.

Istituto Statale d'Arte1, Via Giovanni Boccaccio -- Villa Reale 20052 Monza (Milan) ITALY
Manlio Brusatin describing Carlo Scarpa's house in the essay "La casa del architetto" [1984].
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.[1]
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
The choice of this particular argument, besides the desire to create interesting forms, was fundamentally motivated by the following considerations: - The descriptive precision to which the forms of conch shells lend themselves that illustrate a process that is easily described in geometrical-mathematical terms but that is also capable to generating a variety of forms that is large enough so as not to seem the product of a single matrix;
- Conch shells present a case in which geometrical-mathematical requirements (in this case dictated by the necessity of isometric growth) impose certain rules upon nature;[2]
- The various aspects that this itinerary brings to light concerning modularity, one of the traditional tools of design. The analysis undertaken by means of what we have called 2D and 3D demonstrates how the traditional concept of modularity based on the congruence of parts can be, in some cases, advantageously amplified by means of other geometric transformations.[3]
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).
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: **Gastropods**, which are herbivorous and carnivorous, require a portable shell that permits them to move agilely in search of food. The gastropods are the classic univalve conch shells with a spire that is more or less marked (fig. 1).
**Bivalves**are sedentary filterers that live in muddy environments and that protect themselves from predators by quickly digging down into the mud thanks to the wedge shape of the shell that is formed of two halves that are hinged together. Also, the bivalve conch affords better protection of the soft parts of the body from the abrasive action of the grains of sand (fig. 2).
- The
**cephalopods**are predators for which speed of movement is an essential tool for survival. They have more or less lost their exterior shell, conserving in some cases internal residuals that serve as supports, such as in cuttle fish or squid, or as organs that aid in maintaining a hydrostatic equilibrium, such as in the genus Spirula. There are exceptions such as the Nautilus, which produces a very beautiful exterior shell that serves for hydrostatic equilibrium, or as the Argonaut (fig. 3b), which produces through its tentacles a beautiful temporary external structure, used as a egg carrier.
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.
From Thompson [5] we note the following observations: **growth by addition**: conch shells increase their dimensions by adding new material to what already exists. Thus the forms that we see conserve and contain in themselves all the previous phases of growth. Each adult conch shell conserves at its apex the proto-conch in which it has its origins.**isometric growth**: in their growth conch shells always conserve the same shape. If we take two conches of the same species but of different ages we can see that the one shape is the enlargement of the second. This characteristic, known as isometric growth, permits the mollusc to increase its own size while maintaining the same proportions among the parts of the body housed in the shell (fig. 4).
It has been known since 1638 that the spiral shape of conch
shells has the property of self-similarity during growth (see
For the sake of simplicity, let us imagine the surfaces of
the conch shell as the results of the - Let us consider a reference plane containing the axis of rotation; the directrix figure (which we imagine to be flat) can lie in that plane or it can make with it an angle that is always constant.
- The directrix, in the course of its rotation, increases constantly in its linear dimensions, while maintaining invariable the value of its angles. At every successive turn the figure grows uniformly, that is, the rate of growth of the directrix is constant. The distance between the directrix and the axis of rotation also increases according to the same rate of growth.
- In the course of its rotation, the figure can perform a translational movement along the axis. The vector that describes the resulting translation is directly proportional to the rate of growth of the directrix (fig. 5).
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).[7] 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:
By which
This statement implies the assumption of a local kind of approach. As happens sometimes in the context of the science of complexity, [8] 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.
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.
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).
Applying these considerations to the classes of quadrilaterals we can see what follows (see the table summarizing these in fig. 14).
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.
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.[11]
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.[12] 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).
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.[14] 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:
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
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 attempt to represent a specific type of conch shell identifying -- by means of the languages of various disciplines -- a module and a constructive strategy that is appropriate to that case;
- the construction of models of "impossible" conch shells of particular beauty and formal interest.
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.
[2] This is what Peter Stevens in Les formes dans la
nature [1978] 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.
[3] 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".
[4] François Jacob [1978] 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. [5] In 1917 D'Arcy Thompson's On Growth and Form [1961],
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. [6] Michael Cortie in [Stevens 1978] presents a mathematical
model, visualized by means of computer graphics, of the growth
of mollusc shells. [7] 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]. [8] For example, the study of cellular automata. [9] "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. [10] 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. [11] 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. [12] 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. [13] This restriction as well is intended to exclude
a helical torsion in the aggregated structure. [14] Those wishing to go into this part of the work in
greater depth should contact the authors.
Institute for Lightweight Structures. 1977.
"Pneus in Natur und Technik / Pneus In Nature and Technics",
Vol. 9 of the Stevens, Peter S. 1978. Cortie, M. 1990. The Form, Function, and Synthesis
of the Molluscan Shell. In Mezzetti, G. 1987. Francois Jacob. June 10, 1977. Evolution and
tinkering. Jacob, Francois. 1978. D'Arcy W. Thompson. 1992. H. Meinhardt. 1995. Theodore Andrea Cook. 1978. Richard Dawkins. 1997. Robert D. Barnes. 1985.
L. Curcio
(Mathematics), R. Di Martino (Geometric Disciplines),
L. Gerosa (Laboratory of Visual Communication), C.
Tresoldi (Laboratory for Applied Arts) are professors at
the Istituto Statale d'Arte
(ISA) di Monza.
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