Study of the Effects Created by the Sunlight Passing Through the Rose Windows of Mallorca Cathedral

This paper shows the light effects and geometric alignments created by the sunlight passing through the stained glasses of the eastern rose window in Mallorca Cathedral and projecting on the inner side of the cathedral’s main façade and on the cathedral’s floor. As well as providing more accurate information about these already known light effects, this paper makes use of laser scanning techniques and astronomical and geographical concepts in order to graphically display other novel effects which occur in coincidence with certain religious festivities throughout the year.


Introduction
Throughout history, many buildings have been designed to interact with several celestial objects.Archaeoastronomy is the branch of science which studies these interactions, particularly in constructions dedicated to religious worship.Several examples can be cited which show how the design of certain constructions integrates the movement of stars: For instance, Magli (2016) reveals that the megalithic enclosure of Gobekli Tepe, probably the most ancient in the world, was designed to visualize the orbit of Sirius through the structures' orientation.Also, Catamo and Lucarini (2002) and Sigismondi (2012) describe the gnomon built by Francesco Bianchini in the Basilica of St. Mary of the Angels and of the Martyrs, which makes it possible to observe the meridian passage of the Sun and also of the stars Polaris, Arcturus and Sirius.
The star which has played the most relevant role in the history or architecture is the sun (Heilbron 1999;Torres 2000;Linares 2015).Due to its presence,

Sunlight Effects in Mallorca Cathedral
It is well known, particularly in Mallorca, that every year on the same dates, and almost at the same time, the morning sunlight passes through the giant eastern rose window and projects on the inner side of the main façade right underneath the western rose window, thus forming the celebrated 'Eight Effect' or 'Festival of Light' (Ruiz and Pol 2010).This light effect occurs every 2nd of February (Candlemas Day) and every 11th of November (Saint Martin of Tours) (see Fig. 1).These dates are 40 days and 43 days off Christmas, respectively, and the positions of both projections differ slightly.Also, in the days prior to the winter solstice, the sunlight passing through the eastern rose projects almost exactly on the western rose window, thus creating another light effect which can be seen from outside the cathedral (Fig. 2).
Even though these sunlight effects are widely known and appreciated by the visitors to the cathedral, their astronomical and geometric explanation is not commonly known.Therefore, for the purposes of research, the entire Mallorca cathedral has been digitalized and specific techniques were used in order to show the following time and geometric features, both mathematically and graphically, and make them common knowledge: Thus, in addition to providing more accurate information on light effects which are already widely known, this paper presents some novel effects created by the sunlight inside the Mallorca Cathedral (we consider these effects to be novel because they have been largely overlooked), as well as a graphical analysis of the eastern Fig. 2 Sunlight effect which occurs in the days prior to the winter solstice, and can be seen from outside the cathedral.Images: courtesy of Mallorca Cathedral rose window projection during certain religious festivities throughout the year.All our results are graphically displayed (floor plan and elevation) on the basis of strict astronomical criteria, as described in the following section.

Digital Reconstruction of Mallorca Cathedral
In order to calculate and graphically represent our results, it was necessary to create a digital model of the cathedral.Using a RTC360 Leica scanner, we carried out a laser scanner survey with 1250 stations and determined a cloud made up by 21,000,000 points covering the cathedral close and the space under consideration (Fig. 3).
In order to have a better graphic control of results and incorporate the numerical calculations into the architectural drawings, some parts of the building were outlined with CAD on the basis of the abovementioned survey (Figs. 4 and 5).
After this graphical analysis, the following parameters were established:  • Amplitude of the angle formed between the inner side of the western façade and the equatorial plane in the direction of rotation north-east: = 57.163• sexagesimal degrees.

