INTRODUCTION
The purpose of this paper is to demonstrate an approach and method for constructing perspectival space that may account for many of the distinguishing spatial and compositional features of key Renaissance paintings. The aim of the paper is also to show that this approach would not necessarily require, as a prerequisite, any understanding of the geometric basis and definitions of linear perspective as established by Alberti. In particular, the paper discusses paintings in which the spatial/geometric structure has often defied conventional reconstruction when the strict logic of linear perspective is applied. It specifically examines the spatial construction of four very different paintings in order to explain how the geometry and methods involved may shed light on Brunelleschi's architecture, as well as on some of the questions and issues surrounding the history, origins and nature of linear perspective.

Briefly, I propose that an explanation for the unique compositions and the apparent inconsistencies of many paintings is that their spatial structures have not been generated purely using the logic of linear perspective. I would argue that their distinctive characteristics are not the result of making a projection from a ground plan or constructing a pavimento. Rather they are the result of developing a space using a particular two-dimensional geometric construction -- a matrix -- to create a ready-made framework into which the imagery is then fitted. Further I would say that the imagery within the paintings is sometimes directly inspired by the geometry and imagery of the matrix itself. The matrix contains both a surface grid/pattern and the diminishing proportions that provide the characteristic convergence, and the controlled changes of scale necessary to create the spatial illusion. I will demonstrate that this kind of matrix can be developed simply from the geometric patterns found in pre-Renaissance paintings.

BACKGROUND
The evidence and the ideas that I present here are the direct result of research that involved re-thinking the geometric constructions of two key Renaissance paintings. These are Masaccio's Trinity, (circa 1427), which is thought to be the earliest painting to show a systematic approach to spatial diminution, and Piero della Francesca's Flagellation.[1] These paintings are considered to be particularly significant because of the artists' respective links with Brunelleschi, credited with discovering perspective, and Alberti, credited with the first written account of the theory of linear perspective in 1435. The importance of the Flagellation, (circa 1455?), relates not only to the apparently rigorous application of linear perspective, but also to its distinctive and innovative composition and to Piero's own status as a perspective theoretician. Martin Kemp has also commented on a property of the compositions of both paintings that is of particular relevance and importance for my thesis.[2] He has noted that the artists appear to show complete control over the conjunctions of elements that are spatially unconnected within the paintings. Elements that are on the surface of the paintings or in shallow space appear to coincide with, or complement in some way, elements that are deeper in space. As I will demonstrate, I believe that this phenomenon is a direct result of the type of geometric construction used by the artists, namely a matrix.

My research has also been fuelled by reading James Elkins's The Poetics of Perspective [1994], and his article "The Case Against Surface Geometry" [1991], in which he is critical of attempts to analyse early Renaissance paintings in terms of their surface geometry and perspective. In this article, he raises important general questions about the analysis of paintings and examines the relationship, if any, between surface geometry, perceived harmonious relationships, and perspective constructions -surface geometry defined as being systematic non-illusionistic geometry. He also discusses Wittkower's [1953] interpretation of Brunelleschi's use of perspective in relation to proportion in his buildings. I feel broadly sympathetic to James Elkins's sentiments about the nature of perspective and his reservations regarding reconstructions, and I have attempted to pay heed to his criticisms in my own approach to these paintings. My approach is also informed by my own experience as an artist who uses linear perspective and for whom the matrix of lines implicit in a perspective construction itself acts as a vehicle or medium for the imagination. As a student, I learned the mechanics of perspective directly from Piero della Francesca's De prospectiva pingendi and was intrigued, not only by the nature of perspective projection, but equally fascinated by the potential spatial ambiguity of perspective diagrams, simple geometric forms and patterns.

I will now set out the broad areas that are particularly relevant for my thesis, and then show how they are relevant to the construction of certain paintings. I will look at Alberti's construction, the orthodox history of perspective and some of its assumptions, and the various geometric patterns found in paintings, the significance of which I believe has been overlooked.

