[History of Mathematics] 14/04/08 Jeffrey Gallo Mathematical Perspective The humanistic movement, following the Black Death, sparked an intellectual revolution, which shaped, to a great extent, the ways in which people thought about the world. Artists discovered a new method of painting (i.e. perspective painting), and along with it came a renewal of interest in nature. The Greek doctrine that mathematics is the essence of nature s reality was also revived in the process. The advent of perspective painting also marked a revolution in the way people looked at the world, not just in terms of visual representation, but also from a philosophical point of view. Unlocking the mathematical structure of the organization space was the job of the artist (note: technically speaking, there was no such profession at the time). It is important to note that this concern was not exclusive to the artist, but also to the architect architects were people who built structures in accordance to special mathematical proportions in order to create stable structures. The intention of the artist was not to record nature as it is, but to record it as it looked. The artist was concerned with how to create the illusion of depth on a 2 dimensional surface. The artist soon realized that correct perspective involved a deeper understanding of the mathematical structure of space. It is with this understanding alone that accounts for correct perspective. My aim in this paper is to trace the development of mathematical perspective by drawing on the following topics: A background history in the development of perspective A science of vision The inception of perspective drawing through the work of Felippo Brunelleshi and Lorenzo Ghiberti The mathematics of perspective
Perspective drawing changed the way people thought about the world. This was an important turning point in the history of western culture because people began to explore different ways of knowing and living. This was characteristic of the transition from the scholastic (medieval period) tradition to the humanistic tradition of the Renaissance. A Historical Sketch: The Scholasticism of the 13 th century (Middle Ages) was dominated and controlled by the Church. Focus was placed on the Bible and selected Greek texts, it rejected experimentation and observation. This is a relevant factor in the development of perspective painting because an artist must have an understanding of the organization of space in order to create realistic paintings. The geometrical structure of space is a crucial element when creating the illusion of depth on a 2-D canvas. It wasn t until the humanistic (early Renaissance) movement that ways of knowing in European society changed. We can call this the shift in intellectual life from the Middle Ages to the Renaissance. The transition from the Middle Ages to the Renaissance was caused by (MATH 4400 notes): Awareness of Byzantine and Islam wealth and cultures from the crusades; Discovery and translation of Greek and Arabic manuscripts; Growth of towns and, trade and manufacturing; Close relation between Italy and the Byzantine Empire in the first half of the 15 th century; Loss of power of the Catholic Church and the Reformation (16 th century) The humanistic movement involved the revival of both the Greek and Roman cultures. These cultures were looked upon as the models for living. Prior to this, some Greek ideas were woven into the fabric of religious indoctrination. One example of a Greek idea that was woven into religion was Aristotle s cosmology. The church proclaimed, as did Aristotle, that the Earth was at the center of the universe, surrounded by the eternal,
unchanging, celestial spheres. This was an idea that many people accepted. The famous Italian poet, Dante Alighieri, used this same model as a metaphor for the different levels of spiritual consciousness (see figure 1) as detailed by Dante. Figure 1: Dante s Universe Another ancient Greek idea that was revived was the Pythagorean idea that number was the fundamental aspect of reality and that mathematics is the basic tool for investigating this reality (Lindberg, 32). In other words, nature must be investigated mathematically in order to reveal its hidden structures. This will be a key idea in the development of perspective drawing (see The Science of Vision). The loss of power of the Catholic Church was a significant factor in the transition from the common held notions of objective truth (i.e. divine truth) to the validation of man s lived experiences. The Italian poet Francesco Petrarch (ca 1350) was a major catalyst in this shift because he was the one who placed the human at the center of the universe (Aristotle s cosmology places the Earth at the center of the universe). He emphasized the earthly realities of human existence, that is, he suggested that we should
