Here is Rogier van der Weyden's painting of Saint George and the Dragon, created around 1435.
Its geometry involves decagons and pentagons, as the 72-degree slope of the saint's spear suggests. More specifically, one can begin to understand the composition by inscribing the largest possible circle within the bottom section of the panel, and then circumscribing a decagon around that circle. The rightward extension of the decagon's lower facet forms a perpendicular with the spear. The top vertex of the decagon locates the central tower of the city wall in the distance, while the vertices to the left locate the forehead of the princess, and the end point of the sweep in her dress.
To find the exact location of the spear, one must first draw 45-degree diagonals rising from the bottom center of the panel. The left one parallels the horse's left lower leg, while the right one intersects the joints just above its front hooves. Next, diagonals from the bottom corners of the panel rise to meet the first set, one passing into the horse's rear right leg, and the other passing through the dragon's eye to a point on its back. An arc swung down from the latter point locates the tip of the spear along the previously constructed decagon extension.
The tip of the spear then serves as the tip of a large pentagram whose vertical axis coincides with that of the panel. The spear aligns with the steep right facet of the pentagram, while the corresponding left facet traces the plume descending from George's shoulder, eventually passing through the bottom leg joint of the horse. The horizontal facet of the pentagram passes just over the horse's head, while the shallow facet of the pentagram passes just beneath its nose. Other details can also be seen as related to the pentagram, including the folds in the princess's dress, the crack pattern in the rocks at right, and the top of the horse's rump.
Further details can be added to the picture by constructing lines of 30-degree slope, here shown in green. The green line rising from the bottom left corner of the panel aligns with the forelegs of the horse, and its reflection back to the upper left passes directly through George's eyes and through the corner of his raised elbow. The corresponding line rising from the bottom right corner locates the bottom of the princess's dress on the left margin, and its reflection back to the upper right passes just above George's head.
Finally, in blue, a semicircle is shown in the bottom sector of the panel, describing the area occupied by the dragon at right, and by the horse's tail at left. A horizontal resting on this semicircle frames the horse's body, and the inflection point in the curve of the plume falling from George's shoulder. A higher horizontal through the tip of the large pentagram defines the horizon line. A V-shape with the 72-degree slope characteristic of the pentagram separates the space of the princess on the left, of George in the center, and of the rock outcrop at right; its intersections with the green lines in the upper corners locate the corner of the tower at left and the small tree at right, respectively.
This painting belongs to the National Gallery of Art in Washington, DC. This analysis is based on the image:https://commons.wikimedia.org/wiki/File:Saint_George_and_the_Dragon_Rogier.jpg
Here is Edward Hopper's Nighthawks, painted in 1942.
The overall width to height ratio of the painting is 1.828, or 1 + 2 (√2-1). To see this, one first creates a unit square on the right side of the painting, whose left side coincides precisely with the left margin of the bar. Unfolding its diagonal to form an arc, one finds the right margin of the door on the building in the background, which is thus √2 units from the right margin of the painting, and √2-1 units to left of the original square. Reflecting this arc about its left edge, one finds the left of the painting, which is thus 2(√2-1) units to the left of the original square. Within the original square, a circle can be inscribed, which cuts the diagonals of the square in points connected by a vertical rising along the back of the leftmost patron of the bar. The midpoint of the square is located just between the heads of the couple facing the viewer.
When nested octagons are constructed within the original square, other relationships become evident: The equator of the circle skims just over the patrons' heads and at the tops of the chrome cylinders at right. The left customer’s elbow and the bartender’s eyes, meanwhile, lie on the rays of a large octagon inscribing the main square. Hopper even appears to have used an octagon sequence in order to locate elements such as the vertical mullion of the main bar window and the crease in the wall behind the bartender’s head, as the orange constructions show.
This painting belongs to the Art Institute of Chicago. This analysis is based on the imagehttps://upload.wikimedia.org/wikipedia/commons/a/a8/Nighthawks_by_Edward_Hopper_1942.jpg
This painting shows the worksite around a palace, with workers in the foreground. The palace now rises nearly to the top edge of the painting, which was trimmed, suggesting that the panel was originally at least a bit taller.
