The Corner Tower of Anagni Cathedral: Geometry and Equilibrium

This paper explores the corner tower of the Anagni Cathedral, a Romanesque structure built in the eleventh and thirteenth centuries. The tower, located beneath the baptistery, was likely constructed to support a small chapel with a font. Through digital surveying and geometric analysis, this study examines the tower's structural element and speculates on the ideal stereotomic apparatus and reference models. The paper also delves into the mechanism responsible for maintaining the cantilevering structure’s equilibrium. The tower and baptistery exemplify the role of stereotomy and friction in maintaining equilibrium, with internal tensile forces and unilateral contact between the structure's blocks. This study provides valuable insights into the Anagni Cathedral's structural elements and highlights the importance of understanding stereotomy and friction principles.


Introduction
The Anagni Cathedral is a Romanesque structure built between 1072 and 1104. Additional chapels were added in the thirteenth century. This paper focuses on the corner tower ( Fig. 1) located beneath the baptistery, which was likely constructed to support the small chapel with the font. Using an accurate digital survey and geometric analysis, this study examines the structural element and speculates on the ideal conformation of the stereotomic apparatus and potential reference models. This paper also interprets and explains the primary mechanism responsible for maintaining the equilibrium of the cantilevering structure. The tower and baptistery are excellent examples of the role of stereotomy and friction in maintaining the state of equilibrium, where internal tensile forces are necessary, and the blocks composing the structure interact with each other via unilateral contact. Overall, this paper provides valuable insights into the structural elements of the Anagni Cathedral and highlights their significance in understanding the principles of stereotomy and friction.

Historical Notes
There are not many ancient documents related to Anagni Cathedral that can ensure an accurate reconstruction of the historical phases that led to the current layout of this building. We know that the construction of this complex was commissioned by bishop Pietro (?-1105), prince of Salerno, between 1072 and 1104, and it was carried out starting from the foundations of a previous church dedicated to St. Magnus. In this first phase, the building had a typical Romanesque three-nave structure with alternating quadrangular and circular pillars, a transept, and three apses (Matthiae 1942;Palandri 2006).
Two coeval crypts are located below the transept and part of the nave to the west: the first crypt -entirely frescoed and defined as the 'Sistine Chapel of the Middle Ages' -houses the body of St. Magnus. The second chapel was dedicated to St. Thomas Becket by Pope Alexander III. Both of these underground places were originally accessible from outside the church through an arched passage located on the west side. Today's access is guaranteed by two stairways that lead from the church's aisles to the level below. The building was probably constructed following the Abbey of Monte Cassino model, which was erected under abbot Desiderio's guidance between 1066 and 1072, as confirmed by spatiotemporal proximity and the friendship between Desiderio and the bishop Pietro (Urcioli 2006: 191). Side chapels were added in the thirteenth century (Lauri, Caetani, and the one hosting the baptismal font) on the west area, as well as a monumental staircase (later demolished in 1830) that used to give access to the church from the underlying Piazza Innocenzo III through a new portal walking on a terrace called 'the blessing Loggia.' As mentioned above, the room of the baptistery was also built between these chapels, under which there was a corner supporting vault, placed in continuity with the portico below the blessing Loggia (Fig. 2). The vaults of the portico have incorporated a system of hanging arches on zoomorphic shelves; to this day, they protrude out the intrados surfaces, but at the time they were used to support the above-elements (Matthiae 1942;Piacentini 2006).

Digital Survey
This research has employed laser scanners and photogrammetric techniques to develop the morphological genesis of the vaulted surface of Anagni Cathedral's corner tower. The survey practice involved the placement of targets on the vertical walls supporting the vault and performing four scans with a Leica BLK 360 laser scanner (Fig. 3), followed by a photogrammetric survey via a Fujifilm X-T20 camera. The resulting textured mesh model was scaled and oriented using data from the laser scanner and imported into 3D modelling software for geometric analysis. This study found that the intrados surface of the vault closely resembled an abstract geometric surface arising from the intersection between two cylinders (Fig. 4), demonstrating significant precision of the constructive method despite the monument's age.
The study also examined the similarities and differences between the Anagni vault and a complex architectural element called the Trompe (De l'Orme 1567; Calvo-Lopez 2020), created using stereotomic techniques during the Renaissance, particularly in France. The Trompe is a suspended corner tower that connects interior spaces and two converging walls, typically positioned outside in the corner of the facade. However, the Anagni vault differs from the Trompe because the latter has an intrados surface resembling a conoid rather than a cylindrical surface. The This research sheds light on using stereotomic techniques in architecture and implies a better understanding of the similarities and differences between the Anagni vault and the Trompe. The use of laser scanners and photogrammetric techniques in this research demonstrates the potential of modern technology to investigate and analyze historical architecture.