Astronomical and Geometric Explanation
The graphical results of this paper, presented in section "Results", are based on solving some astronomical and geometric problems.Assuming that the center E of the earth is the center of the universe and also the center of a celestial sphere where the stars are, we need to: 1) determine the position S of the center of the sun in the celestial sphere (apparent right ascension app and apparent declination app ) at every given instant JD , 2) determine the time equation E m at the instant JD , and 3) determine the intersection p L = ∩ r S of plane with the straight line r S passing through C L and having the direction of vector ������� ⃗  E  S .Readers may turn to Martín (1990) and Meeus (1991) to learn in detail the definitions of the celestial positions, the time equation and all the concepts mentioned in this section.
In order to solve these problems, we use the well-known algorithms mentioned in Martín (1990) and Meeus (1991).The initial variable in these three problems is a calendar date.
Thus, we start from an initial variable date: year a , month m ∈ ∩ (1, 12) , day of the month d , hour h , minute m , second s , in Greenwich Civil Time, i.e., GMT + 12 h .Please note that in Mallorca, due to the delays and advancements imposed by the Spanish Government, we must consider the Greenwich Civil Time plus 1 hour in Autumn-Winter or plus 2 hours in Spring-Summer.
Next, the date is converted to Julian Date JD .Readers may turn to Martín  (1990) and Meeus (1991) for definitions and calculations of: CE Gregorian year (from 14 th October 1582 onwards), CE and BCE Julian year (until 4 th October 1582), and Julian Day JD (#) according to the time measurement system proposed by Joseph Scaliger.Meeus (1991) provides the following algorithm to calculate JD when JD > 0

Calculating the Position of the Sun in the Celestial Sphere
Now that we know JD , we can calculate app , which is the apparent declination of the Sun in the celestial sphere.We will use the algorithms in Meeus (1991) with an error of 0.01 ′′ sexagesimal seconds.Even though we do not need such level of accuracy for our purposes, we follow the steps and calculate the following astronomical variables:  (l) , the true obliquity of the ecliptic: = 0 + Δ . (m) app , the apparent declination of the Sun: app = sin −1 (sin sin ) , calculated in sexagesimal degrees.

Calculating the Time Equation
Next, given the date JD , we proceed to calculate the time equation E m .Again, we use the algorithms provided in Meeus (1991) to calculate the following astronomical variables: (a) The sum of the mean value of the aberration in longitude and the correction for reduction to the FK5 system: 0 • .0057183= −20 �� .49552− 0 �� .09033.

(c)
app , the apparent right ascension of the Sun taking into account the aberration and the nutation: app = tan −1 cos sin cos 2 , calculated in sexagesimal degrees.(e) If app ≥ 0 , then: (h) E , the time equation: Since app is in the same quadrant as , the function tan is defined in a particular manner which the reader may find in Meeus (1991).

Calculating the Geometric Projection
Having calculated app and E m based on JD , the required parameters are gathered from subsection "Digital reconstruction of Mallorca Cathedral" in order to carry out the solar projection of C L on the plane and determine the point p L .Cartesian orthonormal reference system ;� ⃗ e 1 , � ⃗ e 2 , � ⃗ e 3 is used, such that a) if points in space have coordinates (x, y, z), then the equation of plane is x=0, b) � ⃗ e 3 has direction towards the zenith, c) � ⃗ e 1 has direction towards C L , d) � ⃗ e 2 is such that the base � ⃗ e 1 , � ⃗ e 2 , � ⃗ e 3 is clockwise (i.e., if � ⃗ e 1 pointed south, � ⃗ e 2 would point west), and e) (0,0,0) are the coordinates of O L , which is the orthogonal projection of C L on .Using the algorithms described in Martín (1990) and Meeus (1991), the following geometric parameters are calculated in order to determine p L . (g) y 1 = (−cos(Height)cos( h)sin + cos( ht)sin(Azimuth)cos ). (h) z 1 = sin(Height).
After the above steps, the coordinates of the projection of C L are determined.Such coordinates are: p L = 0, y 1 , z 1 .The variation of this projection p L over time results in paths T i∕j on the plane .These paths are graphically described below.
Similarly, the intersection q L = ∩ r S is determined.This is the intersection of plane , which is the floor plane, and the straight line r S passing through C L and having the direction of vector ������� ⃗  E  S .In order to do that, the last three steps above f), g), h) are modified as follows: (1) = −35.313sin(Height) .( 2) After the above steps, the coordinates of the projection q L of C L on the floor plane are determined.Such coordinates are: q L = x 1 , y 1 , −35.313 .The variation of this projection q L over time results in paths L i∕j on the plane .These paths are graphically described below.

Results
Using the abovementioned astronomical and geometric algorithms (which were programmed in C++ by the authors for better numeric accuracy), the points p L are obtained which describe the daily paths T i∕j generated by the projection of C L on the plane ; and the points q L are obtained which describe the daily paths L i∕j generated by the projection of C L on the cathedral's floor plane .These daily paths T i∕j , corresponding to the dates listed in Table 1, are graphically represented in Figs. 6, 7 and 8; and the daily paths L i∕j are graphically represented in Fig. 9.
Besides, in order to provide the reader with more information, the previous Fig.6, 7 and 8 show (in orange and left to right) the analemmas for 7.00 am, 8.00 am and 9.00 am in Greenwich Civil Time. Figure 9 below shows (in orange) the analemmas for 8.00 am, 9.00 am and 10.00 am in Greenwich Civil Time.