Here one sees a crucial clue to the overall logic of the composition: the entablature of the palace colonnade aligns with the point where 30-degree lines launched from the bottom corners of the panel converge. More subtly, these sloping lines pass along elements such as the stacked building material and elevated platform in the left and right halves of the painting, respectively.
Here 45-degree diagonals have been drawn upwards from the panel corners to the palace colonnade, and verticals have been dropped from the intersection points, creating squares. The diagonal of the left-hand square passes along the belly of the ox and through its team, and the right margin of the square aligns with the seated worker beside them, but these relationships hardly appear compelling in themselves. The diagonal of the right-hand square, similarly, passes through the head of the horseman seen in profile, and the margin of the square aligns with the head and shin of the white-clad sawyer just to his left, but these could all be dismissed as coincidences.
When arcs are unfolded using the squares’ diagonals as radii, however, other clearer geometrical evidence comes into view: the left arc sweeps along the tilted body of the red-capped worker in the foreground, and a vertical rising from its bottom point describes the right edge of the block on which his companion sits. The corresponding vertical in the right-hand side of the panel passes quite precisely through the tip of the small tripod supporting the seated sawyer.
A small half-circle placed between these verticals rises to visually support the forelegs of the horse on the middle axis of the painting. More importantly, the intersections of the large arcs with the 30-degree lines define the ground line of the palace. Also, when the large orange arcs are expanded to a full 90-degree sweep, their top points establish a level just above the painting’s truncated upper edge, suggesting that this level may have marked the original margin of the panel.
To explore this hypothesis, it helps to begin by drawing yellow semi-circles centered on the points that would have been halfway up the original sides of the panel, coinciding with the endpoints of the line describing the front edge of the palace platform.
When half-decagons are inscribed within these half-circles, many significant relationships reveal themselves. At the lower left of the panel, the radius to the midpoint of the lowermost decagon facet passes precisely along the rod held by the man facing the ox; the 72-degree slope of this rod, in fact, helps to signal the presence of this geometry. The other radii in this zone help to organize details such as the horsemen in the distance, the workers just in front of them, and their shadows, and the same pattern can be seen on the right side of the panel, as well. The horizontal through the lower facet midpoints passes along the base of the block in the left foreground, and along the top of the horse’s shadow at the far right. The corresponding horizontal through the top facet midpoint, meanwhile, aligns precisely with the roofline of the palace. The horizontal through the top corner of the middle facet aligns with the horizon, and the horizontal through the bottom corner of the middle facet aligns with many details, such as the front of the large slab, the front of the molding being moved by workers, the muzzle of the frontally-facing horse, the pattern of rocks just to the right, the feet of the workers carrying the front of the coffin-like box, and on the righthand facet corner, the ear of the transversely oriented horse.
At left, it can be seen that the front legs of the oxen have the same 72-degree slope as their master’s staff. The extension of the decagon’s sloping lower facet passes along the leg of the workman seated on the block, directly across his companion’s inclined shoulders, along the dark materials behind, directly between the heads of the frontally-facing horse and its rider, and eventually along the hills in the right background. The corresponding line starting from the hills at left passes again below the face of the front-facing rider, along the foot of the white-clad sawyer, and through the leg joints and rear hoof of the side-facing horse. The tan ground on which the workers and the palace stand terminates in a horizontal line halfway between the decagon equators and the red line describing the palace entablature.
The inner margins of the palace’s two constituent blocks align with verticals through the points where lines of 60-degree slope launched from the vanishing point on the horizon intersect the green horizontal defining the palace roofline. The left and right margins of the palace colonnade align with verticals through the midpoints of the steep decagon facets. The outer margins of the palace’s second story align with verticals through the points where the orange 45-degree lines cut the extensions of those facets. Raking blue lines connecting the bottom points of the former verticals with the original top points of the latter align with many of the workers in the middle ground including those carrying the coffin-like box, and the wood being sawn. The end of this wooden piece and the hat of the seated sawyer line up with the 45-degree line rising from the midpoint of the panel’s bottom edge; the corresponding diagonal in the left half of the panel, similarly, passes along the corner of the block being used by the red-hatted worker, passing by his head and along the sides of the stone slabs behind him. It is possible to go further into the geometry of the panel, but these steps suffice to determine its overall outlines, and many of its details.
