The 19th century saw the development of topology, a branch of mathematics which would have a great influence on the way in which artists sensed and represented space. 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. We first present Giorgini's beautiful architectural designs. Then, we explain the topological ideas behind six of his sculptures, in a formal mathematical language. Surprisingly enough, four of them turn out to be topologically equivalent. This means they can be continuously deformed into one another, even though their geometric appearances are very different from each other. We illustrate the steps in the deformation process by drawing. Then, we provide a formal proof of the equivalence using basic notions from algebraic topology.

Giorgini's sculptures represent non-orientable surfaces, which are one-sided: one could paint in colour the whole surfaces without crossing its boundary and detaching the brush. The first sculpture we present is Modified Klein, a Klein bottle with a disk removed around the surface self-intersection. Then, we introduce Giorgini Sphere, Giorgini Torus I, and Giorgini Torus III, and demonstrate that, beside being neither spheres nor toruses, the three sculptures are topologically equivalent to Modified Klein. Indeed, they represent non-orientable surfaces with one boundary and non-orientable genus equal to 2. Therefore, they have the same topological type of the Klein bottle minus a disk. We draw eight steps in the deformation process which takes Modified Klein to Giorgini Torus I. Finally, we report on the topological type of Giorgini Torus II, a Moebius strip minus two disks, and Giorgini Torus IV, a sphere with five holes, three of which capped by Moebius strips.