1, Via Giovanni Boccaccio -- Villa Reale
20052 Monza (Milan) ITALY
" a small collection of curious objects, celebrations of natural and artificial forms, decorated a shelf on the wall. Some ingenious wooden puzzles, some shaped colored stones, some rock crystal spheres, some shaped and smoothed conch shells that appear to the eye as almost the natural origin of architecture, in the same way that once every fossil was considered "nature's joke" and thus an inanimate witness to the creation of the world."
Manlio Brusatin describing Carlo Scarpa's house in the essay "La casa del architetto" .
The material forms of the world speak in a language that is a network of simultaneous stimuli, far from the logical sequences that helps us to assign names to things and to rationalize. It is a process in which the sensorial impressions are first extracted from the continuum and then ordered in itineraries formed of questions and attempted answers. To obtain something that is not merely chance from this flux of information is a difficult task but it is one of the principle objectives of a formation in the world of vision and design. Since we undertake this task in the context of didactics, we can say that that each discipline possesses analytical filters that attempt to translate "continuous" reality into a "discrete" whole, an interpretative structure.
Natural objects, given the extraordinary stratification of meanings, lend themselves nobly to investigations, such as this one, of an interdisciplinary nature. Further, the spontaneous beauty of natural forms is something to aspire to; this beauty is born from the intimate relationships between form, material and function where every layer, every color appears essential, a sobriety made of infinite subtleties, which design theorists hold to be a fundamental quality of formal coherence.
The observance of natural forms has always inspired design choices in architecture; from the classic theme of the spiral staircase, articulated in innumerable examples both ancient and contemporary --from the admirable staircase at the Castle of Blois attributed to Leonardo, to that of Gaudi for the Sagrada Familia, to Pei's recent museum in Berlin -- to more subtle and profound relationships between architectural form and natural principles (it suffices to think of the late works of Gaudi based on a vocabulary of static spontaneous forms that implicitly invoke images of natural objects). This kind of research, ever more diffuse, in the course of the twentieth century with the so-called architecture of engineers, has been carried forward, albeit in different languages and with different approaches, by Nervi, Musmeci, Le Ricolais, Candela, Fuller and Calatrava. The German Frei Otto merits a separate acknowledgment, as founder of an interdisciplinary group where architects and engineers work side-by-side with biologists. In terms of research into architecture and natural forms that is decidedly less related to engineering but not for this the less interesting, we recall the work of the American Frank Gehry and the Swiss Jacques Herzog and Pierre De Meuron with their book Natural History.
THE STRUCTURE OF THE WORK
The choice of this particular argument, besides the desire to create interesting forms, was fundamentally motivated by the following considerations:
It should be added that this local kind of approach implied by this work could serve as a concrete introduction to the study of complex systems. This vision inverts the hierarchy of organization of form, introducing the concepts of parallelism and sensitivity that for years has been the patrimony of the more abstract scientific disciplines but that should nevertheless be learned and digested within the context of a design school (to this we shall be dedicating some didactics projects in future years).
THE RELATIONSHIP BETWEEN FORM AND FUNCTION
Schematically we can observe that in all cases the conch shells perform the function of a shield and support for the soft, vulnerable parts of molluscs, even if the behaviour of the various classes of molluscs are rather different. There are essentially two typologies: univalve shells and bivalve shells. Let us consider the classes of subtypes of the conch molluscs:
Engineer or tinkerer? The collection of different functions assigned to the conches of different species of molluscs reminds us of what Jacob said in his "Evolution and Tinkering" . He observes that , from the point of view of human design, nature seems to adopt sometimes an approach to engineering that conceives structure rationally in order to respond to the requirements, sometimes even extreme, of the ecosystem in which they have to operate. But more often nature seems to act as would a passionate tinkerer who finds himself having to adapt to new uses structures that were already conceived for other purposes.
In the case of the conch shells, we can perhaps say that the portable shelters seem conceived by the engineer while the tinkerer then re-adapted them as egg carriers, internal skeletons, or even as floating devices similar to submarines.
Observations and morphological surveys. Conch shells of almost all molluscs, including bivalves, are characterized by an elegant design and symmetry that is based on a cone that turns in a spiral about an axis.
From Thompson  we note the following observations:
It has been known since 1638 that the spiral shape of conch shells has the property of self-similarity during growth (see isometric growth). This implies that the projection of any generating spiral onto an orthogonal plane at the axis of symmetry (in reality, as will be explained below, this is an axis of rotational translation) produces a curve that was studied for the first time by Descartes and defined by him as an equiangular or logarithmic spiral.