A BRIEF DESCRIPTION OF THE ORTHODOX HISTORY OF PERSPECTIVE
Linear perspective, the geometry of which appears to have its origins in the early fifteenth century, has come to be thought of mainly as a tool for painters, enabling them to represent spatial relationships systematically on a two-dimensional surface.[3] As far as geometry is concerned, it is the projection of three-dimensional spatial relationships from a single point, onto a surface.

Figure 1. The general principle of linear perspective

For a painter, this surface is usually, but not necessarily flat, and is known as the 'picture plane'. The discovery of perspective is attributed to the architect Brunelleschi, and it has been suggested that it originated in his desire to understand the mechanism that governs the apparent diminution of architectural elements according to their position and distance from the eye.[4] This knowledge would then have enabled him to control the relationship between the real space within a building and the projected image of that space. The subsequent adoption of linear perspective as a tool by artists/painters appears to have its roots in a general desire to represent or depict the third dimension more accurately or convincingly. The assumption is that artists wanted to solve the problem of creating a more logical, measurable, naturalistic and unified space and that Brunelleschi's discovery provided the solution. However, the precise nature of that solution and the mechanism and date of its discovery remain unclear.[5] Reliefs by Donatello are the earliest works to show a relatively consistent approach to diminution in space, but Masaccio's Trinity, (circa1427), on which it is believed Brunelleschi collaborated, is held to be the first painting to fully utilize the new knowledge of perspective and to show systematic and accurate proportional diminution.[6] It is therefore generally assumed that the method used in the spatial construction of Masaccio's Trinity reflects, or in some way relates to Brunelleschi's original findings.

In 1435, some eight years after Masaccio's Trinity was painted, Alberti described the general principles for creating a regularly diminishing floor grid and provided the first theoretical account of what we now call linear perspective (Figure 2).[7]

Figure 2.

He described it as a projection and explained the relationship between the position of the eye, the cone of vision and its intersection with the picture plane. He described two separate constructions, the information from which is combined to give a receding perspectival grid. Consequently, most analyses of the perspective constructions of Renaissance paintings post 1435, are based on the tacit assumption that the artists would be using Alberti's method in one form or another, or would be aware of the principles underlying Alberti's theory.[8] It is assumed that the artist would initially have created a perspectival grid on the ground (pavimento), or would have made projections directly from an architectural ground plan, having first made a conscious decision regarding the eye level and distance from the picture plane. The receding grid on the floor would then act as a guide to the relative scale of all other elements within the picture. Alberti suggests relating the viewer's height, (3 braccia), to the size of the floor squares, (1 braccia), and also states that a check for correctness is the convergence of the diagonals of the squares on the ground to a single point.

In reality, however, there are paintings that show an approach that could not be considered to be purely Albertian. Many paintings show a floor grid with a recession that appears to be governed solely by the 45° diagonals of the grid squares being drawn towards a point at eye level, often placed at the edge of the painting. This approach is often referred to as the 'distance point' method and these points are known as 'distance points' simply because the distance between them and the central vanishing point is the same as the distance between the viewer and the picture plane.[9] It follows that if the vanishing point for the orthogonals is placed centrally, and the edge of the painting is used as a distance point, then the 'correct' viewing distance is half the width of the painting. It also follows that the angle of view is 90°.

It has been generally assumed that these points have been placed at the edge of the paintings for completely practical reasons. However, Alberti's description of the mechanics of the perspective construction is, in fact, slightly ambiguous and open to various interpretations. Samuel Y. Edgerton demonstrated that Alberti's description could imply that the viewing point should be placed at the edge of the painting, and that the artist subsequently decides the position of the picture plane [Edgerton 1975: 40-49]. The two diagrams described by Alberti are, in effect, overlaid, and under certain circumstances, when the picture plane is placed down the centre of the painting, would result in the distance point and the viewer's position coinciding. The particular properties and wider implications of this construction, however, appear to have gone unnoticed.

The following two figures show two of many possibilities. In Figure 3 the viewer's distance from the picture plane is half the width of the picture, making the angle of view 90°. The resulting transversals are found to be placed at 1/3 and 1/2 the height of the rectangle.

