The Geometry of Thinking

“Complementarity requires that where there is conceptuality, there must be nonconceptuality. The explicable requires the inexplicable. Experience requires the nonexperienceable. The obvious requires the mystical. This is a powerful group of paired concepts generated by the complementarity of conceptuality. Ergo, we can have annihilation and yet have no energy lost; it is only locally lost.“
— R. Buckminster Fuller, Synergetics, 501.13

“…the sum of the angles around each of every local system’s interrelated vertexes is always two cyclic unities less than universal nondefined finite totality. We call this discovery the principle of finite Universe conservation. Therefore, mathematically speaking, all defined conceptioning always equals finite Universe minus two. The indefinable quality of finite Universe inscrutability is exactly accountable as two.”
— ibid., 224.50

“Since unity is plural and, at minimum, two, the additive twoness of systemic independence of the individual system’s spinnability’s two axial poles … must be added to something, which thinkable somethingness is the inherent systemic multiplicative twoness of all systems’ congruent concave-convex inside-outness: this additive-two-plus-multiplicative two fourness inherently produces the interrelationship 2 + 2 + 2 sixness (threefold twoness) of all minimum structural-system comprehendibility.”
—ibid., 1073.11

At the core of both Fuller’s geometry and his philosophy is the idea that “unity is plural and at minimum two.” You can’t draw a circle without drawing two circles, for example. The circle must be drawn on the surface of some system, and it always divides the system into that which is contained by the circle’s concavity, and that which is contained by the circle’s convexity. The moral point of this concept is that nature has no preference for one over the other. In the math, this essential duality appears as a plus two (the additive two) and the times two (the multiplicative two).

Additive Twoness

For any polyhedron, the sum of its vertices and faces is always equal to its number of edges minus two:

Vertices + Faces = Edges – 2

This is Euler’s topological abundance formula. It can also be written as:

Vertices – Edges + Faces = 2

This same “2” occurs in Fuller’s shell growth formulas for the close-packing of spheres: for any symmetrically close packed array of spheres, the number of spheres in the outer shell is always:

nF²+2

where n is a constant endemic to the system, and F is its frequency, or number of subdivisions along any edge. Fuller attributes this excess of +2 to the poles or spin axis of the system, and sometimes called it the system’s “polarity constant.”

Other examples of the additive two:

• A vector has 2 vertices, its starting point and its endpoint.
• If the radius of a sphere is taken to be 1, a line drawn between two sphere centers (a single frequency primitive vector connecting two event foci) has a length of 2.
• A triangle, or any polygon, has 2 faces, the obverse and the reverse.

Multiplicative Twoness

The formula for the number of spheres in the outer shell of radially close packed spheres is 10F²+2. At zero frequency (F=0) there is only the central sphere, the nucleus. But the formula returns a shell volume of 2 spheres.

10(0)²+2 = 2

The primitive idealized sphere in isolation from other spheres has no defined poles, but it does have two surfaces, its convex outer surface, and its concave inner surface. So, in the case of the single sphere (or any whole system) the +2 has a different meaning, the division of universe into inside and outside, and of polyhedral systems into the concavity enclosing a defined space, and the convexity enclosing the in-definite remainder of the finite universe. This is what Fuller called the “multiplicative two.”

In any omni-triangulated structural system, that is, for any polyhedron structurally stabilized through triangulation: a) the number of vertices (“crossings” or “points”) is always evenly divisible by two (2×1); b) the number of faces (“areas” or “openings”) is always evenly divisible by four (2×2), and; c) the number of edges (“lines,” “vectors,” or “trajectories”) is always evenly divisible by six (2×3). For example, the icosahedron has twelve vertices, twenty faces, and thirty edges. The number of faces is evenly divisible by two (12/2=6); the number of faces is evenly divisible by four (20/4=5); the number of edges is evenly divisible by six (30/6=5). This holds for any polyhedron of whatever size or complexity, just so long as its faces (areas, openings) are triangulated and therefore constitute a “structure” by Fuller’s definition, i.e. any system that holds its shape without external support.

A principle of angular topology (attributed to Descartes) states that the sums of all the angles around all the vertexes of any polyhedron (whether or not it is omni-triangulated and a “structure” by Fuller’s definition) is always 720° less than the number of vertices times 360°. For example, the sum of the angles around the four vertices of the tetrahedron is 720° (4×3×60), 720° less than 1440°, the number of vertices times 360°. Fuller interpreted this to mean that the difference between the non-conceptual and conceptual, the indefinite and the definite, or between infinity and any conceptual system is always one tetrahedron, 360°×2, or two cycles of unity, the multiplicative 2. In fact, the tetrahedron as the miniumum system is emblematic of all the foregoing. Its two (non-polar) vertices, four faces, and six edges are the divisors in the formula for omni-triangulated systems described in the previous paragraph.