N-Dimensional Perspective. The Mathematics behind the Interpretation of Ancient Perspective

Different factors appear to have prompted the renewed studies of ancient perspective that took centre stage throughout the nineteenth century and continued in the twentieth century: this study focuses on the mathematical basis of this new approach to the question. The origins of the studies which were to lead to the creation of two factions of what was to be defined “the issue of perspective” appear to have different roots as well as ramifications in several other fields. In fact, the enthusiasm generated by archaeological finds in the area of Vesuvius sparked a radical revision of the geometry and mathematics behind the concept of space. This revision, ostensibly based on ideas by Carl Friederich Gauss, was triggered by the radical, late eighteenth-century review of Euclidean fundamentals after questions were raised regarding his “parallel postulate”. In the last decades of the nineteenth century another event accompanied this radical rethink of the geometry of space: the readmission of Euclid’s Optics as a legitimate treatise following the discovery of the so-called “genuine version” and ensuing mitigation of the author’s responsibilities, no longer rejected as the promoter of a theory of vision based on the rays emitted by human eyes, but accepted in light of the possible breakthroughs created by vision based on angles. This led to a sort of superimposition between retinal vision and curvature of space in geometry (Gauss, Riemann et al.), and this seems to echo with the coeval development of the idea of space-time and its curvature (Einstein). This cultural climate questioned the uniqueness of the perspective method and led to the possibility of identifying spatial and projective concepts in ancient frescoes and in the illusory spaces they were able to inspire, concepts completely different to the ones classified from the Renaissance onwards. The consequences of this renewed approach to mathematics and geometry do not only re-open the discussion about linear perspective and its fundaments, but lead to a renewal in the fields of physics, art and, at the end, architectural representation, thanks to the new possibilities offered by the application of new geometric tools to digital instruments for drawing and modeling.