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.


The beginnings

The first concrete step regarding the creative work in geometry was when I intuitively discovered the special properties of the sixth part of a circle, also known as sextant. It was more an aesthetic drive, than a thirst for knowledge. I observed that this simple shape is kind of a “base ingredient” from which one can build different structures in the plane with markedly harmonious appearance.

It is not an exaggeration to call obsession the strong positive feeling what made me motivated to think about and experiment with this “abstract cell”. 

The primary compositions I identified were represented by three arcs surrounding an equilateral triangle, what can happen in four ways: three concave, three convex, two convex and one concave, one convex and two concave. Starting from the relation with the inscribed triangle I named these four as “-3, +3, +1, and -1“, referring to the disposition of the circular segments which make them different. 

The first case is when three circles are tangent, the “-3” will be the interstice between them. The second and third case appears when three circles are touching the others center, where the “+3” is the central triple intersection and the “+1” are the three double intersections. The fourth case is when two circles are tangent and a central third touches the center of both, where the “-1” represents the two non-intersecting parts of the middle circle. Joining the curved triangles in a symmetrical manner you can reach beautiful patterns, which are the most expressive when painted in a chessboard fashion.

Note: only afterwards I heard about the well known Flower of Life and realized that it’s made from the same units as my compositions. 

The YinYang-like design above is composed of six white and six black +3’s and -3’s, or six +1’s and -1’s, depending on which curve you consider the limit between the two possibilities along each triangle side. As the summation of the convex and concave arcs are the same whichever you choose, the surface area of both the white and the black sectors are the same as it would be using equilateral triangles or rhombs for the partitioning.


Although it’s not directly related, I want to mention here a specific quest of reduced dimensions, but of big personal significance. Namely the idea to found the “most imperfect triangle“. The conception was the result of an inversion rapport with the equilateral triangle, which can be considered the “most perfect” in the category. As there all sides are equal, I concluded that in the reverse case those must be the “least equal”. As a triangle has three sides, there are also three relations: a/b, b/c and a/c. To reach the most unequal complex, even the most balanced of the three ratios must be the farthest possible from equal.

The puzzle was solved by the double rapport a/b=b/c and a=b+c, which leads to a quadratic equation with the solution 1.618033…, known as (surprise!) the golden ratio. The resulting “least perfect triangle” is actually flattened to a straight line section, where two vertices are at the endpoints and the third 1.618033… times closer to one end than to the other.


Later on I moved to the third dimension and tried to figure out what can be the correspondent of the sextant. It was a little disappointing to realize that there is no such a thing, as you can’t combine the concave and convex parts of a sphere’s surface in the same complementary way as in the case of the 2D tilings. Still I was searching for some insight.

The first solid conceived was the one I named “the space between the spheres“, which is the interstice formed by four tangent spheres. As it’s not a completely closed space I concluded that it must be delimited “artificially” along the openings and the correct way to do this is to cut it at the narrowest parts.

At that time I thought that a closely packed bunch of spheres will have solely these tetrahedral spaces separating the neighbors, only later understood that there are two kind of voids (the other one is the octahedral). Actually it’s the same mistake as the presumption that one can fill the space without gaps using solely regular tetrahedrons (including Aristotle). That’s not possible, while it is with a tetrahedral-octahedral combo. 

Following the 2D relations, the second (complementary) idea was to intersect four spheres in a manner that each will touch the center of the other three. This way the central, quadruple intersection will be a spherical tetrahedron. Unlike the tetrahedral interstice, which can’t be considered the genuine 3D representation of the triangular interstice, this tetrahedron delimited only by curved convex faces it’s the exact 3D correspondent of the Reuleaux triangle.

Some years later I heard for the first time about the 3D printing technology and with the assistance of virtual designers, respectively dental technicians soon I could hold in my hands the concrete objects. Consequently I involved myself even more into the creation of interesting solids. 


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.