never deny all the qualities that make us human (this includes original sin). And these feelings must be understood in order to become better people. This is a radically different approach to life. The church encouraged people to strive for the ideal qualities as professed in religious doctrine, but Petrarch turned to the human and emphasized the complexities of human experience. We are to reach the divine through our own understandings and experiences. This was a whole new perspective on life itself! This is also an important factor in the development of perspective drawing. But before we talk more about perspective, we should develop some ideas about the science of vision. This analysis will help us understand how correct perspective can be achieved. The Science of Vision: The works of Aristotle, Euclid, and Ptolemy dominated Greek thought about light and vision. Euclid s theory, for example, was exclusively mathematical. Euclid, in his Optics, developed a geometrical theory of perception of space based on the visual cone. He placed minimal concern for the nonmathematical aspects of light and vision (Lindberg, 308). Euclid s theory of vision posited an extramission theory of light, that is, he believed that rays of light emanates from the eye in the form of a cone; and perception occurs when the rays of light intercept the object. It is important to note that Aristotle suggested an intromission theory of vision, that is, he believed, based on physical plausibility, that light reflects off of objects in space back into the eye (Lindberg, 309). One can see here that there were conflicting schools of thought within the Greek tradition. These conflicting theories were later unified into a coherent theory of vision. The Islamic mathematician and natural philosopher Ibn al-haytham (known in the west as Alhazen, ca. 965-ca.1040) was the one who did this. He rejected the extramission theory of light using the argument that bright objects can hurt the eye from without. Alhazen first recognized that we see an object because each part of the object directs and reflects a ray into the eye. Alhazen used the concept of the light cone from Euclid s work in order to come up with a unified theory of vision (see figure 2). Figure 2: Alhazen s Visual Cone
He believed that light comes from sources of illumination and is then reflected off the object, carrying the image to the eye. I will not go into too much detail on how his theory works, but I will explain its relevance to the development of perspective drawing. Working with Alhazen s geometrical theory of vision, we can explain, for the first time, why an object appears larger when it is close to us, and why that same object appears smaller when it is farther away. Here is a diagram that explains the phenomena (figure 3): Figure 3: Relative sizes of visual light cones
We can clearly see, according to Alhazen s theory, how the cone of rays from the outline and shape of an object varies when it moves away from the eye. And this is the key idea: objects appear smaller when they are farther away because the light cone that enters the eye is considerably smaller than the size of the light cone when it is closer to the eye. It is this, and only this, that accounts for the change in apparent size of the object. We see here that it doesn t matter if one accepts an intramission theory of vision or the extramission theory of vision. The reason for this is because we can still use the same geometrical method to describe how we see objects in space, provided that eye beams, like light, are assumed to travel in straight lines. It is interesting to notice how a simple geometrical description can explain a phenomenon such as this one. The Greeks, on the other hand, were not able to explain why an object changes its apparent size (Bronowski, 179). This is so simple a notion that scientists were unable to explain this for 600 years! This becomes a key idea to understand structure and organization of space. The foundations of perspective lie in the concept of the light cone from the object to the eye. Now we will see how this theory was applied to create the illusion of depth in perspective drawings. Perspective Drawing: Traditional paintings, prior to Renaissance art, reveal to us the societal values of the time. Art was used as a medium to impress religious values upon the people. There was no attempt at capturing the realism of a particular scene. Earthly objects/sceneries were neglected in order to emphasize the transcendental qualities of the divine. This was done by filling in the spaces between religious figures with gold coating (Burke, 58). Paintings were to be symbolic instead of realistic. The message of the time was clear: attention must be turned away from the earth, a dimension where man s spiritual devotion is constantly being tested (maybe a symbol of our original sin). Attention must be turned towards each other and the ways we choose to conduct ourselves in relation to others. But one important concept was left out: man must also define himself in relation to his environment! It is precisely this relationship that characterizes Renaissance art. We must realize that culture (Middle Ages) was stagnant prior to the humanistic movement.