Conclusion
The following considerations can be made about the light effects which are graphically displayed in section "Results".
Considerations about the commonly known sunlight effects: (a) On the 11th of November (St.Martin of Tours) at 8.33 am C L is projected on the vertical straight line passing through the center of R P ; the distance between p L and C P is 13.05 meters.Similarly, on the 2nd of February (Candlemas Day) at 9.03 am C L is projected on the vertical straight line passing through the center of R P ; the distance between p L and C P is 14.19 m.Thus, the 'Eight Effect' is more clearly visible on St. Martin's Day (cf.Fig. 6).(b) On 2nd February at 9.03 am the projection of C L on the plane is exactly at half the distance between the cathedral's floor level and the height of the topmost point in the edge of the rose window R P (cf.Fig. 6).This middle point lies at a height of 20.61 meters.(c) The paths T 21∕12 , T 25∕12 , T 8∕12 and T 6∕1 cross the stained glasses of the western rose window between 8.00 am and 8.30 am.Due to this, the sun rays passing through R L are projected onto R P , illuminating it from the inside of the cathedral.This effect is strongest on Winter Solstice Day at 8:17 am, when the distance between C P and the path T 21∕12 is the shortest (1.31 meters) (cf. and 8).
Considerations about other sunlight effects which are not commonly known: (d) On the 1st and 2nd of November at 8.45 am C L is projected onto the ornamental element located on top of the frieze which crowns the main entrance porch to the cathedral.The mean distance between both projections p L and this element is 0.89 meters (cf.Fig. 8).(e) On 17th January and on 20th January at 9.03 am the projection of C L on the plane is positioned exactly at two thirds of the distance between the cathedral's floor  With regard to the projections of the eastern rose window R L on the cathedral's floor plane , no significant relationship can be observed between the solar paths and the architectural elements.However, it should be noted that between March and June, the projection of this rose window illuminates a larger area of pavement than during the rest of the year.Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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Fig. 1
Fig. 1 Sunlight effect which occurs every 2nd February inside the cathedral.Images: courtesy of Mallorca Cathedral (a) Accurate calculation of the moments in time when the projection of the eastern rose window is positioned underneath the western rose window on 2nd February and 11th November each year.(b) Accurate determination of the day and the moment in time when the projection of the eastern rose window comes closest to the center of the western rose window.(c) Daily path of the eastern rose window center projection on the inner side of the cathedral's main façade during certain religious festivities.Specifically: Epiphany (6th January), Candlemas (2nd February), All Saints' Day (1st November), All Souls Day (2nd November), St. Martin's Day (11th November), Solemnity of the Immaculate Conception (8th December), Winter Solstice Day (21st December) and Christmas Day (25th December).We have also considered the following traditional festivities in the island of Mallorca: St. Anthony (17th January) and St. Sebastian (20th January).(d) As a novelty, particular position of the eastern rose window's projection on the inner side of the cathedral's main façade on St. Luke's feast day (18th October) and St. Alexander's feast day (26th February).

•
Distance between the eastern rose window R L and the inner side of the cathedral's western façade : L = 75.954m. • Height to the center C L of the eastern rose window R L and height to the center C P of the western rose window R P with respect to the central nave's floor level: H L = 35.313m and H P = 34.715m. • Distance between O L , which is the orthogonal projection of C L on the wall , and the vertical axis containing C P : D = 0.141 m.

Fig. 6
Fig. 6 Graphical representation of T 11∕11 and T 2∕2 on the plane

Fig. 9
Fig. 9 Graphical representation of L 19∕3 , L e , L 25∕3 , L 8∕9 , L 15∕8 , L 31∕5 and L 21∕6 on the cathedral's floor plane calculated in sexagesimal degrees.(i) E m , the time equation: E m = 4E , calculated in minutes of time.We know that |E m | < 20 m , i.e., the equation of time is always less than 20 minutes in absolute value.Therefore, if the calculation of |E m | renders a value larger than 20, then we will add or substract a convenient multiple of 24 h .

Table 1
Paths T i∕j and L i∕j calculated for the above dates of the year 2022, Julian Day at 12:00 h