The twelve disciples of Jesus are arranged in four groups of three people each. Jesus is at the center of the painting. His head and hands form an equilateral triangle. In this way, Leonardo alludes to mathematics and geometry: the number 12 results from 2 sides (left and right) × 2 groups × 3 people per group.
The equilateral triangle around Jesus is perceived as especially harmonious. Although the sides of the triangle are more or less precisely defined by Jesus’ head and arms, the position of the base of the triangle is unclear. It may run along the upper edge of the table, or it may be defined by the hands of Jesus (mouseover). In that case, the equilateral triangle would be the two-dimensional image of a three-dimensional triangle projecting into space, with its base at the hands of Jesus and its apex at the doorway in the back wall.
Plato describes how two-dimensional geometric figures outline a three-dimensional form in his “Allegory of the Cave.” In a famous contemporary fresco by Raphael, Leonardo himself was portrayed as Plato (“The School of Athens,” see below). It is located in the Vatican, in the former private apartments of the pope
The painting can be divided into twelve equal squares because it was composed in a 4:3 format. Of particular note is the striking marker at the lower right edge of the picture (lower light-blue dot). Its upper end marks the lower boundary of the grid of 3 × 4 squares — the painting’s aspect ratio. Within the surrounding quadrant, it is just as far from the edge as a visually very similar structure above and to the left of the doorway (blue lines, see “Detail Views”).
It is often overlooked that the three lunettes above the painting also belong to the work (yellow area). They display the coats of arms of the family of the Duke of Milan who commissioned the painting.
The corners of the interior walls of the refectory divide the work horizontally into three equal sections (mouseover). The resulting T-shaped pattern (yellow area) is known in Christianity as the Tau cross or Saint Anthony’s cross. It alludes to the equally sized Crucifixion painting on the opposite wall of the room, in which Jesus and the two thieves are nailed to just such a Saint Anthony’s cross (lower illustration)
The upper three lunettes are laid out in a 2:3 ratio (black and white lines at the top), creating seven equal units. The vanishing point of the perspective lines is located at Jesus’ right temple, not at his eye as is often claimed (yellow lines). At that spot there is still the trace of a nail that Leonardo used to stretch the strings.
The perspective lines of the windows are the only ones that are not axis-symmetric (orange lines). Their angles are 15° and 4° — Leonardo was born on April 15. Brunelleschi died on April 15 (1446), although it cannot be ruled out that the date was later aligned with Leonardo’s birthday to symbolically emphasize the continuity of Florentine creative power. The famous Florentine architect and engineer Filippo Brunelleschi influenced Leonardo in many ways. Among other things, he built the largest dome in the world for many centuries for Florence Cathedral. Leonardo was probably involved in the finishing work as a pupil of his teacher Verrocchio, who made the gilded bronze sphere on top of the dome.
The lower perspective lines divide the bottom edge of the painting into seven equal sections (black and white lines). When hovered over, the origins of the perspective lines are easier to see. The lower doorway was cut in 1652, partly destroying the painting. As a result, two of the six ornamental lines on the floor are no longer visible, but they are known from early copies
Shown is the reconstruction of the pictorial space. The room is twice as long as it is wide. The windows of the back wall are divided into two squares by the horizon line. The doorway is divided vertically by the horizon at the golden ratio. On the ceiling there are wooden beams forming 72 squares. This refers to the sending out of the 72 disciples by Jesus (Luke 10:1). They are arranged in twelve rows of six
Leonardo also paid attention to harmonious proportions in designing the side walls. The rows of windows on the left side are positioned higher than those on the right. The striking dark areas are wall hangings. Their upper edge is twice as far from the upper edge of the windows on the right side as it is on the left side (black and white tiles). The lower edges of the windows are hidden by the figures, but with respect to the windows of the back wall it can be assumed that they too are based on a square form (yellow frames). From the consistent pattern of quartering, the height of the wall hangings can also be inferred.
The tablecloth is likewise divided into four parts. It is horizontally divided by 16 folds (black and white tiles). The spacing of the folds corresponds quite closely to the height of the tablecloth. The width of the tablecloth equals four fold-spacings. In this way, the tablecloth can also be seen as 16 cubes arranged in four successive rows.
Overall, the impression arises that Leonardo wanted to emphasize the three-dimensional tablecloth, moving from the vertically four-sectioned window rows (1-D), through the two-dimensional wall surfaces, to the tablecloth in 3-D. If one adds the squares of the window rows (4 × 6), the four-part blue areas of the side walls (4 × 8), and the cubes of the tablecloth, the total again comes to 72
The schematic reconstruction of the pictorial space
That Leonardo places Jesus in front of the doorway alludes to a parable in the New Testament: “I am the door. Whoever enters through me will be saved; they will come in and go out and find pasture.” (John 10:9)
In addition, there are further geometric relationships. If the round arch above the door is drawn, its center lies at the vanishing point (yellow dot). This circle touches exactly the left and right windows as well as the lower edge of the doorway (yellow circle). It also divides the wall behind it vertically into three equal parts (white lines).
