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Author's Note: I have been studying the work of Buckminster Fuller for a few months now, I have become deeply impressed by the depths of his understanding of the many facets of nature. His book Synergetic's I and Synergetic's II documents the discovery of 'natures own coordinate system'. Synergetic's is subtitled 'the geometry of thinking' suggesting that thought itself reflects natures coordination's. This geometrical magnum opus claims to provide a system for accounting in both the physical and the metaphysical aspects of reality. I feel that the ideas and discoveries put forward by 'Bucky' Fuller through his 57 year 'experiment', will prove to be a major catalyst for future progress in bridging the divide between the sciences and the humanities. What follows is a brief description of my thoughts and findings on the topological aspects of space and number as relating to the systematic approaches laid out by Fuller in his work. In my own explorations I have made some fascinating discoveries that I wish to continue to explore, note that, what follows is written by someone who has only begun to scratch the surface of Fuller's extensive body of work and It's safe to assume that these ideas will be greatly modified and refined as I continue in my explorations.
Topology of 0 Topology deals with the nature of space, mathematics can be described as the science of structure. If you feel that you already know all that there is to know about the number zero, then you're in for a surprise. Through a blend of topology and basic algebra, the language of number will be shown to defy the foundational axioms of mathematics, this connection between number and space will allow us to explain the surface of nothing. Closest packing of spheres In synergetic's Fuller describes a coordinate system based on spaces own basic constraints, a system of closest packed spheres produces a means of describing spatial coordination in the most natural way. The inherit limits of packing unit radius spheres together produce a means to describe the properties of volumetric space. Despite the overwhelming evidence, science does not recognize this natural system of coordination and feels content with sticking to awkward conceptual abstractions in their descriptions of nature. Fuller insisted that the synergetic geometry he described, was how nature operated, as opposed to the mechanical, rectilinear XYZ coordination's and mathematical abstractions still in use today. There are two ways to pack spheres in the most dense, all space filling formation, the Face centered cubic (ffc) and the hexagonal close packing (hcp). The 'ffc' sphere packing system gives us the most symmetrical arrangement, providing a lattice familiar to crystallographers. Chemical bonds, proteins, virus shells etc. have all been shown to correlate with the patterns that occur from 'fcc' closest packing of spheres. Applying the three topological characteristics of polyhedral systems (points, lines and faces) to our sphere packing arrangement, we can say that the sphere centers represent the points or vertices, interconnecting the sphere centers crosses the points of tangency between adjacent spheres (where the spheres 'kiss') represented as vectors or lines and the areas between spheres represent the faces or openings. Now we can represent sphere packing arrangements as polyhedral shapes, with the IVM being the total all space filling representation of our ffc sphere packing. The IVM can be represented as a lattice of rods connecting the centers of spheres of equal radius where every sphere is a nucleus surrounded by 12 others producing the cubeoctahedron polyhedron. The rods define an array of both tetrahedral and octahedral shapes which is known in architecture as the octet truss, the 60degree angles make it the most stable lattice possible. This 12 spheres around one, represents what Fuller called the Vector Equilibrium, it describes a state where all radial vectors and all circumferential vectors are in equilibrium i.e. each of the vectors are of equal diameter.
This can be thought of as the moment where the explosive (radial) and Implosive (Circumferential) forces are completely balanced, this is one reason why the VE represented an important structure in Fuller's synergetic geometry. If you started off by close packing 12 equal radius spheres around one central sphere creating a cubeoctahedron/VE shape, you could continue packing more and more spheres, nesting the spheres in the valleys created by the layer below. If you continue this process you would not end up with a large spherical shape, you would find that you now get a larger more sharply defined (higher frequency) cubeoctahedron/VE shape. Vector Equilibrium growth rate Developing on the topology developed by Euler, Fuller used the equation 10F²+2=N to calculate the concentric growth rates of spheres in the outer shell of the VE. This became helpful for virologists who needed to calculate the nodes on the shell of virus capsids, the virus shell is Icosahedron in shape and the Icosahedron has the same number of 'spheres' as the VE. This elegantly simple formula describes beautifully what could be called the 'shape of space', an arithmetical description of how nature operates in the most economical way.
