Space, the universe that surrounds us, mutates, and is strictly dependent on our scientific concepts. One could say that space mutates following the mutations of our theories. We are in a “topological” concept of space. We are not interested in the construction of geometric “absolutes” but in decoding the possible relationships between them.

 Ancients Greeks wrote the first mathematical texts. Euclid has gathered in his work named “Elements”, axioms and theorems that remained indisputable for hundreds of years. Between 1830 and 1850, Lobachevsky and Bolyai created the first examples of non-Euclidean geometry. Riemann, Stringham and Poincaré explored geometries beyond three dimensions and try to define their qualitative properties.
 In 1884, Edwin Abbott Abbott writes a novel entitled: “Flatland, a romance of many dimensions”. Inside the book, a two-dimensional character, the Square, describes its world, Flatland, and its amazing experience when it finds himself in three dimensions. The story of the Square has contributed to the propagation of the forth dimension.
 The visualization of the forth dimension became a riddle to the mathematic and artistic world. The invention of the computer rendered this task possible and the first animations of hyperstructures were created.
 Architecture is connected with the hyperspace through the notion of Topology. The object of topology is the study of the properties of geometric figures that remain invariant when the latest undergo non linear deformations. Manifolds, such as the Möbius band and the Klein bottle excited the imagination of architects that tried to use their properties in their works. Through this prism, architectural topology means the dynamic variation of form facilitated by computer-assisted design and animation software.
Stephen Perrella introduces the term hypersurface in architecture. He uses the hypersurface to describe the void between the classic dualities of thought (interior/exterior, ground/building). Brian Massumi sees the hypersurface as a “superfigure” that exceeds and generates all visible forms.
 Marcos Novac is regarded as the leading proponent of cyberspace as an autonomous architectural space. He explores the relations and the boundaries between real and virtual spaces. Using complex algorithms extracted from the internet, he uses hypersurface as a form generator and a window to the virtual world.
 Peter Eisenman locates the hypersurface in the folds of architectural form. He tries to redefine architecture, give it a new meaning using the displacement of the body in his buildings.
 Lars Spuybroek and Kas Oosterhuis, in an attempt to attach material character on hypersurface, create architecture that embodies the spectator. They believe that the body can expand it self using the space around it.
 Information has contaminated architecture. Architects must take back the freedom of their creativity by proving the obvious, that architecture is modern by nature. Maybe one translation of the forth dimension is the human thought and a hypersurface is a way of thinking. We live in a world of networks, grids and nodes. A world that promotes architecture without limits, where opposite notions is interactively connected, where each architect can project his unique hypersurface.