Nevena Radojevic

Il disegno di Ognissanti testimonia in modo rilevante l'idea della città concepita da Firenze in epoca gotica. Ognissanti è protagonista di interventi di urbanizzazione condotti con sapienza economica e intendimenti imprenditoriali, con scienza e tecniche di controllo del territorio. Le sue trasformazioni nel tempo annunciano i mutamenti del paradigmi architettonici con i quali la città partecipa ai movimenti della cultura, in epoca manierista e barocca .

The umbrella vault of Pazzi Chapel in Santa Croce church in Florence has been the subject of many, 3d survey based researches, but it’s shape has never been described in satisfactory manner. The descriptions given so far, verified through three-dimensional models, despite whether drew near to geometry, did not give any explanation of building process, that as we know, often guides design choises and principles. In the current research, attempts are made to formulate new hypotheses on the logic and form-finding processes that could have determined the choosen shape of the sail. The existing technical literature and treatises on masonry do not give any information about the building processes in the sources, which is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlating detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. The form of the inner sail is ...

The spherical geometry research which made possible the celestial (time) and terrestian (space) measurements in the past, produced many assumptions, theorems and models. The architecture has followed the scientific research on these topics, erecting with exemplary models. The three domes designed by Filippo Brunelleschi for the Pazzi Chapel, have different geometrical genesis - all linked to a different interpretations of the sphere. The umbrella vault, particulary, reveals a very curious shape. It’s obtained by a three-dimensional transformation of a torus surface, with respect to the fixed point (dome’s oculus) and constant lenght (torus radius), like the concoid of Nicomedes in 2d. This, very particular design choice, may have found the nutrients in scientific thought. In that particular moment, the Ptolemaic model of the universe was showing its weaknesses, and the proposals which would remove them speculates the planetary motions, which generates dynamic figures. The fact that the apparently complex motions of the planets could be actually very simple if observed from another point; could be easily intuited by Brunelleshi. The same rules, as ones used to generated pictures in central projection, were designing the orbits in our sky.

The umbrella vault of Pazzi Chapel in Santa Croce church in Florence has been the subject of many, 3d survey based researches, but it’s shape has never been described in satisfactory manner. The descriptions given so far, verified through three-dimensional models, despite whether drew near to geometry, did not give any explanation of building process, that as we know, often guides a design choises. In the current research, attempts are made to formulate new hypotheses on the logical and design form-finding proceces that could have determined the choosen shape of the sail. The existing technical literature and treatises on masonry do not give any information about the building process in the sources, whitch is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlating detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form.

The paper examines some of the possible ways to generate the coffered domes, where generative grammars are determined by different cultural paradigms. A method applied is the shape generation method, that uses shape grammars, which take shapes as primitive and have shape specific rules, as described by G. Stiny and J. Gips. The generation of the three dimensional shape consists in the definition of a language of two dimensional shapes, the selection of the shape in that language, a specification of a sphere mapping schema and the determination of a location and scale of the shape on the series of concentric spheres of a given size. This work shows that the essential differences between the Pantheon's dome and some " Pantheon inspired " coffered domes are caused by applying different generative grammars, characterized by their cultures.

In the current research, atempts are made to formulate new hypotheses on the processes that could have determined the shape of the sail of the umbrella vault of Pazzi Chapel in Santa Croce church in Florence. The descriptons given so far, verifed through three-dimensional models, despite whether drew near to geometry, did not give any explanaton of building process, which is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlatng detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. The vault is double-sailed and the form of the inner sail (conchoid surface) is obtained by a three-dimensional transformaton of an outher sail (toric surface), with respect to the fxed point (dome’s oculus) and constant lenght (torus radius), like the Nicomedes conchoid in 2d.