This painting is the property of the John and Mable Ringling Museum of Art in Sarasota, Florida. This analysis was based on the image:
Here Perseus appears twice : arriving from the upper left borne by his winged sandals, and again wielding his sword against the sea-monster that threatens Andromeda, who flinches away at left.
The proportions here are very simple, being based on two side-by-side equilateral triangles, which gives the panel a width : height ratio of √3 : 1, which is approximately 1.732 :1. The groups of small figures in the foreground are clustered beneath these triangles, while the flying Perseus seems to skate on the rising diagonal, which also seems to brace the female figure at lower left. The descending diagonal similarly aligns with the instrument held by the dark-skinned figure in the middle of the lower right quadrant. The horizontal midline of the panel aligns with Andromeda’s pubis, the lower tusks of the monster, and the widest area of the watery inlet.
The orange verticals here divide the left and right quadrants of the panel from its middle quadrants. The left one aligns with the structure in the background, with the edge of the water in the middle ground, and with the knee of the woman in the left foreground. The right one locates the front leg of the flying Perseus, and the base of the long instrument held by the woman in the right foreground. The sloping orange lines rising and descending from the endpoints of the orange verticals help to locate other elements of the composition, such as the bodily alignment of the sword-wielding Perseus, the hand position of the flying Perseus, and the knee and dress of the figures in the middle foreground.
The yellow horizontal, which lies at ¾ of the panels height, locates Perseus’s sword blade, and the shape of the landform just to its left. The shallow yellow diagonal just below trace the angle of Andromeda’s flinch, the border of the water just behind the monster’s head, the alignment of Perseus’s shoulders, the right-hand border of the water, and some outlines of the village in the right background.
Andromeda’s head and gaze, and the tip of Perseus’s curved sword, lie along the green diagonal in the strip between ½ and ¾ of the panel height. The green horizontal in the middle of this strip corresponds closely with the horizon line.
The blue verticals stand 1/8 and 7/8 of the way across the panel. They appear to help locate the large standing figures in the foreground. The shallow blue diagonals connecting their lower endpoints to the midpoints of the panel’s sides locate features including Andromeda’s foot, the adjacent coastline, and the body arrangement of the figures in the foreground and right middle ground. These details, combined with the aspect ratio of the panel, demonstrate the fundamental role of this simple armature in defining the outlines of the composition.
This painting is the property of the Uffizi Museum in Florence, Italy.
This analysis was based on the image:
This panel of Mars and Venus has a fairly subtle proportional scheme based, rather surprisingly, on the idea of decagonal symmetry ; several steps will be necessary to see this. The first clue in this direction may be seen near the bottom right corner, where the sharp reflection in the shoulder joint of Mars’s discarded armor has the characteristic 72-degree slope characteristic of decagons. Not only the angle but also the location of this reflection line proves significant.
When one inscribes the biggest possible half-decagon in the panel, with its center aligned with with the bottom edge of the painting, one finds that its right facet coincides with the reflection line in the armor, and with the angle of Mars’s shoulders. The radii of the decagon also seem to pick out various elements in the panel, including (from left to right) Venus’s leg, her shoulder, her child’s hand, the positions of the smaller putti in the background, and the placement of Mars’s right hand and shoulder.
Here, as a preparatory step, two circles have been stacked within the height of the panel.
The line through the center of the upper circle corresponds closely to the horizon line, while the lower horizontal aligns with Mars’s left wrist and chin.
Most importantly, though, green lines through the corners of the decagon intersect the upper yellow horizontal precisely at the left and right margins of the panel, suggesting that its aspect ratio was determined by this construction. Further evidence for the relevance of the green lines comes from the way the descending one aligns with the faces of Venus and her child, Mars’s left leg, and the bottom edge of his discarded armor, while the right one aligns with Venus’s finger, her genital, and Mars’s right knee, arm, and shoulder.
