# Topological Classification of Vittorio Giorginiâ€™s Sculptures

The 19th century saw the development of a new branch of mathematics, *topology*, which studies the properties of geometrical
objects that are invariant under continuous transformations. In the subsequent decades, topology would have a great influence
on the way in which artists and architects sensed and represented space. Among them, the Italian architect Vittorio Giorgini
(1926 - 2010) pioneered the topology-flavoured ideas of elastic surfaces and of form as a dynamic structure.
Beside his architectural works, Giorgini left a few enchanting sculptures, in which he explored the connection between
shape and topology.

In this paper, we first present Giorgini's beautiful architectural designs, which were largely unrecognized in his lifetime,
and they have only recently started to receive the attention they deserved. Then, we explain the topological ideas behind
six of his sculptures, in a formal mathematical language. Surprisingly enough for non-mathematicians, four of his most famous
sculptures - *Modified Klein*, *Giorgini Sphere*, *Giorgini Torus I*, and *Giorgini Torus III* -
turn out to be topologically equivalent. This means that they can be continuously deformed into one another, even though their
geometric appearances are very different from each other. We first illustrate by drawing the steps in the deformation process.
Then, we provide a formal proof of the equivalence using basic notions from algebraic topology: *non-orientable genus*,
Euler characteristic, and the classification theorem for surfaces.

Giorgini designed his three-dimensional figures in-between the late 1960's and the first half of the 2000's. The figures only
existed as bi-dimensional drawings for many years. Then, Giorgini asked David Dainelli and Alessandro Marzetti to carve alabaster
sculptures from his drawings, in their artistic workshop in Volterra (Pisa, Italy), under his direct supervision. Giorgini's
sculptures represent non-orientable surfaces. Intuitively, non-orientable surfaces are one-sided: one could paint in colour
the whole surface without crossing its boundary and without detaching the brush. That would not be possible for orientable
surfaces such as the familiar cylinder and sphere, which are two-sided.

The first sculpture we present is *Modified Klein*, a Klein bottle with a disk removed. The Klein bottle is a well-known
and fascinating example of a non-orientable surface, which not only is one-sided, but also closed (it has no boundaries)
while it does not enclose an interior. As it often happens with physical reproductions of the Klein bottle, Giorgini cut
a disk around the surface self-intersection.

The other three sculptures named *Giorgini Sphere*, *Giorgini Torus I*, and *Giorgini Torus III* are constructed through variations
on the same procedure: start from a sphere (a torus), cut away portions of the surface to make a cylinder, then connect the
top and bottom boundaries of the cylinder with a strip without twisting, to get a non-orientable surface. What would sound
surprising to a non-topologist is that, beside being neither spheres nor toruses, the three sculptures are topologically
equivalent to *Modified Klein*. All four sculptures are non-orientable surfaces with one boundary, and after some calculations
one can verify they all have non-orientable genus equal to 2: being the non-orientable genus a topological invariant,
the four sculptures share the same topological type. In our paper, for illustration purposes, we draw eight steps in the
deformation process which takes *Modified Klein* to *Giorgini Torus I*.

Finally, we report on the topological type of two other sculptures: *Giorgini Torus II*, a non-orientable surface with three
boundaries which has the topology of a Moebius strip minus two disks, and *Giorgini Torus IV*, a non-orientable surface with
two boundaries which has the topology of a sphere with five holes, three of which are capped by Moebius strips.

*Bridges 2020 presentation (.pptx)*

*Deforming Modified Kein into Giorgini Torus I (.mp4)*

Vittorio Giorgini: Architetture d'Alabastro, by Duccio Benvenuti

## Images and movies

## BibTex references

@InProceedings\{GDFC20, author = "Giorgi, Daniela and Del Francia, Marco and Ferri, Massimo and Cignoni, Paolo", title = "Topological Classification of Vittorio Giorgini\^a€™s Sculptures", booktitle = "BRIDGES 2020: Mathematics, Art, Music, Architecture, Education, Culture", year = "2020", url = "http://vcg.isti.cnr.it/Publications/2020/GDFC20" }