1. Vertex recognition and the deficiencies of linear perspective in art
James Geary • University of Kent •
THE PROBLEM: LEONARDO’S PERSPECTIVE
THE SOLUTION: RECOGNITION-BY-COMPONENTS
In order to solve the problem of the Virgin’s book we will use the cognitive theory of Recognition-by-Components, proposed by psychologist Irving Biederman. Biederman argues that we recognise objects by decomposing them into a limited number of simple three-dimensional shapes, or ‘Geons’, such as cones, cubes and spheres. Geons can describe a great number of figures; for example, a sphere on a cone can describe an ice-cream. The small number of Geons makes for easy and thus quick recognition.
The aspect of Recognition-by-Components theory that is of interest here is Biederman’s theory that the recognition of Geons is facilitated by vertices. Biederman found evidence for this by showing volunteers selections from Figure 4. The volunteers identified the objects when shown the middle column than the right-most column.
Biederman classifies vertices into three types: the arrow-vertex (the vertex on an external edge of an object), the Y-vertex (the vertex on an internal edge of an object), and the T-vertex (the vertex that appears in segmentation and occlusion) (Figure 5). By the recognition of vertices, the viewer gains information about the shape of objects, and whether or not an object is in front or behind any other object.
Consider for a moment Leonardo da Vinci and Andrea del Verrocchio’s Annunciation (Figure 1). Note that the painting is formed with perfect linear perspective. We take it for granted that, due to the physical properties and geometry of the eye, linear perspective gives the most lifelike way to depict a scene in three-dimensional space.
If we now turn to Figure 3, Giotto’s Meeting at the Golden Gate, we see that the linear perspective is badly composed. The horizontal lines at the top of the two towers do not line up, and the rusticated blocks at the bottom of the towers do not recede into space. Despite these failings, however, Giotto’s buildings have a solid three-dimensionality, unlike the Virgin’s book in the Leonardo/Verrocchio.
Figure Leonardo da Vinci and Andrea del Verrocchio Annunciation c.1472–1475
In the Annunciation, though, despite the linear perspective being perfectly composed, the Virgin’s book appears curiously two-dimensional (Figure 2).
Figure Giotto di Bondone Meeting at the Golden Gate c.1305
We are thus left with a conundrum: how can the correctly constructed linear perspective of the Virgin’s book fail to provide a life-like three-dimensionality, whereas the buildings drawn with Giotto’s poor perspective succeed?
Figure Drawing of objects with sections left out, to illustrate Recognition-by-Components theory. From Irving Biederman. ‘Recognition-by-Components: A Theory of Human Image Understanding.’ Psychological Review 94 (1987): 115–147: 135
Figure Biederman’s three vertices. Diagram by the author
Figure Leonardo da Vinci and Andrea del Verrocchio Annunciation c.1472–1475 (detail)

2. CONCLUSION: SOLVING THE PROBLEM OF THE ANNUNCIATION
AN EXAMPLE IN ART: HIERONYMUS BOSCH
CONCLUSION: SOLVING THE PROBLEM OF THE ANNUNCIATION
An example of how Recognition-by-Components theory can be applied to art involves one of the early paintings of Hieronymus Bosch (c.1450–1516), Adoration of the Magi (Figure 6, TOP).
The overpainted picture now looks more three-dimensional, but its three-dimensionality is still weak. The only information occlusion provides is that an objects in front or behind another; it does not properly depict volumetric form. The vertices that describe volumetric form are the arrow and the Y, and thus in order to depict volumetric form an artist should use one of these.
How has Bosch failed to include one of the other vertices? We might note that the vertex at the bottom of the gable nearest to the picture plane should be an arrow-vertex, thus providing a depiction of volumetric form. However, this is not the case. The bottom edge of the gable and the bottom edge of the thatch instead meet together to form a straight line, thus leaving the corner of the roof without a defining arrow-vertex. We should note that in fact the two lines discussed do not quite make a straight line. However, that we would perceive the two lower edges of the roof as one line, rather than two non-parallel lines, can be demonstrated by an optical phenomenon known as the ‘hypotenuse illusion’, which we see illustrated in Figure 7. The shape in Figure 7 looks like a right-angled triangle, but in fact its hypotenuse is made up of two separate non-parallel lines; the shape is in fact four-sided.
The technique of the cut-away depiction of the building is successful in its role of aiding the narrative, but is less successful in its depiction of volumetric form. (Bosing, W. (1994). Bosch: Between Heaven and Hell. New York: Taschen.) The notion of the vertex from Recognition-by-Components theory can explain this lack of success.
As we saw in Figure 5, one way that vertices can indicate space is by occlusion. Occlusion occurs when one object partially obscures another object, thus showing that the first object is in front of the second object. Recognition-by-Components theory posits that this is indicated by a T-vertex. We should note that there is only weak occlusion in the Bosch painting, as the ‘overpainting’ of Figure 6 (MIDDLE) demonstrates. Figure 6 (MIDDLE) reinserts the central support of the building, which the artist left out to help form the ‘cutaway’ theme. We can note the changes made in Figure 6 (MIDDLE) make the painting more three-dimensional by increasing the number of T-vertices.
In a similar way we perceive the bottom edge of the roof as one straight line. Thus the gable’s bottom vertex is not best labelled an arrow-vertex, for it lacks the necessary clear angle of an arrow-vertex. Figure 8 shows this further: the top picture shows the roof uncorrected, while the bottom picture has been overpainted with a clear arrow-vertex. Figure 6(RIGHT) shows this in the full painting.
There are many of these vertices in the overpainted picture, including one made with the central support and St Joseph’s arm, one made with the central support and the arm of the green-robed Magi, and a number made with the central support and the folds of the Virgin’s robe.
Figure Hieronymus Bosch Adoration of the Magi c.1500–1550. Additions by the author
TOP: no additions
MIDDLE: with central post added
BOTTOM: with modified gable
The reason for the lack of three-dimensionality of the Annunciation’s Virgin’s book can be seen in Figures 9 and Figure 10. The depiction of the book is similar to Bosch’s roof by being constructed predominately using one straight line. It fails to produce a clear vertex, thus explaining why the three-dimensionality of the book is weak. If we now turn to Giotto’s Meeting at the Golden Gate (Figure 11), we notice that the towers have a range of all three types of vertices, hence providing a strong depiction of volumetric form.
Figure Leonardo da Vinci and Andrea del Verrocchio Annunciation c.1472–1475 (detail), with a diagram describing the representation of space. Diagram by the author
Figure Giotto di Bondone Meeting at the Golden Gate c.1305 (detail), with vertices added. Additions by the author
Figure Hypotenuse Illusion. Diagram by the author
Figure Leonardo da Vinci and Andrea del Verrocchio Annunciation c.1472–1475 (detail), with vertices added. Additions by the author
Figure Hieronymus Bosch Adoration of the Magi c.1500–1550 (detail). Additions by the author