Geometric Operations in Mario Botta’s Architectural Work

This paper investigates the role of primary plane figures, primary solids, and geometric operations in Mario Botta’s architectural works during fifty years of his career. Through the selection of three case studies, the research allows the creative process to be discretized and analyzed from a geometric-mathematical point of view.


Introduction
The objective of the present work is to interpret and represent via graphical analysis the geometric aspects that govern the composition and shape of Botta's architecture. It represents an analytical work at an early stage, conducted on 74 pieces which compare different building models, thus highlighting the role of geometry as a fundamental tool. The process begins with analyzing his work, identifying a series of geometric figures and solids, frequently found in his architecture, with a digital volumetric reconstruction. At this stage, examining the master's repertoire, our work focuses on three buildings: the cymbalist synagogue in Tel Aviv; the single-family house in Riva San Vitale; and the church of San

Preliminary Studies
The study of buildings as products of design requires theories and tools that describe topo-geometric properties as well as the interaction between spatial and geometric systems (Psarra 2017). Architects identify systems by combining geometric forms that give buildings their specific appearance.
The description of form and space is a description of composition as a process of transformation. Formal and spatial patterns interact through geometric properties; the designer's syntactic and semantic rules mediate the relationship between form and function. As Mitchell (1990) argues, the architect's task is to give form to buildings according to their function and then relate the two.
How architecture is given form and experienced in space elicits the question: how can geometric-formal properties of an architectural work be described analytically?
In this paper, the authors attempt, through digital modeling, to compare and classify Botta's architecture and establish relationships between different geometric entitites and their compositional rules. This provides recognition, decomposition, and recomposition of various elements and reconstruction towards possible variants and invariants (Spallone and Vitali 2020).
To understand the compositional principles underlying Botta's architectural projects, it is essential to start with certain assumptions, highlighted by Psarra (1997); this constituted an essential reference to draft the present contribution: • the more geometric properties remain unchanged, the less tension is created between its parts, and structural unity will be achieved to facilitate intelligibility of a single reading; • the composition is dominated by an explicit syntax, established early in the transformation; • the spatial narrative is subject to the formal model.
His architectural production and training were inextricably linked to the rationalist experience of Le Corbusier and Louis I. Kahn, aiming for continuous refinement of simple, elementary forms. This simplification is linked to his views on tradition, to refer to the Modern Movement and the Palladian legacy, from which regulatory layouts and composition of solids were often inspired. Botta intuitively uses geometry as a tool to control form, by which he can describe the connection of his buildings to function and context (Sala and Cappellato 2003).

Geometrical Analysis of Mario Botta's Architectural Works
In Botta's design, the geometric foundation is understood as a source of inspiration and only later as support of the creative process and a matrix of forms. (Studio Botta 2010). The plan represents as it is combined with volume, the starting point, the definition which constitutes a central moment within the composition process.
"The plan defines the volume to such an extent that other elements of the project-the section, the elevation-impose themselves as necessary consequences." (Trevisol 1982: 98).
The process leads to a definition of the project, always starting from the definition of a primary geometric entity, which develops and changes, but always remains legible. Generally, a main volume shapes the whole and smaller volumes are distinct from it. When there are several volumes, their conjunction follows precise rules. Symmetry is the main rule of order (Sakellaridou 2000). The plan and volume form the basis and beginning of the design process, which involves geometric transformations, always related to context, function and light.
Drawing accompanies Botta's design process and becomes a key expression. The ultimate result is often intrinsically linked to initial intentions. However, there is a logical maturation throughout the process, as Botta never denies the principles from which he began, constituting a circular design process. His architecture, as Alessandra Coppa argues, prompts us to attend to the signs used as well as to their changes (Coppa 2007).
Starting from this assumption, the authors present three projects representative of geometric devices that recur as Botta's modus operandi. These are identified by analyzing his whole architectural production, the result of which is explicit in tabular form (Tables 1, 2, 3).
After an initial overall reconstruction of the artifacts and an identification of primary entities, the geometric operations that characterize their tectonics are discerned: Boolean unions (in blue), Boolean subtraction (in red), Boolean juxtaposition (cyan), cut/section (in yellow), and section transformation along a trajectory, called loft (in green) (Fig. 1).