The outer frame of the doorway is laid out in a 3:4 ratio and is divided vertically at the golden ratio. The inner frame of the doorway is a golden rectangle, meaning its width and height are in the golden ratio. Furthermore, the horizon line divides the inner frame of the golden rectangle vertically into three equal parts (white vertical line). Even at this point, the geometric construction is extraordinarily artful.
Leonardo’s universal spirit is especially evident here. The upper edge of the doorway divides the wall behind it exactly at mid-height (middle white horizontal line). The tripartite height of the inner doorframe can be mirrored upward along the white horizontal line. The height of the back wall is then divided into six equal segments (black and white dots). Between the horizontal mirror axis and the upper end of the decorative arch, the segment can be divided exactly in the middle (blue dot) and once more between the lower end of the arch and the mirror axis (orange dot). In this way, nine points can be identified that, with reference to the arch around the door (yellow outer circle), can be understood as radii around the vanishing point (black circles).
This creates the impression of an astronomical sketch. That impression is reinforced by the outer circle, which is missing about 10% to the left and right before reaching the end of the wall (blue areas). Ten percent corresponds to the average difference between the farthest and closest distances of the planets in our solar system from the sun. The planetary orbits are therefore not circular but elliptical (mouseover). Leonardo draws six circles. Until 1781, only the first six inner planets visible to the naked eye had been known since antiquity; the currently known eight planets were discovered thanks to more powerful telescopes. Leonardo developed a reflecting telescope a hundred years before Galileo, with a diameter of several meters, begun no later than 1513 in Rome but never completed. It is unclear whether Leonardo possessed a smaller prototype at the time that might have allowed him novel observations of the heavens
The astronomical sketch from VII can be extended downward by lowering the floor to reveal the lower ends of the circles hidden in VII. Remarkably, this produces precise overlaps of the two planes in the floor area (blue and yellow line on the floor). If the space is divided in depth into four equal sections — each side wall has four distinctive wall hangings — it becomes apparent that the two partially hidden circles from VII rest exactly on the sections of the quartered floor (lower edge of the blue and yellow circles). However, two of these lines are unoccupied.
Two additional circles can reasonably be added. The first is directly in the foreground. Its lower edge is defined by the quartering of the floor. The upper edge of the circle borders exactly on the line at the center of the pictorial space (yellow line at the top). The lower edge of another circle is likewise defined by the quartering of the floor (lower orange dot). The radius of this circle is confirmed by a second point already known from I — a strange shape above the door on the left, which is also repeated in the lower right foreground of the painting.
That this point was chosen deliberately becomes clear when the points of the 5th, 6th, and 7th rings are connected (mouseover). This creates a triangle with interior angles of 108°, 54°, and 18°. The numbers are harmonious: 108° ÷ 2 = 54° and 54° ÷ 3 = 18°. At the same time, all three angles relate to the regular pentagon. In this way Leonardo has masterfully constructed eight circles. This is all the more remarkable as it connects different dimensions (1-D, 2-D, and 3-D)
If the floor has been lowered, the ceiling could likewise be raised. When it is raised by the height of the tablecloth, the half side walls and the back wall change their proportions to a harmonious 4:3 (blue and yellow areas). The side walls as a whole are now 3:8 high — Fibonacci numbers and thus close to the golden ratio (orange lines). All the walls of the room are now in harmonious proportions. The back wall also has the same aspect ratio as the painting itself.
Leonardo worked on exactly two wall paintings in total: The Last Supper and The Battle of Anghiari. The latter was also designed in a 4:3 format and was intended to be as monumental as The Last Supper. When the shape of the decoration of the side walls is taken into account, especially through the wall hangings, the impression of a gallery with open wall space emerges.
If Leonardo constructed the back wall to match The Last Supper, it is reasonable to think he might also have fit his other unquestionably authentic paintings into the same scheme. Arranged by date of creation, The Last Supper appears as the central work chronologically as well. A pattern in the sizes also becomes apparent: a large work was followed by two smaller ones. All paintings are shown in their original size ratios, taking The Last Supper as the largest work. Six paintings are displayed to the left and right of The Last Supper. Two unfinished works — Saint Jerome and The Adoration of the Magi — do not yet fit the scheme (mouseover). The Battle of Anghiari was probably walled over unfinished and has since been lost, though one study has survived
The grand finale – coming soon