*3.3. Internal Reflections*

We have dealt with the question of primary or direct radiative transfer but in order to solve the problem completely, we have to consider secondary energy sources produced by reflections that may hold fair significance in some particular cases such as in this church. In a closed space, the main components of secondary transfer are those produced by the limiting surfaces that, depending on their properties, re-emit some of the light that reaches them. This phenomenon may be treated with the help of a procedure similar to the so-called "radiosity" (or balancing infinite rebounds) in the technical literature, see below [28,29].

Within the process of analysis of an architectural space, it is important to qualify adequately this component of inter-reflections because direct light may not be satisfactory or even wanted and secondary sources may add the required surplus to compensate for such decrement.

All cultures have been aware of the phenomena of reflection at some point or another in their history and they have furnished the interior renderings of their buildings to produce more brightness and to serve better their spatial purposes and dwelling aims. In that way, their chambers consistently enjoyed the boons of nature.

In the case of Saint Louis, the inner surfaces present a lavish decoration enriched by innumerable artworks, including, gilt pads, mirrors, engravings and frescoes. Beyond the liturgical symbolism of these decorations underlies a univocal aim to enhance illumination of the space. Such "enlightenment" undoubtedly refers to spiritual and territorial domains other than the European. Oriental and American pre-Hispanic cultures accrue in the forge of the transition to a baroque era, as we have shown, a period very much informed by the Jesuits as the present authors have studied in other treatises [31].

Particularly, some of the most significant events of the Jesuit mission in Japan feature in this church. Saint Xavier appears in his arrival to the shores of Kagoshima (Kyushu, Japan) in a niche with numerous mirrors, precious stones and other glistening devices. Moreover, presiding the altarpiece of Saint Stanislaus Kotska in the south wall, there is a representation of the three martyrs of Japan (Pablo Miki, Juan Soan de Goto and Diego Kisai) holding palms and embracing the Cross symbol of their faith and torment (Figure 10). Naturally, news of the grea<sup>t</sup> martyrdom of Nagasaki soon diffused in Spain. Most of the missionaries who travelled to the Asia departed from the ports of Seville and the maritime city of Cadiz. A rather curious phenomenon related with the former is that the medallion representing the three martyrs of Japan always remains in backlight.

**Figure 10.** Interior of Saint Louis of the Frenchmen: (**a**) niche of St. Xavier; (**b**) representation of the three martyrs of Japan. Source: Almodovar-Melendo.

Returning to technical matters in discussion, we have made a summary of the necessary algebra to treat this complex problem. In the first stage, we have considered the illuminance of each surface as a final average, acquired after a high number of reflections (Figure 11). As we mentioned earlier, this procedure is akin to the procedure of balancing infinite rebounds of radiation in an enclosed volume and is amply used for thermal problems though not so often in lighting, where more detailed distributions of energy may be required.

**Figure 11.** Typical volume of the central space in churches composed of a cylinder and a spherical cap used to find the reflected component in illuminance exchanges. Source: Cabeza-Laïnez.

For a given set of surfaces as defined in the figure above, the involved mathematics summarizes as follows: the total component is made of the direct light plus the reflected light.

$$E\_{\rm tot} = E\_{\rm dir} + E\_{\rm ref},\tag{8}$$

Thus, if we are able to create two matrices *Fd* and *Fr* with the elements described as follows:

$$F\_r = \begin{bmatrix} 1 & -F\_{12}\rho\_2 & -F\_{13}\rho\_3 \\ -F\_{21}\rho\_1 & 1 & -F\_{23}\rho\_3 \\ -F\_{31}\rho\_1 & -F\_{32}\rho\_2 & 1 \end{bmatrix} \tag{9}$$

$$\begin{aligned} F\_{\mathbf{r}} = \begin{bmatrix} F\_{11}\rho\_1 & F\_{12}\rho\_2 & F\_{13}\rho\_3 \\ F\_{21}\rho\_1 & F\_{22}\rho\_2 & F\_{23}\rho\_3 \\ F\_{31}\rho\_1 & F\_{32}\rho\_2 & 0 \end{bmatrix} \end{aligned} \tag{10}$$

On the understanding that surface 3 represents the floor plane. *Fij* are the corresponding configuration factors from surface *i* to surface *j* and *ρi* is the coefficient of reflection of surface *i.*

Then we could easily establish a relationship between reflected and direct illuminance.

$$F\_{\mathbf{r}}E\_{\mathbf{r}} = F\_{d}E\_{d\mathbf{r}} \tag{11}$$

For instance, the first line of the matrix product above gives:

$$E\_{r1} - E\_{r2}F\_{12}F\rho\_2 - E\_{r3}F\_{13}\rho\_3 = E\_{d1}F\_{11}\rho\_1 + E\_{d2}F\_{12}\rho\_2 + E\_{d3}F\_{13}\rho\_3. \tag{12}$$

And extracting the reflected component at surface 1, we would obtain,

$$E\_{r1} = \rho\_1 F\_{11} E\_{d1} - \rho\_2 F\_{12} (E\_{d2} + E\_{r2}) + \rho\_3 F\_{13} (E\_{d3} + E\_{r3}),\tag{13}$$

Expressed in worlds the former means that, the light reflected on surface 1 is the total received from surfaces 1, 2 and 3 multiplied by their reflection coefficients and their configuration factors (in the case of 1 being curved we would deal with *F*11, an auto-factor). As we have seen that total light is the sum of direct and reflected light or once again,

$$F^{-1}{}\_{r}F\_{l}E\_{r} = F^{-1}{}\_{r}F\_{d}F\_{d\prime\prime} \,. \tag{14}$$

$$E\_r = F\_{rd} E\_{d\prime};\ where\ F\_{rd} = F^{-1}{}\_r F\_{d\prime} \tag{15}$$

In this situation, the resultant component of illuminance is a sum of a direct component and a reflected one. That is, the amount of light, which a particular surface receives from the primary source and a second component consisting of the reflections, received from all the other surfaces in unison. In this way, the problem is solved from a mathematical point of view.

Thus, if we know the direct illuminance for each surface we are able to predict all possible reflections and add them to the first one.

The reflective coefficients are often difficult to find in ancient veneers. In some cases, we had to determine them at our laboratory by constructing several replicas of the wall materials and measuring it by means of spectroscopic procedures.

Such coefficients were checked in situ whenever possible by direct monitoring (bearing in mind that the church has been closed and restored for ten years). The richer decorations correspond with the lower part of the church and do not add much quantitatively to the illumination of the space, their effect is mainly qualitative and so to say subjective.

In addition, it is interesting to notice that, the former expressions discussed, give the average value of illuminance for the whole surface (the entire drum of the church for instance), thus minimizing errors.

In the unlikely event that we would need to know a point-by-point field distribution, minor adjustments would be required for each case or usually we could divide the problem into an adequate number of sub-surfaces.
