Geometry in Art; John the Baptist / © 1974 -2006 Franz Gnaedinger, Zurich,, fg(a), fgn(a) / provisional version in freestyle English


gia01.JPG  gia02.GIF  gia03.GIF // kouros1a.JPG  kouros1b.JPG  kouros1c.JPG  kouros1d.JPG  kouros2a.JPG  kouros2b.JPG  kouros2c.JPG  kouros2d.JPG  kouros2e.JPG  kouros3a.JPG  kouros3b.JPG  kouros3c.JPG  kouros3d.JPG  kouros3e.JPG  kouros3f.JPG


gia04.JPG  gia05.GIF  gia06.JPG  gia07.JPG  gia08.JPG  gia09.JPG  gia10.GIF  gia11.JPG  gia12.JPG  gia13.JPG  gia13a.GIF  gia14.GIF  gia15.GIF  gia16.JPG  gia17.JPG  gia18.JPG  gia19.JPG  gia19a.JPG  gia19b.JPG  gia20.GIF  gia21.JPG  gia22.JPG  gia23.JPG  gia24.GIF  gia25.JPG  gia26.JPG  gia27.JPG  gia28.JPG  gia29.JPG  gia30.JPG  gia31.JPG  gia32.JPG  gia33.JPG  gia34.JPG  gia35.JPG  gia36.JPG  gia37.JPG  gia37a.JPG  gia38.JPG  gia39.JPG  gia40.GIF  gia41.JPG  gia42.JPG  gia43.GIF  gia44.GIF  gia45.GIF  gia46.GIF  gia47.JPG  gia48.GIF  gia49.JPG  gia50.GIF  gia51.JPG  gia52.GIF  gia53.GIF  gia54.JPG  gia55.JPG  gia56.JPG  gia57.JPG  gia58.JPG  gia59.JPG  gia60.JPG  gia61.JPG  gia62.JPG  gia63.JPG  gia64.JPG  gia65.JPG  gia66.JPG  gia67.GIF  gia68.GIF  gia69.GIF  gia70.GIF  gia71.JPG  gia72.JPG  gia73.JPG  gia74.JPG  gia75.JPG


(By the end of 2005 the British company Fulcrum TV contacted me on behalf of a documentary on a London picture attributed to Leonardo da Vinci: John the Baptist and Christ as Infants, dubbed The Lost Leonardo. I answered their fourty-four e-mails, patiently, at length, taking all my time, and compiled a CD for them. In January 2006 I got two panettone from Italy, then – nothing. Complete silence. Did a professor intervene? Here you are with the material of my CD for Fulcrum TV, improved and enlarged. But still in my own personal freestyle English, on which I hold a copyright, so please no one dare copy my mistakes  ;-)




Geometry in Greek and Renaissance Art


Poseidon from Cape Artemision, Athens:  gia01.JPG / Geometry:  gia02.GIF  gia03.GIF / Text:  poseidon.htm // Kouros from Tenea  kouros1a.JPG  kouros1b.JPG  kouros1c.JPG  kouros1d.JPG // Getty Kouros, in my opinion the work of a young pupil of the master of the kouros from Tenea (photograph courtesy J. Paul Getty Museum)  kouros2a.JPG  kouros2b.JPG  kouros2c.JPG  kouros2d.JPG  kouros2e.JPG // Kroisos from Anavyssos, in my opinion a work of the mature master of the Getty Kouros  kouros3a.JPG  kouros3b.JPG  kouros3c.JPG  kouros3d.JPG  kouros3e.JPG  kouros3f.JPG


Leonardo da Vinci, Neptune and Seahorses, Windsor:  gia04.JPG / The geometry of this drawing is based on the golden section:  gia05.GIF / The red circle holds a dramatic group:  gia06.JPG / Concentric circles reveal the dynamic of this composition:  gia07.JPG / The drawing may have been a cartoon for a mural in the saloon of Agostino Chigi’s villa in Rome:  gia08.JPG


Raphael, Galatea, Rome:  gia09.JPG / Geometry:  gia10.GIF  gia11.JPG  gia12.JPG  gia13.JPG  gia13a.GIF / The fresco may originally have been planned as a mural for the saloon of the summer Villa of Agostino Chigi, called Farnesina:  gia14.GIF


Leonardo da Vinci, Isabella d’Este, Paris / Leonardo, drawing of a woman’s profile in a circle:  gia15.GIF / Isabella d’Este, circle of the head, arc of the shoulders:  gia16.JPG / Reconstructing the original format 4:3, grid 4 by 3 large or 8 by 6 small units. The diameters of the circles and of the arc measure 1, 2, 3 and 4 units:  gia17.JPG  gia18.JPG


