**1. Introduction**

Famous buildings from the past and the present have brandished their excellence in the respect of lighting control. Their thresholds were praised from an artistic point of view for the way in which solar radiation penetrated their interiors [1]. It is said that these buildings scattered the light when it was excessive or augmented it if scarce. Inside their vaults the light itself has been termed by poets or architects as quaint, bland, dim, uplifting, ruthless or even monosyllabic. Consequently, knowledge of daylighting has interested many architectural theorists including Siegfried Giedion, Le Corbusier's colleague, who pointed out:

"It is light that induces the sensation of space. Space is annihilated by darkness. Light and space are indissolubly summoned." [2] (p. 445)

Even though architectural theoreticians have celebrated the excellence of daylighting in relevant buildings, they carried out very few assessments of such illumination. In fact, the architectural literature has usually shown a high degree of inaccuracy regarding its most revered buildings (Figures 1 and 2). Consequently, they have produced an undesired effect, the inability to preserve or transmit the benefits of daylighting in historic halls, thus provoking inadequacy of environmental measures and supplementary means of lighting.

We would agree that we require objective evaluation of performance in ancient buildings. However, such evaluation of lighting is difficult and cumbersome due to the unpredictable behaviour of visitors and personnel and the variability of weather conditions. Henceforth simulation is an appropriate tool to evaluate the potential of the building in sustainability terms, given that we take certain caveats into account [3].

Following such concern, we have been involved, for a number of years, in the development of a mathematical model to simulate the radiation or more specifically illumination in a physical or architectural medium. To this end, the authors expand a series of algorithms stemming from the algebra of configuration factors [4]. This mathematical model extends the radiation properties of diffuse sources to all kinds of luminous exitance from building surfaces irrespective of their shape (a significant novelty to our knowledge) and including the reflected component, even for curved geometries, for the first time in history. The surfaces are therefore treated as radiative emitters by virtue of the generalized law of the projected solid angle [5,6]. The concept of surface source allows for the inclusion of direct solar radiations since it is independent of the sky condition. The simulation of the aforementioned objectives can be fulfilled by entirely geometric or formal factors and thus appropriate to the architectural profession.

When we are discussing a radiant phenomenon in which a sufficiently diffuse energy is transmitted, we intend to know how the energy will be propagated or distributed so we can improve our architectural design with respect to such distribution. In this regard, illumination manifests itself through fields of a fundamentally vector nature [7]. Therefore, our main objective is to discern the behaviour of these fields in their unaltered state. On this topic, we have incorporated important contributions by Yamauchi [8], Moon and Spencer [9,10] among others.

Nevertheless, modifications of architectural features have the potential to alter substantially the field of study. In this regard, one of the main problems in environmental sciences applied to architecture has been to determine in which ways the existing physical fields are transformed due to buildings' features and towards which direction our design should be oriented in the search for a sounder transmission of climatic events [11]. In other words, how the design and architectural forms could be improved to achieve an optimal and coherent distribution of natural energies or at least of supplementary energy in the case of retrofit of heritage.

In this regard, the proposed mathematical model allows us to determine the illuminance vector at every point of the space under study and hence to immediately obtain the flux lines in the radiant field caused by any architecturally conceived form. This procedure has been validated in dozens of projects and hundreds of radiation measurements around the world [12–23], see Figure 3. Among them, we would like to present some random cases of the following: in Rome (Italy), the Church of Sant'Andrea by Bernini, Sant'Ivo by Borromini and the Pantheon. In Paris (France) the National Library by Henri Labrouste. In Kyoto (Japan), the Buddhist temple of Ryoanji. In Salvador (Brazil) the Church of Sao Bento, in India the temples of Ajanta and Modhera and in Korea the temple of Seokguram (Gyeongju). These field works have been completed roughly between 1996 and 2018.

**Figure 1.** The Roman Pantheon, lighting simulation conducted and validated by Joseph Cabeza-Laïnez.

**Figure 2.** Plans and Views of Chaitya number 26 at Ajanta (India) showing illuminance levels. (3000–50 lux). Simulation conducted and validated by Joseph Cabeza-Laïnez.

In this manner, such remarkable spaces for Universal Architectural History have been thoroughly and painstakingly analysed and we ought to say that with the encouragemen<sup>t</sup> of no one.

**Figure 3.** The Seokguram Buddha temple at Gyeongju (Korea). Simulation of a hypothetical aperture conducted and validated by Joseph Cabeza-Laïnez.

### **2. A Brief Outline of the History of the Church**

The Church of Saint Louis lies in Sevilla in the southernmost point of Spain (latitude 37◦22 N), (Figure 4). It was erected by local builders between 1699 and 1731 under the guidance of Jesuit architects and hypothetically among them Angelo Italia. The building was so influential in the urban layout that the former King's Street, in which the church is located, changed its name to the current denomination of Saint Louis' Street.

**Figure 4.** View of the Dome decorated with frescoes from the inside and the lantern and main eight windows. Source: Almodovar-Melendo.

Despite the discussion on the authorship of the project, an important participation in the works of the famous Sevillian master Leonardo de Figueroa has been reported, albeit in collaboration with other artists. Its Greek-cross disposition is rare in Spain and probably relates to the Italian tradition showing a clear parallel with the church of S. Francesco Saverio (St. Francis Xavier) in Palermo (Sicily), where the Jesuit Angelo Italia was the chief architect. In fact, resemblances can be traced to other important religious constructions, featuring designs by Carlo Fontana, Bernini, Rainaldi or Borromini, like the church of Santa Maria in Monte Santo or Santa Agnese in Piazza Navona. The church has been considered as one of the most outstanding temples erected by the Society of Jesus of all times and comparable to the ancient wonders of antiquity as ziggurats et cetera [24].

This church is oriented according to the four cardinal points in a way that the main entrance is located to the east and the main altar to the west and, in this manner, revolves the usual Christian disposition where the main façade must face west. Many artworks manufactured by grea<sup>t</sup> masters of the Sevillian Baroque can be found in the interior of the church [25]. The altarpieces, which exhibit a grea<sup>t</sup> profusion of gilt surfaces and other glistening and reflective details, were mainly carved by Pedro Duque Cornejo and the author of the majority of the frescoes was Domingo Martinez. The iconographic program refers to relevant evangelizers and founders of the Society of Jesus like Xavier depicted in his arrival to the shores of Japan and Loyola. An altarpiece dedicated to Saint Stanislaus Kostka was carved in the south wall, while another dedicated to Saint Francis Borgia appears in the north side of the temple (Figure 5).

**Figure 5.** Altarpiece dedicated to Saint Francis Borgia, detail of the church. Source: Almodovar-Melendo.

Referring to the girth and typology of the building, the width of the cupola is of 13.5 m and it rests on a drum 14.85 m high, that presents eight tall windows with a total area of 6.3 square meters each. It is interesting to notice that the ratio of glazing to lateral enveloping area is lesser than 10% and incomparable to modern buildings made of curtain walls or even fully glazed (Figure 6).

**Figure 6.** Virtual reality depiction of the central nave of the Church. Source: Gonzalo Pulido.

### **3. Description of the Mathematical Model**
