Nexus Network Journal

Among the architectural and mathematical treatises that flourished during the Renaissance period, Leonardo’s codices deserve special attention. They are not didactic treatises, arranged in several books that must be read from the first page to the last, but information about the scientific research in the Renaissance flows from their pages, full of sketches and notes as from an endless font. The reader always bumps into something new or unexpected when going through the drawings, whichever codex or whatever page he is exploring.

Leonardo’s mathematical notes bear witness to a work in progress and allow us to look directly into the mind of the writer. In Leonardo we find two of the three fundamental classical geometric problems: the duplication of the cube and the quadrature of the circle. While Leonardo is extremely familiar with two-dimensional geometry problems, and proposes playful graphic exercises of adding and subtracting polygonal surfaces of all kinds, he is still unable to solve the problem of the duplication of the cube. Numerous pages testify of the attempt to rise above planar geometry and reach the realm of the third dimension, but Leonardo always bumps against the limits of quantity calculation possibilities of his age.

Often in Leonardo’s drawings of octagons, precise geometric constructions were lacking; the master’s approach was freehand. The author seeks to learn if Leonardo’s sketches can be put to the rigors of strict geometric construction, and still be viable as accurate renderings of octagonal geometric spaces with his own geometric constructions of those same spaces.

Leonardo invented a new technique of representation which combines the building plan and a bird’s-eye perspective of the whole into a single system. Bird’s eye perspective may have developed out of cavalier perspective, and instances pre-dating Leonardo can be found, but not used in the same way as he employed it. Though not pre-axonometric, Leonardo took advantage of axonometric representation’s capacity to construct/deconstruct an object into its component parts in order to clarify fitting and functioning. This paper investigates the originality of the technique and special relationship with his research on centrally-planned churches, while examining it in the context of contemporary developments and architects.

Leonardo da Vinci used geometry to give his design concepts both structural and visual balance. The paper examines aesthetic order in Leonardo’s structural design, and reflects on his belief in analogy between structure and anatomy.

Leonardo’s drawings of grids and roof systems are generated from processes best known from ornamentation and can be developed into spatial structures assembled from loose elements with no need for binding elements. His architectural plans are patterns based on principles of tessellation, tiling and recursion, also characteristic of the reversible, ambiguous structures which led to Leonardo’s further inventions in structural and mechanical design as well as dynamic representations of space in his painting.

In recent times, the ambiguous structures in the art of Joseph Albers, the reversible and impossible structures of M. C. Escher, the recurring patterns and spherical geometry of Buckminster Fuller and the reciprocal grids in structural design of Cecil Balmond display a similar interest. Computer models and animations have been used to simulate processes of perceiving and creating ambiguity in structures.

Ideas similar to Leonardo’s for lattice structures can found many later practical applications (Buckminster Fuller’s domes, the Zome geometry of Steve Baer from the Whole Earth days, the Tensegrity structures based on the sculpture of Kenneth Snelson, as well as the Catalan vaulting traditions of Gaudi and the Guastavinos.

“Dynamic symmetry” is the name given by Jay Hambidge for the proportioning principle that appears in “root rectangles” where a single incommensurable ratio persists through endless spatial divisions. In Part One of a continuing series [Fletcher 2007], we explored the relative characteristics of root-two, -three, -four, and - five systems of proportion and became familiar with diagonals, reciprocals, complementary areas, and other components. In Part Two we consider the “application of areas” to root-two rectangles and other techniques for composing dynamic space plans.

This paper contextualises, describes and discusses a student project which takes a particular exploratory approach to using mathematical surface definition as a language and vehicle for corational design co-authorship for architecture and engineering. The project has two authors, one from an architectural and one from an engineering educational background. It investigates the metaphorical and operational role of mathematics in the design process and outcomes.

When contemporary scholars read and interpret historical architectural treatises, one dimension that is frequently lost from the original works is their moral or ethical values. For example, when John Ruskin argues that there are right and wrong geometric shapes in architecture, he may be talking about the difference between mechanical Euclidean curves and freehand quasi-logarithmic ones, but he is, more importantly, also talking about lines that are not simply geometrically but morally correct as well. Similarly, the texts of Vitruvius and Alberti contain rules for the ideal construction of architectural form, but this rightness is first and foremost a moral or ethical quality that is symbolically embodied in a geometric construction. However, while the geometric construction of historic and modern architecture remains the subject of considerable research today, the ethical dimensions in the same architectural techniques tend to be forgotten or are often considered irrelevant to the work. One reason for this shift away from a consideration of the moral dimension in design is, as Levine, Miller and Taylor [2004] observe, that the concept of an ethics of architecture has become increasingly problematic over the last few centuries. This is because throughout the late nineteenth and early twentieth centuries philosophers persuasively argued that only human actions can have ethical connotations. Inanimate and non-sentient objects such as architecture are assumed to be without innate ethical capacity because they do not necessarily shape human responses, actions or behaviours. While in recent years the anthropocentric focus of ethics has been successfully challenged and broadened to include consideration of animals, plants and ecosystems, manufactured or synthetic objects have remained largely excluded from the ethical domain.

As a key figure in English architectural history, Inigo Jones has already been the subject of considerable scholarship and publication. Christy Anderson’Sincerely yours, book, however, ably demonstrates the value of giving further consideration to well-studied and well-known figures. Whilst ostensibly a text about Inigo Jones, the underlying theme of the book is the evolution of an architect in seventeenth-century England, his persona, his training and his legacy. This is a fascinating notion, and one that reveals a wealth of information regarding the availability of mathematical and architectural texts and the pursuit of knowledge in the early seventeenth century.