How mathematics creates meaning in architecture

Abstract

Mathematics is about relationships, repeatability, and nested structures. Regular ordering affects us viscerally because human perception relies upon information reduction through symmetries. Random (disorganized) information becomes too much for us to process, which generates anxiety. Architectural elements are visible shapes, and they need to be combined, compared, counted, grouped, and juxtaposed. This is what our brains do automatically. We subconsciously analyze and process the information presented in any composition using mathematical relations. We perceive our world by grouping adjoining geometrical elements, via symmetries, into larger wholes. We make our way in the world thanks to a mathematical process for making sense of our environment. Basic symmetries have a profound effect on composition and design. Some elements have the same size and shape (oriented in the same way, or reflected, or rotated), and are aligned horizontally or vertically. Their repetitions are regularly spaced; otherwise there is no symmetry. Scaling symmetry is something entirely distinct, and links components visually when we see magnified or reduced versions of the same thing. This self-similarity at different magnifications is a basic feature of a ‘fractal’. Scaling symmetry is a dominant feature in traditional and vernacular architectures, and is one reason those quite different form languages have meaning for us. Mathematics also relates components of a whole via their relative number and size. The universal distribution law says: “In a complex system, there are few large objects, more intermediate-size objects, and many smaller objects, roughly in an inverse-power relationship”. The number of elements of different sizes we perceive at the same time should be inversely proportional to their size. These requirements influence architectural composition to have an “ordered” appearance that echoes traditional and vernacular styles.