[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 13th 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 15th

century;

Loss of power of the Catholic Church and the Reformation (16th 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 AXi Yi B. This section creates

the same impression on the eye as the rectangular floor.

The line Xi Yi is formed when the plane defined by the line XY and the point E cuts

the plane ADCB. And we know that the line Xi Yi is straight because the intersection of

two planes is a straight line. How do we know that Xi Yi 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 Xi Yi . 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 Ai X and BYi 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 XYi do not have equal spacing because the

lines AXi and BYi converge to P. In addition, the line Xi Yi is shorter than the line AB

because it is closer to P. But Xi Yi 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 Z1 and Z2 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.

[WORKS CITED LIST]

1 Bronowski, Jacob. The Ascent of Man. Little, Brown and Company. 1973.

2 Burke, James. The Day the Universe Changed. Little, Brown and Company. 1985.

3 Kline, Morris. Mathematical Thought from Ancient to Modern Times. Oxford University Press. 1972.

4 Kline, Morris. Mathematics for the Nonmathematician. Dover Publications Inc. 1967

5 Henderson, David. Experiencing Geometry. Pearson Prentice Hall. 2005

6 Lindberg, David. The Beginnings of Western Science. University of Chicago Press. 1992.

7 Wirtz, Rolf. Art and Architecture: Florence. Konemann. 2005.

8 Kemp, Martin. Science of Art. Yale University Press. 1990.