SERGIO MUSMECI'S " FORMS WITH NO NAME" AND "ANTI-POLYHEDRONS"

SERGIO MUSMECI'S " FORMS WITH NO NAME"

AND "ANTI-POLYHEDRONS"

CAPANNA Alessandra, (I)

Abstract. SERGIO MUSMECI (1926-1981) is one of the most daring and unconventional engineers born in the twentieth century; he was master equally of music, astronomy, aeronautics, mathematics, and philosophy, all of which informed his structures, whose shape was determined by the spatial distribution of static actions. Musmeci thought that he could reach the expression of "modernity" through the Science. His activity is the answer to those who believe that the studies concerning the relationships between architecture and mathematics are much too theoretical. His works are substantially of two types. The first one is referred to a "geometry of the continuous", as in the constructive technique of the lightweight stressed skin structures and improved in what he called "forms with no name" elaborated for the bridge on the Basento river. The forms he conceived can be truly defined as three-dimensional because they are endowed with a different kind of curvature and orientation in space, in every point. As a consequence, the construction of spatial forms can be directly connected with the concept of structural minimum. The other subject of Musmeci's researches concerns those aggregated structures that are the expression of a geometry of discontinuity, represented by the crystallographic conformation of trussed structures. In 1979 Musmeci exhibited full-size models representing various aggregated structures illustrating those "enigmatic and sharp space frames", geometrically constructed through the same formative process of regular and irregular polyhedrons and their reciprocal transformation from one into the others. His studies on polyhedrons culminated with the definition of the anti-polyhedron, a potentially unlimited and undetermined geometrical figure, even though it was generated from regular figures. Key words and phrases. Architecture Mathematics Polyhedrons Structural minimum Mathematics Subject Classification: Primary 00A67 52B99; Secondary 97M80 01A70

1. Genesis. Looking for the shape, this is the question.

Awareness means to unveil the (correct) shape.

Musmeci, explaining his aesthetic and philosophical theories, illustrated the scientific and mathematical basis of his researches bringing to light the historical origins and the processes for

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obtaining ever greater knowledge from which his architectures come. His point of view was that science is a tool for ideas, thought and invention, therefore for design, and it is not only a means to verify structural strength. Science must lead us to discover the best geometry for that particular static (or dynamic) condition, without using surplus of materials analyzing stress limit states of the structure. As a matter of fact, he began this research with the solution of a scientific problem in which he was engaged since he was a student: the determination of the arch-limit shape. "Its equation is y = log cos X (a part from some multiplicative constants depending on the resistance of building materials)" he stated, and it looks like a very extended parabola. This curve has some very interesting properties, particularly that the angle between the median axis of the parabola and the horizontal line is proportional to its abscissa, that is, the distance from the vertical axis. The limit span of this arch is that distance corresponding to a 90° angle. 2 Strains are not unknown. Structural minimum and its geometry. Musmeci thought that the building science, so careful and accurate in checking structural safety, was not enough to find the correct figurative solution; he believed that «strains are not unknown (quantities)», the real unknown factor is the shape of a structure, so, strains must be calculated from the specification of their shape on. On the contrary, he accounted as belonging to the past Nineteenth century period even those beautiful shapes born by Pier Luigi Nervi’s and Riccardo Morandi’s genius. During that period the construction of ribbed structures conforming domes and vaults was obtained through automorphic transpositions of symmetries through the rotation or the translation of plane figures, so that one can consider those volumes almost two-dimensional because it is their generative matrix. He had the positive ambition to create shapes which have not yet named and to conceive structures in which each portion had its specific shape. Forms that can be truly defined as three-dimensional because they are endowed with a different kind of curvature and orientation in space, in every point, in view of the fact that in space the intensity and the direction of strength at each point will be different. Different forms for different situations, that is complex forms of modernity.

Fig.1 – Bridge on the Basento river – historical photo

So is the bridge over the Basento river, the main entrance to the city of Potenza. It looks like a thin line slightly inclined to the city, supported by a complex three-dimensional shape, and while holding the belt road, rotates and moulds itself, and finally settles to the ground almost disappearing in the vegetation. The concrete structure is modeled after an alternation of concave and convex to


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form four arches, alternately slightly touching the ground or the floor under the driveway, as leaning on the tips of the fingers of one hand. Musmeci stated that his bridge on the Basento river was not settled like an arch (in terms of strucural template), but like a vault, or better saying an equally compressed reinforced concrete membrane, 30 cm thick (excluding ribs along the edges) that was intended to be used as a walkway for pedestrians, meeting requirements about minimal structure, a mathematical and technical problem he examined at the end of the book La statica e le strutture, he edited in 1971.

Minimal surfaces. Structural minimum and its geometry.

