While irregular cubes subdivide into dissimilar six-sided polyhedra, any tetrahedron, regular or irregular, will always subdivide into similar tetrahedra and octahedra.

The perimeter of any rectangle defined by the section plane of a regular tetrahedron cutting perpendicularly through its edge-to-edge axis is a constant equal to 2 times its edge length.

Any tetrahedron, regular or irregular, may be sliced parallel to any one or more of its faces without losing its basic symmetry. Or, as Fuller observed, “only the tetrahedron’s four-dimensional coordination can accommodate asymmetric aberrations without in any way disrupting the symmetrical integrity of the system.” (Synergetics, 100.304)

The pulling outward or pushing inward of one or more faces of the tetrahedron accomplishes the same transformation as uniform scaling would achieve. This accommodation of asymmetrical aberrations while preserving its symmetrical integrity allows for the linear translation of the its center of gravity without actually moving the tetrahedron.

The illustration below shows the same transformation, but with the two actions performed simultaneously, i.e., the pulling outward of one face occurs at the same time and at the same rate as the other face is pushed inward. This is essentially equivalent to a linear translation of the tetrahedron. We can accomplish any translation to any coordinates using this method, something that is possible with no other polyhedron.

The tetrahedron may be turned inside out by pushing a vertex through the opening of its opposite face.

Each consecutive (non-oscillating) inside-outing produces a unique orientation of the tetrahedron, and results in an infinite or near-infinite number of helices or helical translations. See also: Tetrahelix.

The tetrahedron may also turn itself inside-out by rotating three of its triangular faces outward from their common vertex like the petals of a flower. The process may be halted before the vertices rejoin on the opposite side, with the result being an octahedron rather than another tetrahedron. In the octet truss network, the octahedra occupy the spaces between tetrahedra. In the isostropic vector matrix, the octahedron defines the space between the spheres which in turn are defined by positive and negative tetrahedra sharing a common vertex at the center of the vector equilibrium (VE). It seems appropriate, therefore, to conceive of the octahedron as a positive and a negative tetrahedron turned inside-out, as the illustration below demonstrates.

The tetrahedron is uniquely ambidextrous. By this, I mean that we can model the transformation from the positive to the negative tetrahedron non-destructively, i.e., without violating any of the principles that determine its structural or symmetrical integrity. To demonstrate, we need the tensegrity model.

As I’ve said elsewhere, anything in the universe (both physical and metaphysical as Fuller would insist on saying) that may be called structural is structured on tensegrity principles. The structural polyhedra (the tetrahedron, octahedron, icosahedron, and the geodesic polyhedra derived from them) can all be modeled as both polyhedral and spherical tensegrities. But only the tetrahedron may pass through its spherical phase and reverse polarity, with its vertices defined by either a clockwise or counter-clockwise tension loop. The remaining polyhedra are locked into a clockwise or counter-clockwise orientation, and cannot, non-destructively, reverse themselves. See also: Dual Nature of the Tetrahedron.

The transformation from a positive, or clockwise tetrahedron, to a negative, or counter-clockwise tetrahedron, is what spontaneously drives the jitterbugging of the isotropic vector matrix. See Jitterbug.

Fuller, taking his A and B quanta modules as inspiration, elaborated on their constancy of volume by continuing the progression out to infinity. Ultimately, we have something indistinguishable from a line, but with the volume of of the original tetrahedron. Fuller regarded this as a model of the photon. While we may imagine a photon traveling through space and time from its origin to its destination, the photon, from its own point of view, is already there.

A similar constancy is observed when we orient the tetrahedron on its edge-to-edge axes. The highly skewed half-octahedra in the illustration below each have the same volume as the original tetrahedron.

This construction produces the rational-volume tetrakaidecahedron and pyritohedron that align with the distribution of nuclei in the radially close-packed spheres of the isotropic vector matrix. See also: Formation and Distribution of Nuclei in Radial Close-Packing of Spheres.

Tetrakaidecahedron

Begin with a 2√2 by √2 rhombus. Mark the vertices of the long diagonal A and B, and the vertices of the short diagonal a and b. On AB, mark points ³√(√2/2), about 0.8908987, in from each end. Lines perpendicular to AB and passing through these points bisect the edges of the rhombus at A1, A2, B1, and B2. (See illustration.)

Replicate the rhombus, rotate 90°, and separate (along the center face normal vector) by √2. Mark points as above, substituting C c for A a, and D d for B b.

Connect the vertices to and create faces: BCcB; BDcB; bCBb; aDBa; ACdA; ADdA; aDAa, and; bCAa. Measure the distance, r, from A to either A1 or A2, and, using this length as the radius, scribe arcs from A on each of its adjacent faces. Repeat for B, C, and D.

The precise value of r is ³√(√2/2)×√5/2, or about 0.996055.

The scribed arcs locate the peaks of the tetrakaidecahedron’s four larger pentagonal faces at A3, B3, C3 and D3 (see illustration).

Each of the tetrakaidecahedron’s four larger pentagonal faces occupy a plane defined by the three vertices identified so far. At corner B, the plane (the blue disk in the illustration below) is defined by B1, B2, and B3. To locate the remaining two vertices, draw a line from the midpoint between B1–B2 to B3, and mark a point on this line ³√(√2/2)×√5/3 from the midpoint, or ³√(√2/2)×√5/6 from B3. A perpendicular through this point parallel to B1–B2 intersects BC and BD at B4 and B5, completing the pentagonal face (see illustration).

Repeating the steps for each corner A, B, C and D, produces four pentagonal faces in addition to the two hexagonal faces defined in the first step.

The points surrounding the remaining empty space should all lie on the same plane and define the eight smaller pentagonal faces of the Weaire-Phelan tetrakaidecahedron. A line bisecting their faces from the peak to mid-base should be of unit length, or, in terms of the isotropic vector matrix, one sphere diameter.

Pyritohedron

Begin with three, intersecting and mutually perpendicular 1 x 2 rectangles all sharing a common center. To construct a pyritohedron that will complement the Weaire-Phelan tetrakaidecahedron to fill all-space, the rectangle’s short edge will be ³√(√2/2) and the long edge will be twice that length (see illustration).

The twelve pentagonal faces of the pyritohedron are identical with the four larger pentagonal faces of the tetrakaidecahedron, and they are constructed using a method nearly identical with the procedure for the four larger pentagonal faces of the tetrakaidecahedron described above. Each face plane is defined by the the short edge of one rectangle, and the nearest corners of the rectangle perpendicular to that edge.

This pyritohedron has a rational volume of 24d³ in unit tetrahedra, where d is is the diameter of the unit sphere in the isotropic vector matrix or the height of the tetrakaidecahedron’s smaller pentagonal face. Its cubic volume of 4α³, where α is the length of the pyritohedron’s long edge. This volume should be identical with that of its companion tetrakaidecahedron. See Pyritohedron Dimensions and Whole-Number Volume, and Tetrakaidecahedron Dimensions and Whole Number Volume.