The quest for the tetrahedral centrum-continuity

A big chapter of my plannings was dedicated to the perfectioning of a particular kind of solid, what in a nutshell can be described as a tetrahedral structure, which has the center point on its surface, while there is the maximum possible continuity through this point.

The first and simplest member of this long lasting “dynasty” was a special case of intersection between four spheres, therefore strongly related to the first two self-designed 3D shapes (see the previous post).

This new one was represented by the combined six double intersections (including the four triple ones) formed by the spheres arranged in a manner to have one single common (quadruple) point. Actually it’s the 3D representation of another pattern formed by sextants, which is the one when three circles are meeting in one common point.

In the rightmost part of the above image we can notice one continuity through the center represented by the vertical line (in fact a circle arc), partially masked by a lobe. There are six arcs of this kind on the surface of the solid, intersecting in the middle. One can think that these are also sextants, but the truth is they are somewhere between 1/5 and 1/6.

The next step was to reach more continuities, while keeping the former one unaltered. There are two ways to exceed the initial setting: one is to have concentric continuity inside the “triple petals”, the other is to reach evenness in the intersection of the edges.

In the case of the “concentric type” there will form four funnels in the place of the petals, as the initial generatrix is used for the creation of surfaces of revolution, while in the “vertex type” case the meeting in the four triple points will happen in the form of 0 (zero) degree vectors related to the tangent plane as the six edges are part of a sphere’s surface (109.47 degree circle arcs).

The following development was to change the generatrices of the concentric type’s funnels in a manner to eliminate the outer sectors by reducing them into a single point, thus transforming into vertices . Of course, the central continuity remains (see the image below).

From this shape originated two others with even higher cumulated coherence: in the first one’s case appeared new continuities perpendicular to the edge midpoints, while in the second one’s case the vertices disappeared, reaching evenness by the zero degree vectors. Theoretically this second one can also be considered the evolution of the vertex type level 1 shape.

While different, at the same time both are constituted by the intersection of four surfaces of revolution like their predecessor, only having distinct curvatures, each of these calculated to reach the specific property improvement. The first one still has discontinuities in the tops (dents in this case), while the second one remained with the sharp edges. The strength of one is the weakness of the other and vice versa.

The “final level” was reached by a shape, which possess the advantages of both while lacking their deficits as mentioned in the previous paragraph. I surmounted the dilemma with the construction of “wavy funnels“, by following an undulating revolution path of threefold symmetry with the generatrices, therefore preserving both the edge-transverse and the vertex evennesses unchanged.

Nonetheless, it can’t be considered a finalized, “archetypal” solid as it still remained with edges, even if less marked ones. But it’s likely the closest possible approximation of the original idea.

Actually, I was suspecting from a pretty long time, that what I’m intuitively searching for is an already existing shape named Roman surface, which can be considered the 3D correspondent of the regular trifolium and has many structural similarities with my creations, except that it’s self-intersecting.

What I’ve tried is to attain the same complete surface continuity without the self-intersecting property, using only one single central passage, which finally turned out to be impossible. With all the respect to reality, we can still call the level 4 result “the possible maximum” in this context.


Honorable mentions

There are some secondary designs which show up marked esthetic appearances in the form of YinYang-like complementarity.

Two of them resulted from the reduction of the edge-transverse/ concentric type of level 3 shape: one can be considered 1/3, the other 2/3 part of this solid. The first is kind of “pseudo hyperbolic paraboloid“, while the second one has similarities with the Enneper surface and the cross-cap, depending on the view.

However, unlike the mentioned three, my shapes are all derived from joining different sections of some surfaces of revolution, therefore can’t be considered genuine continuous structures.

As expected, intersecting two pieces of type 1 will result in type 2 (compare the first and middle images from the upper row with the rightmost image from the lower row). The convex parts of both originated from the outer side, while the concave ones from the inner side of the funnels of which the level 3 solid is constituted.

One of the main cross-sections of the first shape is a quadrifolium as it can be seen in the below image. On the right side is a symmetric combination of two transverse-edge/ concentric type of level 3 solids, which surprisingly still has the center point on its surface (reachable from twelve directions).

Another design was inspired by the former 1/3, changing the two “onion dome” curvatures to reach complete tangency (180 degree vector meeting) in the center point, resulting in an oppositely-doubly folded torus appearance. Their close relatedness can be appreciated the best by looking at the principal cross-sections.

While the generatrices are halves of Bernoulli lemniscates, it’s still only a “pseudo-homogeneous” surface, having the same quartette partitioning (180 degree segments in this case) as the previous two.

During the time when I started to experiment with the sinuous surfaces of revolution, the specific maneuver which finally was utilized in the creation of the level 4 continuity-expansion, I created a few other models with partial coherence and varying degrees of complementarity.

The first solid in the below image is constituted of two, the second one of four and the third one of eight intersecting “wave surfaces”.

While the first one show up constant fluidity, it’s not tetrahedral, the second one is tetrahedral, but has no continuity (or tangency) in the concave triple meetings and the third one with octahedral symmetry, built on a frame formed by six lemniscates doesn’t really have any complementarity effect as the concave parts became masked by the convex ones (partly valid for the first one too).

Still decent compositions.


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