The Mathematics of Painting the Birth of Projective Geometry in the Italian Renaissance Graziano Gentili Luisa Simonutti and Daniele C. Struppa

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The Mathematics of Painting:

the Birth of Projective Geometry in the Italian Renaissance

Graziano Gentili, Luisa Simonutti and Daniele C. Struppa

«Porticus aequali quamvis est denique ductu

stansque in perpetuum paribus suﬀulta columnis,

longa tamen parte ab summa cum tota videtur,

paulatim trahit angusti fastigia coni,

tecta solo iungens atque omnia dextera laevis

donec in obscurum coni conduxit acumen.»

Titus Lucretius Carus, De rerum natura, IV 426-431

Abstract. We show how the birth of perspective painting in the Italian Renaissance led to a

new way of interpreting space that resulted in the creation of projective geometry. Unlike other

works on this subject, we explicitly show how the craft of the painters implied the introduction

of new points and lines (points and lines at inﬁnity) and their projective coordinates to complete

the Euclidean space to what is now called projective space. We demonstrate this idea by looking

at original paintings from the Renaissance, and by carrying out the explicit analytic calculations

that underpin those masterpieces.

Keywords. Renaissance, Piero della Francesca, painting, perspective, analytic projective

geometry, points and lines at inﬁnity.

1. Introduction

The birth of projective geometry through the contribution of Italian Renaissance painters is a

topic that has originated a large and very interesting bibliography, some of which is referred to

in this article. Most of the existing literature dwells on the evolution of the understanding of

the techniques that painters and artists such as Leon Battista Alberti and Piero della Francesca

developed to assist them (and other painters) in creating realistic representations of scenes.

These techniques, of course, are a concrete translation of ideas that slowly germinated and were

only later completely developed into a new branch of geometry that goes under the name of

projective geometry.

The point of view that we are taking in this article, however, is to strengthen the linkage

between the pictorial ideas and the mathematical underpinnings. More to the point, the entire

architecture of prospective painting consists in realizing that the space of vision cannot be

represented through the usual Euclidean space, but requires the inclusion of new geometrical

objects that, properly speaking, do not exist in the Euclidean space. We are referring here to

what mathematicians call improper points and improper lines or, with a more suggestive term,

points and lines at inﬁnity. Unlike most other studies, for example [4] and [12], we use here the

approach and the terminology from analytic projective geometry, rather than the proportion

0The ﬁrst author was partially supported by INdAM and by Chapman University. The second author was

partially supported by ISPF-CNR and by Chapman University

arXiv:2210.13295v1 [math.HO] 24 Oct 2022

theory from Euclidean geometry, by introducing the notion of projective coordinates. Just as

the birth of projective geometry was stimulated by pictorial necessities, we show here how the

language of this new geometry can be applied to those necessities.

There are two main reasons for this approach. On one hand, we believe the projective

terminology allows a simpler way to treat the technical task at hand, namely the identiﬁcation

of the technical processes that a painter needs to represent a scene. But there is a deeper reason:

perspective is not simply a technique; rather it is a radical change of perspective (pun intended)

on what space is. In order to formally perfect the process of representation, the mathematicians

had to introduce new objects, new points, new lines, new planes. It is by introducing these

objects that mathematicians were able to create a logically consistent view of the pictorial

space, that allowed them a formally unimpeachable process through which what we see can

be translated into what we draw. The new line, plane, space (which are now the projective

line, the projective plane, the projective space) resemble (and contain) the old Euclidean line,

plane, space, but perfect the nature of their properties. So, for example, while in the Euclidean

plane we say that any two distinct lines intersect in a point unless they are parallel, in the new

projective plane we can say that any two distinct lines intersect in a point, without exception.

Projective geometry is not just a new and useful technique, it is a radically diﬀerent way of

representing the space around us.

We should add a couple of notes for the reader. Projective geometry is born of the necessity

to understand the phenomenon of apparent intersection between parallel lines, and most of our

article is devoted to this aspect. However, once the mathematics is clear, projective geometry

allows the study of much more complex situations. For example, the same techniques that we

illustrate in our article, can be utilized to determine how to represent the halo of a saint, or

the shadow of a lamp against the wall of a church. This topic goes beyond the purposes of this

article, but we did not want the reader to think that projective geometry exhausts its role with

the study of points and lines. We should add that, like it often happens in mathematics, the

theory of projective geometry and its developments has taken a life of its own, and it is now one

of the most fertile and successful ﬁelds in all of mathematics.

To begin our analysis of the evolution of the prospective point of view in painting, we will

look at a few paintings from the early renaissance. Speciﬁcally, in section 2, we will look at two

Tuscan painters: Giotto, whose worldwide fame rests on his fresco cycle in Padova (where he

depicted the life of Jesus and the life of the Virgin Mary), and possibly (attribution is disputed)

on his frescos in Assisi (where he depicted the life of San Francesco), and the equally important

Duccio di Buoninsegna, whose Maestà is visible at the Duomo in Siena.

Giotto was considered, at the time, the greatest living painter, and he is usually credited with

being the link between the Byzantine style and the Renaissance, and the ﬁrst to adopt a more

naturalistic style. Giotto was an attentive observer of reality, as we can see by looking at the

faces and ﬁgures in his paintings, but because of the lack of appropriate technique, his approach

to architecture appears a mixture of artiﬁcial and fantastic.1In this section, we consider some

of his works, as well as Duccio’s paintings, to highlight both their early understanding of the

need for new ideas, as well as their insuﬃcient clarity on what those ideas would need to be.

Section 3 is devoted to the mathematical description of the process that is necessary for a

faithful representation of a three-dimensional scene on a canvas. This section is where we are able

to introduce the basic ideas that will lead to the projective space. How Leon Battista Alberti

and Piero della Francesca understood such ideas is the subject of Section 4, where we go back to

the original texts, and paintings, to illustrate the way in which the theory of projective geometry

was applied in these more advanced works from the Renaissance. To be precise, we will show

that in fact Leon Battista Alberti did not fully justify his technique (costruzione legittima), and

so we have an example of a process which seems to work, while its own developers are not yet

1The reader is referred to [18], [26] for a careful reconstruction of the path from natural to artiﬁcial perspective

in the Middle-Ages and Renaissance. See also the bibliographies [19], [21], [25].

fully aware of its theoretical justiﬁcation. The last Section, before our ﬁnal conclusions, inverts

the process, so to speak. Instead of discussing how to use geometry to represent a scene on the

canvas, we will take a painting as a starting point, to reconstruct what the scene that the painter

had in mind must have been. This is an interesting exercise, not only for the mathematician, but

for the art historian as well, since this reconstruction can help shed light on some interpretation

issues, as we will discuss in more detail in the section.

2. Early steps: Giotto (1267-1337) and Duccio di Buoninse-

gna (1255/60-1318/19)

If one takes a look at any of Giotto’s frescos, the ﬁrst thing that jumps to the eye is a really

distorted sense of distances, positions, and sizes of the elements of the pictorial composition.

As we see below (ﬁgure 1) in a fresco that represents San Francesco who chases away the devils

from the city of Arezzo, the buildings have odd angles, the ﬁgures are too big, and it looks like

we are watching the scene both from the top and from the side (note how we see the side of the

walls surrounding Arezzo, but also the buildings inside the walls themselves). What is going

on?

The answer to this question lies in the fact that Giotto is one of those painters who found

themselves in a moment of epochal transformation. A moment in which painters understood

that the way we see objects, and the way objects are, do not coincide. More speciﬁcally, when

we think of a table, when we touch a table, we deal with a rectangle. This is what most tables

are, and if we close our eyes and simply touch the table, we perceive a rectangle. Opposite sides

are parallel, and the angles between contiguous sides are right angles (ninety degrees). But when

we look at a table, or when we try to draw a table, something completely diﬀerent appears. Now

the angles become acute or obtuse, depending on where we are looking, and the parallel sides

may not appear parallel anymore.

Figure 1. Giotto, La cacciata dei diavoli da Arezzo, scene from “Storie di San Francesco”,

(1295-1299), fresco, Basilica Superiore di Assisi.

So, the painter has to recognize a complex shift: if the table has to look right, it has to

be drawn wrong. Instead of a rectangle, something else has to be drawn, in order to trick the

viewer’s brain into recognizing a properly positioned table. If the painter were a mathematician,

he would recognize that there are two geometries that conﬂict with each other: the geometry

of touching (the geometry of sculpture), and the geometry of seeing (the geometry of painting).