Osservando la quadratura sulla volta della chiesa di San Matteo (affrescata tra 1717 e 1719 da Giuseppe e Francesco Melani) si notano alcune difformità. Né i basamenti né le cornici degli ordini disegnati seguono del tutto l’andamento di quei reali, ma a volte sembrano “salire” o “scendere” rispetto alle cornici della chiesa. Le cornici che reggono le balconate in aggetto, come le loro balaustre, assumono una forma molto insolita. Queste due curve, quando osservate dal punto di vista privilegiato, non si percepiscono né come circolari né come ovali, e in più, non sono nemmeno parallele tra di loro. Le rette verticali (perpendicolari al quadro) sembrano curvare in alcuni punti.
Il presente studio si pone di esaminare alcuni punti “incongrui” di questa quadratura sulla base di un rilievo integrato (scansione laser e fotogrammetria), concentrandosi su due principali passaggi creativi:
1. Come si fa a “progettare” il bozzetto piano per uno spazio così irregolare? L’architettura pensata per la volta segue veramente la geometria irregolare della chiesa (trapezio scaleno), oppure usa vari accorgimenti per adattare un modello a pianta rettangolare sulla base esistente?
2. In che maniera avviene il trasferimento del bozzetto piano sulla superfice complessa della volta?
In quanto riguarda la progettazione del bozzetto, l’altezza ricostruita dell’architettura dipinta sembra fornirci il primo indizio. Essa è pari alla distanza che sta tra il punto di vista e l’imposta della volta. Questa particolare altezza si rileva come molto “comoda” dal punto di vista operativo in un metodo (per fare bozzetti) descritto dal geografo veneziano Cristoforo Sorte. Tale metodo comporta l’uso di un modello in scala, di ciò che si vuole dipingere, ed uno specchio piano posto alla base del modello. Se l’occhio è posto all’altezza del modello nella fase di disegno l’altezza del architettura dipinta sarà la stessa nella ricostruzione (ovviamente in scala). Un’altezza maggiore invece obbligherebbe l’occhio ad entrare dentro il modello (posizione scomoda).
Il secondo passaggio, cioè il trasferimento dal bozzetto piano sulla superficie voltata, in linea di principio, potrebbe avvenire in due modi:
-Il primo sarebbe una specie di proiezione del bozzetto piano posto all’imposta della volta sulla superficie curva dal punto di vista, come suggerito da Andrea Pozzo. Questo metodo (se fatto dal vero e non sul modello in scala), non creerebbe le difformità del disegno dovute alla geometria irregolare della volta. Osservate dal punto di vista tutte le difformità svanirebbero.
-L’altro metodo possibile potrebbe essere quello di costruire un corrispondente modello in scala della volta, proiettare sul esso la griglia del bozzetto e riportarne lo sviluppo delle linee meccanicamente su fogli di carta. Una volta ingrandita la griglia, si prosegue con il disegno di altri elementi principali dell’architettura da dipingere, direttamente sui cartoni ingranditi.  In questo caso, le difformità della volta reale, che forse non sono state rilevate e riprodotte nel fare il modellino, risalterebbero fuori e sarebbero evidenti quando osservate dal punto di vista.
Nel caso di San Matteo, si rilevano alcuni segni che diventano improvvisamente “regolari” nella proiezione sviluppata. Questa operazione farebbe riferimento ad una cultura molto attenente alla cartografia e ai problemi legati alle superfici non sviluppabili, alla quale non interessa più soltanto l’inganno ma vuole mettere altre pratiche in atto.

The umbrella vault of Pazzi Chapel in Santa Croce church in Florence has been the subject of many, 3d survey based
researches, but it’s shape has never been described in satisfactory manner. The descriptions given so far, verified through
three-dimensional models, despite whether drew near to geometry, did not give any explanation of building process, that as
we know, often guides a design choises. In the current research, attempts are made to formulate new hypotheses on the logical
and design form-finding proceces that could have determined the choosen shape of the sail.
The existing technical literature and treatises on masonry do not give any information about the building process in the
sources, whitch is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlating detailed
surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the
hypothesized form.

The spherical geometry research which made possible the celestial (time) and terrestian (space) measurements in the past,
produced many assumptions, theorems and models. The architecture has followed the scientific research on these topics,
erecting with exemplary models. The three domes designed by Filippo Brunelleschi for the Pazzi Chapel, have different
geometrical genesis - all linked to a different interpretations of the sphere. The umbrella vault, particulary, reveals a very
curious shape. It’s obtained by a three-dimensional transformation of a torus surface, with respect to the fixed point (dome’s
oculus) and constant lenght (torus radius), like the concoid of Nicomedes in 2d.
This, very particular design choice, may have found the nutrients in scientific thought. In that particular moment, the
Ptolemaic model of the universe was showing its weaknesses, and the proposals which would remove them speculates the
planetary motions, which generates dynamic figures. The fact that the apparently complex motions of the planets could be
actually very simple if observed from another point; could be easily intuited by Brunelleshi. The same rules, as ones used to
generated pictures in central projection, were designing the orbits in our sky.

The subject of the research is the umbrella vault in the Pazzi Chapel in Santa Croce church in Florence. In some previous researches done by the author, attempts were made to formulate new hypotheses on the form-finding processes that could have determined the shape of umbrella's vault sail. The analyses were done by correlating detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. The hypothesized form of the inner sail is obtained by a three-dimensional transformation of a toric surface, with respect to the fixed point (dome's oculus) and constant length (torus radius), like the conchoid of Nicomedes in 2d. The vault is made by the inner sail (the conchoid surface) and the outer sail that is not visible in this moment (the hypothesized toric surface). The results were verified by overlapping the hypothesized inner sail's form with the laser scanner data, and only 4% of points were out of the 3 cm range. This very particular form, even though it explains the building process in a very satisfactory manner, is not actually something that the visitors can see (with the naked eye). Actually, the conchoid surface (that is a solid central projection of torus), could be perceived like a transformation of torus only under certain conditions. First condition: the inner conchoidal sail's surface should be mapped by the elements that represent the projection of the torus's peculiar elements (circumferences for example) onto the conchoid surface from the fixed point (oculus). These irregular spatial curves, that represent the regular curve's transformation, can be perceived as if they were circular if we look at them from the transformation center. The second condition would be the transportation of the observer's view point in the center of oculus (not accessible point). The exhibition design (work in progress for the Opera of Santa Croce) involves the use of projectors for the 3d mapping of the sails and use of a plane mirrors system that can move the observer's view point as if he was in the cupola's oculus.