Celestino Soddu.

Architect, Professor of Architectural Design

Generative Design Lab, Department of Architecture and Planning.

Politecnico di Milano University






When we look at clouds transforming themselves with the help of the wind we interpret them following our subjectivity and own cultural references. We use the same transforming logics, the same anamorphic logical approach for building our subjective system of codes that we use for recognizing and facing the events surrounding us. 

This system of decoding, unique because it fits our subjectivity, mirrors our creativity and imprinting as artists and designers. Generative artworks spring from this subjective morphogenetical engine. Forms are only a subsequent step. 

Investigating on these transformation logics and the subsequent anamorfic approach we can construct our generative engine reaching a strong identity and recognizability of our works. 

One of the most interesting fields for investigating these logics is the generative engines coming from moving from different dimensions, from 2D to 3D and back, using different perspective approaches. 

Each physical city, mirroring their own Ideal City and the multiplicity of its visionary variations, will be, finally, not clonable but unique, unrepeatable and unpredictable, natural and harmonic. 


1. Forms, identity, recognizability and morphogenesis 

We recognize a form if we identify it as similar or, however comparable to those already experienced. We recognize only a form through the memory of other forms. But they are not singly remembered forms, neither sequence of different events. The memory is structured in system built activating a logic that reflects our subjectivity linking and associating different forms by identifying particular aspects of them. We could affirm that each of us builds and identifies a proper morphogenetic code in associating different forms together. This code is direct expression of our way of seeing, of our cultural references, of our identity.  

There are many and different ways of experiencing and of recognizing the physical world. Some people identify and build the code through the logical-geometric reconstruction of the process of realizing the forms, other people approach it by recognizing, structuring and following the subjective satisfaction of particular needs, from practical to aesthetical or symbolic requests. Everyone identifies and progressively sharpen during their life a series of recognizability codes, a structure of species that fits their own subjectivity and that progressively identify themselves following the increasing of own experience and of own culture. Everyone, therefore, has a unique approach in recognizing and appreciating the events surrounding him. Besides, each person identifies, following his personal way, what appears as normal (inside the species) and what as exceptional. Every system of subjective codification allows people, however, to share identifications with other people and to find the possibility that each form simultaneously belongs different species. Looking at a stool, we could affirm, for instance, that the object belongs to the "chairs" species, with the exception to have not the back, or that it belongs to the "tables" species, with the particularity to have a reduced dimension, or to the "staircases" species with the exception to have only a step...   

If we consider the field of creativeness and of design, the investigation on species and on possible approaches activated by each people to build their codes of recognizability of forms is very useful to explain the logical structure of creation. This investigation expressly identifies different creative ways bringing to the conceptual creation of the idea and its structure and the specificity and identity of the creative approach of each one.   

For instance, let's take a pyramid: a physically existing pyramid like the Pyramid of Cheope. Each people have a memory of this form and he associates it to other forms. If we call these forms as "pyramids" all people could agree, because an individualized geometric common concept exists for all people with the name of pyramid. Instead, if we want to go ahead, each of us could also associate the pyramid of Cheope to other events whose logical structure belongs to the species that each of us built for recognizing the pyramids. These logics can conceptually be very different, and they allow us to produce groups in which it is possible to overlap similar events. Some examples. A first logical approach could be defined by considering the pyramid as a solid cube from which a whole series of pieces has been removed through plain cuts. This vision for subtraction of the three-dimensional solid is the characteristic vision of the sculptor. Michelangelo affirmed that the statue already exists in the stone; it is only necessary to remove what is superfluous. If this logic is adopted for building a species of forms, any form that can be identified as what remain of a cube after cutting away some pieces, it belongs to the same species of the pyramids.   

A second logic could be constructed starting from a plain matrix that produces spatial events. A square with two diagonals, when the center and intersection of the two diagonals is "lifted" and moved in the space produces pyramids of which that of Cheope is only one among endless possibilities.   

A third possible logic still could be born from the cube. If the superior face is magnified or reduced smaller, this transformation produces a whole series of solids where the pyramid is the moment in which the superior face is reduced to zero, while the trunks of pyramid and the hourglasses are the moment when the dimension of the superior face is positive or negative.   

