Here is Gauguin’s "D'où Venons Nous?”. A geometrical analysis will show the origins of the composition and the underlying structure.
The red lines create diagonals across the whole painting. Note that the reclining woman at left leans on the rising diagonal, which also defines the lighter colored ground at left of center, the hand of the central nude with the turned back, and the branching of the tree at top right. The falling diagonal defines the changing ground color at left of center and the small child’s leg and loincloth at the lower right.
A circle filling the picture’s full height at its center sweeps around the sleeve of the larger child in the foreground, past the rear and elbow of the central nude with the turned back, and through the fruit being grabbed by the main figure near the top.
Starting from the left, two squares are made, each inscribed with the “X” of its diagonals. The figures in the middle ground, walking from the left, stop in front of the vertical margins of these squares.
Next, in orange, the same step with the squares is applied, moving from the right this time. In this case, it is the idol that abuts the margin of the square. The outer pair of red and orange verticals divide the composition into a triptych of sorts, particularly in the upper half of the painting. (Note that the steps in this analysis proceed in the spectral order of the rainbow: first red, then orange, and in subsequent slides yellow, green, blue, and violet.)
In the left wing of the “triptych,” the steeply falling yellow diagonal passes through the head and body of the reclining woman, whose bracing arm is defined by the yellow vertical halfway across the left “wing.”
In the upper left half of the painting, the yellow horizontal defining the horizon line passes through the intersection of the red and orange diagonals near the idol.
In the middle panel of the “triptych,” the yellow diagonal falling to the orange vertical defines the sloping feet of the idol, the pink feet of the woman in the middle ground, and the ankle joint of the main figure.
In the right panel of the “triptych,” the dog at right leans its paws on the falling diagonal, while the second woman of the seated group rests her elbow on the rising diagonal.
The navel of the idol can be found by drawing a short green horizontal from the intersection of the orange vertical and the falling red diagonal of the painting until it hits the rising red diagonal of the first square. The green vertical descending from the navel subdivides the idol’s legs, but not its head, which is displaced left.
The bright green vertical stands .707 of the way from the left margin of the painting to the right; this is the ratio of a square’s side to its diagonal, which is an interesting relationship to invoke, since the painting is not square.
A green “X” through the bright green vertical defines a green vertical against which the large child in the foreground reclines. The cats climb the rising diagonal of the “X,” while the falling diagonal defines the tilt of the second woman’s head at right, the placement of her hand, and the head/torso junction of the small child.
In blue, a circle is drawn at the center of the painting, so that it is tangent to the green vertical against which the older child leans. This circle intersects the diagonals reaching out from the painting center in points that define the blue vertical axis of the main figure, which passes through his navel and along the line separating his thighs.
The blue vertical further to the right, tangent to the original red circle, forms a bookend of sorts for the rear of the first woman in the group at right.
In the left half of the painting, the rising violet diagonal to the top center of the painting intersects the descending red diagonal at a point 2/3 of the way up the painting, defining the height of the brown land at left, and 1/3 of the way across the painting, defining the back of the figure near the idol.
In the middle of the painting, violet lines of 60-degree slope descend from the top of the main figure’s axis. The left line passes along his forearm, arriving at a prominent crease in the sleeve of the large child by the cats. The right line passes along the main figure’s other forearm, along the back of the central seated nude, and along the arm and leg of the first woman in the right-hand group.
A violet line ascending from the base of the blue vertical defines the elbow, wrist, and face, respectively, of the three right-hand women. Voila…
Here is Carpaccio’s painting of Saint George and the Dragon.
Add red lines from the left corner at 75, 60, 45, 30, and 15 degrees off horizontal. Note that the steepest one, at left, is an axis around which the dragon’s tail oscillates. The vertical dropped from the top of the 60-degree line passes through the right corner of the square-planned tower in the middle ground. Crucially, the vertical dropped from the top of the 30-degree line passes through the point where the lance intersects its crosspiece.
Take the lower left corner as the center of a very large octagon whose right-hand facet coincides with the red line dropped from the top of the 45-degree line. A line of slope 22.5 degrees from the lower left corner of the painting intersects the red vertical dropped from the 30-degree red line, thus locating the precise point of the lance/crosspiece intersection.
Taking that intersection point on the lance as a center, create two dodecagons; one whose lower facet aligns with the bottom of the painting, and a smaller one whose upper facet aligns with the top of the painting. The lance, with its 15-degree slope, passes through corners of these dodecagons. The tree foliage at the upper right also lies along this axis.
The right margin of the painting may be found by dropping a 45-degree diagonal down from the uppermost left corner of the small dodecagon down to the bottom margin of the painting; this operation thus sets the overall aspect ratio of the composition. A line of 15-degree slope descending to the left from this same upper point passes along the shadows on the rocky hills about a third of the way across the composition. A horizontal through the lance intersection point aligns with the ground level of the hills on the far right.
This painting is the property of the Scuola di San Giorgio degli Schiavoni, Venice.
This analysis was based on a color-enhanced version of the following image:
Here is Charles Bouleau’s analysis of a miniature from the Tres Riches Heures of Jean de Berry, made by the Limbourg Brothers early in the fifteenth century.
Here Adam and Eve are shown in the Garden of Eden, whose circular perimeter was surely drawn with a compass.
More interestingly, Bouleau showed that the height of the Gothic fountain of life in the image could be found by unfolding the diagonal of the square framing the garden’s circular border.