Piero della Francesca, Baptism of Christ, London:  gia19.JPG / Geometry, a square grid of 12 by 12 units, an upper circle, a lower circle, a pyramid ascending to the center of the upper circle:  gia20.GIF / Geometry applied to the picture. The center of the upper circle, zenith of the lower circle, and top of the pyramid, is marked by the dove’s beak:  gia21.JPG  gia22.JPG (An aside. Kenneth Clark, B.A.R. Carter and James Elkins tried in vain to find a geometry underlying Piero’s painting, and so they concluded there is no surface geometry in Renaissance art, which, unfortunately, has become an almost insurmountable dogma of art history:  gia19a.JPG  gia19b.JPG)


Andrea del Verrocchio and Leonardo da Vinci, Baptism of Christ, Florence, in the original format of a square:  gia23.JPG / Geometry:  gia24.GIF


Leonardo da Vinci, Burlington House Cartoon, London / Circle of arms:  gia25.JPG / Reconstructing the original format:  gia26.JPG  gia27.JPG


Raphael, Madonna with Child, a drawing, and the painting in Zurich / Drawing, based on a lovely and simple geometry:  gia28.JPG  gia29.JPG  gia30.JPG  gia31.JPG / The painting in Zurich is based on a more elaborate and complex geometry; here just the painting:  gia32.JPG


Leonardo, Madonna of the Rocks, first version, Paris:  gia33.JPG / Thomas Brachert found that height and width of this charming painting by the young master Leonardo stay in the golden ratio. The top of the head of Mary, and the top of the head of the angel divide the height in the same ratio. Starting from this insight, I added a grid of the golden ratio:  gia34.JPG


Raphael, La belle jardinière, Paris:  gia35.JPG / Geometry, format 3:2, golden grid:  gia36.JPG


Leonardo, Madonna of the Rocks, replica, London:  gia37.JPG / The figures are closer together and slightly bigger than in the first version. They form a group that is held together by a circle. Its area halves the large square, whose upper side marks the height of the start of Mary’s part:  gia38.JPG


Leonardo, John the Baptist, Paris:  gia39.JPG / Original format 4:3, grid 4 by 3 large units, or 8 by 6 small units. The axis of the eyes lies on the half-diagonal that joins the middle of the left margin with the upper right corner. The axis of the face lies on the diagonal that joins the bottom right corner with the upper left corner. The half-diagonal and the diagonal stand perpendicular to each other. Here just these two lines in the grid of 8 by 6 units:  gia40.GIF