Geometric analysis. We sharpen the studies undertaken thus far by citing a geometric definition of the morphological characteristics of the conch shell by Cortie . Not wanting going into the details of his mathematical model with its sixteen variables, we limit ourselves to an effective description of its fundamental characteristics.
For the sake of simplicity, let us imagine the surfaces of the conch shell as the results of the rotation of a plane figure (a directrix curve that represents the shape of the opening out of which the mollusc comes) about an axis according to the following procedure:
Respect for these conditions produces a solid (which from now on we will call a shell) having as its directrix the shape of the opening and as its generator the helixes which, when projected onto an orthogonal plane, produce, as mentioned, logarithmic or equiangular spirals (fig. 6).
At this point the mathematics teacher -- who has already introduced the measure in radians of the angles and the representation of point by means of polar coordinates -- completes the investigation introducing the description and the equations for spirals (both Archimedean and logarithmic).
This description is the point of departure for our work. What has just been discussed imply the principle on which our research will hinge:
This statement implies the assumption of a local kind of approach. As happens sometimes in the context of the science of complexity,  we can describe the global shape solely by means of given module and those adjacent to it.
In order to clarify this kind of approach, it seems best to take as the point of departure the study of a situation that is more accessible in terms of three-dimensional intuition. We will deal with one that, perhaps a little pompously, we have called a 2D model, referring to a hypothetical conch sell in a two-dimensional world. Studying this model helps us to better understand the logic behind the isometric growth that will become the nucleus of a 3D model that approximates the dynamics of form that are actually observed.
THE 2D MODEL
The aggregation of quadrilaterals. Let us begin our itinerary by observing that any convex quadrilateral -- when joined side by side with similar copies of itself -- produce a covering  of an area of the plane in which all the vertexes of the paving belong to two distinct families of logarithmic spirals having the same eye but opposite rotational directions. This is a procedure for paving the plane with modules that have the same shape but are of different sizes (fig. 10 shows examples of how, from any given quadrilateral it is possible to develop a covering of the plane).
We can confirm that
In the next section we will deal briefly with the classification of all quadrilaterals, and then analyse the relationships established between each quadrilateral and the family of spirals that it produces.
Classification of quadrilaterals. We have distinguished
four parameters, a, b, d, f, expressed
in the terms of angular size, whose variations allow us to identify
any family of quadrilaterals -- obtained from the intersection
of two triangles -- and to distinguish every single possible
case (see fig. 11 for a description in intuitive terms).
In particular let us consider how the variation of the four parameters characterizes the categories of convex quadrilaterals (recall that, according to this description, when an angle is equal to 0 the two straight lines that go to it are parallel).
Quadrilaterals and pavings. In light of the parameters for the classification of quadrilaterals, we reopen discussion on the relationships between them and the coverings that they produce. Considering that "each quadrilateral corresponds to a covering", we can identify the two fundamental relationships between a quadrilateral and the covering it generates:
Applying these considerations to the classes of quadrilaterals we can see what follows (see the table summarizing these in fig. 14).
Parallelograms, which have values of a and b equal to 0, produce systems of spirals that are straight lines and thus the covering corresponds to a paving of the plane of modules that are congruent. In this case the eye of the system can be considered as infinitely far (a point at infinity).
Trapezoids, which have values of a and b equal to 0, produce systems of spirals in which the sides not parallel to the module are aligned with the eye. In the case of isosceles trapezoids the system of spirals collapses into a system of concentric circles and radial straight lines that pass through the eye.
Generic quadrilaterals, having values of a and b that are not equal to 0, produce -- generally -- systems of spirals in which the sides of a module are not aligned with the eye.
Pavings of quadrilaterals and 2D conch shells. From
this overview of coverings produced by similar quadrilaterals
we reach the object of this section, which is that of producing
shapes that are 2D models of hypothetical conches in two-dimensional
space. From a conch shell, even one in 2D, we expect, in the
first place, a serrated spire, that is, that between one turn
and the next there are not residual spaces or overlaps. We note
that the serrated spire is a primary element of identification
of a conch shell.
In order to produce the model of the conch shell in 2D, a quadrilateral must tile such that the whole tiling is the result of the spire of one strip only, as we see in fig. 15 (1, 11). The image in fig. 10b, for example, shows 12 strips in one direction and 9 in the other; the lower one of fig. 14c shows 6 and 12; that on the left in fig. 15 shows (5, 5). Now let's describe a geometric construction that allows us to determine the quadrilaterals, whatever they may be, modules that can construct a paving like that in fig. 15b, characterized by the presence of only one strip in one of the two directions. We will not treat in more depth this topic, since its development would require the use of tools that are beyond the simple basic geometry used in this work. It is interesting to note how this aspect of the 2D model leads us to reflect on geometric problems related to the phylotaxis, opening new horizons to quasicrystallography.
