I introduced rigid tensegrities in the article "Le strutture tensegrali" published in L'architettura naturale 10 (2001):

"Deresonated tesegrities structures were Fuller's major interest in the last years of his life. Increasing the frequence of subdivision of the principle polyhedron decreases the distances between the struts and gives importance to the thickness of the struts themselves, which can be dimensioned in such as way so as to permit adjacent struts to touch. This thickness can be calculated taking into consideration the value of the respective geodesic arch. In this way the structure is without resonance, since the struts are no longer hung but touch and can be bolted at their tangent points.
The tension force otherwise visible in the preceding models becomes invisible in this type of structure. There is an evident reduction of the materials, that is, the number of struts of different lengths is reduced. In fact, thee are only two different struts for a non-resonant 4v, in contrast to the eight struts necessary to construct the equivalent geodesic" (p. 62)

In rigid tensegrities the dynamic quality that permits the structures to oscillate in their position of initial equilibrium is blocked (or rendered non-resonant). Increasing the frequency of subdivision, the central corners near the struts are modified and tend towards a form that is less acute and nearer to spherical. The struts try to touch each other and can then be fixed with nuts and bolts which take the place of the tension cables. The resulting structure will be stronger and subject to very few torsion forces. In this way tensegrities are changed from resonant to rigid, subsequently consolidating into geodesic structures.

Observing a rigid tensegrity, anyone would be led to the conclusion that the underlying struts support those above, and that the structure works in compression as happens in traditional structures in which the bolts serve to prevent lateral sliding, thus giving rise to a mistaken idea of the dynamics of the system. In reality, the structure is pushed towards the exterior by a hidden tension system that recalls the latent explosion of a soap bubble, with the difference that in these structures the superficial external membrane is supported by tension forces that derive from the membrane itself.

Another interesting family of structures that are useful for deciphering the grids drawn by Leonardo in the Codex Atlanticus is that of the so called reciprocal frames. Reciprocal frames are three-dimensional structures in which the module of departure must contain at least three beams (the triangle is the first manifestation of a minimal surface) arranged so as to form a closed circuit. Each element supports reciprocally the other. These structure permit the realization of all possible forms, achieving final configurations that are surprisingly stable.

Reciprocal frames are able to withstand considerable loads. The eventual failure of a single element can endanger the whole system, as happens in general in synergetic structures. They can be constructed rapidly and with local materials, so that they are particularly appropriate in emergency situations.

We have at our disposal a significant number of architecture treatises from the Renaissance period from Alberti on, written by the greatest Italian and foreign artists, but the preliminary notes necessary for the redaction of those books were all lost. We only have the final compositions, printed and illustrated in order to offer a didactic edition.

Leonardo da Vinci, on the contrary, never wrote a proper treatise, but left to the posterity a huge quantity of manuscript papers that were gathered into codices after his death. Except for the Trattato di Pittura, none of the codices is specifically thematic. Therefore, the notes about theoretical mathematics problems are scattered among the Atlantic, Arundel, Madrid, etc… codices.

Despite the difficulty of having a complete vision of Leonardo's mathematical knowledge, his notes, being confused and somewhat unordered annotations not yet selected or arranged for a publication, bear witness to a work in progress and allow us to look directly into the mind of the writer. While the real treatises only issue the solved problems and the certified rules, in Leonardo's manuscripts we find other questions that sometimes reach a conclusion, sometimes not, giving us very precious informations regarding to the proceeding of the mathematical research in the Renaissance period.

It is striking to see how the three fundamental classical geometric problems where then still present in the minds of the scientists. In Leonardo we find two of them: 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, which all derives more or less directly from Plato's theorem about the duplication of the square and Pythagoras's theorem about triangles, 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 he always bumps against the limits of quantity calculation possibilities still limited in his times to the manipulation of round numbers and simple fractions. The accepted value for p for instance, (3+1/7), was still the one established centuries before by Archimedes. No progress in this domain had been made since classical antiquity. The three-dimensional problems not solvable by the means of the graphic tools needed an arithmetic and algebraic support which still had to come. Leonardo was not a mathematician and did not provide these tools himself.

May we therefore assume that the research in the field of mechanics (drawing, design and construction of machines) is the challenge to overcome the difficulty and practice three dimensions manipulations… or even four dimensions when movement is involved?

My concentration would be more on the practical applications for building over the years (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, with excursions into what I see as definitely linked with Leonardo's lattice, the Catalan vaulting traditions of Gaudi and Guastavini brothers.

Leonardo's use of geometry in his design concepts provides structural stability, but it can also lead to an assumption that it provides visual stability as well. Vesna Petresin will examine the aesthetic order and its effect in Leonardo's design concepts, particularly the plan of a roof system as appearing in the Codex Atlanticus drawings.

From an architectural perspective, the relation between plan and volume in Leonardo's architectural projects is particularly interesting. His sketches of grids and roof systems demonstrate an aesthetic order derived from composition processes in ornamentation, plaiting and weaving. These overlapping plan patterns can be translated into a three-dimensional structure assembled from loose, straight elements, when the middle point of one element is connected to the endpoint of another one.

The symmetrical, yet ambiguous pattern of the roof plan implies principles of tessellation, tiling, recursion, restlessness / repose; these create a feature that is characteristic for perception of reversible structures - a symmetry break, also frequently used by Leonardo in his multi-perspective approach to landscape and portrait painting.

The growing, symmetric pattern can be used as plan for a construction of a dome, a sphere, a cylinder, a column or other structures. It may have been derived from Leonardo's interest in problems of continuous interlace as demonstrated in his publication of a series of knots called 'The Academy of Leonardo da Vinci' as a response to complex interlacing patterns adapted by Italian craftsmen from Islamic examples. The structural principles of knots, plaits and puzzles led to further inventions: 'Codex Atlanticus' displays similar patterns in wheels, details of wooden structures, chain members,
barriers and bridges.

