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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