Geometric construction of a serrated spire. Let us imagine that we wish to produce a conch shell, that is, a serrated spire of modules starting with any convex quadrilateral, which, as we see in fig. 16a, produces an open spire.
We can find for every quadrilateral in the row the point of intersection of the diagonals, and we then join to them the centres of adjacent modules and those of the modules that are found one in front of the other in the successive turns of the strip in fig. 16b. Once we have identified a quadrilateral we can develop it into a row that is sure to produce a serrated spire or from which we can create a paving of the plane. In fig. 17 we can see a collection of 2D conch shells produced in this way; we see also that they can be interpreted as a projection on a plane of 3D conch shells.
In the next section we will verify a strict analogy between the characters of the systems in two and three dimensions, almost a simple translation from 2D to 3D. To us, these correspondences appear particularly meaningful:
This part of the work was made possible thanks to the use of an interactive computer support such as Cabri II. By means of the creation of a macro-construction it is possible to quickly join the similar quadrilaterals of the right dimensions. Without the use of the computer this task would be almost impossible to manage graphically.
THE 3D MODEL
The idea is that of arbitrarily collocating in the space between two similar polygons (which for simplicity we will think of as regular) and then to unite by line segments the points that we will identify as corresponding vertexes (in the box we will see this operation conducted on two regular hexagons).
The only restriction that we impose is on the free choice of the positions of the sides in space and that of the absence of reciprocal rotation between them. That means that setting side-by-side their planes of appurtenance of the two bases of the module (as if to close the pages of an imaginary notebook), they form couples as the result of a homotheity.
The figure generated by this simple operation is the module (for a more precise definition of its construction, see the box below).
Aggregation. We proceed with the construction of our 3D model by making, by means of a similarity, some copies of the module. The relationship of this similarity will be the same as that between the dimensions of two sides of the module. Proceeding in this way we will produce a series of modules of decreasing sizes. The modules will then be aggregated (one over the other) so as to make couples of coinciding bases with the same dimensions, having taken care to make the vertexes correspond. With this procedure of aggregation we will obtain a spatial structure with a spiralling movement whose characteristics will depend completely on the parameters that determine the form of the module.
Parameters of the module. At this point in the work, after having set forth the basic operations of construction and aggregation we shall see in an effective way -- modifying the critical factors of the shape of the module -- how they can reproduce, in an essential form the dynamics that can be observed in nature.
It is possible to demonstrate that with any choice whatsoever of the following parameters one obtains a module that is capable of generating by aggregation a solid that can be inscribed in a shell, that is, a surface having helixes as the generatrixes:
Parameter A directly influences the velocity with which our shell turns on itself about the axis. It is the parameter a traditionally indicated by biologists (see fig. 6).
Parameter B influences the width of the cone that, when it turns, generates the shell (if we imagine that Parameter A is 0, then our shell would be a cone that is more or less pointed, according to the value of Parameter B).
Parameter C, on the other hand, influences the inclination of the sides with respect to the axes, during the rotational translation.
Parameter D is the vector that controls the movement of the two sides in a direction parallel to the hinge of the module and therefore at the axis of rotational translation of the structure. This parameter is that which in reality distinguishes conch shells that are "pointed", such as that of the Turritella, of those "flat" ones such as the Nautilus or the Ammonites. As can be easily intuited, the difference lies in the presence of the translation along the axis (very evident in the first case, and practically nonexistent in the second).
Some graphic examples. Figures 18, 19, and 20 show some "conch shells" side-by-side with the modules that produced them. We can appreciate the effectiveness of these reproductions, which allow us to intuit how, modifying the basic shape, it is possible to simulate many dynamics of the natural forms and beyond. One of the most stimulating aspects of this work is that the notable resemblances between the images produced and the natural objects lies in the close analogy of the constructive principles and not in an attempt at imitation. In fact, some of these images propose forms that, even though the procedures have been correctly followed, generate forms that are rather improbable for a real conch shell.
Fig. 18. The construction of a conch shell that is similar to a gastropod. This shell is characterized by: Parameter A, a marked inclination between the sides; Parameter B, a low ratio of homotheity; Parameter C, a marked inclination of the sides with respect to the axis; Parameter D, the absence of translation with respect to the axis. The ridges of the lateral faces of the module highlight the non-planar quality of the quadrilaterals of which it is made
Fig. 20. An "impossible" conch shell produced by the module in the foreground
The built models. Following the studies undertaken we realized a series of models in metal and Plexiglas that are shown in Figs. 21 and 22. The intent was to represent some of the more meaningful categories of the undersea world of mollusc shells.