John the Baptist and Christ as Infants, London  gia41.JPG / There are several versions of this motif by Leonardo. Only the one in London contains a geometry, which, moreover, is precisely observed. I examined the geometry of the picture in 1984, then again in December 2005, this time on the basis of a large digital photograph, which allows me to confirm the basic features of the geometry found in 1984. Some shapes fit even better, while some proportions from 1984 have to be modified. / Here you see the painting in the original format 6:5, a musical format:  gia42.JPG / The grid of the painting measures 6 by 5 large units, or 12 by 10 small units:  gia43.GIF  gia44.GIF / Large central circle, diameter 5 large units, radius 5 small units, periphery marked by twelve points of the grid 12 by 10:  gia45.GIF  gia46.GIF / Here you see the circle applied to the picture. John places his right forefinger on the breast of Christ. The center of the circle, marked by a red dot, is just above the foremost joint of the right forefinger:  gia47.JPG / Diagonals of the grid 12 by 10 explain the position of two feet:  gia48.GIF / Diagonals applied to the painting:  gia49.JPG / Large triangle of arcs, standing inside the large central circle. The lower corners are given by two points of the grid 12 by 10. The radii of the arcs measure 4 large or 8 small units each:  gia50.GIF / Large central circle and triangle of arcs applied to the painting. The upper corner of the round triangle is marked by a small overhang of the rock:  gia51.JPG / Small circle of the arms. Its radius measures 1 large unit. Its center divides the width of the original format into 2 plus 3 large units, and the height in a ratio that involves the golden number g = 0.6180339… The ratio of the shorter to the longer part of the height equals (2g + 1) / (2g + 2) = 0.690983...:  gia52.GIF  gia53.GIF / Large central circle and small circle of the arms, applied to the painting. The center of the small circle is given by the lips of Christ:  gia54.JPG / The golden section plays a crucial role in several above compositions. In the case of Poseidon from Cape Artemision the height of the figure and diameter of the circle measures 1 Ionic fathom, 36 large units or 144 small units. The nipples and glans divide the height of the figure in the golden section. Height of head 144, of the nipples 89, of the glans 55 small units. These numbers belong to a golden number sequence that is named for Leonardo Fibonacci but was already known in antiquity: 1 1 2 3 5 8 13 21 34 55 89 144 … The grid of Leonardo’s Neptune with Four Seahorses measures G+2+G+G+2+G by G+2+2+G units, the ratio of height to length equals G + 2 / 2G + 2 (yielding 0.690983…). The reins held by Poseidon and the bared teeth of the rearing horse divide height and length of the original format in the golden section. In the case of Raphael’s Galatea the top of the woman’s head divides the height of the format in the golden section. In the case of Leonardo’s Burlington House Cartoon the right eye of John and the vertical line of his cheek divide height and width of the original format in the golden section. In the case of Raphael’s study for a Madonna there is a hidden cross of four lines which divide width and height of the original format in the golden section. Leonardo’s first version of the Madonna of the Rocks and Raphael’s Belle jardinière are ruled by the golden ratio. And here, in the London picture of the two infants embracing, the eye of Christ divides the height of the original format in the golden ratio, while his member and the tip of his forefinger placed on John’s throat divide the width of the original format in the golden ratio:  gia55.JPG / The geometry of the painting goes along with its meaning, as you shall see in the next chapter. / Conclusion. The large digital photograph in the tiff format, here rendered in jpeg, allowed me to re-adjust the geometry I discovered in 1984. The longer I work on this photograph, the more I admire the lovely plants: ivy, bright anemones beneath Christ’s feet, dark violets beneath John’s feet, and further plants surrounding the boys. I see the master’s hand in it all. However, the painting may be unfinished. When comparing it with the fist version of the Madonna of the Rocks one may miss a final shading of the children. Another sign of the unfinished state may be the missing ivy strings. The ivy leaves hover like butterflies – and thus appear on the lovelier to me:  gia56.JPG




Art, Man’s Creation, and Nature, God’s Creation


Andrea del Verrocchio and Leonardo, Baptism of Christ, Florence:  gia23.JPG  gia23a.JPG / In this painting by Andrea del Verrocchio, young Leonardo painted his first figure, the angel on the left side, and the waterfalls above him. Andrea was so amazed by the skills of his pupil that he allegedly never again carried out a painting, leaving that kind of work to his most talented pupil. / Strangely, the angel looks up to John, while he should look up to Christ in the center of the picture. How come? Have a closer look at the angel:  gia23a.JPG He observes John quite critically. In a similar way Leonardo may have observed his teacher Andrea: critically, already feeling superior, and it seems that the kneeling angel is about to raise and stand up …


May it be that Leonardo saw himself in the angel, and his teacher in John the Baptist? Yes, it may well have been so, not only on the psychological but also on the symbolical or philosophical level. John the Baptist announced the arrival of Christ: He that cometh after me is mightier than I (Gospel according to Matthew, Chapter 3, 11). Art is humankind’s noblest creation, and the task of art, Leonardo believed, is to praise nature, God’s creation. The painter who praises nature in his work conveys a similar message: John the Baptist had been telling his followers that the one who comes after him is mightier than he, and in a similar way nature is mightier than my art --- more splendid and complete than any work by any human being can ever be …


Leonardo, Madonna of the Rocks, first version, Paris:  gia33.JPG / The infant John has found the entrance into a cave, where, to his marvel, he finds Mary, an angel, and the infant Christ. Mary seizes little John, draws him near and presses him on his knees, while blessing her son, who, on his turn, is looking at John and blessing him by raising his hand. / Leonardo, meanwhile a master of the fine arts, having opened a workshop of his own, sees himself in young John. As a boy he had discovered the entrance to a cave, and now he shows himself via his alter ego John the Baptist entering a cave. He feels attracted by the beauty, fertility and secret of nature, symbolized by Mary, and by the very essence of nature, symbolized by Jesus. / Why does the angel point toward John? While Mary takes up little John into the sacred circle of the holy group, the angel looks at us and points toward John, symbol of young Leonardo the painter, who carries out his work with his hand: eye to eye, hand to hand … While Mary symbolizes the beauty of nature, the angel symbolizes the beauty of art, and while Mary is taking up John into her sacred circle, the angel, silently smiling at us, invites us to join the very same circle that is still open to the foreground …


John the Baptist and Christ as Infants, London:  gia41.JPG  gia42.JPG / Here you see little John on the left side, embraced and kissed by little Christ in the center. Leonardo, present in John, is feeling favored by nature represented by Jesus. However, there is a fine symbolism that places nature above art, Jesus above John. Christ appears in the middle, John on the side. You can see the member of little Christ, while the one of John is hidden, meaning that nature is fertile, while art is sterile – a painted anemone may be as pretty as can be, yet it is just painted, sterile, there will be no seeds that spread and grow new flowers. Have a closer look at the left hand of Christ: he places his forefinger on the throat of John, thus indicating that art can make nature speak. Which is the very task of art: in itself sterile, yet able to praise nature. While Christ places his forefinger on John’s throat, John places his forefinger on the breast of Christ, indicating nature’s being, nature’s heart. Nature is, while art praises nature using a visual language … / If you despise painting, Leonardo wrote, you despise a fine piece of philosophy that considers all qualities of nature.


Leonardo, Burlington House Cartoon, London:  gia27.JPG / In the Burlington House Cartoon John has grown, as Leonardo has advanced in his mastership. We see him being taught by little Jesus, hence by nature in the arms of Mary, symbol of beauty, fertility and the secret of nature.


Leonardo, Madonna of the Rocks, replica, London:  gia37.JPG / When painting his replica of the Madonna of the Rocks, Leonardo, a mature master on the zenith of his career, could no longer render himself as a child, and so all that remains of the juvenile masterwork is the religious shell, which is why conventional nimbi appear in the picture. The figures are larger, they form a self-sufficient circle, we are no longer invited to join them, the angel does not longer look at us. He looks over to little John, critically, yet with a secret pleasure ––– much in the way Leonardo may have looked at himself when painting his replica and remembering the former days when he was a young master, and so this painting is not just a replica, it has become a new testimony of how Leonardo felt as a painter … This angel is showing one of the prettiest faces Leonardo ever painted:  gia37a.JPG


Leonardo, John the Baptist and Nymph, missing / Windsor:  gia59.JPG / John the Baptist arrives at the bank of a river, with one hand he points over his left shoulder, with the other one down into the ground. This drawing was kept in Bergamo, but is missing since World War II:  gia57.JPG / A nymph stands or rather hovers over a river bank, smiling at us in an ethereal, almost unearthly way, before a mysterious landscape with big flowers, and pointing to the right side:  gia58.JPG / The two drawings may have been small cartoons for a pair of murals on the opposite walls of the former audience room in the Belvedere of the Vatican, where Leonardo lived when at Rome. The two murals on opposite walls could have represented John the Baptist, hence Leonardo as painter, nearing the end of his career, and having a glimpse at the beyond, the nymph standing on the other side of the river. When seen from the side, in passing the gangway, there is a remarkable optical illusion: John is now pointing across the room, and the nymph into the depth of her mysterious landscape:  gia59.JPG / We know little of what Leonardo did when in Rome, just that he got a commission by the Pope, whereupon he, Leonardo, went to his laboratory and experimented with a new varnish. The Pope was upset by Leonardo’s reaction and exclaimed: This man will never finish anything, he starts with the end instead of the begin! Leonardo should have drawn sketches, instead he experimented with a varnish, the last layer to be added to a finished painting. But it makes sense to experiment with a varnish if you wish to paint the beyond and hide it behind a secret mist achieved by means of a special varnish, and this would have been the last example of the famous sfumato invented by Leonardo …


Leonardo, John the Baptist, Paris:  gia39.JPG / In his last painting, Leonardo shows John the Baptist as a grown up man, before a dark background. He smiles at us, his eyes are overshadowed, yet his front is bright and clear. With his right hand he points toward the sky; with his left hand – consider that Leonardo was a left-hander – to his own breast. While the right arm is completely visible and the hand very bright, the left arm is only partly seen, and the hand in the shadow. John looks at us as if he wishes to tell us something, yet as a figure in a painting he can’t really speak, so he speaks to us in the way Italians do, by means of his hands and gestures:


   God created the world,

   whereas I, Leonardo,

    born into God’s world,

    now soon departing,

    have seen his work.

   Much of what I saw,

   did I study, draw, describe,

   and paint. Yet my work is

   incomplete, and only a shadow

   of nature, God’s brilliant creation.

