The Young Borromini and the use of geometry


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Ancient History & Civilisation

The Oratory of St Filippo Neri makes clear his break with Renaissance architecture in its revival of individual, structural and decorative elements. The masterpiece in this regard is the Church of St Ivo alla Sapienza , with its complex ground plan. With his fellow designers Bernini and Pietro da Cortona He remains one of the top three Baroque architects and an important contributor to the emergence of Roman Baroque art of the seventeenth century. See also Vignola , Born Francesco Castelli, the son of a stonemason, near Lake Lugano in Switzerland, he is better known by his adopted name, Borromini.

He started out as a stone mason, but at the age of 20 moved to Rome, where he worked initially with his relative, the architect Carlo Maderno on St Peter's Basilica in Rome.

Following Maderno's death he and Pietro da Cortona continued working on the Palazzo, under the supervision of Bernini - an experience which contributes to his lifelong jealousy. As a young man he was a great admirer of the architecture of Michelangelo , as well as the forms of Roman architecture whose geometric principles he manipulated to create a unique, if unconventional, personal style.

In , he was given his first major commission: the church, cloister and monastery of San Carlo alle Quattro Fontane. After this he was appointed architect for the dome and facade of the Palazzo Sapienza and Sant'Ivo alla Sapienza He was originally recommended for the latter project back in , by none other than Bernini.

Nexus Network Journal. This paper is about drawing ovals using a given number of certain parameters.

What Borromini Might Have Known About Ovals. Ruler and Compass Constructions | SpringerLink

New constructions are displayed, including the case when the symmetry axes are not given. Many of these constructions make use of a recent conjecture by Ragazzo, for which a Euclidean proof is found, thus suggesting it might have been known at the time Borromini chose the ovals for the dome of San Carlo alle Quattro Fontane. A geometric proof of the same conjecture—as well as constructions—in the more general case of eggs and polycentric curves is the subject of the first part of this same research Mazzotti, a Euclidean approach to eggs and polycentric curves, The oval of the impost of the dome of San Carlino and the fourth construction by Sebastiano Image left by Margherita Caputo, used by permission.

The problem of realising what artists and craftsmen knew about ovals at the time when they made use of them is still an open one.

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About Borromini, Rome

The core of this paper is a conjecture by Felice Ragazzo about the points where the arcs of circle of an oval meet, which I prove by means of Euclidean geometry, and which I use to generate a whole series of oval constructions. Architecture and mathematics historians are those entitled to go deeper in the subject. Only Euclidean geometry was used in order to be consistent with the above presented historical framework.

The use of this conjecture gives way to a classification of oval constructions given any three parameters chosen in a proper way, both with and without indications on the directions of the axes, some of which are new. Freeware Geogebra has been extensively used for this purpose. Properties of the Connection Locus " is devoted to the main theorem and to the properties of the set of points where the arcs of circle can meet—the Connection Locus. In "Constructing Ovals. Issues Still Open" I present a systematic list of different constructions of ovals, as well as issues still open.

Rome day 4 (continuation 2)

I have posted all of the constructions described on the internet. In the historical framework presented in the Introduction to this paper, I thought it more interesting to pretend Descartes and the other mathematicians responsible for the invention of the coordinate system had not been born yet. Formulas derived via analytic geometry covering the whole topic of ovals will be the subject of my next paper. O as the intersection of the two symmetry axes. A and B as the points of intersection between the quarter-oval and the horizontal and vertical axes. K and J as the centres, respectively, of the small and big circles, with radii r 1 and r 2 , whose arcs form the quarter-oval.

J , K and H are co-linear. In any given rectangle there is only one inscribed ellipse the dotted curve , and an infinite number of ovals. To be able to inscribe ovals in rectangles we want to be able to smoothly connect arcs of circles. Image Ragazzo , reproduced by permission. I first wrote the proof of this conjecture, then managed to extend it first to eggs and then to polycentric curves. So it has become a corollary of one of the theorems contained in my preceding paper Mazzotti The quarter oval resulting from a choice of H on the Connection Locus of the given rectangle.

Adirect application of Theorem 2 in Mazzotti Summing up these properties of point R for quarter ovals we have the following. We are now ready to show how nearly any oval can be drawn if enough parameters are known. Properties of the Connection Locus". First of all we will deal with ovals with known symmetry axis lines. Let O be the symmetry centre.

Constructions 21 and 22 are to my knowledge totally new and justified by mathematical arguments. The first six parameters are independent of one another, and any choice of three of them—constrained within certain values—will determine a unique oval. The different oval construction nos. In the following constructions some parameters are subject to constraints, but there exist always feasible values.

What Borromini Might Have Known About Ovals. Ruler and Compass Constructions | SpringerLink

Precise limitations—via analytic calculations—will be listed in my next paper on the subject. Construction of Oval 8 —given a , h and j —using compass and ruler remains at this moment unsolved, although there is graphic and algebraic evidence analytic formulas will be displayed in my next paper of the unicity of the corresponding oval. Solution to any of these cases would probably lead to the solution of the other three. K is the intersection between the axis of AH and OA. Construction of Oval 10 —given a , j and m —using compass and ruler remains at this moment unsolved see further remarks for the construction of Oval 8.

Construction of Oval 11 —given b , k and h —using compass and ruler remains at this moment unsolved see further remarks for the construction of Oval 8. Construction of Oval 13 —given b , k and m —using compass and ruler remains at this moment unsolved see further remarks for the construction of Oval 8. H is the point on the line JK with distance h from OB. All the above constructions are justified by Theorem 1 and by the properties of ovals listed in "The Euclidean Approach to the Study of Ovals". Construction 22 is to my knowledge also a new construction, building the quarter-oval given the centres of the arcs and the ratio p between the half axes.

Limitations for the value p have been found via analytic calculations, they are therefore the subject of my next paper. I will now consider the problem of drawing an oval on a plane without any given symmetry axes.


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We now have three more degrees of freedom and so a much bigger variety of combinations. The CL and the constructions used in the first part of this section are the tools to tackle this problem. Limitations conjectured for the parameters involved are maybe the most interesting part.

I consider constructions , and to be the most innovative ones. To my experience an oval in a plane is determined by six independent parameters, although limitations do occur. After a period of professional difficulty, the architect impaled himself on a sword and died a few days later. All who contradict the official religion are banished.

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