page_title: Mathematics and architecture
section_title: Renaissance
hierarchical_section_title: Mathematics and architecture / Secular aesthetics / Renaissance
caption_attribution_description: Santa Maria Novella, a church in Florence, Italy Own photo - photo made by Georges Jansoone on 12 October 2005
context_page_description: Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
  In ancient Egypt, ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology.
context_section_description: The first Renaissance treatise on architecture was Leon Battista Alberti's 1450 De re aedificatoria (On the Art of Building); it became the first printed book on architecture in 1485. It was partly based on Vitruvius's De architectura and, via Nicomachus, Pythagorean arithmetic. Alberti starts with a cube, and derives ratios from it. Thus the diagonal of a face gives the ratio 1:√2, while the diameter of the sphere which circumscribes the cube gives 1:√3. Alberti also documented Filippo Brunelleschi's discovery of linear perspective, developed to enable the design of buildings which would look beautifully proportioned when viewed from a convenient distance.
  The next major text was Sebastiano Serlio's Regole generali d'architettura (General Rules of Architecture); the first volume appeared in Venice in 1537; the 1545 volume (books 1 and 2) covered geometry and perspective. Two of Serlio's methods for constructing perspectives were wrong, but this did not stop his work being widely used.
  In 1570, Andrea Palladio published the influential I quattro libri dell'architettura (The Four Books of Architecture) in Venice. This widely printed book was largely responsible for spreading the ideas of the Italian Renaissance throughout Europe, assisted by proponents like the English diplomat Henry Wotton with his 1624 The Elements of Architecture. The proportions of each room within the villa were calculated on simple mathematical ratios like 3:4 and 4:5, and the different rooms within the house were interrelated by these ratios. Earlier architects had used these formulas for balancing a single symmetrical facade; however, Palladio's designs related to the whole, usually square, villa. Palladio permitted a range of ratios in the Quattro libri, stating:
  There are seven types of room that are the most beautiful and well proportioned and turn out better: they can be made circular, though these are rare; or square; or their length will equal the diagonal of the square of the breadth; or a square and a third; or a square and a half; or a square and two-thirds; or two squares.
  In 1615, Vincenzo Scamozzi published the late Renaissance treatise L'idea dell'architettura universale (The Idea of a Universal Architecture). He attempted to relate the design of cities and buildings to the ideas of Vitruvius and the Pythagoreans, and to the more recent ideas of Palladio.
caption: Facade of Santa Maria Novella, Florence, 1470. The frieze (with squares) and above is by Leon Battista Alberti.
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