   And if you lose a great artist in me,

   don’t be sad; just look at nature,

   consider the work of the greatest

   artist, the splendid creation by God.


As an apprentice, Leonardo painted the angel in Andrea del Verrocchio’s Baptism of Christ. Now, at the end of his career, he paints John. Yet the scene of the last painting, announcing Christ, immediately precedes the one of the first painting, baptizing Christ. Thus, Leonardo closed the circle, and also, if you like, considered the possibility of a new one …




Leonardo, Raphael and Giorgione


When Leonardo was painting his replica of the Madonna of the Rocks he might have been teaching Raphael and Giorgione, his informal pupils, telling them what he had in mind when he had been painting his first version of the Madonna of the Rocks. Both painters would then have honored their teacher: Raphael by painting his mural School of Athens in the Vatican, which made his fame in Rome, and Giorgione by painting his 3 Philosophers.


Raphael, School of Athens, Rome:  gia72.JPG  gia73.JPG  gia74.JPG  gia75.JPG / The passageway in the center of the painted architecture has a similar form as the frame of the Madonna of the Rocks: a standing rectangle topped by an arch. The main figure, Plato, shows the features of Leonardo. Plato/Leonardo raises his right hand in a similar way John the Baptist does, and in his left hand he holds Plato’s Timaios, the famous dialogue wherein the Greek philosopher explains, among other things, that all matter is composed of geometrical shapes that are ruled by numbers. Meanwhile Aristotle, by the side of Plato, is holding his right hand in the same way Mary is holding her left hand in both versions of the Madonna of the Rocks:  gia73.JPG / In Leonardo’s juvenile masterwork there is an angel at the right side, looking out of the picture, inviting us to join the holy group:  gia33.JPG / At the right margin of the School of Athens Raphael painted himself: looking out toward us, inviting us to join the school of philosophers, poets, singers, painters, sculptors and architects … He has an angelic face, and resembles the angel in Leonardo’s painting, even more Leonardo’s study for that angel:  gia74.JPG / Giorgione’s enigmatic painting of the 3 Philosophers may be a homage to the hypothetical lessons Leonardo gave him and Raphael at Florence, telling them what his first version of the Madonna of the Rocks really means, and about his discovery of a cave when a boy … Now we see an entrance to a cave on the left side, and three philosophers on the other side, from left to right: Giorgione, sitting, looking toward the entrance of the cave; Raphael, standing, in the middle; Leonardo, standing, on the right side, looking and pointing toward the entrance of the cave, holding a pair of compasses and a sheet with geometrical / astronomical drawings:  gia75.JPG




Similar geometries


The drawings of John the Baptist and the Nymph by Leonardo, and of a Venus by Raphael, may show the central figures of a planned mural each, and composition is based on a similar geometry containing a standing ellipse of the format 4:3 in the center. The upper focus of each ellipse is marked by an eye, while the height of the lower focus is marked by a foot:  gia57.JPG  gia58.JPG  gia59.JPG  gia60.JPG  gia61.JPG  gia62.JPG  gia63.JPG  gia64.JPG  gia67.GIF  gia68.GIF  gia69.GIF  gia70.GIF / Raphael, study of figures in a standing ellipse:  gia13a.GIF / On the ceiling of the entrance hall of Agostino Chigi’s Villa, the so-called Farnesina, are two frescoes painted by Raphael’s school. In the center of the right painting you see a young man who strongly resembles John the Baptist in the above cartoon by Leonardo:  gia65.JPG / In the center of the left fresco appears a woman who is pointing toward the right side. If you look at her form the right side, however, she is pointing across the table – the same optical illusion we encountered with Leonardo’s Nymph:  gia66.JPG  gia58.JPG  gia59.JPG / Raphael’s Venus may have been the central piece of a cartoon for a fresco in the enlarged saloon of Agostino Chigi’s villa (Farnesina), however, it was not to be. First, Raphael would have had to give up his plan of painting the Galatea in the saloon – out of whatever reason –, and so he painted a part of his composition in the (then open) garden loggia. Now he would have had to give up to paint his Venus fresco in the enlarged saloon – out of whatever reason –, and instead he painted his a charming Venus cycle in the entrance hall, under the long frescoes that show the pointing young man and woman in the respective center, presumably by a pupil after drawings by the master. Here one of the Venus panels by Raphael himself:  gia71.JPG


Geometry in Renaissance paintings can be used for restoring original formats, attributing a composition to an artist, finding out an original composition among different versions of the same motif, and, in some cases, for interpreting a picture.