For mollusks, spiraling into a permanent shelter can be done in a number of ways. All are variations of a general process that we try to reproduce.
Artificial intelligence claims to imitate human intelligence; it surely advances, but slowly. In biology, projects to computer model a cell while respecting all its complexity are underway, as difficult as they are. Fortunately seashells, at least their limestone part, are better copied.
Research into the shape and surface designs of seashells now makes it possible to write programs of a few dozen lines which, transmitted to 3D printers, guide them to manufacture many and varied models. The quality of the objects produced in this way is such that it is hard to believe that they are only the projection of a mathematical formula of which a few parameters are varied. Francesco De Comité, from the Cristal laboratory in Lille, is an expert in this computer science which imitates living things and which should be considered as a mathematical art.
A history dating back to the 19th century
The regularity of the shape of seashells has long struck people, and the idea that considerations of pure geometry could give the key to their beauty has, since the beginning of the nineteenth century, given rise to research. This is how Henry Moseley, professor at King’s College, Cambridge, England, explained in an article from 1838 the first elements of this mathematization of living things:
“There is an observable mechanical uniformity in the structure of shells of the same species; it suggests that the generating figure of the new layers increases in size at the same time as the spiral-shaped chamber develops according to a simple geometric law. By considering the operculum of certain classes of shells, the determination of this possible law seems possible. Continuously enlarged by the animal as it grows, as the construction of its shell progresses, the operculum gradually widens the spiral chamber. “
The mathematical model which is deduced from these remarks will hardly be modified afterwards. It is described in Box 2 with modern notations.
Logarithmic spiral growth
Some aspects of the biological mechanisms of growth, although quite basic, are not easy to fully justify in practice. Indeed, why, for example, does growth remain proportionately constant, and not weaker and weaker and stronger?
The resulting shape is a logarithmic spiral in space. It is the only spiral which, when magnified, remains identical to itself. This remarkable geometric property is perhaps an evolutionary advantage, or at least a facilitation in the adaptation of the animal to assume different sizes. However, this property is not easy to justify, since it is not general in the animal world: an adult is not a child twice the size or more; it has a proportionately smaller head.
These geometrical-mathematical questions are now part of biology, and this thanks mainly to a famous book whose centenary has just been celebrated: On Growth and Form (the French title is Forme et Croissance), by the Scottish artist D’Arcy Thompson. , a work whose first edition appeared in 1917.
The book argues that biology overestimates the role of evolution by neglecting physical and mechanical constraints, which are determining factors in the organization and forms of living organisms and their components.
Richly illustrated, this book examines the wings of insects, the feathers of birds, the more or less coiled horns of mammals, jellyfish, and plant seeds. It deals carefully with the physics and the delicate geometry of cell contacts and cell aggregates, the strength and flexibility of bones. He studies the geometric structures of different species of fish and makes connections, introducing a theory of transformation which is now useful for computational methods in the field of pattern recognition. He considers the alveoli of bee hives and, on this occasion, the filling of space with polyhedra. There is no shortage of wonder at the snowflakes, the fascinating skeletons of radiolarians (unicellular organisms with siliceous skeletons, part of marine plankton) and, of course, seashells.
D’Arcy Thompson’s work had a great influence in biology by putting physics in its rightful place; and indirectly mathematics: physical laws create forms that can only be understood by solving systems of equations that are sometimes quite complex, and therefore by implementing the mathematical tools of analysis.
Shapes built by a 3D printer from a unique formula
Let’s go back to our seashells and consider the simplest model that describes them. Does it lead to satisfactory reproduction of known species? Until recently, computer tools could only envision models of seashells in computer memory or projected onto a plane, the plane being a terminal screen or a sheet of paper. 3D printers, whose technical developments have accelerated, offer the possibility of going further by transforming equations, found or assumed, into objects that we take in hand, that we weigh, that we examine with care. .. and put against his ear to “hear the sea”!
Remember that 3D printers work from a mathematical definition of the object that we want to obtain, a definition that is expressed in a computer program. These machines sometimes produce objects that cannot be obtained by molding techniques or by conventional cutting and machining (for example a cube whose interior would be hollowed out to leave an empty space that would have the shape of your head). But not everything is possible.
This is because the various parts of the shape sculpted by the printer or fabricated layer by layer must be spatially bound if one wants an object that retains its rigidity. Above all, the intended object must not contain too thin parts that the printer would be unable to create, or which would weaken the final result too much.
The shapes produced by Francesco de Comité (see boxes) were all obtained from a single formula translated into a program and which only includes a few parameters that he varied by exploiting the mathematical model presented in box 2. These geometric parameters are mainly those which determine the three-dimensional spiral described by the center of the operculum during the growth of the shell, to which must be added the parameters of the curve corresponding to the shape of the operculum as well as some data to specify the patterns decorating the surface of the shell.