Scriptures were the source of all knowledge and the church censored new ideas. As a result, experimentation and observation were rejected. These were contributing factors that lead to the incorrect use of perspective. However, in the Renaissance, the depiction of the real world became the goal. And so artists began to study nature in order to reproduce it faithfully on their canvases. But the same problem remained, that is, painters had to find a way to represent the three-dimensional world on a two-dimensional canvas. This brings us to the first pioneer of perspective painting, Filippo Brunelleshi. Linear perspective was invented by the Italian architect/sculptor Filippo Brunelleschi (1377-1446), shortly before 1413. He developed the first mathematical rule for getting the correct perspective (Henderson, 297). He applied these rules when he designed buildings in Rome, including the Florentine Baptistery (see figure 4). Figure 4: Brunelleshi s Florentine Baptistery (a view from the Campanale) His plans were drawn in a way that appeared realistic. We will see in mathematics of perspective drawing how Brunelleshi applied perspective techniques to design his building. The East Façade (figure 5) has giant gates called the Gates of Paradise (1425-52).
Figure 5: The East façade with Gates of Paradise These gates were created and designed by Lorenzo Ghiberti. Ghiberti dedicated 27 years of his life to work on these gates. Ghiberti covered the gates with 10 square reliefs of various biblical episodes. Let s look at the panel depicting the story of Joseph (figure 6). Figure 6: Lorenzo Ghiberti s story of Joseph (detail from the Gates of Paradise)
Ghiberti creates on this panel an effective sense of spatial depth through the landscape and architectural backdrops, which are based on detailed calculations of perspective. Ghiberti presented the architectural backgrounds and landscapes on the basis of the rules of perspective so that they appear exactly as they would in real life. The height projection of figures gradually decrease as they diminish in size and recede into the background. Now we will investigate further into the mathematics of perspective The Mathematics of Perspective Drawing: Now we will consider the problem of how to impress a 3 dimensional scene (in this case the floor AXYB) (figure 7) onto a 2 dimensional screen (ADCB). If one were to paint a scene, the canvas must contain the same section that a glass screen placed between the eye of the painter and the actual scene would contain. It is important to note that the artist cannot look through the canvas at the actual scene, so the artist must work with theorems that inform him how to place the objects on the canvas so that the painting will contain the section made on a glass screen (Kline, 219). The basic idea in perspective drawing is the principle of projection and section. It is a principle that was created by the mathematical genius Leone Battista Alberti (1404-1472). These are the concepts we will work with in the following example. Figure 7: Geometry of projection and section (the images of two horizontal parallel lines AX and BY which are perpendicular to the screen meet at a point V on the screen)
Consider an observer at point F. He sees in front of him, through a screen defined by ADCB, a rectangular floor tile AXYB. The line EF is the height of the observer and E is the point where all the lines of light meet (i.e. in the eye). The projection is where all the lines of light from the scene meet. Now, if we look again at the screen ADCB, we see that the screen itself must contain a section of that projection this is what a plane passing through the projection would contain. The lines from E to the points on all four corners of this rectangle creates a projection of which EA, EX, EY, and EB are typical lines. Now if the plane ADCB is placed between the eye and rectangle, the lines of the projection will cut the plane and outline the quadrangle AX i Y i B. This section creates the same impression on the eye as the rectangular floor. The line X i Y i is formed when the plane defined by the line XY and the point E cuts the plane ADCB. And we know that the line X i Y i is straight because the intersection of two planes is a straight line. How do we know that X i Y i and XY are parallel? If we imagine a vertical plane passing through XY, then the plane defined by the point E and the line XY will pass through this new vertical plane at XY and the screen (or plane) defined by the line AB and the point E. Since both planes are parallel to each other then the plane defined by the line XY and the point E will cut the screen in a line parallel to XY, that is, a plane which intersects two parallel planes intersect them in parallel lines. Hence XY is parallel to X i Y i. But we also know that XY was any horizontal line parallel to the screen or canvas. Likewise, the image of a vertical line, which is parallel to the vertical screen must appear as a vertical line on the screen or canvas. Here are two results: Image in the screen of any horizontal line parallel to the screen or canvas must be drawn horizontal All vertical lines must be draw vertically But how do account for the lines that meet at V on the screen? Suppose that AX and BY are two parallel, horizontal lines in an actual scene (the scene being a rectangular tile on the ground). We can assume that these two lines are
