Description
A tetrahedral space-filling solid with twenty four identical hyperbolic faces, built on a frame given by an ensemble of joined convex and concave tangent arc triads.
It’s the space closed by such sides, which represent the gradual-symmetric transition between the outer and inner parts of two centrally interconnected identical funnel surfaces with the biggest possible curvatures.
*****************************************************************************
The idea
It is based on an inversion rapport with the saddle surfaces. While the main curves of the latter have their vectors in the pole perpendicular to the vertical axis, the ones of the solid in question must be tangent to this. Therefore its constituents can be considered kind of “inverted monkey saddles” as in both cases the main curves are “S” shaped, the major difference is given by their positioning in relation to the axis. To keep it simple, in the below figure I will use 90 degree circle arcs instead of parabola sectors to represent the general leaning of the two curves.
The monkey saddle is an undulated surface similar to the hyperbolic paraboloid (“pringle shape”), but instead of two it has threefold symmetry. It contains three “S” shaped big curvatures (blue) and three straight lines (red), while the rest of the structure is given by a fluid transition between these two. All the radial intersections have their vectors in the pole perpendicular to the vertical axis.
In the right of the below image is a horn torus with both the outer and inner surfaces partially visible. If we take into consideration only the generatrices situated on the upper half, we can identify three convex arcs (green) converging to the center, which are part of a hypothetical vortex type surface of revolution, resembling a funnel interior or “black hole”. If we look only to the lower half there are also three, but this time concave and divergent arcs (blue), which are part of a horn type surface of revolution, resembling a funnel exterior or spike. As can be easily concluded, one is the inversion of the other. The “S” curves here are the a+a-, b+b- and c+c- half green-half blue arcs, which are all tangent to the vertical axis.
How could we create such a shape, which forms a uniform transition between these inner and outer funnel surfaces? Theoretically, in a sense this would be the opposite of the saddle surfaces, as there is a “crossing” between the plain convex (outer dome) and plain concave (inner dome) structures. This “ambivalent funnel” would be our solid’s base motif. Let the details of this task for later and move farther with the big lines.
The other priority is to obtain tetrahedral symmetry, while reaching the biggest contrast between the convex and concave constituents, but also preserving the exact 50-50 ratio between them. Thus the range of possibilities narrows considerably and bobs up the idea of the single correct solution.
*****************************************************************************
Construction
1. The frame
Regarding the formerly mentioned “biggest contrast”, the limiting factor will came from the side of the concave components, as the four axes of the tetrahedral setting will not allow to exceed a certain curvature. Theoretically the convex generatrices could make much bigger deflexions, including the particular case, when they will form continuities with their peers from another side.
Despite I am always looking for extremes, at first I didn’t noticed the special connection between the concave tangencies along the axes and the corresponding convex curvatures, as the latter didn’t seem to reach any concrete limits (continuity or tangency) at those angles. Only later I realized that these positive deflexions are limited by the edges of a hypothetical rhombic dodecahedron, therefore they are representing the maximum possible bending inside the tetrahedral tessellation frame. As a matter of fact, the imagined solid will also have space filling properties, like the ones I’ve described in the previous articles.
Now let’s determine the exact nature of this arc. In short it will be embodied by the Bézier curves of the given rhombic dodecahedral structure (edges and axes). This by the way entirely overlaps with the contrastoid’s frame, thus it can be imagined as a derivative of the former composition. Actually both are resulting from the farther development of the tetrahedral tessellator, the difference is that in case of the contrastoid the convex and concave generatrices transformed into an angled segment which overlaps with the frame, while the new solid’s components will represent the evenly rounded version of this (see the above image).
In this respect, the tangential cohesion’s generatrices are intermediates between the tetrahedral tessellator’s and the contrastoid’s, not only because of the spatial positioning between the two, but also regarding the mutually shared properties: it is even like the tetrahedral tessellator’s straight lines, but crosses the origin perpendicularly like the contrastoid’s inner segments.
As compared with the elementary tetrahedral tessellator, the common trait is that both have smooth curvatures, but the origin crossing is quite different (horizontal vs vertical). Also the deflexions will be contrary and more significant (109.47 degrees vs 35.36 degrees), thus the biggest convex arcs will be where the other has the biggest concave ones, while the biggest concave arcs where the other has the biggest convex ones.
Note: Every tetrahedral space filler have those specific O’ points on its surface, at half edge length from its center. Actually the solids have much more in common, this includes ten vertices (four tetrahedral, six octahedral) joined by twelve edges and an additional twelve straight lines arranged in radial triplets in the interior. Their volume is also equal, exactly r^3.
2. The surface creation
Let’s return now to the “vortex fusion”. From both the inner and outer funnels we have symmetrically disposed, parabolic arc triads, which are fluidly joining in the origins and these generatrices will remain the only constituents what we will keep from the surfaces of revolution. The “S” shaped curvatures will divide the frame into twenty four identical parts, each delimited by one convex, one concave and one straight sector.
Regarding the surface creation, the smooth transition between the convex and concave arcs will be given by the minimal surfaces bordered by the formerly mentioned three different sectors.
This way the convex arc will gradually transform into the concave one by doing a 60 degree revolution around its fixed inner endpoint, while the outer endpoint will follow the edge of the rhombic dodecahedral frame. Then the concave one viceversa and after six transformations inside the skew hexagon the kaleidocycle is closed. The same combination of maneuvres will happen with the generatrices associated with the other three skew hexagons of the tetrahedral setting and the solid is finalized.
*****************************************************************************
Relatedness with 2D shapes
Based on the conclusions stated in the description of the elementary tetrahedral tessellator, the tangential cohesion’s 2D correspondent must be a certain kind of “distorted triangle”, which forms a tessellation derived from the regular triangular tiling.
The edges of these triangles are deviating from the original straight lines the same way as the tangential cohesion’s kaleidocycles from the minimal surfaces bordered by the skew hexagon: both are reaching the maximum possible even curvature inside their limiting cell (convex generatrices), respectively fixed by the axes (concave generatrices). In case of the 2D shape this cell is the regular hexagon, while in case of its 3D version the rhombic dodecahedron.
As in the plane the setting includes also an orientation of the distorted triangles (clockwise or counter-clockwise) we can say that the genuine 2D pair of the new 3D tessellator in question are both directions together. This can be easily represented on the two sides of a transparent paper or by mirroring.
If we want to relate the concentric structure of the ambivalent funnels with those of the monkey saddles, then the below linear representation will show us the essence of this rapport. Here the fluid red curves of the “a” version illustrates the peculiarities of the saddle surfaces, while the green, “onion dome” pattern of the “b” version (the 2D equivalents of the alternating ridges and trenches) those of the funnels.
And last but not least the meaning of the solid’s name in short: “tangential” as the convex and concave generatrices are all osculating to the axes (their vectors are meeting in 0 degree angles) and “cohesion” as because of the significant fragmentation it has a big surface area in relation to the volume, therefore the adhesion between the cells of this honeycomb is strong.
*****************************************************************************
*****************************************************************************