This painting is the property of the Gemäldegalerie in Berlin, Germany.
This analysis was based on the image:
This panel is 2.828 times as wide as it is tall, which is a proportion of 2√2 : 1. Geometrically, this proportion results from unfolding the diagonals of two squares, which are here placed side by side. It should be noted that the foliage in the upper left and right corners of the painting actually follow the arcs of this unfolding. The right margin of the right square traces the foreleg of the dog, and the diagonal follows the axis of its nose. The left margin of the left square, similarly, passes through the shoulder of the satyr, and the falling diagonal traces his forearm, and the upper arm of the nymph. The other diagonal in this left square traces through her face and shoulder, while the comparable diagonal in the right square passes along her foot. This painting is the property of the National Gallery of Art, London, England.
This analysis was based on the image:
This painting depicts the story of the titan brothers Prometheus and Epimetheus. At left, Epimetheus kneels before the clay man that he had made, which the god Jupiter destroys, being angered at the titan’s presumption to the creation of life. In the middle of the panel stands a more successful statue made by Prometheus, who converses at right with the goddess Minerva, who then takes Prometheus aloft to seek the spark of life.
The statue’s head stands at the vertex of the converging 45-degree diagonals rising from the panel’s bottom corners. The body of Jupiter and the building behind him are fixed between the left diagonal and the arc rising to the statue’s head; on the right half of the panel, the arc may similarly have located the tree in the background, the placement of Prometheus’s head, Minerva’s hand, and the plant cluster at the bottom margin. The shallow red diagonals have an inclination of 18 degrees, the angle that one finds in decagons, a figure that Piero used in other paintings. The one rising from the lower left corner traces Epimetheus’s knees, the clay man’s ankle, the tip of the table leg to the left of the statue, and the angle of Prometheus’s arms, before passing along Minverva’s chin. The line descending from the left margin passes by the base of the tree at left, paralleling Jupiter’s shoulders, Prometheus’s knee, and Minerva’s foot.
When 45-degree lines are launched from the intersections of the red lines with the left and right panel margins, points are located along the top margin of the panel from which other longer orange lines are launched to the bottom corners of the panel. These long lines cross just above the statue’s genitals. The orange line rising from the lower left corner passes along the leg of the clay man, through its genitals, and along the border between Jupiter’s red robe and blue mantle. The short diagonal at upper right passes along the garland of fabric trailing from the airborne Prometheus and Minerva. The long orange line descending from the statue passes through Prometheus’s elbow and along the border between the red and blue parts of Minerva’s costume.
Here yellow lines of 30-degree slope have been added through the lower corners of the panel. These lines converge at the level of the statue’s knees, aligning also with the heads of Jupiter, Prometheus, and Minerva. The yellow line rising from the lower left corner passes through the clay figure’s knee, along the bottom of a cloud bank, and along the backs and through the faces of the flying Minerva and Prometheus.
Here a green horizontal has been drawn through the points where the sloping red lines meet the left and right margins of the panels. This horizontal aligns with the tops of the buildings behind Jupiter, and with the top of Prometheus’s raised hand. The structure immediately behind Jupiter stands just above the point where the red diagonal intersects the yellow horizontal. Green diagonals launched from the center of the panel’s bottom margin pass along the edges of the statue base, and through the torsos of Jupiter and Prometheus, before intersecting the yellow horizontal at points aligned with a house at left and with Prometheus’s elbow at right. The vertical rising from that elbow also passes by the feet of the flying pair. Finally, the steep green lines departing from the lower corners of the panel are inclined at 72 degrees, the complement of the 18-degree red lines introduced earlier. The left of these green lines passes through the hand of the kneeling Epithemeus, and along the back of the small climbing figure behind him, which is his future self. The corresponding line at right passes along the leg of the flying Minerva, and along the adjacent garland. The overall composition thus involves lines based on the slopes of the equilateral triangle (30 degrees), the square (45 degrees), and the pentagon/decagon (18 degrees and 72 degrees).