But this must have been incredibly diﬃcult for Giotto and his contemporaries back in the

fourteenth century. This diﬃculty explains why his frescos appear so odd, and why the angles

in the buildings that he depicts are so un-lifelike. It is because Giotto understood that right

angles do not always appear as right, but in fact they need to be depicted as acute or obtuse.

But he did not grasp, for example, the fact that parallel lines don’t always appear parallel. The

unrealistic sizes of the ﬁgures in his frescos are a consequence of a similar cognitive dissonance.

Giotto realized that objects that are closer to us appear larger than objects at a distance. But

he lacked the mathematics to ﬁgure out the precise proportions that should be used. As we will

see in Section 4, it will only be with Leon Battista Alberti (1404-1472) that a precise method

to address this issue will be developed.

This conﬂict is quite apparent in another great contemporary of Giotto, namely Duccio di

Buoninsegna. In his Maestà, there is a section where Duccio paints a Last Supper (ﬁgure 2).

Figure 2. Duccio di Buoninsegna, L’ultima Cena, scene from the back of the “Maestà”,

(1308-11), tempera on wood, Museo dell’Opera del Duomo, Siena

The central object in any such painting is the table, and when we look at this representation,

we have the impression that the plates on the table are on the verge of falling on the ﬂoor. The

reason for such an impression is that the table is represented not as a rectangle (Duccio like

Giotto realized that this would not have worked), nor as a trapezoid (which is the correct

representation). Rather, it is a parallelogram, in which the right angles are eliminated (as

they should), but the parallelism among sides is preserved, thus oﬀering a totally inadequate

representation. One should also look at the ceiling and the beams in the ceiling itself. In the

room, such beams are clearly parallel, and we know (we will get into more details later) that

parallel lines must be represented as converging to a point. But, as we see in the modiﬁed

picture below, while Duccio understands this, he seems not to know that all lines parallel to

each other must converge in the same point. Instead, as we see (ﬁgure 3), the internal beams

converge on the ﬁgure of Christ, while the external beams converge on the table. The outcome

is a ceiling that is clearly wrong.

Figure 3

These two examples are not oﬀered to demean these great painters, but rather to suggest

how complex it must have been for those living in the XIII and XIV century, to realize how

to go from Euclidean geometry to projective geometry. As we will see in the next sections,

this process will lead to the understanding that a fundamental mathematical truth is hidden

somewhere. And it was because of a few artists with great mathematical background, that this

was ﬁnally understood. Before we get to that point, however, we will take a brief mathematical

detour.

3. The mathematics of perspective

It is an interesting challenge to illustrate in modern terms how the eﬀort to understand vision

and painting leads to a new geometry, that mathematicians call projective geometry.

First of all, notice that the main approach of a painter to this problem does not discuss how

the eye and the brain allow us to see the surrounding world, but only how the light and the

colors reach the eye: in fact this is the environment in which a painter can mainly intervene.

It is therefore reasonable to think the eye as a point of our 3-dimensional space, and set its

position as the origin Oof a system of Cartesian coordinates (x, y, z).2

We can then base our study on the experimental fact that light rays essentially propagate

along straight lines in space, and hence assume that the vision relies upon what all the inﬁnite

rays entering Obring to the eye.3Every ray entering Obrings a colored point, which comes from

the object being viewed (possibly the sky). Therefore each ray entering the origin contributes

to the vision with a colored point. Of course Obrings no contribution to the vision.

Let us now imagine to be able to insert, between the observer, and the object which is

observed, a canvas, possibly a transparent one. Then, it is clear that the ray that joins the

object to the observer will intersect the canvas in one point and one alone. That point, with

the color that the object has, becomes like a pixel on the canvas, and the entirety of the pixels

that are generated in this way, is a faithful representation of the object itself. Note that if we

2This is clearly a simpliﬁcation that does not model the anatomical aspects of vision.

3This interpretation of vision, a revolutionary one in the Renaissance, was due mainly to ibn Al-Haytham.

also known as Alhazen, a well-known authority of the 11th century. Indeed, this visual theory was based on his

Book of Optics (Kitab al-Manazir ) [3]. On this topic see [5], [6], [7].

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The Mathematics of Painting the Birth of Projective Geometry in the Italian Renaissance Graziano Gentili Luisa Simonutti and Daniele C. Struppa

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