The straight and the twisted

Let's imagine a plane with only two highlights, A and B; Obviously, your elementary Voronoi diagram will be the Mediatriz of the AB segment. If it is the plane of a town where a and B indicate the location of their pharmacies, the Mediatriz of AB is the boundary between the respective zones of influence of both pharmacies (assuming that people come to the one closest to their home).

In the case of three non-aligned points, the diagram is formed by the perpendicular bisectors of the sides of the triangle whose vertices are those points (if the points are aligned, the diagram is reduced to two parallel lines that delimit three adjacent fringes). And, in general, the diagrams are constructed by plotting the perpendicular bisectors of the segments that join contiguous points, perpendicular bisectors that, when cut, produce a tessellation of the planto such that all the points of each zone are closer to their point of reference than of any Another of the generator points in the diagram.

In nature, the diagrams of Voronoi appear everywhere, from the spots of the giraffe to the honeycombs of the bees, which also manage to optimize the performance of the wax as a building material of their cells. But then, why are our shelves and lockers almost always rectangular instead of hexagonal? The answer is hinted at the next paragraph.


Lo recto y lo torcido

We tend to think linearly, and conceive the same time as a straight line, like the lines of scripture. And our vision--and perception--of space is basically orthogonal. Perfectly understandable thing, since gravity pulls vertically from us (defines verticality, to be exact) and horizontality gives us a stable basis. As Le Corbusier says in his poem of the Right angle:

Erect on the ground plane

of understandable things,

Cons with Nature a

Solidarity Pact: It's the right angle.

The right angle is our pact of solidarity with nature, also governed by the horizontal-vertical binomial, although in it the angles do not usually appear in their elemental nudity, as occurs in human constructions, from a building to a box of shoes.

But nature is much more complex, and we are sometimes surprised by unforeseen twists and twists. Kekulé had to dream the circular structure of benzene, because his Cartesian mind was stubborn in seeing it as a rectilinear chain. And the double helix of the DNA demanded by its discoverers a mental pirouette similar to the geometric pirouette of the nucleotides in the polymer sine.

And, as we saw last week, a new and unexpected geometric twist has just opened up a promising avenue of research. It was believed that epithelial cells were prisms, such as mosaic tiles (we tend to think of mosaics as flat objects, but the tesserae have a considerable thickness). However, a team from the University of Seville formed by Luisma Escudero, Clara Grima, Javier Buceta and Alberto Márquez has shown that the epithelial "tesserae" are not prisms or truncated pyramids but escutoides (so called in honor of Squire), bodies Geometric that Clara Grima describes thus: "The escutoide, technically, is obtained from perpendicular segments to all the layers [of the epithelial tissue] comprising between the apical layer (the one above) and the basal layer (the one below). To do this, a set of points (seeds) are chosen in the apical layer, for example. The segments perpendicular to the apical layer are plotted in each of these seeds. In each layer between the apical and basal, each segment will produce an intersection (a new seed); To these new seeds we calculate diagrams of Voronoi in this layer (similar to how it is done in the plane, but you have to adapt something techniques). Now pasting the regions of Voronoi (which will be polygons) corresponding to all the points of the same segment you get a Escutoide ".

I invite my sagacious readers to reflect on this fascinating new geometric object. and to share their reflections.