A series of decagons and half-decagons also help to govern the composition. When a half-decagon is placed with its center on the left margin of the panel and aligned with the yellow horizontal, one sees that its most steeply descending ray parallels the body of Epimetheus, and that its next three rays meet the knee, elbow, and shoulder of the clay man. Its rightmost facet below the painting’s equator aligns with the torso of Jupiter. The equivalent facet of a half-decagon centered on the right margin of the panel aligns with the torso and left leg of the standing Prometheus, while the airborne Prometheus and Minerva fit into this decagon’s upper diagonal sector. A slightly smaller full decagon centered on the yellow horizontal and with its top facet aligned with the top edge of the painting has right facets aligned with the forearm and torso of the standing Prometheus, while the endpoints of its bottom facet locate objects such as the lump of clay in the left foreground, while corners of its upper diagonal facets locate the figures in the clouds. Most importantly, though, its center coincides with the left knee of the standing statue, and the V-shaped line from that center to the endpoints of its top facet embraces the statue’s torso, while aligning with its left leg and left forearm. A similar but even smaller decagon centered just above the statue’s genitals, where the orange diagonals cross, produces an analogous V-shape whose faces pass along the left side of the statue’s torso, and intersect the statue’s pointing right hand. The lower left corner of this decagon coincides with Jupiter’s left shoulder. All of these relationships serve to underscore the importance of the decagonal symmetries already introduced by the red 18-degree lines in the first step of this analysis.
This painting is the property of the Alte Pinakothek, in Munich, Germany.
This analysis was based on the image:
This painting depicts six nymphs tending to Vulcan, who had been hurled from Olympus.
The panel nearly but not perfectly square, being slightly wider than it is tall, as one can see from the fact that narrow strips border the sides of the circle inscribed within its top and bottom margins. The circle suggestively traces out a number of elements in the painting. Starting at the left center and moving clockwise, these include the side of the left nymph’s basket, details of the foliage in the trees , and most obviously the arm of the nymph at lower right. A square inscribed within this circle contains the faces of all of the nymphs, even though the heads of three of them protrude from this frame. Vulcan seems to kneel on the bottom of the square, whose top edge corresponds to a prominent break in the cloud bank above. The diagonals of the square appear to govern the pose of the central nymph and her companion to the upper right. The diagonal rising from left to right also aligns closely with a band of shadow in the grass at lower left.
When an octagon is circumscribed around the central circle, it can be seen that the radii to its corners serve to locate many elements in the composition. Starting at the top left center and moving clockwise, these include the eye of the nymph at left, the shape of the trees at upper left, the placement of the tree and landforms at upper right, the shoulder of the rightmost standing nymph, her foot position, the line of her kneeling companion’s gaze, the placement of Vulcan’s right foot, the angle of his right arm and face, and the points on the two nymphs at left where their bare legs emerge from their garb.
The overall shape of the panel is set by adding to the sides of the octagon narrow strips whose width equals one third of the interval between the octagon and its inscribing square, as the small yellow circles indicate.
When green diagonals are extended from the endpoints of these circles’ radii, one finds an armature that helps to establish other elements of the composition. The uppermost left diagonal, for example, traces the treebranch at left, the hand of leftmost nymph, and the pattern of the adjacent tree’s foliage. The upper right diagonal, similarly, traces not only the wing of the upper bird, but also the gaze of the two standing nymphs at right.
Finally, this system can be extended by using the full width of the panel, rather than just the largest square contained by it. So, example, the rightmost nymph steps on the ray of an octagon extended to this margin of the panel. Also, when 45-degree diagonals are launched from the corners of the panel, and reflected around the red diagonals of the large square, a blue-bordered X results, whose central diamond coincides with the chest of the central nymph. Her head fits into the upper left branch of the X, while here upper arm and left leg fig into the lower right branch. The lower left branch of the X aligns with her shoulders, Vulcan’s eye, and the face of the second standing nymph from the right. Further details can be found within this framework, but those cited already should suffice to demonstrate the relevance of this overall armature for the composition.