Selected Case Studies
Design of residential buildings, concentrated in the early years of his career, provided a means for experimenting with the potential of variations in elementary forms of the square and circle as well as the cube and cylinder. In the 1979 monograph (Battisti and Frampton 1979), Botta collects in a single sheet the genetic patterns of twelve buildings designed in that period.          , 1990-96) The single-family house in Riva San Vitale (1971)(1972)(1973) has a square plan on which the parallelepiped volume is set, subject to boolean subtractions generating voids (Fig. 1a.3) and interior terraces (Fig. 1a.4) that relate to context. The building plans are set on a square base that varies from floor to floor. This figure recurs on all floors. We found it scaled several times along the diagonal to define the staircase body, the terrace, and volume at the entrance level. The skylight is obtained by subtracting a cylinder from the roof slab (Fig. 1a.5). A linear element, the footbridge, is inserted here, creating a plan with an orientation (Fig. 1a.6).
Clarity of volume and symmetry are elements that we can also find in the Cymbalista Synagogue and Jewish Heritage Center in Tel Aviv (1998), enhanced with symbolic value. The solid parallelepiped base with a flat profile, containing the entrance lobby and service spaces, symbolizing the human condition. At the same time, the vertical axes of the two symmetrical truncated towers represent the quest for the divine.
Analyzing the plan, it is possible to identify an axial pattern set at the center of the towers and draws cuts of the outer envelope ( Fig. 1b.1). At the volumetric level, we see a series of operations: three parallelepiped volumes are subtracted ( Fig. 1b.3) from the base parallelepiped ( Fig. 1b.2), which define the entrance and square bases of the two symmetrical towers (Fig. 1b.4). A loft operation links them to roof circumferences (Fig. 1b.5), where the square slab is juxtaposed and inscribed in the edge circumference ( Fig. 1b.6). The composition takes on a totemic aspect, the Botta's rationalism allowing for an abstraction of his work, symbolically aiming at the circle's squaring.
The church of San Giovanni Battista in Mogno (1986-96) represents another project in which there is a geometric transformation of the plan at different levels. The church space is tiny, and the geometric synthesis between figures of the rectangle, ellipse, and circle is composed with a symmetric balance.
It represents the ascent from the human dimension (space of the basement) to divine perfection (circular roof). Again, the center of the plan's volume represents the construction's starting point. The diagonals of the rectangle and square, rotated at 45°, pass through it ( Fig. 1c.1). The rectangle represents primary footprints of volume subtraction (Fig. 1c.3). Then, the rectangular-based volume of the apse is added (Fig. 1c.4) and the volumes of the entrance, the apse and of the side niches are subtracted (Fig. 1c.5). The cylindrical volume with an elliptical base is then cut by a plane inclined to 45° at its axis ( Fig. 1c.6), forming a circular roof. This circle is finally linked to the rectangle-shaped setting wall with a loft operation. Two arches are juxtaposed to the main solid, balancing the composition.

Conclusion
Our graphical analysis highlights and compares the geometric compositional processes of Botta's architecture.
The presented classification has been composed for multiple purposes, can originate different interpretations and indexing regarding the evolutions and transformations of geometries. It is therefore a dynamic tool, capable of generating different fields of information, depending on the adopted reading key. It can generate considerations about the evolution of design poetics over time, about the relationship between geometry and function or between geometry and scale of intervention.
Parametric digital modeling tools can be useful for analyzing architecture with a geometric matrix by highlighting recurrences, variants, and invariants, and offering interpretative keys to the architectural design. The research offers broad perspectives in the Visual Programming Language landscape, to implement the geometry under study in the algorithmic system of Grasshopper (McNeel Rhinoceros plug-in).
Given these developments, we discern broad prospects for additional work. With this, it is plausible to investigate different aspects of the process: in the input phase for the generation of masses, in the processing phase for the geometric operations performed, and in the output phase for a mathematical/geometric analysis of assumed formal variations.