10F²+2=N Frequency represents the subdivisions along any vector edge, you can think of this as the spaces between spheres in closest packed spheres. For example, if you rack together a triangle of billiard balls, you can count the spaces between each ball along the edge of the triangle, if there are five balls in each edge, then the spaces between the balls would be the frequency, in this case it would be four. Frequency is always one less than the number of balls along a shapes edge. You can also think of this as representing how often the shape has been subdivided, like halfway cuts along each edge, Fuller called this the frequency of modular subdivisions. By invoking this method we are using what we call, multiplication by division, starting from the whole and proceeding to subdivide unity. Frequency also correlates with the number of layers of spheres surrounding the central, or nucleus sphere in the VE configuration. This proved helpful for me in visualizing the sphere packed shells. This interior counting of frequency is very much like how we evaluate the age of trees, by looking at a slice through the trunk and counting the concentric rings outwardly from the center. In a very meaningful way, this method is giving us the frequency as the age of the tree counted in years. The graphics below show the ways in which the frequency of VE's can be read. By using the equation 10F²+2=N you are calculating spheres on the surface, or outer shell of the VE. The second powering of the F² gives us the frequency growth rate at the surface level, just like in the equation E=MC², the second powering is giving us the speed of radiation not at a linear rate, but 'growing' at an omnidirectional surface rate. Fuller pointed out that first powering related to vector edge growth rates(i.e. linear), second powering related to surface or area growth rate and third powering was the volumetric growth of the system. The plus 2 in the equation is present in all topologies that are dealing with points or vertices of self enclosing volumetric systems. As Fuller pointed out, the plus 2 can be thought of as the poles of the system, without which the system would not close back on itself and would instead, 'trail off to infinity'. I think that this is very much related to, what Fuller called 'the principle of angular takeout', which was originally noticed by René Descartes. Briefly explained; "In every polyhedral system, the sum of the angles around all the vertices is exactly 720 degrees less than the number of vertices times 360 degrees, or (360° × V)  720°. True for the tetrahedron, true for the crocodile. In Fuller's words, every system has exactly 720 degrees of 'takeout.'" ~ Quoted from "a Fuller explanation" Finally the multiplication of the 10 at the start of the equation gives us our starting point, as the first frequency of the VE shell has 12 spheres in the outer shell I am speculating that the 10 puts a minimum frequency set to one. I see the 10 as representing one complete 'cycle' in the growth rate, relating to the one complete cycle of the first frequency, with it's 12 spheres around one. I had been playing around with this equation shortly after I got interested in Bucky's work, I arrived at a similar realization to Fuller's in relation to a 0 frequency VE. Soon after coming to my own conclusions, I found Fuller's own studies on the 0 frequency VE. He pointed out that by setting the frequency value to 0, the plus 2 of the equation always made the VE have a minimum value of 2. "415.10 YinYang As Two: Even at zero frequency of the vector equilibrium, there is a fundamental twoness that is not just that of opposite polarity, but the twoness of the concave and the convex, i.e., of the inwardness and outwardness, i.e., of the microcosm and of the macrocosm. We find that the nucleus is really two layers because its inwardness tums around at its own center and becomes outwardness. So we have the congruence of the inbound layer and the outbound layer of the center ball. 10F² + 2 A new perspective. When I experimented with this fascinating equation, I started off on a different route than that of Fuller. I need to explain some of the logic that I used when I played around with the VE shell growth rate equation. The first frequency of the VE has twelve spheres touching one central sphere, equating the cubeoctahedron with the ability to have room for a nucleus got me thinking, what polyhedral shape would constitute a sphere packing arrangement with no room for a central sphere? The answer was simple, a tetrahedron would constitute a zero nucleus volumetric system because it is the minimum volumetric system possible. Look again at the way we can evaluate the value of frequency with the shapes ability to have a unit radius sphere beneath the surface or shell. Visualizing the tetrahedron as four spheres packed closely together, I was convinced that this represented the volumetric shape with a zero sphere nucleus. Using a single sphere by itself would constitute a single vertex and therefore would not be a volumetric system. Being unaware of Fuller's experimentation with the formula, I am still convinced that using the tetrahedron as a VE with a zero nuclear sphere arrangement is justified.
Then I plugged in the known outer sphere shell value of four, into the equation for a zero nucleus shape to see what happened. But, not before removing the 10, which, as I touched upon earlier, must correlate to the presence of one central sphere and relates to the one complete cycle of the first frequency, so, as we're not accounting for a central sphere, we can remove to multiplicative 10 for now. 10F²+2=N Placing in the known values for the zero nucleus and the four 'outer' spheres, we get Therefore 0²=2 I immediately stood up, it felt like a thousand thoughts went rushing through me all at once, what a rush !! This went against everything we've been thought, this challenges our fundamental understanding of basic algebra. I eventually came down from the excitement of that early realization and got together some strategies for explaining what had happened. There was never a thought that this might be wrong, my intuitions made it clear, this was how it really was, in a very real sense, I was never more sure of anything else in my life. The logic of how I arrived at this, was and still is, telling me that this is the case, 0² is 2. But there was never a feeling that I had made some 'new' discovery, it felt like this was something that was staring us in the face, it was already known, and that deep down, subconsciously, it was always known. Needless to say, the implications of such a fundamental behavior of number, would require a lot more than a simple explanation on my part. Although I have already made some progress on applying this new perspective and found some more interesting characteristics and correlations within other number behaviors. I can however, relate it to the insights gained from exploring some of Fuller's work. Fuller's experimenting with the VE at a frequency of zero showed that the inherent twoness of the central sphere related to its convex and concave surface. From my experiment with the VE equation, I can say that just like the second powering of the speed of light in E=MC² describes a spherical surface, 0²=2 is showing that the 'surface' of 'nothing' has both a convex and a concave, i.e. it has twoness. The difference is subtle but profound, Fuller looked at the topological properties of a conceptually zero shape and showed that it had a twoness, whereas I ended up looking at the function of number in basic arithmetic and found that zero multiplied by zero was not zero, it was two. Echoing Fuller's words, it seems obvious: "Unity is plural and at minimum two". I would suggest at this early stage that the plus 2 in the topology equations may have some relation to this hidden property of second powering in relation to volumetric polygons, for example, the equation 10F²+2=N would read 10F²=N, if the 2 was already accounted for in our second powering of the frequency. My inspiration for starting these explorations into such outerlimits of science and metaphysics came from a researcher and broadcaster named Fintan Dunne, his ability to see through the 'smoke and mirrors' of reality is an inspiration to many. Soon after I became familiar with Fintan's 'Science of Meaning' at Treeincarnation.com, I remember him suggesting that "space itself, is enantiomorphic", exhibiting a fundamental mirroring, having both an inside and an outside, essentially two. I never could have predicted that this fundamental nature of reality, would reveal itself to me in the simplicity of the nature of number itself. 0²=2 simply states, that the surface of 'nothing', in a very real, arithmetical sense, has an inherent twoness at it's surface, it has it's In'side' and it's Out'side'. In conclusion, this simple topological experiment is asking a fundamental question, could our understanding of this basic property of arithmetic be wrong, have we misunderstood the Mathematics of Nothing? Justin Lawless ~ January 21, 2007


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graphic based on Menger Sponge by Paul
Bourke. Copyright applies. Use only with permission.
