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. 


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