:

Now, bisect the line AB and transfer that distance to BC:

BC is one-half AB, and is perpendicular to AB.

The difficulty arises when we try to precisely find the point E! We can never quite get a perfect location. We're always off just a little bit in practice from our theory. Perhaps this is the reason Nature seems to prefer working in whole number increments.

OK. Now we will use the intersecting sphere pattern to produce (as best as I can here) the 5 regular polyhedra -- tetrahedron, octahedron, cube, dodecahedron, icosahedron.
We will take out some of the spheres for clarity, otherwise the drawing would be too busy.

First, the tetrahedron and the cube . Here we show 2  tetrahedrons contained within a cube:

The first tetrahedron is in yellow.
The vertex at O either comes out of the screen or goes into the screen depending upon how you look at it. The second tetrahedron is outlined in green and blue. You can see the cube surrounding the tetrahedrons (look at vertices Q,V,O, L0. That is the top face of the cube. The bottom face is at O,Y,D0,G0). The cube is unstable and if you build one with the edges hinged, it will collapse. A cube with 2 tetrahedra inside of it fully braces and supports it.
The tetrahedron is the simplest regular solid in the universe and has been used as the basis for constructing a photon. Fuller used it as the basic quanta in the construction of the universe.
However, we can see from the pattern that the circle and the sphere are prior to it. All things which persist must be curved, as we have pointed out before in earlier essays. Perfectly straight lines cannot exist in nature because they do not persist. Therefore, all polyhedra (3 dimensional solids) are actually relationships between the nodes. The faces and the edges don't really exist except in our minds, as a way to help us see how nodes of energy come together.
As we saw at the beginning of this essay, the sphere comes from the circle and the circle comes from the decision of consciousness to occupy a location and rotate a point of awareness around itself. Therefore, the cretaion of all Forms proceed from the circle and the sphere.

Next, the octahedron: (the orange sides are to the front. The blue sides are at  the back of the figure)

The octahedron again fits precisely into the pattern.
 

Now, the icosahedron:

It looks like the points fit nicely into the pattern at all of the sphere centers, but this is not so. QY, for example, is divided at L in extreme and mean ratio, and QV is actually divided in EMR as well, but this pic doesn't show it, because QV actually travels down further into the sphere than I can show here. This is true for all of the vertices of the icosahedron, except for the top and bottom POINTS at Q and D0. All of the coordinates of the icosahedron involve Ø.

Even though the Icosahedron does not fit directly into the pattern, this Phi based Platonic Solid can be generated from the Octahedron or the Cube, which fits perfectly.  The Icosahedron can be generated from the Octahedron by dividing its sides in Mean and Extreme (Phi) Ratio:

The Icosahedron can be generated from the Octahedron which fits perfectly into the Pattern.

Why does the icosahedron have problems fitting into the sphere pattern? Because it is  composed of pentagons!
All of the faces of the icosahedron are equilateral triangles (for example, NPL) but you will notice that it can also be considered to be made of pentagons. JKHIG is a pentagon, for example, and so is OHRNG. In fact, the Icosahedron is made up of pentagons AND equilateral triangles!
The ratio of the radius of the icosahedron to the length of its side involves Ø. More on that when we analyze the icosahedron in detail.
The icosahedron has a pentagonal sub-structure, and so it is one of those polyhedra associated with life. The icosahedron is always striving toward perfection, toward that perfect division into EMR, and so contains the internal dynamic tension common to all biological life.
Fuller has shown that if you remove the central sphere in an arrangement of closest-packed (tangent) spheres called the vector equilibrium or cube-octahedron, the icosahedron can form nicely from all of the sphere centers. He called this the 'jitterbug' transformation. Perhaps this is how life gets the idea of how to make the pentagon.

Finally, the dodecahedron:

The intersection of the LINES of communication between sphere centers create more POINTS which can be used to build the dodecahedron.
These lines divide themselves into EMR, but never of course perfectly. You can see that the vertices of the dodecahedron are not at easily identifiable places within the structure.
The dodecahedron is composed entirely of pentagons. It is therefore the most dynamic of all of the regular polyhedra. There is more energy within the structure, more tension, as it constantly strives for the perfect division.
The ratio of radius of the sphere enclosing the dodecahedron to the length of the dodecahedron's side also involves Ø.  Somehow this structure must be intimately associated with biological life.

The Dodecahedron can also be formed indirectly from pattern by using the Cube, and it's vertices can be found by orienting 5 Tetrahedrons, like so:

The main idea is that we can make any possible geometric shape, either directly, or indirectly, from the intersecting sphere pattern. (I have just shown the 5 regular solids here).
Even though the dodecahedron and the icosahedron must always remain slightly imperfect (or any shape which has  Ø within its makeup), by altering the radius of the spheres and superimposing  interlocking sphere patterns one over the other, all of the forms we observe in existence can be constructed.

(0)  Geometric drawings done using R. Parris's free 'peanut' software at http://math.exeter.edu/rparris/default.html
(1) GIF animation by Ed Stephan (stephan@cc.wwu.edu)   from U.S. Naval Observatory page at:
(2)  http://tycho.usno.navy.mil/vphase.html
(3) ref "The Ancient Secrets of the Flower of Life" by Drunvalo Melchizedek.
This is a very interesting and informative book, but rather on the wild side. I enjoyed the geometry in it, but I can't say I subscribe to some of the way-out theories presented as explanations for earth history. You will have to judge for yourself the validity of the information. I highly recommend it as a way to loosen up your mental horizons, and to make you question  the  dogmas of present day science. The book is Part One of a series of lectures describing background data to a very powerful meditation, which I have had very good results with.
(4) from http://www.dnaftb.org/dnaftb/ Section 15 of  'DNA From the Beginning'.
(5) from http://mathworld.wolfram.com/Torus.html, "Eric Weisstein's World of Mathematics"