Geometry in Art; John the Baptist / © 1974 -2006 Franz
Gnaedinger, Zurich, www.seshat.ch, fg(a)seshat.ch, fgn(a)bluemail.ch /
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
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,
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,
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
Leonardo, Madonna of
the Rocks, first version,
Raphael, La belle
jardinière,
Leonardo, Madonna of
the Rocks, replica,
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,
John the Baptist and
Christ as Infants,
Leonardo,
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,
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.