Figure 3.

In Figure 4, the viewer is 1/4 the width of the picture from the picture plane, and the resulting transversals are found to be placed at 1/2, 2/3, and 3/4 the height of the rectangle.

Figure 4.

These forms of the Albertian construction, where the viewer, the picture plane and grid units along the base are in a defined relationship to each other, are particularly important. Under normal circumstances regular surface grids and perspective diminution do not mix, but in this particular case they do. The resulting divisions within the rectangles follow simple ratios, creating simple harmonic grids based on the reciprocals of whole numbers. It is known that both Piero della Francesca and Leonardo da Vinci were interested in and researched these specific relationships [Wittkower 1953].

For these relationships to hold, the distance from the eye to the picture plane must be the same as the length of the grid units along the base (Figure 5).

Figure 5.

Figure 6 shows another way of developing the same divisions, and Figure 7 shows that the transversals of a perspectival grid drawn within such a rectangle divide along their length into 4, 5, 6, 7 etc, equal parts.

Figure 6.

Figure 7.

Alberti's text can be interpreted as describing a general principle of projection, with no predetermined measured relationship between the plan, the picture plane and the viewer. The construction that results lends itself to the drawing of a simple rectangular space, based on a rectangular floor subdivided into squares.

Edgerton's interpretation is more specific and results in a construction which contains very particular geometric and number properties. Additional orthogonals are easily inserted where the transversals touch the sides of the rectangle, resulting in a space that is not confined by two dominant orthogonals from the two lower corners of the rectangle (see Figure 7). The construction fills the whole rectangle and is consequently more evocative of larger and more open spaces, a quality that can be seen in the under-drawing of Uccello's Nativity (ca. 1450, Soprintendenza alle Gallerie, Florence).

Equally important is the fact that it is a construction that can be developed without any regard for the concepts of 'projection' and the 'picture plane'. I suspect, for reasons that will become apparent, that it is this type of construction and the simple harmonic relationships within it that are the key to Brunelleschi's architecture and his involvement with the development of the geometry of perspective.

NOTES
[1] For Piero's Flagellation see [Wittkower and Carter 1953]; For Masaccio's Trinity see [Field 1997], [Field, Lunardi and Settle 1988], [Aiken 1995], [Kern 1913], [Schlegel 1963], [Janson 1967], [Coolidge 1966], [Polzer 1971], [Sanpaolesi 1962: 42-53] and [Cristiani-Testi 1984]. return to text

[2] Kemp writes: 'In the Flagellation, the artful ambiguity is less developed, but there is no question that Piero is sharply aware of surface interplays such as those between the sharply silhouetted light and dark forms inside and outside the praetorium. I think it is true in general to say that the greatest perspectivists - we may think of Masaccio, Piero, Leonardo and Saenredam as among such - have not only exhibited complete mastery of the construction of space, but have also shown a heightened awareness of the shapes of forms when projected on to the flat surface of the painting' [Kemp 1992: 32]. Speaking of Domenico Veneziano, Kemp writes: 'His St Lucy Altarpiece not only contains a virtuoso display of advanced perspective but also exploits a marvellously cunning series of visual conjunctions which compress elements at different depths into an interlocked composition' [Kemp 1992: 35]. return to text

[3] The date for its discovery is suggested by different authors to be anywhere between 1409 and 1425. It is complicated by the fact that the term 'perspective' was originally linked to the general area of optics, and only later took on its more specific meaning. return to text

[4] Wittkower acknowledges that '…he had to invent painters' perspective since the two-dimensional projection was the only mathematical way of determining the relation between distance and diminution' [1953: 288]. However, any building containing regular intervals and a fixed height throughout would achieve some of the qualities that Wittkower is claiming for Brunelleschi's buildings, but he goes on to say that 'it was only during the Renaissance that everything was done to make the perception of a harmonically diminishing series a vividly felt experience'. The elevation of a building is parallel to the picture plane and so is unaffected by perspective. There must be something else that distinguishes the image formed by a Brunelleschi interior, from the image formed by any other set of objects placed at regular intervals in space. This is an area that was originally dealt with earlier in Argan [1946]. return to text