We can identify a fourth possible logic considering the pyramid as a solid generated by the following facets of a half-sphere. If we progressively divide the half-sphere in triangles we produce a whole series of solid that, departing from one almost-half-sphere with a large number of triangles we reach the square based pyramid as next to last step before the tetrahedron.   

A fifth logic, that we could call zigurrat, considers the pyramid as an overlap of squared based prisms. The range can start overlapping two prisms and can go ahead increasing the number or prisms when each one becomes more and more thin.   

But we could continue imagining the pyramid as one individual of a species in which each event is inside the progressive transformation from the cone to the triangular based pyramid or to the triangle itself, imagining this last event as a pyramid with the base constructed with two or only one side.   

Which is our subjective logic for declaring that an event is a pyramid? If we find us in front of the pyramid of Cheope anybody have no doubts because this pyramid is a common point of reference. But if we are looking at an event whose form is not so axiomatic, because it contains, for instance, also some curved surfaces or it has not peak, and so on, who will identify it as pyramid, even if particular and on the boundaries of this species? Perhaps only who conceives the pyramid as an event inside the progressive transformation of a half-sphere, or of a cone. The other people will associate the form to other morphogenetic species.   

Every form, when losing its geometric axiomatic aspect, that is when no one can incontrovertibly associate it to only one geometric species, becomes anamorphic shape. It becomes a form changing meaning, changing character, changing "species" according to the subjective approach of the observer. In other terms it is possible to associate each complex form to different species of events using different logics and association codes. These differences define the subjective recognizability, the matrix of reference based on own personal cultural background. We could define these approaches to forms as anamorphic interpretations, as the results of subjective anamorphic logics.    

These approaches normally happen when we are fascinated by the contemplation of the clouds, interpreting their forms in multiple ways; or when we find human expressions in the form of some rocks or in the plot of a carpet.    

The anamorphic logical approach is different from anamorphosis because it doesn't happen through an artwork, where the author stratifies meanings, but it happens when we look at complex forms, also natural forms, using our memory and our subjective codes of recognition. The anamorphic logical approach is the creative speculation on possible different readings of the existing form and of the possible variations of its image, meaning, and structure. Each of us implements this approach with the awareness not of the ambiguity but of the stratification of possible affiliations to different species, to different functions, to different aesthetical, symbolic and functional structures. In this sense, the anamorphic logical approach can be considered one of the bases of the creativeness, of the design imprinting and of the style. Following their subjective logic, each artist makes his artworks recognizable. His approach to interpreting forms is an essential part of the identity of his idea.   

Generative Art has discovered the anamorphic logical approach as one of the possible motors able to produce endless possible events through the activation of codes, of morphogenetical logics. Generative Art also discovered the not-eludeble strength of species. Generative artists need to realize their creative identity in the endless generated artworks. Each artwork, also unexpected, must be recognizable not only as single results but as belonging to a species, to the artist's identity and style.  If not Generative Art will be confused with Random Art. Nothing is so far and different as Generative Art and Random Art. 


2. The passage from a dimension to another.   

The field of reference is the relationship between the three-dimensional form and its two-dimensional image in its manifold variations. But we could consider also the image and its possible forms, in its manifold interpretative variations. The "generative" reciprocity between the form and the image of the form where every form "produces" a plurality of images and where each image "produces" a plurality of forms, in an endless spiral, is one of the principal fields of construction of the Generative Art, of the art that was born from expressing ideas as morphogenetic logic.    

First of all, a difference of dimension can exist between the form and its image. Often this difference exists by considering the form as a three-dimension event and its image as a two-dimension representation. But this is only one of the possibilities. We can get a 3D representation from an event having a lot of dimensions or we can increase the dimensions of the representation in comparison to the dimensions of the event, when we, for instance, try to represent the image of a jewel pending from the neck of a noblewoman in a seventeenth-century portrait by building a three-dimensional object that interprets the image of the painting. In this case only one of the possible two-dimensional representations of the constructed 3D event will fit the original image.   

