Tetrahedral divergence


The general idea of this shape was conceived much earlier, not long after the invention of the tangential cohesion, to which (despite the big visual discrepancy) is closely related. Actually it shares all its big “S” shaped curves, though nothing more. While the previous one joined the convex and concave arcs in the most direct way (through minimal surfaces), the new one is a little more complicated.

Here, during the revolution inside a kaleidocycle the neighboring arcs will not approach each other, merging halfway along the horizontal plane, but quit the opposite: they will diverge until both will overlie with the axis of rotation. The joining will be represented by the fact that the divergence will finally reach 180 degrees, when the two will merge into the same line, but from opposite directions. Their single common point is the center of the kaleidocycle, contrary with the overlapping. The below image is a figurative representation of the rapport between these two situations: in “A” case the blue and green triangles share a common edge, while in “B” case those edges are only situated on the same line.

To reach even more interesting correlations, we need to go back to the elementary tetrahedral tessellator. This new shape is kind of “complete inversion” of that solid in relation to the rhombic dodecahedral cell’s four rhombohedra. In case of the tangential cohesion the inversion was only partial, represented by the vertical “S” shaped generatrices taking the place of the horizontal ones, while the halfway joining between two neighboring curves were overlapping in the horizontal plane at every 60 degrees revolution. Here instead, also the transformation along the revolution movement (the surface creation) is “inverted”, thus not only the big “S” curves, but anything between will be tangential to the vertical axis.

In case of the elementary tetrahedral tessellator every vertical plane touching the kaleidocycle’s center will intersect the surfaces along horizontal tangencies, while in the case of the tetrahedral divergence all these will became vertical tangencies. Let’s see the concrete details of how all this will manifest.



1. The frame

To get in line with the previous conception, at every 30 degrees revolution inside the kaleidocycle, one big curve (half of the “S” shape) will need to transform into the vertical axis. The exact length of this straight sector will be the tetrahedral radius of the rhombic dodecahedral cell, while everything between the two must remain inside this spatial unit, at the same time continuously touching its limits.

This action will divide the space twice as more as in the case of the tangential cohesion, thus there will form not 24, but 48 identical frames. Each of them will be represented by three segments of which one is the half “S” curve, one the straight (half) axis and one is the path what the moving endpoint of the first will draw while transforming into the second. Let’s identify the exact nature of this third segment.

Its equivalent in case of the tangential cohesion (but also the tetrahedral tessellator, elementary tetrahedral tessellator and the contrastoid) is half edge length of the skew hexagon frame (kaleidocycle outer limit).

Here instead, the path will be not straight, but another Bézier curve, portrayed in the upper 2 images in green. In the first image the inner end (A) will need to be tangent to the 30 degrees shift (related to the revolution), while the outer one (B) to the big diagonal of the concerned rhombic face (related to the constant rise, or descent). In the second image you can see a cluster of 12 green curves inside one of the four rhombohedra (one kaleidocycle), also the “S” curves marked with blue, respectively the axis of revolution in red.

The image above shows the side-view of one of the 48 minimal surfaces (“half leaf”) with the three limiting segments colored according to the previous description.

2. The surface creation

After the frame of the base pattern is ready, the next step is identical as in case of many other previous shapes: it will be represented by the minimal surfaces bordered by the three edges (here two béziers and one straight segment). Inside one kaleidocycle there will be 12 minimal surfaces, 6 in the upper half and 6 in the lower one (see the below image).

Every two will share a common half S curve (left and right sides), this pair resembling a “leaf”. There are 3 leaves up and 3 down, arranged in symmetric antiprismal composition: related to the vertical plane, where one triad has a leaf, the other has the gap and vice versa. The top view resembles a flower of life base motif, while the complete spatial disposition is a 3D version of this pattern with kind of a tesseract effect.

The inner four kaleidocycles (4×3 leaves) will confine a certain space shaped like an 8 pointed star with 4 shorter and 4 longer limbs, while the outer four will be separated hyperbolic surfaces without any volume. That’s before joining with other tetrahedral divergences according to the rhombic dodecahedral honeycomb structure, where each outer leaf triad will join other 3, theoretically repeating the mentioned 8 pointed star pattern into infinity. The overlapping is along the Bézier curves situated on the surface of each rhombic face, the final result being a certain 3D mesh following the tetrahedral molecular geometry.


Relatedness with 2D shapes

The below image shows the plane equivalent of the tetrahedral divergence shape (teal color). The hexagonal tiling is the 2D correspondent of the rhombic dodecahedral honeycomb, where the flower-like pattern represents the gaps, while the interwoven mesh in between is the plane variant of the spatial net composed of the previously described 8 pointed stars (here 6 pointed ones).

In each hexagonal cell there is one six pointed star (inside the red triangle) and 3 separated thirds of other stars of the same kind (outside the red triangle), which will be centered along 3 symmetric corners of the initial hexagon. The infinite net formed by them follows the triangular tiling.