Leonardo believed universal principles of mechanics and anatomy had to be adopted to create dynamic structures. Due to a temporary inaccessibility of his drawings and an absence of any built work, his influence on architecture was minimal, yet some of the 20th century art and architecture display similar interests. In his series 'Trotz der Gerade', Josef Albers demonstrates that even the simplest elements allow inverting permutations rich in ambiguities. In M. C. Escher's watercolour series, an optical
illusion of a reversible structure is achieved using a looping, knot-like structure. Buckminster Fuller uses recurring patterns and spherical geometry to create light, flexible structures such as the 'Transegrity sphere'.

Computer modelling and animation are used to simulate processes of perceiving order and creating ambiguity in some of Leonardo's designs such as the pattern of the roof plan, as well as to illustrate the generation of possible three-dimensional structures.

For the Leonardo 2003 Project, I would like to present a study on Leonardo's abundant use of the octagon in his drawings and architectural renderings. Specifically, I want to focus on Leonardo's applications of the octagon:

• in his studies and sketches of the centralized church, and for which we can find influences specifically from Brunelleschi, as well as from other 15th Century architects working with this type of religious structure;
• in his almost obsessive and frequently repetitious drawing of octagonal shapes and forms in his notebooks throughout his career;
• in his project for a pavilion while with the Sforzas in the last part of his period in Milan.

I also plan to work with members of our group to see if there may be a way to develop the modules to accommodate the square root 2 and the theta rectangles. The application of the modular units, so far, have been within the square and its gridwork, but as the octagon has traditionally been used in the development of both the circle and the square, I believe that this shape will be an interesting challenge in terms of linking the two dimensional surface to the three dimensional forms we are planning to generate. The hands on experience will also hopefully provide us with more insight as to why the octagon held so much fascination for Leonardo as one of the ultimate geometric expressions of grandeur and practicality in spatial organization, design, and development.

In 1989 I made a drawing of a net on a cube. The net consisted of 12 lines/elements and they were connected in a way that I recognised a couple of months later in the da Vinci drawings. I don't know which moment impressed me the most: my own discovery of a very simple and powerfull connecting system or the discovery of the da Vinci drawings, which implied
that my own discovery was in fact a rediscovery. What we see on the da Vinci drawings are some examples of roof constructions built with a lot of straight elements (see p. 899v in the Codex Atlanticus). These drawings can be 'translated' into the following definition: On each element we define four points at some distance of each other'. So we get two points somewhere in the middle and two points somewhere near the ends. To make constructions with these elements we may
only connect a middle point of one element to an end point of another one. My first drawing in 1989 was the start of a serie of drawings and models. Out of the simple definition of the elements, I designed a lot of different patterns that I could use for my so called + - - + structures. Domes, spheres, cilinders and other models were made.

Let me explain the "+ - - +". On a bar, which I use for the construction, I define 4 points, two at the ends, these are the "+", and two somewhere in the middle, the "-". At the attached pictures structure-1b and structure-1d I've added the + and -. In a construction each "+" of one bar has to be connected to a "-" of another bar. I think you will understand this from the pictures.

Structure 1b	Structure 1d

When you now compare the roof construction pictures with the da Vinci drawings you will see it's the same. There were so many possibilities that I couln't imagine that no one else had ever found this. And after a lot of searching in libraries I only found one source: the drawing of Leonardo da Vinci. In the last 13 years I have studied the possibilities and I've made a lot of objects, based on this idea. I plan to bring some of them with me to Vinci next summer and hope I can convince you all of the importance of the Leonardo structures.

My contribution to the Leonardo project was a careful study of Leonardo's page 899v in the Codex Atlanticus, a comparison of the transcription of Leonardo's caligraphy (sometimes written in mirror script, other times upside down on the page) with the original document, a digital image identifying each block of text on the page and relating it to its transcription, and finally an English translation of the text. This makes available to all scholars working on the project an accurate text, allowing new and correct interpretations of Leonard's intentions.

For me what is remarkable about Leonardo's architectural drawings, such as MS 2307, fol. 5v., is the combined use of the building's plan and a bird's eye perspective of the whole. In a way this is like a personal synthesis of Vitruvian ideas on architectural representation, as for him it was apparently enough to use only two kinds of dispositio, as Vitruvius says, the so-called ichonographia (plan) and scenographia (perspective).

We may pose some questions concerning this choice and this could be a way to put together some topics for presentation and further discussion:

1. It is evident that the kind of perspective drawing he uses contains implicitly the front elevation - ortographia - but are there other reasons to consider this elevation unnecessary to express his architectural ideas?
2. Why doesn't he care much about Alberti's prescriptions on architectural representation (later confirmed by Rafael's letter to Leo X) reintroducing, instead of a section, a perspective drawing very near to what we may call a parallel perspective?
3. Is the plan exactly what he needs to display the mathematical structure of the building and that perspective a way to control the 3D geometry of the whole and the spatial articulation of its parts?
4. Could this kind of perspective, utilized also to display his famous mechanisms, be considered in fact as pre-axonometric drawings? What is its role in the creative process of form? Or is this only a persuasive way to show results?
5. Can we relate Leonardo's architectural drawings to the ones he made for the polyhedron in Luca Pacioli De Divina Proportione?
6. How relevant are the connections with Francesco di Giorgio's centralized city plans drawings shown in his treatise?
7. Is it possible to build bridges between all these fields and Leonardo's structural investigation concerning dome construction?
8. Could this option suit conveniently a mathematical will to build centralized spaces shaped by simple combined geometric volumes symmetrically disposed around a central axis?

For now there are 8 questions. As the investigation proceeds I hope to add more topics.