perpendicular to the screen. Now if we draw lines from E to each point along AX, then these lines, that is the projection, will lie on one plane for the point E. The point E and the line BY determine another plane. And we know that these two planes will cut the screen, but we don t know exactly where they will be situated on the screen. So we know that the intersection of two planes is a line and this is drawn as A i X and BY i on the screen. As one looks farther out along the parallels AX and BY, the lines of sight will become more horizontal. If the eye follows AX and BY to infinity, he lines from E tend to merge into one horizontal line that is parallel to both AX and BY (line EV). This line from E will pierce the screen at the point V. The point V is an imaginary point where the lines AX and BY seem to meet at infinity, but we know that no such point exists in the real scene. Even though the lines AX and BY are parallel, the eye gets the impression that they do meet. The point V is called the principal vanishing point. Now if AX and BY are horizontal lines perpendicular to the screen, then (another result): All horizontal lines that are perpendicular to the screen must be drawn on the screen or canvas as passing through the vanishing point V. Furthermore, the distance between AB and XY are the same because they are parallel lines, but the corresponding images AB and XY i do not have equal spacing because the lines AX i and BY i converge to P. In addition, the line X i Y i is shorter than the line AB because it is closer to P. But X i Y i corresponds to the distance XY in the actual scene, which is farther from the screen than AB is. Therefore, lengths that are farther from the screen must be drawn shorter than the equal lengths closer to the screen. So in order to achieve correct perspective (another result): Lengths that are farther away from the observer must be foreshortened. All the mathematical results mentioned above create the illusion of depth on the canvas. As we can see here, the problem of how to create a realistic drawing can be solved by applying a mathematical system. Let us take a look at the methods Brunelleshi employed
to design the Baptistery. The following diagram shows the viewpoints that Brunalleschi used in his design plan. Figure 8: Diagrammatic reconstruction of Brunelleshi s demonstration of the Florentine Baptistery As a professional architect/designer Brunelleshi had to sketch this building on a two dimensional surface in order to create a realistic vision of the design. Brunelleshi used two-point perspective in his sketch to achieve this. The points Z 1 and Z 2 are the two vanishing points he used for the sketch. Here is a more detailed look at how Brunelleshi created depth in the foreground (figure 9). The point V is the vanishing point that runs along the line of sight at F. Figure 9: Foreground of Brunelleshi s plan
The following figure shows where he stood, relative to the structure, as he envisioned the baptistery. Figure 10: Diagrammatic plan of Brunelleshi s situation for his perspective demonstration of the Florentine Baptistery It is clear that the correct use of perspective was important for design purposes. The only way to visualize the design in plan view is to re-create the design using lines perspective. Brunelleshi used this system, but only later was it mathematized into a coherent theory. Conclusion: The advent of humanism marked a shift in intellectual thought. The authority of the Church was diminishing and people began to look at the world in a different way. With this came new approaches to life, new possibilities and new meaning. The artist was no longer concerned with the absolute truths of religion, he was looking for something more, a better understanding of God s work (i.e. nature). This motivation
propelled the artist into the domains of science, where they confronted the mathematical language of nature. The essential difference between the art of the Renaissance and that of the Middle ages is the incorporation of the third dimension. The realistic rendering of space, distance, and forms by means of a mathematical system of perspective became characteristic of Renaissance art. Through this system, the process of seeing was rationalized. All this was achieved by the artist s aim to uncover the secrets of nature. This approach broke away from the scholastic tradition in the sense that that artists began to study and observe the mathematical structures of space. The Renaissance artist was a scientist in the proper sense because he shared the common goal of science, and that was to better understand the workings of nature. The line that separates the arts and the sciences today was a lot less prominent during the Renaissance. The Renaissance artist, as did the scientist, searched for the truths of nature. And so mathematics became the language of nature, just as the Pythagoreans believed. Mathematics enabled painting to reveal the structure of nature. This marked a key turning point in man s understanding of the structure and organization of space.
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