"To analyze the influence of mathematics on architecture from antiquity up to the present day, the difference between the criteria that is the basis for the design process must be recognized. Such criteria are the geometry, the structure's function, the loadbearing behaviour of the structural elements and of the structure, the manufacturing technique and the choice of materials, and interior and exterior lighting and decoration. Lately new ecological and economical criteria have been added.

Although all these criteria are well-known, each era assigned different meanings to them. While modern man understands the term "function" as purely meeting the primary task, e.e., a bridge as a means to cross a valley, or walls and a roof as protection from external influences, the definitions of past eras far exceeded this understanding. A structure not only had to be of material usefulness psychologically beneficial and intellectually fruitful influence.

Religious structures before the Industrial Revolution were to form the connection between man and divinity. Geometry was an important tool, as was the choice of forms and proportions that relate the structural elements to each other. However, the choice of the geometry and the measurements was limited by the technically feasible and the loadbearing behaviour in the selected structural elements and the chosen materials, which in turn are closely connected to the choice of manufacturing procedure. The manufacturing procedure also dictated the form of the structural elements and consequently the geometry and the overall appearance of the structure.

Mathematics combined with technology and mechanics is a part of every design criterion. Each era applied this very differently to the structures, based on cultural understanding."