Fig. 23. Conch shell similar to some of the gastropods characterized by a high level of translaton of the sides along the axis, a marked inclination and a high ratio of reduction between the sides
Fig. 24 Conch having triangular sides, a marked translation along the axis, a strong inclination between the sides and, and like all the models built, the absence of inclination of the sides with respect to the axis
Fig. 25 Tatcheria mirabilis
All of the models were based on a module with octagonal sides.
With the aim of alleviating the problem of the side faces of
the modules that, as mentioned, are curved, we chose to avoid
realizing them by connecting the octagonal sides to a continuous
metal structure. This choice highlights the shape of the load-bearing
structure based on a planar logarithmic spiral and allows us
to underline the fundamental role of the this shape, which dictates
the rhythm of growth in space as well.
Future didactic work could develop in at least two directions:
The relationship between possible/impossible is a particularly stimulating theme for research into the fundaments of linguistics such as the one presented here, and is one of the specific themes of research undertaken in accordance with scientific criteria.
Translated by Kim Williams
 This is what Peter Stevens in Les formes dans la nature  defines with the effective expression "space tyranny" , that is, those cases in which spatial imperatives, mostly of a topological nature, impose some of the formal choices. return to text
 Effectively substituting, within the theory of the tiling of the plane, congruence with other geometric transformations such as affinity or homology gives rise to curious phenomena of visual illusions, one example of which is "false axionometry", which we discuss in the section "Some graphic examples". return to text
 François Jacob  was the winner of the Nobel prize for medicine with Jacques Monod, author of Chance and Necessity, a fundamental thesis on the natural philosophy of contemporary biology. return to text
 In 1917 D'Arcy Thompson's On Growth and Form , a bible of morphological studies of nature republished several times throughout the world, has influenced generations of naturalists, architects, engineers and scholars of shape in the broadest sense of the term. return to text
 Michael Cortie in [Stevens 1978] presents a mathematical model, visualized by means of computer graphics, of the growth of mollusc shells. return to text
 The value a of the angle that is characteristic of this spiral shows a particular taxonomic value, tending to remain constant in the course of the evolutionary history of many species; see [Meinhardt 1995]. return to text
 For example, the study of cellular automata. return to text
 "Side to side" means that the side of a module has to coincide entirely with that of the module adjacent to it. In the case of a tiling with rectangles, an example of side to side tiling is that of a grid made of intersecting straight lines, with a tiling that is not side to side is that, for example, in which the rectangles are arranged like bricks in a wall. return to text
 By covering we mean a tiling of the plane in which a point on the plane can belong to more than one module; by paving we mean rather a tiling in which a point on the plane can belong to only one module, the final possibility being that the articulation (or packing) is one in which a point on the plane may not belong to any module. return to text
 All constructions which by their nature are developed through a chain of local steps, pose the problem of the multiplication of the margin of error at every successive operation. return to text
 The presence of a rotation between the sides should be excluded since it would induce in the final structure a helical torsion that isn't present in real conch shells. return to text
 This restriction as well is intended to exclude a helical torsion in the aggregated structure. return to text
 Those wishing to go into this part of the work in greater depth should contact the authors. return to text
Institute for Lightweight Structures. 1977. "Pneus in Natur und Technik / Pneus In Nature and Technics", Vol. 9 of the Mitteilungsreihe des Instituts für leichte Flächentragwerke, Institute for Lightweight Structures, Stuttgart. (In particular we cite the section dedicated to conches: "Growth despite hardening").
Stevens, Peter S. 1978. Les formes dans la nature. Seuil Evreux.
Cortie, M. 1990. The Form, Function, and Synthesis of the Molluscan Shell. In Spiral Symmetry, Istvan Hargittai, ed. World Scientific.
Mezzetti, G. 1987. L'uomo. Dalla natura alla scienza. La Nuova Italia.
Francois Jacob. June 10, 1977. Evolution and tinkering. Science 196: 1161-1166.
Jacob, Francois. 1978. Evoluzione e bricolage. Turin: Einaudi,1978.
D'Arcy W. Thompson. 1992. On Growth and Form. Cambridge: Cambridge University Press. (First edition 1917).
H. Meinhardt. 1995. The Algoritmic Beauty of Sea Shells. Berlin: Springer-Verlag.
Theodore Andrea Cook. 1978. The Curves of Life. New York: Dover Publications. (First edition 1914).
Richard Dawkins. 1997. Climbing Mount Improbable. Penguin Books.
Robert D. Barnes. 1985. Zoologia: gli invertebrati. Padua: Piccin.
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