[5] Manetti, thought to be the author of the biography The Life of Brunelleschi, describes Brunelleschi as the inventor of perspective. Manetti claims to have first hand knowledge of the painted panels that were used by Brunelleschi to show perspective It is from Manetti's description that numerous attempts have been made to understand the precise nature of these panels which no longer exist. The main problems are understanding exactly what Brunelleschi did or did not know at the time, what it was he was actually demonstrating, the exact nature of these panels, and how he made them. These problems have generated a multitude of explanations ranging from the panels being made from measured plans and elevations, through to the idea that Brunelleschi was in fact demonstrating paintings made using a camera obscura. See [Manetti 1970]; [Tsuji 1990]. return to text

[6] The suggested collaboration between Masaccio and Brunelleschi is usually on the grounds of the style of the architecture depicted in the painting. return to text

[7] [Alberti 1972]; [Alberti 1966]. The original Latin version, De pictura, was published in 1435. Alberti's description of the mechanics of creating a perspectival floor is brief and incomplete as it was probably not written for artists. The Italian version, Della pittura, the text of which is not a direct translation from the Latin, was published in 1436. return to text

[8] A reconstruction of the perspective structure of a painting makes certain assumptions, the most important being that the painting actually contains a rational space. Working backwards, a rational ground plan can then be reclaimed from the painting, and it is often assumed that the artist must have started with such a ground plan. The most well known reconstruction of this type is Wittkower and Carter's analysis of Piero's Flagellation, where the outcome was a ground plan and elevation together with suggestions for the significance of the various geometric and numeric relationships perceived to be in the plan, the elevation and the painting itself. There are several basic problems with this approach, the main one being that linear perspective is a self contained geometric system with its own internal logic and it is possible to fabricate a rational illusory space without knowing or specifying the real nature and dimensions of that space beforehand. It is through the identification of an object of known size and position that the rest of the space becomes measurable, and although a ground plan can usually always be generated, it does not logically follow that the artist necessarily started with a ground plan. It also assumes that the geometric perspective construction is always separate from, and takes priority over, all the other compositional and spatial devices available to the artist. return to text

[9] The two vanishing points on the horizon at which diagonal 45° lines in the horizontal plane meet, are known as distance points. They are the same distance from the central vanishing point as the viewer is from the picture plane. If within a picture, a horizontal square parallel to the picture plane can be identified, extending the diagonals to the horizon will give the distance points. The distance of the viewer to the picture plane is then known, and it becomes possible, by working backwards, to create a plan of the space within the picture. It is debatable whether the correct viewing distance was of any importance to the early users of perspective. It becomes more important when the game playing potential of perspective is realised. There can also be a conflict between this 'ideal' viewing distance, the physical distance that the artist is from the surface while working, and the distance from which the painting is normally seen. For instance, I think that within Piero's Baptism of Christ in the National Gallery, London, Piero may have generated a perspectival grid using a diagonal to the edge of the painting, making the 'correct' viewing distance half the width of the painting. Standing very close to the painting, one eye shut, level with the horizon, the space within the painting alters dramatically, making sense of the proportions of the figures in the background. return to text

ABOUT THE AUTHOR
Richard Talbot is a visual artist who makes large scale drawings involving geometry and perspective. He studied Astronomy and Physics at University College London, and then Fine Art at Goldsmiths' College, University of London, and at Chelsea School of Art. In 1980, he was awarded the Rome Scholarship in Sculpture and spent two years at the British School in Rome during which time he travelled widely throughout Italy and also in Egypt. He continues to exhibit, and currently teaches drawing and sculpture part time at The City and Guilds of London Art School, as well as being a visiting artist at several other art schools and universities in the UK. In 2002 was awarded a Rootstein Hopkins Foundation grant. His drawings can be seen at http://www.richardtalbot.org .

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