This study analyses the possibility of making a new bond for the construction of the stone domes, according to a particular geodesic tessellation with five-fold symmetry. The bond configured in this study is obtained according to the geometry given by a suitable spherical polyhedron that, in dividing the dome into parts that are repeated equal to themselves throughout the hemispherical surface, reduces the number of invariant ashlars. The polyhedron chosen is a spherical disdyakis triacontahedron or a spherical regular dodecahedron whose regular pentagonal faces (Fig. 4d) were divided into ten triangles through the 5 lines of symmetry of the regular pentagon (Fig. 4f). Each of these triangles is the minimum portion in which the sphere is divided into equal and symmetrical parts (Wenninger 1999: 3–15), and, through analysis of the various possible tessellations of the sphere, a configuration is chosen into which it can be divided, and which may prove effective for its construction in cut stone. With radial projection was obtained the spherical dodecahedron, and the tessellation of this minimum triangle is repeated symmetrically onto the whole sphere surface, determining the bond of the stone dome according to precise and interdependent static, geometric, formal and typological requirements (Fig. 4c). The configuration chosen for the subdivision of the minimum triangle TA and its symmetrical triangle TB (Fig. 4), which determines the stereotomic bond of the dome in cut stone, is based on the 5-fold symmetry, respecting the geometric properties of the polyhedron from which it originated, and is derived from J. Kepler’s “Aa” tessellation (Fig. 4a) found in his Harmonices Mundi (Kepler 1619: 58–59). It is the same geometry that characterizes the atomic structure of quasicrystals (Fig. 4b), which has never been applied to the architectural construction of domed spaces in cut stone. Indeed this 5-fold symmetry is present in nature in the atomic lattice of quasicrystals with icosahedral symmetry (Takakura et al. 2007: 58–63) discovered in 1984 by Dan Schechtman, who received the Nobel Prize for Chemistry in 2011 for this discovery, and it is proportionate according to the golden ratio because it is constituted by the juxtaposition of regular pentagons from which the tessellations coded by Roger Penrose derive.
This well-connected and well-proportioned configuration is harmonic, deriving etymologically from the ancient greek άρμονία (harmony) (Folicaldi 2005: 29), meaning “connection, concord, proportionate structure, agreement” (Montanari 2003: 324), and increases the stability of the atomic lattice; in addition, the interweaving, according to five-fold symmetry, hinders the process of dislocation due to the sliding of its sections, thus preventing the deformation of its materials, and resulting in greater hardness and structural strength at break.
The transposition of this type of five-fold tessellation in stereotomic dome architecture has considerable advantages in structural terms, as it improves the interlocking of ashlar tiles and generates a highly resistant texture, and in formal terms, as this particular harmonic geometric composition, involving the golden ratio, determines a strongly expressive embroidery.
In the choice of the tessellation and the design of the bond, where it is possible to calculate the number of element-type invariants, one must consider their deformation when projecting on the sphere and making architectural choices that simultaneously involve its planar and spherical design, topologically equivalents or homoeomorphic ones. In fact, what is equal in the plan is not equal on the sphere: pentagons in Fig. 4e are equal in the planar pentagon (Fig. 4d) but all different in the spherical pentagon. Instead, the ashlars composing nonagons and decagons respond to the symmetry rules of the triangle, optimizing their number.
The control of this complex geometry is made simpler by modern three-dimensional infographic modelling technology tools that, compared to the traditional geometric tracing method, allow us to visualize the geometric division on the plane and the corresponding structural tessellation in space, and to verify the equivalences and symmetries of the parts in which it is subdivided with visible security of the actual accuracy of the design, without resorting to mathematical calculations and more complex geometric construction rules.
The bond defined according to this tessellation (Fig. 5a) is particularly effective for the geometry that optimizes the number of invariant ashlars, for the new strong aesthetic definition and for its static interlocking. The ashlar invariants of this dome (Fig. 5a) are 34 and are from 25 cm to 40 cm long. The diameter of the dome is 10.06 metres long at the extrados, and the thickness of ashlars is 22 cm. This research study describes the method by which the stereotomic definition of the dome and the construction of a maquette (Fig. 5c-5d) is possible, as well as the method allowing me to determine the phases of bond assembly and to take its static efficiency into consideration.
Compared to the methods used in past, existing infographic CAD/CAM software allow the direct transfer from ideal to real, from a three-dimensional model to rapid prototyping machines and a numerical control machine and robot, simplifying the production process. From the infographic model, in fact, it was possible to prototype ashlars in PLA material with the Ultramaker2 machine for the construction of a maquette at a scale of 1:14.37, with a dome diameter of 70 cm at the extrados. The height of the face of ashlars at the extrados ranges from a minimum of 1.5 cm to a maximum of 4 cm. After manually numbering and classifying all the prototyped ashlars (Fig. 5b), the maquette was constructed in five working days and the assembly process was simplified by construction drawings prepared for that purpose in which same-type invariant ashlars have the same colour, and the same number and belong to the same layer in infographic software.
Mounting took place without centring through the traditional construction technique using a trammel that was obtained through a small wooden rod equal in length to the radius of the sphere inscribed in the dome, with one end positioned in the centre of the dome and the other in the faces of the ashlars, which allows for the definition of perfect sphericity of the intrados. The construction of the maquette is very useful to confirm the interlocking achieved by the bond and to understand the most effective laying rules.
This research analysed the static behaviour of this new bond and its holing limits through the subtraction of ashlars that compose nonagons and decagons. Static analysis was done through the computer modelling, showing that the structural form is verified, also through the appropriate subtraction of particular ashlars, thereby unloading the structural form, as shown in the maquette. The forces’ distribution is carried out along the discharge arches composed of the pentagonal and hexagonal structural elements, as illustrated in Fig. 6.
To achieve a dome configured in the way that is presented in this study, but with a smaller diameter and the same size of ashlars, it is necessary to repeat the design process with few tiles in the minimum triangle, the smallest part in which it is possible to divide the sphere into equal and symmetrical parts.
Another dome is configured in this study (Fig. 7a), with the diameter of the extrados of 7.12 m, and 17 or 18 types of ashlar invariants. To verify the feasibility of the cutting work and the assembly of the bond, a stone prototype (Fig. 7c, d) of this smaller dome was made by the société SNBR at Sainte-Savine-Troyes, in France, using Robot ABB. The scale of the stone prototype is 1:5.74, and the extrados diameter measures 1.24 m. The prototype was further simplified since the decagons and the nonagons are made up of only one ashlar, because they are smaller than at real scale; in fact, they measure approximately 10 cm in length and are 4 cm thick (Fig. 7b). Ashlars were made in pierre de Lens, cut by the robot in sixty hours; they were numbered and then placed on polyurethane centring, which was prepared in four hours by the robot and was transferred by engraving the design of the bonding project. The execution of this design is not indispensable to dome construction, as demonstrated by the PLA maquette, but it was helpful in speeding up assembly time, which took place on two working days by two people, including cleaning of the prototype after assembly.