The last post concluded with a simple abstract expression of a deeper 3D transformation between fundamentally different plane types – the square and the triangle. In either 2 or 3 dimensions, the transformations are attended by circular arcs, with such rotations intersecting in an equilibrium manifested by the *equilateral triangle.*

In the square (rectilinear) plane of the CBScase scenario of a fused gable delineating an equilateral triangle is depicted above by setting the angle omega () ; or for the slope of the gable . In either case, those angles are an identical: approximately 55°.

Conversely, one could set phi and delta equal to each other in the fusion formula to arrive at the same 55° -

The gist of both approaches is that upon fusing the tri-wings to a slope of 55°, the slope of the spread tri-wings slope is also 55°. Again, the angle made by the tri-wing edges against the innate rectilinear lines of the slope fused to is the 60° of an equilateral triangle.

Recall that the tri-wings themselves – prior to their wall plane-conforming bisection – are a pair of equilateral triangles themselves, independent of latitude, always. Thus by the symmetry of the configuration, the remaining triangle projecting over and beyond the wall plane is also equilateral, and the 4 triangles meeting edge-to-edge is a tetrahedron.

The tetrahedron constitutes the alpha form, the first 3D form built in the original assemblage of spheres, and it was its not-so-obvious rectilinear aspects that guided the more deliberate rational accretion of spheres. After reaching fruition in the cuboda, exploration of that form’s unique intrinsic pattern revealed how innate octahedra are always bridged by tetrahedra, a condition meaning that each and every cubodal triangle (inside and out) represents an interface between the two forms.

With square/triangle fusion and transformation, the tetrahedron finds additional accommodation. If the cuboda is oriented to situate on any of its squares, the tetrahedron finds the 55° slope of an oriented triangle to fuse to, a matchup that forces the triangular interface to configure itself to satisfy the aforementioned condition.

In such case, orthogonal lines from each tetrahedron finds parallel lines in the other, a reality that presents the rudiments of a 3D rectilinear grid. Conversely, if the cuboda is oriented to situate on a triangle, the tetrahedron finds 55° fusing accommodation on the sloping square of the budding octahedron.

Like the octahedron’s square/triangle transformation, its matching form – the tetrahedron – finds its underlying rectilinear reality expressed of itself by the four 90° intersections characterizing the unique orientation that places its 2 essential lines in parallel with a plane of consideration, and by the apparent square delineated by the form’s 4 sphere center points projected onto that same plane.