Bouleau also demonstrated that the width of the fountain was set by the intersection of two pentagons inscribed within that circle. In the lower-left of the image, moreover, Adam’s foot locates the corner of the pentagon with the horizontal base.
Bouleau extends that baseline to the right, as shown in green, and then connects it back to the tip of the fountain.
The intersection of this green diagonal with the circle’s equator, he claims, sets the width of the Gothic portal through which the first couple exit, while the blue diagonal to its top sets the sloping pose of the expelled Eve.
This page is folio 25v of the Très Riches Heures du Duc de Berry, which is MS 65 from the Musée Condé in Chantilly, France.
This particular image is from:
Here is the Visitation scene painted by Piero di Cosimo.
The inner field of the painting is a perfect square, flanked by narrow strips on the bottom and sides. The center of the square locates the clasping hands of the Virgin and Elizabeth.
The center of the square also defines a level, identified here with the orange horizontal, that describes the edge of the architectural platforms on which the small figures in the middle ground stand. From the ends of that horizontal, orange lines of 30-degree slope are now launched.
Then, from the points where those orange lines intersect the top of the panel, yellow verticals descend, which frame a yellow circle, in which a yellow square can be inscribed in a quadrature operation. Note that the bottom facet of the square aligns with the back of the platform on which the Virgin and Elizabeth stand.
The front of the platform corresponds to the bottom tip of a green equilateral triangle whose baseline corresponds to the equator of the square. A similar triangle above this equator helps to locate the heads of the two women as they lean towards each other, with their wimples sloped at 60 degrees from the horizontal. Their heads and the Virgin’s halo fit into the almond-shaped figure that circumscribes the paired triangles. One odd detail about this painting is that the horizon line of the seascape in the deep background between the women is not actually horizontal. Instead, it falls gently from left to right.
It is tempting to imagine that this might have been because the artist connected the wrong lines in his geometrical armature. Here, blue lines of 30-degree slope depart from the equator of the panel, within the red frame, and one can find the skewed horizon line quite precisely by connecting the point at left where the rising blue line intersects the yellow vertical with the point at right where the falling blue line intersects the side of the green triangle.
This painting is the property of the National Gallery of Art in Washington, DC.
This analysis was based on the image:
That image has now been superseded by:
Here is the Broadway Boogie-Woogie by Piet Mondrian.
Charles Bouleau claimed that Mondrian used such armatures governed by the Golden Ratio to define his compositions. Unfortunately, his demonstration of this principle was far less plausible than many of his other analyses, so we will be walking step-by-step through this connection.
Here is the unfolded half-diagonal section of the square painting, so as to form a Golden Rectangle.
The successively smaller squares constructed along this rectangle’s diagonal locate the heights of the red lines shown at left, each of which forms the top margin for one of Mondrian's streetlike color strips.
The orange rectangle at upper left is actually a Golden Rectangle in terms of proportion; its bottom edge aligns with the lower red horizontal, and its right margin defines one of the main verticals in the painting.
An arc swung up from the lower right corner of the painting through the intersection of this vertical and the second horizontal rises to height .441, a level through which is a green horizontal, which again forms the top edge of a color strip. The green rectangle and square above this level locate the right edges of two vertical color strips.
The blue square and rectangle within the green square serve a similar function on a smaller scale.
A few more constructions based on diagonals locate features such as the large blue rectangle in the painting’s upper right quadrant, and the horizontal color strip immediately above it, at height .842. With its insistently rectilinear articulation, Broadway Boogie Woogie might seem to represent the very antithesis of compass-based circular thinking. To the extent that its proportions were based on those of the Golden Rectangle, though, it contains circles below its surface.
This painting is the property of the Museum of Modern Art in New York.
This analysis was based on the image:
Here is an anonymous Italian Renaissance painting of an ideal city painting now preserved at the Walters Gallery in Baltimore. Its width is considerably greater than its height.
More specifically, its width is 2.828 times its height, or two times the square root of two. This can be seen by unfolding the diagonals of two squares set on its outer edges. Note that the right margin of the left square aligns with the central axis of the amphitheater in the distance.
Several key elements in the painting can be found by tracing lines of 30-degree and 60-degree slope upward from its outer lower corners. The right margin of the amphitheater aligns with the vertical where the 30-degree line cuts the left arc, and the left margin can be found by reflection. The baptistery at right fits into a nearly equivalent frame, although it is compressed slightly to the right.
The front corners of the palaces in the foreground can be found by first drawing a yellow horizontal through the point where the 30-degree orange line cuts the axis of the amphitheater, and then drawing yellow verticals through the circled points halfway between the intersections of this horizontal with the 45- and 60-degree lines.
The rear corners of the palaces are exactly halfway between the panel margins and its center, as the green verticals indicate, and the green horizon line can be found at the level where the rising green lines of 30-degree slope cut the orange margins of the amphitheater and the baptistery.
The top edge of the right-hand palace can be found by running a blue line from the vanishing point to the top of the yellow vertical describing the right-hand palace corner. From the point where this blue line cuts the orange line of 60-degree slope, a blue vertical can be dropped to find the axis of the column in the foreground, and a similar construction also works in the left half of the panel, even though the palace there is shorter.
The perspectival lines in the pavement can be found by connecting the vanishing point to already constructed points in the foreground, and a similar scheme located at the front edge of the platforms on which the palaces stand.
This painting is the property of the Walters Art Museum in Baltimore, Maryland.
This analysis was based on the image: