## Imagination and visualization of geometric and topological forms in space: about some formal, philosophical and aesthetic features of mathematics and physics

**Luciano Boi
**

*École des Hautes Études en Science Sociales, Paris, France*

Our talk is aimed at studying some aspects of the imagination and visualization of geometric and topological forms, like non-orientable surfaces, knots and links. The objective is to show that this study may represent a logic and philosophical powerful method allowing for describing and explaining new mathematical, physical and perceptive properties of our surrounding space. We aim at showing that some “basic” operations like*cut* and *glue* can be composed in order to get more complex constructions or structures, such as *connected sum* and *boundary* *surface*, which show the existence of new mathematical properties. We will study some of these complex structures in relation with the processes of *embedding* and *immersion* of some families of surfaces and spaces. These properties may be elucidated thanks to the concepts of *homeomorphism* and *isotopy*. The most relevant point, from the topological and philosophical points of views, is that two objects may have the same “form” and therefore correspond to (at least) two different graphic images. This fact shows first of all that the *equivalence of forms* has a topological meaning much more important than the simple* equivalence of images*. Thus, we will clarify this formal and physical difference with respect to two families of objects or surfaces, the first being knotted and the other being unknotted. In fact, the knotted-like form is a property that essentially depends upon the kind of three-dimensional space in which these knotted or unknotted objects or surfaces are imbedded. Our hypothesis is that the study of objects and of the spatial environment in which they allow for different types of deformations is deeply correlated with the understanding of the dynamic transformations and the new emergent properties and behaviours of these objects and spaces. This is a point of paramount importance for our deep philosophical understanding of the different structures of space, which can be grasped only if we develop a dynamic and relational vision of space and its structures.