Structural Analysis
The vault in Fig. 5 is more challenging since the Anagni vault is made of smaller and rougher stone elements. Hereafter, we will focus on a qualitative description of the mechanism that can be used to interpret and explain the equilibrium of such a cantilevering structure. This structure is a perfect example of the role of stereotomy and friction in maintaining equilibrium.
The stone blocks do not run from taut to compressed parts of the masonry (Fig. 6a). They are assumed to be in unilateral mutual contact and capable of exploiting small tensile forces in their "long" direction-orthogonal to the main thrust forces. The compressive thrust forces act transversely to the stone joints, allowing the employment of tangential friction forces that turn out to be pull forces inside the stones in the long joint directions. The equilibrium solution assumes that two spatial Linear Arches, called Γ 1 and Γ 2 , exist inside the masonry (Fig. 6). These arches spring from the walls and produce thrusts, interacting with each other through tensile forces transmitted to them by the transverse stones. We consider these spatial Linear Arches-essentially spatial thrust lines-as 1D structures inside the masonry. Some authors recently introduced this kind of compressive line network , and it was also discussed in Carlo Olivieri's PhD thesis (Olivieri 2021).
We introduce a Cartesian reference system O; x 1 , x 2 , x 3 with the axis x 2 located along the cord of the arch from the springings, the origin O in the midpoint of the cord and the axis x 3 along the vertical (Fig. 6a). We start by assigning the shapes Γ 1 , Γ 2 of the linear arches Γ 1 , Γ 2 into the horizontal plane = {x 1 , x 2 } (Fig. 6b). The weight of the superstructure, pavement, and filling is lumped in the two arches based on the "influence areas." Ω 1 andΩ 2 depicted in Fig. 6a. Known vertical forces represent this load −q 1 , −q 2 applied as loads per unit projected length along x 2 . The two arches are assumed to interact through mutually distributed tensile forces p, −p, which have zero components in the direction x 2 : p = {p, 0, q} (Fig. 6c).
Therefore, the linear arches Γ 1 , Γ 2 are acted on by vertical loads −q 1 + q, −q 2 − q and by transverse forces acting in the directionx 1 :p, −p . The equilibrium of the linear arches Γ 1 , Γ 2 can be decomposed into two planar equilibrium analyses concerning the archesΓ 1 , Γ 2 , the first one in the horizontal plane = {x 1 , x 2 } , and the second one in the vertical plane = {x 2 , x 3 } . The equilibrium problem can be reduced to the following system of four ordinary differential equations.
in which g 1 x 2 , g 2 x 2 , f 1 x 2 , f 2 (x 2 ) are scalar functions describing the linear arches Γ 1 , Γ 2 , q = p f 2 −f 1 g 2 −g 1 , and H 1 , H 2 are two arbitrary constants representing the projection of the thrust along the x 2 axis. We note that this system of four odes has Fig. 6 Schematic view of the Linear Arches Γ 1 andΓ 2 of their projections Γ 1 andΓ 2 on the horizontal plane π, and their "influence areas" Ω 1 and Ω 2 : a down-top view; b plan view; c axonometric view; d equilibrium solution five unknowns ( g 1 x 2 , g 2 x 2 , f 1 x 2 , f 2 x 2 , p x 2 ) and there exists a class of equilibrium solutions, among which the most favourable can be identified through optimization. The solution shown graphically in Fig. 6d gives the following values for the thrust forces: H 1 = −48.0kN , H 2 = −32.1kN , and of the maximum pulling stress: p = 9.6kN∕m . On the safe side, on assuming as the least stone slenderness ratio the value b • ∕l • = 0.217 -b • , l • being the two surface dimensions of the stones (Fig. 6a) -, we can estimate the maximum tangential stresses produced by p and acting on the long stone joints as = 2pb • ∕l • = 4.18kN∕m , and for the normal stresses produced by the thrust H 2 acting on the same faces: = H 2 L = −33.7kN∕m . The ratio between these two stresses: f = − ∕ = 8.06 has to be compared with the friction coefficient between the stones, = tan = 0.839 (corresponding to a friction angle of 40°). The value of f is much greater than , indicating that the compressive forces are widely sufficient to sustain -through friction -the tangential stresses produced by the tensile stresses generated by the cantilevering arch section.

Conclusions
The vault covering the portico at the base of the corner tower is an impressive display of geometric complexity and morphological precision, given the time of its construction. It is possible that the builders took inspiration from previous solutions that had already been evaluated and tested in terms of static and stylistic efficiency. This is one of the few examples in which the role of tensile forces sustained by friction assumes (together with a clever and attentive cut of the stones) a crucial role in the equilibrium.
This study presented an accurate geometric restitution of the structure, and a simplified structural analysis was carried out using the equilibrium method. Preliminary findings indicate that the structure is in a stable equilibrium state with a large geometrical safety factor. Further work will provide this equilibrium analysis with a kinematical analysis based on the Distinct Element Method (DEM) and 3D analyses based on PRD (Olivieri et al. 2022). These efforts will involve a more comprehensive understanding of the vault's structural behaviour and contribute to our knowledge of the innovative construction techniques used in this period (Iannuzzo et al. 2021).
fellow working on topics about History of representation and advanced technologies for Architecture. He has lectured in conferences in academic institutions in Italy and abroad, and has participated to national (PRIN 2010(PRIN -2011  Contractor, where, for two years, he was involved in relevant international projects. During his PhD, he developed new strategies for the assessment and for the form-finding of complex curved structures using the membrane theory and an extension of the classical Thrust Line method to special structures called Linear Arch Static Analysis. This new methodology opens new possibilities to look at the complex topic represented by curved masonry and concrete constructions. He also investigated the dynamics of masonry arches subjected to ground motion both from a theoretical and experimental perspective. His research is currently directed towards the definition of new optimization approaches for purely compressive shapes under seismic actions and new strategies for using low-carbon material blocks to construct these structures. As a co-founder partner of KEIKO Cultural Association, he has organized and participated as Teaching assistant at the International Summer School on Historic Masonry Structures since 2018. In 2022 he joined the Form Finding Lab of Princeton University.