If we like that the result of this moving through different dimensions can be considered totally acceptable, it would be necessary that each point of the form corresponds to one point of the image and that the structure of the form-system will have the same topological logic then the image-system. This obviously is not possible in the passage from a dimension to another. The "perspectiva artificialis" of Piero della Francesca is only one of the possible two-dimensional representations of three-dimensional events. With this approach a lot of information are lost. The inverse run, from the perspective representation to the three-dimensional event is, in fact, only a reasonable hypothesis. This passage could be considered as acceptable only if we built this three-dimensional event on the base of a lot of further knowledges (what we don't find in the image) as the point of view used in the representation. If we don't know previously it, we could identify it only through a subjective interpretation; therefore every interpretation "produces" different forms.  

More. We can reconstruct only what we see and not what is behind or what is inside the represented events. As Florenskji said, the perspective image represents only the skin of the three-dimensional event approaching the three-dimensional event to the two-dimensional representation. But, also with this consideration, the bending of the skin won't be ever sufficiently represented in the plain sheet of the sketch. The relationship between bending of the skin and plain sheet can be compared to the relationship among Euclidean geometry and not-Euclidean geometries.   

Not only. We have to operate a further interpretation choosing among the different techniques of perspective representation that we suppose could have been used for producing the two-dimensional images. These techniques are manifold and we could synthesize them in three types, each of which links the form to its image in different way: 

1. Perspective - 1*1.    

Perspective with only one point of view and only one direction of the look. The observer and the represented event are faced.    

It is the "perspectiva artificialis" of Piero della Francesca: only a point of view, therefore only one eye and not two, and only one direction is considered. This direction becomes also the point of central escape in the geometric construction of the image. 

Piero della Francesca, “Flagellazione”. If you reconstruct the represented space, as L.Ragghianti did, you will find a very long space, different from your expectation. 

2. Perspective - 1*N.   

Cylindrical and spherical total perspective: these perspective technique considers only a point of view but manifold directions of the look, up to cover 360 degrees in horizontal (cylindrical perspective) or also in vertical (spherical perspective). The observer is the center of the system.   

The curved perspectives follow the naturality of the vision. In fact, if we are inside a space, for instance a rectangular room with the parallel walls and with the plain ceiling, and we look toward a side we will see all the parallel sides to the constructed image converge toward a point (the fire). Then, if we turn the eyes and we look at the opposite wall, we realize that the same lines converge toward another point, opposite to the first one. Quickly turning our look from one side to the other, we could realize that these parallel lines converge in two points of the image that we are building in our mind. A bundle of parallel straight lines converge in two points only inside a non-Euclidean geometry system. The amazing aspect is that if we pass from a perspective built inside a Euclidean geometric system to a perspective built inside a non-Euclidean geometry, as spherical geometry, the mathematical representation of the transformation, the algorithm representing the passage from 3D into 2D becomes, mathematically, very beautiful being possible to represent all through the measure of the angle. I have experimented these non-Euclidean total perspectives twenty years ago. These experimentations and the algorithms that I wrote for building and representing the "total perspective" are at the base of my generative software. They configure a generative engine able to generate endless possible results starting from a single image. (C.Soddu, "L'immagine non Euclidea" non-Euclidean image, 1987, Gangemi Publisher) 

Generated Castle by C.Soddu represented in Total not_Euclidean perspective in two different views, the first one with horizontal sight and the second inclined, using the software designed by the author.


Generated Castle realized with rapid prototyping using 3D STL model directly generated by Argenia and the Cstle of the other images represented with anamorphic total perspective using the software designed by the author.


3. Perspective - N*1.    

Reverse perspective by Florenskji. This approach considers a multiplicity of points of view, the two eyes and their various possible motions, and only one target of the look.    

The represented event is the center of the system.   

This perspective intends to contain in one two-dimensional image the multiplicity of different visions. The practice construction of this perspective approach can be realized through an interesting conceptual overturn that I have experimented in my software. If the target of the look is unique and the points of view are different we can capsize the total perspective, that has only one point of view and different targets, setting the point of view in the target and the directions of the look in a lot of "eyes". The realized images could be assimilated to a representation of the skin of the object seen by the inside. The reverse perspective has been identified and explained by Florenskji looking at the Russian icons. Being sacred representations the fundamental choice is setting the represented event as center of manifold looks. In these two-dimensional images the representation of the face of the Saint is, according to my hypothesis, represented as seen by the inside of its same head. Since, as Florenskji affirms, we represent only the "skin" of the physical event we can capsize the face. Its projection on a sheet will result similar to the representation in reverse perspective of the Russian icons. In other terms I like to affirm that the reverse perspective is the overthrow of the spherical total perspective and not the overthrow of the" perspectiva artificialis" of Piero.  


Russian icon with Christ represented in “reverse” perspective.

The passage from a dimension to another, and particularly from the three-dimensional to the two-dimensional events through different perspective logics, but above all the reconstruction of the object 3D using different perspective-visual logics introduces fields of variation owed to different factors inherent in the dimensional transformation and in the type of used representation. These fields of variation belong to the subjective interpretation of the image or better, to the interpretative reconstruction of the parameters that could be used for the production of the image, and of the reconstruction of the parts that are not represented because not in sight because behind or inside to the volume of which the skin is represented.   

The hypothesis of reading an image decoding it through the perspectiva artificialis when instead it had been built through the Florenskji reverse perspective can produce unpredictable forms. For instance a cube could be reconstructed as pentagonal prism. This happens because, with the inverted perspective, the two opposite faces of a cube are represented as "in sight" together with the face in front of the observer. The reverse perspective of a cube is able to show three faces in sequence because you can see the cube both from left and from right. This happens everyday when we look at a very small cube and we approach it to the eyes. An eye sees the right face and the other the face on the left. The resultant image is the synthesis of the two sights. The mental image reconstructs a cube representing three consecutive faces. If we look at this representation with a canonical Euclidean perspective approach we need to suppose something different from a cube. The space "behind" appears too much ample and the re-constructive interpretation of the three-dimensional form can bring us to imagine more than a hidden face, for instance two, and therefore to generate an acceptable reconstruction of a prism with five or more consecutive sides. The cube, through these following passages of dimension (3D - 2D - 3D) is turned into a pentagonal prism.   

These transformations are born from our interpretations: It is a "natural" construction of generative motors that mirrors our creative identity, our cultural references.   

The idea of an architect doesn't base on forms but on transformations. This is a transforming approach able to see the existing world as a dynamic world, and able to generate visionary scenarios and their endless variations. The generative engines are the structure of the designer's idea. They work on morphogenetic codes fitting the oneness of the approach; they are the anamorphic logics that allow the designer to generate endless visionary worlds by mirroring, in their multiplicity, the design idea.

Generated castle by C.Soddu, represented in elevation and in two different “reverse” perspective, using the software designed by the author  following the Florenskji approach.    


3. Construction of generative morphogenetical processes: subjectivity and variations.

The identity of an artwork exists if people can recognize it as belonging to a species. So, if we like to build the identity of our artworks we need to identify its species and to realize it designing an artificial Dna. This approach is Generative Art:  building a series of logics of transformation able to generate endless possible results recognizable through the morphogenetical paths used for their creation and through the reference to possible anamorphical logics belonging to our creative and cultural identities.     

The results, in terms of quality and extended appreciation, are the best where the anamorphic logics produce answers pertinent to different subjectivities, therefore where the generated complex system don't give only the possibility to be understood as axiomatic structure of a shape or of a function but its complexity performs the availability to subjective and unpredictable uses. This usability is realized and appreciated when the suggestions, the logics of use and the aesthetical appreciation of each user is related to the complexity of the designed system and to the potential anamorphic interpretations that this complexity makes possible. 

Not only. The identity have to belong to a species without denying, rather strengthening the identity as individual, as unicum. It brings to consider that the design of morphogenetical paths rather than of shapes doesn't remove anything from the final results identity but strengthens then, especially because of the parallel presence of "variations".  As happens for the music, from Bach to Mozart and to jazz. Variations are built consolidating different forms in different moments, but these results are reciprocally congruent because of the common morphogenetical paths that, from the detail to the whole, are at the base of an idea. These "endless" variations could seem aesthetically less strong and functionally, less recognizable than only one result chosen because considered the best at the end of the optimisation of the form-function relationship. This approach is misleading. The affiliation to a species, with the possibility of mirroring each result in the infinity of the parallel variations, creates two congruent layers of recognizability and identity that are strengthened one each other: the identity of the species and that of individuals.

Two variations of ”Cordusio” project by C.Soddu. All these 3D models are fully working in the field of functional and structure system and are completely generated by Argeněa.

“Cordusio” project by C.Soddu. the aim of this architectural projec was to fit the Futuristic cultural reference of Milan buiding an architecture able to fit the Milan identity as represented by the cordusio square in the beginning of last century. This because the twenties of ‘900 were really important in construicting the “idea” of Milan. 2005.

This approach find the quality also in the oneness conjugated with the recognizability. For instance the oneness of a painting of Van Gogh is also appreciated through the possibility to recognize it as painted by Van Gogh, as belonging to a species with unique characters and unrepeatability. This happens in the appreciation of the Nature where the multiplicity of the variations mirrors the multiplicity and oneness of every one. 

This process of appreciation happens not only on the aesthetical layer but also in the aspects more directly related to the functionality. The use of the object becomes "intuitive" really because linked to subjective runs of appreciation and recognizability. As, for instance, sometimes happens in the structure of software interface. For using a function, manifold "logical" runs are designed mirroring different and subjective possible approaches. 



Four more visionary variations of “Cordusio” project for Milan, realized by the author usig “Argeněa”, his generative software. Milan, 2005

4. Generation versus Cloning.   

In a production process of individuals belonging to a species, the copy doesn't exist and, we could say, it is not possible. We are able, in fact, to copy an object, to reproduce it until the least details, but we are not able, with the same tools and with the same philosophy to copy a species. This because the anamorphic logic that has been activated during the design of the species is, for its own nature, different from the anamorphic logics of whom analyses an object, or also a series of objects belonging to the same species. The subjective and interpretative component is so strong and involves the passage from a conceptual system to a plurality of physical events that is not possible to "reconstruct it"; it is possible only "redesign it".  A  "generative" design is not reproducible departing from the results. It is possible to produce only "clones" of single variations. The only certainty that we can acquire is the feasibility of the generative project, because someone has already realized it. We cannot copy it. We can recreate it only ex-novo using the personal subjectivities and interpretative ability. But it will be another project, however.


5. The generative city and the visionary worlds   

The future of the ideas and their realizations is a city living of unique events, unrepeatable and anamorphic events able to answer in pertinent and recognizable way to a plurality of citizens with their unique identity. But also a city that progressively discovers its own identity approaching to the Ideal City that is in the mind of who lives and designs it. As each ideal city and the visionary worlds representing it, also every physical city could be, surprisingly, unpredictable, not homologable neither clonable and therefore, finally, natural and harmonic.




[1] Celestino Soddu, “L’immagine non Euclidea”, (not-Euclidean image), Gangemi Publisher, Rome 1997

[2] Celestino Soddu, "Generative Design / Visionary Variations - Morphogenetic processes for Complex Future Identities" in the book Organic Aesthetics and generative methods in Architectural design" edited by P. Van Looke & Y. Joye in Communication&Cognition, Vol 36, Number 3/4, Ghent, Belgium 2004

[3] C. Soddu, "Meta-Code, Identity’s Borders" in "Generative Art 2004", proceedings of the International Conference GA2004, Aleadesign Editor, December 2004

[4] C.Soddu (interview to), "From gene to design: Prof. Soddu and his Generative Design", pag. 6-13, Magazine "Designer" n. 1, 2004, Changsha, Hunan , China

[5] C. Soddu, "变化多端的建筑生成设计法" (Generative Design), article in the magazine "Architect", December 2004, China.

[6] C. Soddu, “Milano, Visionary Variations”, Gangemi Publisher, Roma, May 2005.

[7] C.Soddu, “Generative Art in Visionary variations”, “Art+Math=X” proceedings, University of Colorado, Boulder, 2005.

[8] C.Soddu, “Visionary Variations in Generative Architectural Design”,article in Chepos magazine number 003, TU/e, Eindhoven 2005.

[9] C.Soddu, E.Colabella, “A Univesal Mother Tongue”, Leonardo Electronic Almanac Volume 13, Number 8, August 2005