The fluid contrastoid

Another solid with similar conceptual background preceded the tangential cohesion. In short, both are based on the “ambivalent funnels” discussed in the previous chapter, the main difference is that while in case of the former the number one priority was to maintain the 50-50 rapport between the convex and concave parts, in case of the latter it was the connectedness (continuity) between the constituents.

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Construction

1. The frame

As my purpose is to generate a closed system (solid with volume) it is inevitable that the convex generatrices will be longer than the concave ones, otherwise their coherence could attain only linear or planar results as in the case of the corrugated cardboard or the egg box. In a tetrahedral arrangement this is impossible. Then what could be preserved at the level of complementarity?

Maybe is possible that the shorter concave sectors be the exact inverse of those segments of the convex ones, which represent the central parts of the funnels. Yes, this way the elements can be combined well and at the same time turns out soon that there are only two right solutions to the problem.

Note: the special case when the origins of the four funnels are overlapping in the solid’s center falls out as there the convex parts would mask the concave ones. 

a) The original method: We know that the four origins are situated on the vertices of an imaginary regular tetrahedron and the radial curves which starts from those points are belonging to well defined planes (3×4=12 in all, but reduced to six because they are overlapping in pairs). As the tetrahedral axes form 109.47 degree angles and the funnels main curves are coplanar in twos with the axis pairs, there are only two possibilities when the neighboring arcs convex ends will form continuity, in other words they will join without a fault. That will be attainable using certain kind of elliptic arcs.

The below image represents one of the two solutions, when the joining will happen inside the 109.47 degree axis angle (the other one would be the 70.52 degree complementary angle).

In this case the necessary curves are segments of a special kind of ellipse, whose main characteristics are the sqrt(2) ratio between the big and small diameter, respectively the equality between the latter and the distance between the two focal points. This can be obtained in the simplest way by intersecting a cylinder with a 45 degree plane. 

In the below image we can see the ellipse (left) and the structure formed by joining the convex ends of the neighboring arcs (right), with the origins O1 and O2, and the appearance on the opposite poles of O1′ and O2′ after mirroring the upper half to the lower one to close the curve. The green convex arcs belonging to the painted segments are exact complementaries of the blue concave ones, marked the same way. We get six “open lemniscate” shapes like this one after fluidly joining all the generatrices of the four funnels in the convex ends (J1, J1′). These united pairs will be sorted out along the twelve edges of a stella octangula (stellated octahedron).

The concave ends will mirror the continuity in the origins (O1, O2), creating alike attachments on the other side (O1′, O2′). This way the former tetrahedral symmetry will be doubled, transforming to octahedral, consequently there will form eight interconnected funnel frames. Here the concave generatrices will be simultaneously part of two funnels, while the convex ones, beside their main role inside the structure will also represent the boundaries of other funnels. Actually every tierce will be part of two different funnels (overlapping). 

The other solution will be represented by the more elongated eight shaped curve (unpainted ellipses) perpendicular to the vertical one, which we were describing before. As can be seen, in this case the curvatures are starting more gentle from the origins and even if they are compensating in part at the joinings, in case of identical center-origin distance the used perimeter is slightly bigger (sqrt(2) in function of the radius), hence less “ambitious” than the first method. So we stay with the efficiency. 

b) The cube method: The same result can be achieved more straightforward by slicing a cube along one of its diagonals (intersecting with a 45 degree plane), sliding the resulting rectangle in his own plane simultaneously in all four cardinal directions by exactly one length, than drawing tangent ellipses around each new projection. Thus the four ellipses will be tangent on the corners of the initial rectangle.

In case of the two rectangles obtained by the sliding along the shorter edges we keep only the ellipse arcs situated outside of the tangencies, while in case of the other two we keep only the arcs situated inside the tangencies. This way the four curves (two bigger convex and two shorter concave) will form the “open lemniscate”. Doing this operation with the other five diagonal rectangles of the cube you will get the complete frame of the wanted solid. 

This can be imagined as if you take a cube and bend its edges inward in a manner that the initial 90 degree vectors in the corner points will change to 0 degrees (tangent to the symmetry axes), then bend the diagonals of the cube’s faces outward in a manner that the initial 60 degree vectors in the corner points will change to 360 degrees (tangent to the symmetry axes from the opposite direction). This way you will got a complete edge continuity through the vertices, given by the joining of the convex and concave arcs created before. 

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2. The surface creation

The six tangent “pseudo lemniscates” will divide the frame into twenty four identical parts, each delimited by one smaller concave and two bigger convex arcs (J1O3’O1 in the below image). The convex parts have the same size and are longer than the concave ones, but they are including the exact complementary of the latter in their structure.

None of the three curves of a cadre are coplanar. 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 ellipse 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 trace a curve identical to the arc itself. 

Regarding their directions related to the axes, the minimal surfaces of a given funnel composition (kaleidocycle) can be sorted into two antagonistic triplets, as we can see in the below image: the greens are the “clock-wise team”, the reds are the “counter clock-wise team”. However, as every minimal surface is simultaneously part of two funnels, their directions are ambivalent, it depends on which connection are we take into consideration.

The solid can be painted in a chess-board fashion, following a simple rule: after choosing a color for one face, every neighbor (sharing a convex or concave arc) must have the other color. Because of the overlapping, in four funnels (O1, O2, O3, O4) the greens will be the clock-wise, the reds the counter clock-wise ones, while in the remaining four (O1′, O2′, O3′, O4′) inversely. 

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Homeomorphism with the contrastoid 

This shape shows upfront similarities with my older creation, hence their namesake. It can be perceived kind of a further development of this, during which the axes are contracting to the midpoints, therefore the straight edges are curving inward and are creating the convex generatrices, while the concave ones are forming between these points from the simultaneous upflexions of the axes from the center, overlapping in twos. At the same time it preserves many basic properties of this solid: octahedral symmetry, kaleidocycle disposition of the minimal surfaces and the dual inherence of the tierces. 

The differences between them can be sorted in two main categories in function of the regularity. On the one hand the shape with the rhombic dodecahedral cadre represents a higher level of symmetry summation. The 90 degree hyperbolic paraboloid faces of the contrastoid are symmetry compages on their own. Due to this and to their specific arrangement one has the exact same view from the center to the triple vertices as it has vice versa.

On the other hand the new creation has more connections with respect to the edges, as the generatrices are merging fluidly (no corners). Thus the angled triple edge meetings will turn into funnel centers, while the quadruple ones to rounded attachment points. Unlike the contrastoid, which actually isn’t a genuine solid, but a composite of six identical pieces joined together along the edges, its fluid version is a compact structure.

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Honorable mention

Following the same logic of combining the antagonistic tangencies in a different setting (also based on the rhombic dodecahedral frame) resulted in the composition with tetrahedral symmetry below. Here the positive generatrices are not fluidly merging into the negative ones, but the two are parts of perpendicular planes.

This was kind of a “bold project” as the essences of all the four main curved 3D shapes (outer dome, inner dome, outer funnel, inner funnel) were present in its structure. For a short period of time I believed that the convex (left image) and concave (right image) positionings will be perfect complementaries, consequently the solid will have tessellation properties, but it turned out that this was a false presumption. Nonetheless, the shape still has an aesthetic appeal.

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Tangential cohesion

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. 

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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.

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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 B├ęzier 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.

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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.

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