After the episode of the cuspid surface, professor
was charged by the Milan Triennale Exhibition to arrange the mathematics section. He asked me to create a few knots to be put on exhibit at the Triennale. So I happened to discover the inner beauty of those rules that paved the way to the most advanced evolution of modern geometry. Such an adventure is all the more strange because knots, just like
T.E.S.T., the hypercube
and other solid geometric figures in a four-dimensions space, were revealed to me under a curious description: that of the soap-bubbles, i.e. the extreme surfaces, where the maximum result is achieved with the minimum effort. I see them very closely related to the archetypes.
::.. Pierelli's Knots..::
pencil moves. It leaves marks on paper. It never traces the same line
twice and in the end it returns to its point of departure. The result
maybe complicated, maze-like, imaginative and it always produces a
particular phenomenon: the plane is divided into two sections - one
pertaining to the points, contained in the line and the other pertaining
to the external points, precisely as if the line were simply a circle, and
in fact with continuity it can always be reduced to a circle.
Pierelli makes hi pencil move in space. We see the marks it makes. The
marks of this metallic luminosity, of this dense ray of light that rises,
curves round, enters into itself, fexits and enters again and then returns to its point of departure. It is stile a closed line. But now we are in space and the line is annulled. We do not succeed in loosening the knot with continuity unless we make cuts in it, unless we use violence on the material. We become aware of a complexity that frightens us. Things are no longer as they were on the plane. We do not manage to reduce the line to a simple circle with continuity. And Pierelli's knots become ever more
complex, insoluble, mysterious.
Then a new idea appears. Little by little a surface emerges from the knot. The surface completes itself and the knot is its raised edge, its border. We have filled the space circumscribed by the knot with a thin patina and this extends itself in a natural manner within these confines, bending harmoniously to fill the space within the knot in an unexpected and curious way.
Like soapy water that expands within this constricting line, tense with the effort of filling the smallest possible area. The unexpected form of this "soap bubble" in its fascinating naturalness is like a revelation to us. We understand more, we even understand better the structure of the knot that alarmed us at first. Our eye, our intuition, the technique of mathematics more easily dominates and classifies the form of these surfaces. If the surface has no holes and is of the same type as that delimited by a circle, then our knot can be loosened.
This jumping of a dimension is among the methods most frequently used in modern Geometrical research. A form whose dimensional aspect seems complicated to us will often appear simpler, paradoxically, if interpreted within the framework of an extra dimension. For this reason both mathematics and physics utilize even very large hyperspaces for the study of complex phenomena taking place in three-dimensional space.
The minimal surfaces associated with a knot are an example of this process and represent forms that, due to their "radically extreme" nature, naturally appear to be constructive elements of a new geometry that is no longer composed of triangles, squares an golden sections.
The sculptor Pierelli has, in his poetic imagination, captured this figurative richness, this dimensional jump that opens doors upon a geometric universe that is extraordinarily rich and yet, paradoxically, more intelligible.