In mathematics we know that minimal surfaces are surfaces with a mean curvature of zero, including, but not limited to, surfaces of minimum area subject to various constraints that is minimizing surface area. The project for this particular bridge is a development of previous designs: for “Tor di Quinto” viaduct in Rome, on the Tevere river, and for a viaduct in Calabria. For the Basento bridge the minimal surface is unique and develops for a length of 280 mt. “The shape is determined by outlines designed in such a way as to exclude any bending stress. The calculation results have been checked with models consisting of soap blow bubbles and highly stretched rubber and finally verified with static tests over a metacrylate model and over a micro-concrete model constructed in Bergamo by ISMES (Experimental Institute for Models and Structures) (in Sergio Musmeci, Pordenone 1979 p. 22).

Fig. 2 – Basento river soap blow bubble check it

is the minimal surface, set from this peculiar edge frame.

Fig. 3 – Scherk surface shaped with a soap film:

The whole structure, with its 4 spans of 70 mt. each, crosses the tracks of Potenza railway station, the river and two roads inside the industrial area near the city and resisted Irpinia and Basilicata huge earthquake in 1980 November 23 (6,89 Mw - Richter magnitude scale)



Fig. 4 e 5 – Study-models for Tor di Quinto viaduct (1959) and for the bridge over the Astico river(1956) 3 Intriguing polyhedrons. Origins of spaces frames. The other subject of Musmeci's researches concerns those aggregated structures that are the expression of a geometry of discontinuity, represented by the crystallographic conformation of trussed structures, consisting of a set of framework that follows, in strict accuracy, a structural space arrangement according to which usually identical elements are assembled. Trussed structures are the best known among discrete systems in architecture; they consists in a set of bars whose modulus is a prism. The most widely used, because its non-deformability, is the one presenting triangular faces. These space frames are light and flexible and generally work better than other structural systems. Musmeci began to study this system with the aim to test an innovative material, provided with high resistance to the strains. A particular type of concrete produced with the addition of synthetic compound polymer. In addition his intuition to try out some new configuration of the most important and complex parts of the trussed structure being the nodes, produced the remarkable result. Musmeci decided to eliminate the nodes replaced by empty polyhedrons upon whose immaterial faces bars final sections converge. Fixed joints soldered together with concrete. The most important characteristic is that the shape of any single bar arises from the condition that each node does not involve coupling of elements not related with the whole geometrical shape.

These kind of structures can be arranged in various way depending on the polyhedron node (in Musmeci’s lexicon: “poliedro di nodo”) from which they are originated. Regular polyhedrons in 3d space are only five (tetrahedron, cube, octahedron, dodecahedron and icosahedrons): as Lewis Carrol said, it is a small number so provocative. Could it have to do with the fact that perfection is so rare? Musmeci discovered that there are only three regular polyhedrons (tetrahedron, cube, octahedron) and two semi-regular (triangular bipyramid and rhombic dodecahedron), among the 13 known in 3d space, fitting the role of the node. Another intriguing small number indeed. He also noted that polyhedron node can join both prismatic and anti-prismatic bars If in geometry, a n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles, in a framework it is composed of prismatic or anti-prismatic identical bars (Fig 6)



. Fig.7 – Detail of octahedral framework system with anti-

prismatic bars.

Fig. 8 – Detail of octahedral framework system with

prismatic bars. Antiprisms are similar to prisms but the bases are twisted relative to each other, and the side faces are triangles, rather than quadrilaterals.

Anti-polyhedrons. Musmeci’s definition.

Musmeci called anti-polyhedron all those structures made with vertex, edges and faces just like regular polyhedrons, but supplied with an excess of polygons around each vertex. As a matter of fact frameworks made with prismatic or even anti-prismatic bars are anti-polyhedron (…) In Bukminster-Fuller dome, for instance, each node has 6 bars except for few that has 5 (those icosahedrons originating the volume of the dome. (…)In the same way, if we insert 7 bars in anti-polyhedrical nodes … we can obtain the generative points of the inversion of curvature. And if anti-polyhedrical nodes have increasing reciprocal distances starting from the origin of that specific point, we can obtain a framework shaping a saddle surface like a Hyperbolic paraboloid (“Parametro” n. 80, 1979, p. 30)

