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.


The contrastoid


An ensemble made of 90 degree hyperbolic paraboloids built on a rhombic dodecahedral frame, presenting unusually high degree of symmetry.



I modified the four sextets of the formerly discovered solid named “tetrahedral tessellator” in a way, that while the frame remained unchanged, the concave generatrices (trenches) will lean into the center and at the same time the convex ones (ridges) will rise twice as high.

This combined maneuver will cause the double bending of the isosceles triangles (their inner vertices will elongate), as a result the “star points” (the centers of the sextets) will transform into (half)axes with the length identical with the solid’s edges. Concomitantly, the former tetrahedral symmetry will turn into octahedral.

The starting point of the idea was, that I tried to maximize the contrast between the positive and negative generatrices, while preserving the tessellation property unaltered. First I was thinking of arcs, but meanwhile realized that the most extreme case will be given by some angled line segments, where the concave generatrices will overlap with the symmetry axes, with their corners reaching the shape’s center. The construction resulting from this action will be indeed kind of “limit case” as it will narrow to zero thickness along all the four axes.

The following practical steps were directly targeting the construction of 90 degree hiperbolic paraboloids as I understood in the imaginative way, that will be the result of the triangle’s transformation (see also the construction of the tetrahedral tessellator).

In the below image we can see that the O’ point of the initial polyhedron turned into the D’O section (half axis) of the saddle surface composition.

Since we took as much from the inside as we added to the outside, the volume of the contrastoid is equal with its predecessor, thus half of the rhombic dodecahedron’s volume as well (interestingly exactly r^3) and at the same time its edges are overlapping with the catalan solid’s edges. 



It has twenty four bent faces, twenty eight edges (twenty four convex, four concave) and fourteen vertices (eight triple edge meetings, six quadruple edge meetings). Two of the four margins of the hyperbolic paraboloids are situated on the edges of the rhombs, while the other two along the axes. The saddle surfaces are meeting in the solid’s center, each touching the nucleus with one end of its convex generatrix. The other endpoint of the convex arcs are found at the quadruple meetings of the rhombic dodecahedral frame. Both endpoints of the concave generatrices are situated in the triple edge meetings. 

Along the four axes of the shape the bent surfaces are forming kaleidocycle-like sextets on both sides (eight), while each one simultaneously belongs to two different axes. The hyperbolic paraboloids sharing the quadruple edge meetings are forming closed spaces. These shapes are owning tessellation properties, namely if they are laid on top of each other than they will completely fill the three dimensional space.

The contrastoid can be defined also as an ensemble of six “3D tiles” of this kind, joined along their edges. The shape of the gaps between the tiles are the exact negative mold of these, thus the compositions joined along the rhombs will form such a spatial network, where the mentioned tiles will alternate with the gaps in a fifty-fifty rapport. 

The saddle surfaces are meeting in four different ways:

1: In two edges: There are two subtypes: convex and concave meeting. In the first case one endpoint of each convex generatrix are touching in the solid’s center, forming 0 (zero) degree vectors. In the second case one endpoint of the concave generatrices are touching in the triple corners, forming 360 degree vectors. 

2: In one edge: This has also two subtypes, in one case we can identify a single straight line across the two neighboring surfaces. 

3: In two vertices: The endpoints of the convex generatrices are shared, these are meeting in the center, respectively in the quadruple corners. 

4: In one vertex: This has three subtypes. In all cases one endpoint of the convex generatrices are touching in the center. In one case the two convex generatrices are forming a smooth curvature, in the second case the outer endpoints are found on the opposite quadruple corners (polar disposition), while in the third case the faces are situated on the two opposite sides along the axes joining the triple corners. 


Relatedness with 2D shapes

Starting from the correlation between the regular hexagon and the rhombic dodecahedron, the contrastoid’s 2D mate must be the radial triad of the alternated triangular tiling, as it is showed in the darker sector of the below image (maroon or olive).

Just as the hexagon’s surface is double of the triangle triads that make it up, the volume of the rhombic dodecahedron is also double of the contrastoid, namely the shape of the voids are identical with the shape of the six 3D tiles that make up the composition. Actually there are twelve pieces of “void halves”, these laid on top of each other in pairs will make up the negative molds of the ensemble’s six cells. The 3D equivalent of the hexagon’s three diagonals are the four axes of the rhombic dodecahedron. 

The contrastoid has an extremely complex symmetry. Both the whole composition and its components have many symmetry planes. Beside the direct mirroring, the 90, respectively 60 degree turns are generating the pattern repetitions. 

My assumption is that it’s the structure with the highest degree of formal complementarity.

The meaning of its name is “antagonistic shape”, which refers to the harmonic interlacement of the parabolas forming the convex and concave extremes (tangencies).  


About the project


I have a lifetime affection for geometric shapes, regarding their inherent aesthetics and complex connection between the constituents. Symmetry, ratio, complementarity and coherence are at the base of formal beauty in general. For me geometry is kind of “compressed knowledge” in the sense it can offer you spontaneous understanding, sometimes accompanied by deep, highly satisfying experiences. I am consciously avoiding the mainstream terms “sacred” or “spiritual” with the purpose not to be confused with pseudo-sciences. 
The intent of this blog is to share and give an adequate description of some solids with interesting properties, which happened to be discovered by me. The more detailed descriptions will belong to the ones considered as highly “archetypal” or finalized, some others will be mentioned in a more summarized way. Though I’m not a mathematician, as accuracy is a serious issue here, the presentation will be mostly technical with only a few “personal intrusions”, like names and certain conclusions. On the whole the target is the blending of science and art, where both can be appraised separately, while also forming a symbiotic conglomerate. 
With the fast development and propagation of the 3D printing technology, the emerging new domain known as “mathematical art” is becoming more and more popular. Being an intuitive visualizer, I never start with raw algebraic formulas. As a generality, the simplified process of the inventions is a sudden inner cognition represented by some vaguely contoured idea, after what is a pretty long rational interval, when I try to figure out the exact connections, followed by the isolated logical steps to reach and synchronize those and finally the “translation” of the latter to concrete maneuvres for the 3D modeling guidance. 
My name is Orbán Sebestyén Zsombor (Zéró), a freethinker and independent explorer from Transylvania and this is my second blog, preceded by The Exiled Weatherman.