# Explorations in the Geometry of Thinking

R. BUCKMINSTER FULLER

6 Strut Tensegrity

3O Strut Tensegrity

609.00 Instability of Polyhedra from Polygons of More Than Three Sides

609.01 Any polygon with more than three sides is unstable. Only the triangle is inherently stable. Any polyhedron bounded by polygonal faces with more than three sides is unstable. Only polyhedra bounded by triangular faces are inherently stable.

610.00 Triangulation

610.01 By structure, we mean a self-stabilizing pattern. The triangle is the only self-stabilizing polygon.

610.02 By structure, we mean omnitriangulated. The triangle is the only structure. Unless it is self-regeneratively stabilized, it is not a structure.

610.03 Everything that you have ever recognized in Universe as a pattern is re-cognited as the same pattern you have seen before. Because only the triangle persists as a constant pattern, any recognized patterns are inherently recognizable only by virtue of their triangularly structured pattern integrities. Recognition is as dependent on triangulation as is original cognition. Only triangularly structured patterns are regenerative patterns. Triangular structuring is a pattern integrity itself. This is what we mean by structure.

610.10 Structural Functions

610.11 Triangulation is fundamental to structure, but it takes a plurality of positive and negative behaviors to make a structure.

For example:

• always and only coexisting push and pull (compression and tension)

• always and only coexisting concave and convex

• always and only coexisting angles and edges

• always and only coexisting torque and countertorque

• always and only coexisting insideness and outsideness

• always and only coexisting axial rotation poles

• always and only coexisting conceptuality and nonconceptuality

• always and only coexisting temporal experience and eternal conceptuality

610.12 If we want to have a structure, we have to have triangles. To have a structural system requires a minimum of four triangles. The tetrahedron is the simplest structure.

610.13 Every triangle has two faces: obverse and reverse. Every structural system has omni-intertriangulated division of Universe into insideness and outsideness.

Fig. 950.31

Fig. 950.31

Tetrahedra and Octahedra Combine to Fill Space: Regular tetrahedra alone will not fill space, but when four tetrahedra (A) are grouped to define a larger tetrahedron (B), the resulting central space is an octahedron (C).

Therefore tetrahedra and octahedra will combine to fill all the space. If the volume of the smaller tetrahedron is equal to one then the volume of the larger tetrahedron is eight, i.e. edge length two to the third power (2 × 2 × 2).

When we double the linear dimension of a figure we always increase its volume eight-fold.) If the volume of the large tetrahedron is eight the central octahedron must have a volume of exactly four, while the small tetrahedra each equal one.

The volume of a pyramid is 1/3 the base area times the height. Therefore the 1/4-octahedron (D) has exactly the same volume as its corresponding tetrahedron, further proof that the regular octahedron has exactly four times the volume of a regular tetrahedron of the same edge length.