Spatial Analysis of Leonardo's Last Supper
We don't need to have any doubts about the size of Leonardo's extraordinary spirit. Although complete body his work is not known, we must always admire how he managed to excel in so many different areas. Many of the Leonardo's essential works were created in Milan at the court of Lodovico Sforza also known as Lodovico Il Moro. Lodovico was Leonardo's contemporary and a perfect patron for the great artist. He gave Leonardo the opportunity to employ his creativity not only as a painter, but as an engineer, architect and military inventor. Around 1495 he received a commission from Storza to decorate Dominican monastery Santa Maria delle Grazie and Leonardo di ser Piero da Vinci set about what would become one the most powerful works of art of western culture -The Last Supper.
Leonardo described himself as a man without education. His great advantage was that he did not rely on scholarly books and ignored the authority. Anything that interested him, he probed thoroughly by himself and because he was not to be influenced by previous authors his results were far-reaching. Leonardo was preferring empiric experience, based on a proper and versatile examination of nature's hidden mysteries. For Leonardo there was no higher authority than that of his own eye, which he characterized as the window of the soul. His pragmatic mind was telling him that science and art was very closely interrelated, because both were empirically accessible to the eye.
At that time, activities associated with manual work, painting including, were considered inferior and Leonardo did as much as he could to elevate his beloved art of painting to become the respectable, liberal discipline. Other mental artistic disciplines such as logic, rhetoric, grammar and geometry belonged to the liberal arts and such activities were deemed worthy. Men such as Leonardo and later Raphael proved that art of painting belongs to the liberal arts and manual work it carries is as necessary as writing in the literature. The contribution of logic and geometry caused that aristocracy's attitude towards the art of painting began to change.
The secret perspective system
There are not many images from which we can decipher if Leonardo knew the secret perspective system. In his early painting The Annunciation he works quite creatively with space, however, he still uses linear perspective. All orthogonal lines are properly align to the vanishing point, but the lateral point are too close. The tiles in the foreground thus bends too much and it looks like the Virgin Mary must hold the lectern in order not to slide down the slant.
Regarding the Last Supper analysis show completely different level of space arrangement. The mural is painted on the end wall of the refectory of Monastery of Santa Maria delle Grazie and I suppose that one of Leonardo's first dilemma was to organize its proportional relationship with respect to the refectory own space. Leonardo created the effect that the room in which Christ and the apostles are seen was an extension of the refectory. Nevertheless, the painted space of the Last Supper appears slightly awkward with the refectory space because the edges of the fictional room overlap and don't align with the corners. It is one of the reasons why to some the room behind Jesus may appear as a trapezoid.
13 figures has to to fit between 15 braccia of the walls. From practical reasons Leonardo had to make the figures proportionally larger and they are approximately 1.6 times larger than real life figures. The table is so long that Bartholomew on one side and Simon on the other side barely squeeze in, but Leonardo managed with grace to fit them into the given space.
The scene depicts the moment when Jesus announces to his companions, that one of them will betray him. Christ's words bring a stir among his comrades. Despite agitated gestures and emotions that his speech caused, there is nothing chaotic in the scene. Everything blends to natural and graceful symphony of movement, which is divided into four groups.
Leonardo utilizes the inherent properties of the number twelve. There are four hangings on each side wall and three windows at the end wall. There are six men on each side of Christ, who seems to be alone and resigned to his fate. Above his head at the ceiling there are six rail divisions. Multiplication of all these numbers gives us sum of twelve. Twelve is very important number in Judaism, Christianity and Islam and is also found in some older religions and myths. There are twelve signs of Zodiac. There are twelve months of the year and twelve hours of the day and of the night. In terms of geometry twelve is a pentagonal number. The dodecahedron, mysterious Platonic solid, has twelve faces. Regular cubes and octahedrons have twelve edges, while icosahedrons have twelve vertices.
Leonardo decided herein for frontal composition, which is fully justified due to the placement of the painting on the refectory's rear wall and efficiently integrated the scene as if it was a part of the spacious room.
The painting's lower edge is about 2.5m high above floor level and the viewer has to look up. There is only part of the floor and no tiles, which would help us to determine the lateral points. Nevertheless, if we know how he constructed the geometrical scheme everything falls together, and space suddenly starts to be tangible.