Fig. 9 bridge over the Niger river



Roma 1979. Piazza S. Salvatore in Lauro. Exhibiting space frames. In 1979 Musmeci exhibited full-size models representing various aggregated structures illustrating those huge, enigmatic and sharp space frames, that are systems geometrically constructed through the same formative process of regular and irregular polyhedrons and their reciprocal transformation from one into the others. His studies on polyhedrons culminated with the definition of the anti-polyhedron, a potentially unlimited and undetermined geometrical figure, even though it was generated from regular figures. At S. Salvatore in Lauro square, in the very middle of the city of Rome, Musmeci exhibited the only five possible spatial systems with the joint shaped voids - just described - and bars all equivalent. According to Musmeci, the significance of the system was in determining the so called “coordination number” of each of the five systems, that is the maximum number of bars meeting in a node (or joint). The easier is the cubic system, whose “coordination number” is 6 and one of the more complex is the rhombic dodecahedic with “coordination number” = 12; the octahedric system (“coordination number” = 8) the nodes are situated on the vertex of an ideal cube whose diagonals are antiprismatic bars. The same characteristic has the tetrahedric system whose nodes have the same spatial disposition of carbon atoms in a diamond crystal. So it is the shape for the project of Oikos pavilion in Bologna (work completed by his wife Zenaide Zanini after Musmeci’s death).

Fig. 10 – Tetrahedral framework system with anti-prismatic bars.



4 Consequences, (partial) conclusions and open questions. To sum up, any structure can belong to one of those two main systems: the continuous or the discontinuous one, but also to a combination between them The former are those structures made with concrete or else with plastic materials, the others are trussed structures and space frames. Musmeci applied the minimal structure concept to both, as we can realize in the bridge over the Basento and in the space frames. The fact that in Musmeci’s researches the role of geometry is no more exploited as an instrument for controlling architectural form, but for its liberation, is clearly highlighted in the projects presented, but also - and even much more - in the project for an helicoidal skyscraper which consisted of a succession of warped wings developed on the layout of the logarithmic spiral. It is a structure which considers the use of high-limit steel cables placed in accordance with helicoidal spirals in the space. It has an aerodynamic form and structure inspired by high technology, but it also assumes a strong organic metaphor referring itself to the extremely effective endurance due to the relationship between bone and muscle: a compressed central core and external tensioned fascias. Three warped wing anchored to three 565mt-high mast elements define a form minimizing the wind load.

A rose is a rose is a rose. Back to Aristotle's Law of Identity? Fig. 11 – Helicoidal skyscraper

Everything that exists has a specific nature. Each entity exists as something in particular and it has characteristics that are a part of what it is. "This paper is about Sergio Musmeci’s structures and is 8 pages long." To have an identity means to have a single identity; an object cannot have two identities. Or not? In the particular Musmeci’s figurative researches generated from one subject matter: minimal surface, one can say that a singular condition can generate one and only one entity/identity, such as for the skyscraper, but also that different conditions (continuity of concrete structures and discontinuity of trussed frameworks) can reach the same conclusion, each with its identity. Are they or not very close one another, if we imagine the framework of the bridge over the Niger river, covered with the foreseen precast shell, to the Basento bridge? And if in the Aristotelian logic, the law of identity is the first of the so-called three classic laws of thought that states: if an object is the same as itself: A → A (if you have A, then you have A), then Gertrude Stein’s a rose is a rose is a rose meaning "things are what they are" is a popular version of the statement of the law of identity. Deleuze's main philosophical project in the works he wrote prior to his collaborations with Guattari can be briefly summarized as a systematic inversion of the traditional metaphysical relationship between identity and difference. Traditionally, difference is seen as derivative from identity: e.g., to say that "X is different from Y"



assumes some X and Y with at least relatively stable identities. To the contrary, Deleuze claims that all singular identity is effect of difference. Does it means that When all is said and done, a thing is what it is

Suggestions required Selected bibliography

There are very few publications about Musmeci, so Musmeci's research is often hardly known. With the exception of a very rare book edited by Sergio Musmeci himself in 1979 and of a monographic number of the italian review “Parametro”

[1] Sergio Musmeci, Pordenone 1979 [2] Sergio Musmeci o delle tensioni cognite, in “Parametro” n. 80, 1979 (monographic

number) and some interesting presentation of single projects on magazines, three more hints: [3] M. Nicoletti, Sergio Musmeci.. Organicità di forme e forze nello spazio, Torino 1999 (brief

monography) [4] M. Guccione (a cura), Il ponte e la città. Sergio Musmeci a Potenza, Roma 2004 [5] R Capomolla, Le "forme organiche strutturali". Materia e spazio nelle opere di Sergio

Musmeci, in “Rassegna di architettura e urbanistica”, n.121/122, gennaio-agosto 2007, pp.135-148 e 171-172

Current address Alessandra Capanna, prof. arch., Ph.D "Sapienza" Università di Roma – Dipartimento di Architettura e Progetto, via Flaminia 359, 00196 ROMA tel. +39 06 32101244, mobile +39 335 6709838 e-mail:[email protected]

SERGIO MUSMECI'S " FORMS WITH NO NAME" AND "ANTI-POLYHEDRONS"

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