As for the shape of the operculum, the following passage shows that D’Arcy Thompson not only understood the abstract principle of construction, but that his expert naturalistic eye could identify the parameters that characterize a species of shell:
“The generating curve [of the operculum] takes on a variety of shapes. It is circular in Scalaria, Cyclostoma and Spirula; it can be considered as a portion of a circle in Natica or Planorbis. It is triangular in Conus or Thatcheria, and rhomboid in Solarium or Potamides. Very often, the generating curve takes more or less the shape of an ellipse: in Oliva and Cypraea, the major axis of the ellipse is parallel to that of the shell; […] and in many well-documented cases, such as Stomatella, Lamellaria, Sigaretus haliotoides and Haliotis, the major axis of the ellipse is oblique to that of the shell. This generating curve becomes almost a half-ellipse in Nautilus pompilius and a little more than a half-ellipse in Nautilus umbilicatus, the major axis in both cases being perpendicular to the axis of the shell. The shape of the curve rarely lends itself to a simple mathematical expression. […] In certain ammonites, those of the “Cordati” group, […] the generating curve corresponds almost exactly to a cardioid of equation r = a (1 + cos q).”
Computer difficulties and surface drawings
To have a mathematical definition that the 3D printer understands, it is not only necessary to take care of defining the outer surface of the shell, but also its inner surface, which is of the same type. The two surfaces must be linked by operating in such a way that the delimited volume is without holes, never too fine and does not require greater precision in detail than that provided by the printer.
This construction of the volume therefore requires work that is not reduced to writing the formulas proposed by the geometric analysis described in Box 2. Using a classical method, Francesco De Comité obtains the solution by transforming the surfaces into a very large one. number of attached triangles, triangles which are also useful for coloring the surface. The size of the program, when stripped of comments and reduced to its essentials, is 4000 characters (or bytes), or about a large page.
Beyond the shape of the seashells, which is fairly well mastered, a second challenge presents itself to the modeler, that of the colored patterns on the surface of the seashells.
A first method of reproducing them consists of capturing them by a photographic process and plating them on the surface of the artificial shells that are programmed. This will of course give quite good results, but it is a form of cheating which will not allow to obtain a short mathematical model, since it takes a lot of information to memorize the image, copied pixel by pixel, of the surface of a real seashell.
Fortunately, more satisfactory methods seem possible. They are based on the idea that the shell comes from the slices of the operculum that are built successively. The radius increases and these slices are colored according to simple rules, which determine the coloring of each new slice according to the previous ones. Most often, the color of a point of slice n + 1 will be deduced from that of the point of slice n located at the same place and of its neighbors. Sometimes, to determine the pattern of slice n + 1, it will be useful to use the patterns of several slices before slice n.
The British Conrad Waddington and Russell Cowe adopted this hypothesis in 1969, followed in 1984 by their compatriot Stephen Wolfram, the famous creator of the computer algebra software Mathematica. They assumed that the dots were colored according to the rules of a one-dimensional cellular automaton. This type of mechanism, devised by the Hungarian-born mathematician John von Neumann in the 1940s, is the simplest conceivable to define deterministic interactions between discrete objects (see Box 3).
The cellular automata hypothesis, which for a computer scientist is the easiest to program, does not seem to be based on precise biological considerations, however. At least as a test, it deserved to be explored and what it yields vaguely resembles what is seen in nature. One reason for this limited success is that cellular automata are often used with two colors, which is insufficient to accurately reproduce the complexity of shell pigmentations.
A hypothesis closer to biology is inspired by the view that the British mathematician Alan Turing developed in the last years of his life to describe mathematical models of morphogenesis. Adopted by the German biologists Hans Meinhardt and Martin Klingler, it was taken up by Francesco de Comité.
The idea is that of a short distance activation combined with a more prolonged inhibition determined by two substances, an activator and an inhibitor. The concentration of each of them depends on their concentrations at each point in the previous stages of construction. The model is continuous and is expressed by partial differential equations (differential equations where the functions depend on several variables).
Here is an example of these equations.
Without going into detail, let us state that a (x, t) and b (x, t) are the concentrations of activator a and inhibitor b. They depend on the time t and the position x of the colored zone on the contour of the operculum, assimilated to a curve. The coefficients Da and Db are the diffusion parameters; the higher their value, the faster the diffusion of substances. The coefficients ra and rb characterize the decay rates of the concentrations a and b (for more details, see the magnificent book by Hans Meinhardt, The Algorithmic Beauty of Sea Shells, Springer, 2009, page 23).
Aesthetic games and new challenges
After trying to imitate the reality of the biological world as well as possible and having discovered and exploited some of its secrets to make shapes that mimic its creatures, it is fun to play with the resulting model. We then produce shapes as beautiful as the real ones, but which do not exist in nature.
However, nature, which did not play at making all these pseudo-shells that the mastery acquired allows to create, has explored other more complex avenues … which today escape modelers. In particular, the shells of the Muricidae family (which includes the genus Murex), used since ancient times to extract purple, show regular spikes.
These structures which escape the general mathematical model used for the other shells require complements and will undoubtedly cause a complication of the current programs. Francesco de Comité reflects on how we could overcome the obstacle created by these exceptional and recalcitrant seashells. If he succeeds, let’s not believe that the work of modeling reality will then come to an end: the living world is not limited to shells and its inventiveness will not be easily reduced to a few mathematical formulas, whatever the care. and the intelligence that we will devote to it.