All orthogonals aims to the vanishing point - the Jesus's temple, which is placed in the middle of the picture. By transferring the area of the picture, we arrive at the base square from which all geometric operation unfolds. Mathematician Luca Pacioli in Divina proportione (which his friend Leonardo illustrated) described this as a technique that subdivides any given scene into a sequence of squares. Not as a flat surface grid, but squares laid perspectivally to the depth. Leonardo used as a basic unit of measurement Milanian braccio and set the side of the basic square at 4 braccios. 4 * 0.59 equals 2.36m. 1 Milanian braccio equaled 0,59 cm.(1)
Let us get back to the figures size and how Leonardo dealt with the problem of proportion. The universal arithmetical tool of literate Italian people in the Renaissance was the Rule of Three, also known as the Golden Rule and the Merchant's Key.(2) Today we would represent the relationship as, for example 8/5 = 12/?. The Rule of Three is in geometric proportion. The first term stands to the second term as the third stands to the fourth, so 8/5 = 12/7.5. To find out dimension Leonardo used for its base square we have to retrace his steps with the help the Rule of Three. We do not know yet the measurement of the table or the tapestries, but we know the average height of an adult at that time was around 168 -170cm or 67inches(3). A person of this height would be around 130cm if seated.169/130 = 1,3. If sitting Simon on the right side of the table measures 207cm(4) than 207/130 = 1.6. By the Rule of Three he would be around 270 cm standing (207 * 1.3). 270/169 gives us again ratio of 1,6. By the same Rule of Three we estimate the width of the room or the base square. We just need to careful measuring the horizontal abscissa exactly where the vertical is positioned. In this case, Simon sits (or stands) right between the first and second square. I do not give an example, because everyone gets a different measures according to their screen set up. However, in case of sitting person the ratio should be in around 4.1 to the width of the room or 1.8 to the side of the base square. 130 cm or 2.2 braccia * 4.1 = 9 braccia. The width of the room is 9 braccia, and the length of the base square is 2.2 braccia * 1.8 = 4 braccia.
To confirm the width of the room we can divide the width of the room by side of the square. This proportionality works anywhere along the orthogonals and its ratio should be around 2.25. Then we multiply side of the square by the ratio of 2.25. Thus 4 braccia * 2.25 equals 9 braccia or 5.31 meters.
The foundation of the whole scheme, the tiers points, which we attain through the geometrical method of the Renaissance perspective system, help us to determine the back side of the foreshortened square. You can consider these squares as lying on the floor and exactly six of them (Leonardo again uses count of six) fit to the end of the room. Now we can determine the length of the room by simple math. Since the floor starts in the middle of the first square we apply the count of 5,5 squares. 5.5 * 4 braccia equals 22 braccia or 5.5 * 2.36 equals 12.98 meters of the floor's length from the edge of the picture to the end of the room.
The tiers point help us to foreshorten not only floor tiles, but also the ceiling coffers. As you can see all ceiling diagonals converge to the tiers points and therefore we can assert that the coffers are squares. Subsequently we can calculate their size by projection method or simply by dividing the room's width 9 braccia by 6 rows. 9/6 = 1.5. The side of a single coffer is 1.5 braccia and 10 of them fits to the upper edge of the base square.
Then by the Rule of Three we can also determine the height of the room.
Room's width/height = 1.5 then 9/1.5 = 6 braccia.
Similarly, we will find out that the tapestry panel on the walls measure 4 braccia. Between them, gaps with the niche measures 1.5 braccia. Leonardo set the height of the tapestry according to another sacred geometry ratio. The ratio 1,414 or square root of two. Thus 6/1,414 equals 4.24 braccia.
To determine the individual objects within the picture such as table is more delicate. Here we are dealing with an elevated area and to find out the value we must proceed by raising the vertical lines from 45° orthogonal to the front and back edge of the table (green dashed lines). Then by projection from the tiers point we can determine the size of the table. Again we apply the Rule of Three. Dividing the front side of the base square by the projected segment give us ratio of 2. Then 4 braccia divided by 2 equals 2 braccia. So the width of the table is 2 braccia or 118 cm. The length of the table is 7.5 braccia or 442cm.
The fact that Leonardo was known for his love of symmetry is evidenced by the overall design of the painting. Notice how the floor is divided into the (now faded) yellow stripes, which are carefully proportioned by the Golden Ratio. The orthogonals aims from the crossing of the circle and the base line to the vanishing point and are an integral part of the sacred geometry. Fig.1
The visual paradox
The main problem lies in overall perception of the space depth. The visual paradox lies in the fact that although the figures are bigger than life (as apostles deserve to be) the space they occupy is smaller than the room we are looking at them from.
Martin Kemp in his book The Science of Art says: "Leonardo's space looks logical but actively resists unequivocal translation into an actual space."2Kemp The most viewers perceives room's space not as deep as it is. This is also caused by the relatively wide viewing angle of little over 47°, but mainly due to the narrow angle 24° of the orthogonal formed by the floor and the wall. The vertically rectangular shape of the image further enhances it. Combination of these two factors distorts the perspectival space with the corresponding effect on foreshortened objects, in this case tapestries. At last, we see only a very little of the floor, rest of it is blocked by the table and our visual and perspectival references are made up mainly by the side walls.
The case of uncut door
Lavinia Mazzucchetti stated that in 1652 the mural was damaged and the crust was loosened when an existing door was worked on. Later, few authors declared that the door was widened and heightened and some even suspected the monk of cutting the whole new doorway through the mural. According to my calculation the door must have been there from the beginning just about the same size as today and Leonardo must have taken it into the count while composing the geometrical scheme of the Last supper. Therefore, even at the mural's creation, neither he nor the monks could see the Jesus' feet.
Leonardo was known for leaving projects unfinished. In many previous cases, his restless mind was already planning another project, but this time he had completed his mission. Although The Last Supper is his only surviving painting containing secret knowledge, he was among the first, if not the very first (that is for further research) to successfully apply the principles of the Renaissance perspective to a work of art.