This painting is the property of the Wadsworth Atheneum Museum of Art, in Hartford, Connecticut. This analysis was based on the image:
Here is the so-called Brera Altarpiece, painted by Piero della Francesca, showing the condottiere Federico da Montefeltro in adoration of the Virgin and Child, surrounded by saints. Piero della Francesca is famous for his writings on perspective and solid geometry, but this composition was actually governed by a simple two-dimensional geometrical scheme whose logic would have been familiar to any compass-wielding Gothic architect or early medieval craftsman. To see this, start by considering a few details of the painting. First, its sides have been slightly trimmed, and the elbows of John the Baptist at left, and of Andrew at right, thus appear slightly truncated. Second, these two saints hold their attributes in ways that serve as geometrical markers, which is particularly conspicuous in the case of the Baptist’s staff. Both the staff and Andrew’s book are aligned at 22.5 degrees from the vertical.
Together they thus describe the converging sides of a triangle with an interior angle of 45 degrees, which can be understood as one eighth of a regular octagon. The baseline of this triangle likely corresponds to the original width of the panel, as the red verticals here indicate. A related construction correctly gives the height of the panel.
Here, at the top of the image, five facets of an orange octagon are centered on the apex of the red wedge and framed by the red verticals. The upper corners of the octagon’s lateral facets locate the top edge of the panel. In the bottom half of the image, an orange circle fits within the square left over beneath the octagon’s lateral facets, and a square can be inscribed within that circle. Notice that the vertical facets of the square locate the corners of the pilasters on the wall behind the saints, and that its top facet locates the bottom edge of the pilaster capitals.
Next, another quadrature step inwards can be performed by inscribing a yellow circle within the square, and a yellow square within the new circle. Note that the Virgin’s blue cloak just fills the arc of the yellow circle, that the right lower corner of the yellow square locates Federico’s elbow, and that the left yellow vertical defines the midline of Saint Jerome’s bifurcated garment.
One large green triangle, sharing its baseline with the yellow square, has its upper vertex in the Virgin’s left eye socket, since she inclines her head slightly to one side as she gazes at her child. The child’s body lies along of 30-degree slope departing from Federico’s elbow, which also locates the elbow of the angel to the left of the Virgin. The comparable point on the right locates the hand of Saint Francis. In the upper section of the panel, meanwhile, octature creates a new green horizontal just above the orange one, by inscribing arcs within the lateral wedges of the orange octagon. This new green horizontal matters because it serves as the geometrical baseline for the painting’s upper architectural structure. It coincides with the top edge of the entablature on either side of the main niche.
By constructing nested blue octagons and semicircles centered on this level and framed by the orange uprights, moreover, both the location and the thickness of the arch over the main niche can be determined. The large blue arc circumscribing the orange octagon and sweeping through the Virgin’s face also locates the eye level of most of the saints. That relationship is only approximate, but the bottom point on this arc also serves quite precisely as the point of convergence for the perspectival system defined by the converging lines of the entablatures, as the blue raking lines along the entablatures indicate.
This painting is the property of the Pinacoteca di Brera in Milan, Italy. This analysis was based on the image:
Here is Piero di Cosimo’s depiction of the Virgin and Child adored by saints including Elizabeth of Hungary, at left, and Catherine, at right.
The panel is a perfect square. Note that a circle inscribed within this square swings tantalizingly along the side of the cloud at upper left and the putto’s wing, at right. This circle can be understood as just the first in a sequence of compass-drawn quadrature steps
The base of the orange square inscribed within the red circle coincides with the edge of the lowest step on the Virgin’s throne, and the left and right midpoints of that square locate the hands of Saints Peter and John, who flank the composition. Here also rays have been added subdividing the composition into 24 equal wedges; note that these align with elements including the outer visible tiles in the floor in front of the throne.
Continuing inwards, further concentric circles divide the saints into ranks of proximity to the Child
Only Elizabeth and Catherine, for example, have faces within the green square.
And only Catherine’s hand penetrates into the blue circle that frames the child.
The bend of her fingers aligns with the same violet square that frames the child’s leg, and her fingertips penetrate just beyond the circle that frames his shoulder.
The horizon line can also be readily found within this framework, as can the curving top edge of the Virgin’s throne.
This painting is the property of the John Museo degli Innocenti, Florence, Italy. This analysis was based on the image: