Pantheon, Lumen in Splendore
Celestino Soddu, Director of Generative Design Lab, Politecnico di Milano, IT
Enrica Colabella, cofounder of Generative Design Lab, Politecnico di Milano, IT
Abstract. This paper talks about the experience of an operative research, developed in the last twenty years, regarding the implementation of a software able to imitate, using a dynamic chaotic system, the creative process of the architectural design path.
The designed algorithms, that operatively manage the creative design path, are a usefuf support for the human design activities. Argenia is a generative software able to generate endless sequence of results in 3Dmodels. These are unpredictable variations of the same architectural concept, of a subjective Idea of Architecture defined as an artificial DNA.
The main point is the role of two particular operative fields used for the construction of the dynamic generative systems of Argenia: the translation of events from Euclidean geometry to spherical notEuclidean one, and viceversa, and the dimensional upgrade, that is the translation from 2D into 3D and 4D and viceversa.
Using these two fields for experimentations, operative results were really interesting and important in the construction of the generative engine. They worked very well in the dynamic notlinear system performed for Argenia. The reason is that these two fields involve subjectivity and interpretation: main aspects of a creative process.
This paper treats about the discovery that one of the most important architectures in the history, Pantheon, contains, magistrally represented, these two field and their use in the creation of artificial spaces.
Key words. Generative Design, Euclidean / NotEuclidean Geometries, Perspectives artificialis, total 360 degree, Florenski reverse, Harmony, Natural/ Artificial DNA
Mathematics Subject Classification: Primary 01A99; Secondary 9703.
1 The starting point of a creative process
At the beginning of a design process we perform a void as representation and a full as concept. This void can be reported by a sphere represented with a perspective view by its inside. The full is its specificity that is not represented by results/forms but by attributes defining its possible characters, by adjectives describing the aims to reach and the context. Attributes and adjectives built as codes of transformation, algorithms able, all together, to define an artificial Dna.
The beginning of generative process is the orientation of the hidden system. The sphere, until now not visible by its inside with a perspective view, suddenly becomes visible, its representation seems to be born from nothing, but it is only the passage from an event without orientation to one oriented. This is the first generative action.
The further design developments are nothing else than progressive and multiple transformations making the system more and more visible and complex. Trasforming it progressively into an architecture, beautiful, stately, technologically attractive, fantastic architecture. A generative process imitating what happens in Nature from the seed to the adult.
Geometry and Mathematics are the specific fields of this creative moment, because interpretation of Nature is the main creative moment. Transformations are easily representable as algorithms, and this is the most immediate and controllable way to conceive transformations, also before knowing on what and when they will be applied.
The Patheon is the oldest and the most impressive representation of this complex process.
We can see it looking at the floor of Pantheon, that is curved as a portion of a very large sphere, and to the hole on the top that is the only windows where the sun can enter.
The floor, shaped with a sphere tangent to the Pantheon sphere, orients the system from bottom to top. The sun, shaping its image of light, an ellipse, orient the dynamic system in the fourth dimension.
2 Geometries and numbers: 7 and 3,4,5, codes of Harmony
Moving from infinite to finite numbers, translation from Natural to Artificial. Suddently, when we discover the curved floor and the sun enter in the starting sphere, the spere is oriented and divided in two parts, the upper part, that we will discover managed by the number 7, and the lower part managed by the human complexity, there is the number 3, number 4 and number 5: the more famous rightangle triangle, the symbol of human construction as code of Harmony.
Redrawing Pantheon some years ago (1996 for the Italian Polygraphic Society, Zecca dello Stato), not only in a linear system of measuring but defining possible geometries we discovered that we can identify many stratified possible divisions used in the design of side spaces.
We can easly identify the division in eight parts because the center of each little temple occupies a point of an octagon. But if we cansider the corner of the temples of both parts of the entrance we discover that is the side of a decagon. More, if we condider the space of each four coloums we will fing a side of a twelvesided polygon, as also the distance from these coloums and the center of the entrance.
What is amazing is the actuality of this space, its ability to stimulate the search, in every person who crosses it, a personal point of view, to answer in this way to the unpredictable needs of each person, to be a boundless continuum of possible subjective “discoveries” . And one hardly realizes that the inside is double. An original layout of the attic internal facade exist together with a more recent layout that completely transforms the structure of the inside front. The idea is so strong and explicit that the two layouts, despite their strong difference, appear only as two possible and interchangeable scenarios of the same design idea.
It’s the main concept of variations. Variations don’t destroy the architectural identity but enhace it, and Bach teaches.


Interior with different attico’s setup, curved perspective, Pantheon with Bernini ears 
Ever Bernini, building above the Pantheon the famous ears, then demolished, has not done anything else other than to produce a further possible scenario, a further variation of the same idea of Adriano, of his code.
3 From Sphrical perspective to “Perspectiva artificialis”



Views from the center of Pantheon with spherical perspective,and with “perspectiva artificialis” 
If we are inside we look at Pantheon using our natural spherical perspective. We cannot think, if we are in the centre of this space, that all spaces around are different, neither that we see them in different dimension. We see them all with the same measure and we appreciate the differences following the architectural concept. But if we will try to take a photo or, better, if we try to draw a perspective view using “perspectiva artificialis” we discover that the system is represented with spaces on each side more large then spaces in front. And the sequence is curved. This because, with “Perspectiva artificialis”, we measure the distance using a flat surface as interface. And spaces in each side are more closed to us then space in front. In the same time we discover the curve as the main represented element. The sun, shaping its image of light, an ellipse, orient the dynamic system in the fourth dimension. Curve that appeared hidden in a spherical perspective. The result is that, going to the side spaces and the entrance we cange completely our way to look at the space, translating our notEuclidean reference to an Euclidean one. It’s not a case that the floor of the entrance is flat and not curved as the floor inside.
This continuous moving from one to another perspective system enhances the production of our interpretations of architecture. We design in our mind variations of Pantheon each time we visit this architecture. This “translations” were used in Argenia for managing the complexity of architectural forms.
4 Moving from notEuclidean to Euclidean geometry: translation from Natural to Artificial
The two main spaces, the spherical inside and the entrance follow different geometries, not only because on the need of different perspective approach. This passage through different geometried is clearly recognized by each visitor. More, looking at the pronaoentrance and at the side spaces from the center of the inside space you can easly recognize the transformation of approach because, suddently, you have one vanishing point for parallel sides instead of two: you can see the infinite represented as far from you. Inside the infinite was near to you.
5 From total (360°) curved perspective to Florenskji reverse perspective
If we are in the centre of Pantheon space, and we look at the dome, we discover that the dome is shaped with ambiguous repeated (28 times, 7x4) events. Each of them is like a sequence of frames, one inside the other.
You cannot read the 3rd dimension since something happened: the bright ellipse of the sun light, entering by the round hole in the top, will show you the shadows and, with them, the 3rd dimendion of each shape.
More, before the light you don’t understand if the sequence of frames is going to the inside or to outside. Because the grown is oriented following the sight line. And something incredible can happen. With the sunlight the 3D objects reverse their position. For one moment you can understand how if the Pantheon in a Florenskji reverse perspective, when all you are seeing in the inside is the skin outside. Reverse perspective of Florenskji is a spherical perspective of an interior, like the interior view of the skin of a face, viewed from outside. This possibility to generate unpredictable but fascinating representations and, following that, subjective creation of objects was mainly used in the generative software Argenia. In this way each event becomes a generative machine able to generate unpredictable events following a wide range of possible interpretations.
6 Idea as a code
This Idea, concretized as a mathematical / geometrical code, realizes a cryptic writing that will emerge when functional/aesthetical/symbolic demands change. An idea is timeless, appreciable as, endless reading variations following multiple interpretations.
The Pantheon is timeless. This idea was by Adriano. It doesn't matter if it has been realized by transforming an existing building or if it is a new building. And it is not important who initially realized it and who transformed it. This idea is strongly identifiable by discovering its possible codes: the omnidirectional structure as a initial concept of the universality and the relationship with nature, the orientation, the inside perspective structure that connects man to the built space and through this, the universe to the environment but, above all, the overlapping multiplicity of possible geometries that, hidden in the apparent simplicity of the construction, manage the progressive transformation of the space and its increase of complexity. These overlapping geometries stem from the concept of Harmony as the awareness of manifold subtended relationships that build a pattern whose sense is discoverable by each man in a different but calibrated way of feeling himself a craftsman, inside the natural complexity of the possible.
7 The passage from a dimension to another
The field of reference is the relationship between the threedimensional form and its twodimensional 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 Generative Art, of 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 threedimension event and its image as a twodimension 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 seventeenthcentury portrait by building a threedimensional object that interprets the image of the painting. In this case only one of the possible twodimensional 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 formsystem will have the same topological logic then the imagesystem. This obviously is not possible in the passage from a dimension to another. The "perspectiva artificialis" used by Piero della Francesca is only one of the possible twodimensional representations of threedimensional events. With this approach a lot of information are lost. The inverse run, from the perspective representation to the threedimensional event is, in fact, only a reasonable hypothesis. This passage could be considered as acceptable only if we built this threedimensional 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 threedimensional event approaching the threedimensional event to the twodimensional 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 notEuclidean 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 twodimensional 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:
7.1 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 vanishing point of central escape in the geometric construction of the image.

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


7.2. 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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" nonEuclidean image, 1987, Gangemi Publisher)



Generated Castle by C.Soddu represented in Total not_Euclidean perspective with inclined sight, using the software designed by the author. The generated Castle realized with rapid prototyping using 3D STL model directly generated by Argenia and the Csstle of the other images represented with anamorphic total perspective using the software designed by the author. 
7.3. 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 twodimensional image the multiplicity of different visions. The practice construction of this perspective approach can be realized through an interesting conceptual overturn that Argenia experimented. 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 twodimensional images the representation of the face of the Saint is, according to inquiring 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 we can affirm that the reverse perspective is the overthrow of the spherical total perspective and not the overthrow of the" perspectiva artificialis" of Piero.
The passage from a dimension to another, and particularly from the threedimensional to the twodimensional events through different perspective logics, but above all the reconstruction of the object 3D using different perspectivevisual 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 reconstructive interpretation of the threedimensional 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.


Russian icon with Christ represented in “reverse” perspective. 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 
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.
References
[1] C.Soddu, P.A.Rossi, "Il calice di Paolo Uccello uno e senza limite", (the goblet of Paolo Uccello, one and without limit) with a referee of C.L.Ragghianti, in "Critica d'arte" magazine n.8 1986
[2] C.Soddu, "L'immagine non euclidea" (the not euclidean image), Gangemi publisher, 1987.
[3] C.Soddu, "Citta' Aleatorie", (random cities), Masson Publisher. 1989
[4] C.Soddu, E.Colabella, "Il progetto ambientale di morfogenesi", (the environmental design of morphogenesis), Progetto Leonardo Publisher 1992
[5] C. Soddu, "Argeněa, Artificial DNA and Visionary variations", in "O sentido e o universo digital", proceedings of SIGRADI, the VII congreso igeroamericano de grafica digital, Unisisod Nov. 2004
[6] C. Soddu, "变化多端的建筑生成设计法" (Generative Design), article in the magazine "Architect", December 2004, China
[7] C. Soddu, “Milan, Visionary Variations”, Gangemi publisher, Rome 2005 (book)
[8] C.Soddu, “Generative Art in Visionary Variations”, in Art+Math=X proceedings, University of Colorado Boulder, 2005.
[9] C.Soddu, E.Colabella, “A Univesal Mother Tongue”, Leonardo Electronic Almanac Volume 13, Number 8, August 2005
[10] C.Soddu, “Argenia, Artificial DNA and Visionary variations”, The Journal of designing in China, summer 2005, Hong Kong
[11] C.Soddu, “Argenia in timeless gencodes”, proceedings of ARTMEDIA IX, dept. Philosophy, University of Salerno, 2005
[12] C.Soddu, “Gencities and Visionary Worlds”, proceedings of 8th Generative Art Conference, Politecnico di Milano University, Milan, 2005.
[13] C.Soddu. “Generative design, a swimmer in a natural sea frame”, proceedings of the 9th Generative Art conference, Politecnico di Milano University, Milan, 2006.
[14] C.Soddu, E.Colabella, “Argenia” PNCM conference proceedings, Daugavpils University, Latvia, 2007
[15] C.Soddu, “Endless interpretations, infinite in the mirror” in the GA2007 proceedings, Milan, 2007.
Current address
Celestino Soddu, professor, registered architect
Enrica Colabella, registered architect
Generative Design Lab. DiAP, Politecnico di Milano, via Golgi 39, 20133 Milano IT. Celestino.soddu@polimi.it enrica.colabella@polimi.it www.generativeart.com www.soddu.it