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ISSN 2533-2899 https://doi.org/10.26375/disegno.2.2018.13
The Perspective.
A Matter of Points of View
Federico Fallavollita
Introduction
The starting point of the research was a lesson for chil-
dren aged between 8 and 14 years under Unijunior. “Uni-
que in Italy, Unijunior was born with the ambitious goal of
bringing the youngest children closer to study important
subjects, using simple and familiar tools to the child such as
practical experience, play and entertainment. Unijunior sti-
mulates the curiosity of the child by leveraging the natural
instinct of exploration that spurs him to know the world,
finding answers to his endless questions and fulfilling his
priority needs” [1].
The occasion was an opportunity to gain some expe-
riences of perspective and stereotomy. To illustrate the
properties of the perspective and, at the same time, to
illustrate some characteristics of the human perception
of space, a plastic of polystyrene was designed, which was
then realized by means of a numerical control wire cutting
machine.
The intent is twofold: the first, to explain to children,
through the design of the room, what the drawing of ar-
chitecture is and how it differs from the commonly un-
derstood drawing; the second, equally important, consists
in demonstrating the illusory power of perspective, which
derives from the modalities of visual perception.
The first part of the article, briefly describes the lesson
given to the children, with some other insights on the
theoretical issues addressed, which obviously could not
be included in the lesson. The second part describes the
design and construction of the Ames room.
Abstract
This article presents a study that preceded a lesson for the Department of Architecture of Bologna held to children between eight
and fourteen years of age, , as part of an initiative by the University for the dissemination of science. The study presents some expe-
riences of perspective and stereotomy. Among the objectives of the research is to explain to children, through playing, the illusory
power of perspective and, therefore, the deceptions to which the human perception of space is subjected. For this purpose a small
Ames room was built, studying its decomposition in parts, in a sort of contemporary stereotomy.
In the first part of the article, the lesson given to the children is briefly presented, as well as some insights on the theoretical issues
addressed, which obviously could not be included in the lesson. In the second part, of the study, design and construction of the Ames
room are described.
Keywords: Ames Room; Descriptive Geometry; Mirror; Perception; Stereotomy
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Synopsis of the Lecture for Children
The original title of the lesson for children (The world as
it is and the world as it appears) is inspired by methods of
representation of architecture.
As is known, these methods are used to accurately descri-
be and transmit forms in space and also allow us to study
the mutual relations and the properties of these forms.
The graphic methods are: representation in double or-
thogonal projections, axonometry and perspective. These
methods have been complemented by digital representa-
tions: numerical and mathematical. We will not talk about
the latter, because their peculiar characteristics do not
change the substance of the topics covered [2]. The first
two methods, i.e. the so-called parallel projections, serve
to describe the ‘world as it is’. The third method, or per-
spective, serves to describe the ‘world as it appears’.
In fact, the architect uses the plan, the section and the ele-
vation to design and measure space. He also uses axono-
metry to understand the relationship between volumes
and the mechanism of relationships between shapes in
space. Parallel projections serve, therefore, to control the
metric and formal aspects of space in an environment that
is isotropic and homogeneous as space itself.
But man does not see three-dimensional forms as they are:
man sees space through the filter of perspective, that is,
of projection that transforms the three-dimensional world
into the images collected by the eye [3]. And so it beco-
mes vital to be able to describe space as it actually appears
to the human eye and for this purpose the architect em-
ploys perspective.
To explain what a perspective is, just think of a photo-
graph, and the functioning of the eye is analogous to that
of a camera. However, it would be wrong to think that
the phenomenon of vision is limited to these passages of
optical-mechanical nature, because, as we shall see shortly,
it is the brain that processes the images collected by the
eye and it is in the brain that the deceptions of perspective
are produced.
In truth, the perspective question we have just touched
is very controversial. We can say that there are two main
distinct schools of thought.
The first considers perspective (linear and relief) only a
scientific method to produce images (static or dynamic)
that have the right to exist only as a product of human
ingenuity, but which are not able to evoke the perception
of space.
Fig. 1. Pablo Picasso, Guitar with violin, 1913. Violin by Antonio Stradivari,
technical drawing.
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The second, on the other hand, considers perspective as a
scientific method capable not only of describing the forms
in space, but also of describing them in such a way as to
evoke, in the observer, the natural vision.
In other words, according to the first school, the perspecti-
va naturalis, i.e. the vision, and the perspectiva artificialis, i.e.
the perspective representation, are distinct and in conflict,
because vision is conditioned by peculiarities, such as the
curvature of the retina, that the perspectiva artificialis does
not simulate (but the question would take us too far and
refer to the bibliography for further study).
The second school, on the other hand, affirms that the
perspectiva artificialis, discovered in the Italian Renaissance
and developed to this day, is one and corresponds to the
natural vision of man, being perfectly able to imitate it in its
many forms, also through stereoscopy and the dynamics of
cinematographic images.
In theory, an experiment could be carried out that would
put an end to the dispute, but we are currently unable
to put it into practice [4]. The experiment would consist
in taking two photographs from a certain point of view.
The first natural photograph should be taken by our eye
and proposed in the mental processing of the observer.
The second artificial one could be taken through a camera
or built, geometrically, using the perspective method. The
author is convinced that the overlapping of the two pho-
tographs, the natural and the artificial ones, would be per-
fect. Something very similar to this experiment was done
by Filippo Brunelleschi in the early years of the fifteenth
century [Fields 2005].
The difference between the two schools of thought is
not a minor difference, as it involves a different vision of
the history of representation and architecture. There is an
important literature on this subject and the well-known
essay Perspective as Symbolic Form of Erwin Panofsky of
1927 could be considered the progenitor of the long
dispute [Panofsky 1961].
Today, the most recent studies on human visual perception
affirm that man interacts with his surrounding reality throu-
gh the five senses but sees reality through his own brain
[5]. In other words, we never see the world as it is but we
see it as our brain reconstructs it, comparing the images it
receives from the eye with the models it has memorized in
the evolutionary age [6].
The science that describes the methods of representation
is descriptive geometry. This name is due to a French
engineer and mathematician of the revolution, Gaspard
Monge [7]. For years this discipline has been taught
in schools and universities by engineers and then
mathematicians. In the last forty years, however, especially
in Italy, descriptive geometry is studied and taught only by
architects and engineers. We do not want to elaborate
Fig. 2. The mirror image and the ‘twin game’.
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Fig. 3. The real world where our ‘alter ego’ is and the reflected world where our ‘twin’ is.
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here the historical motivations of this change: let’s just say
that mathematicians have lost interest in the drawing and
power of vision, even if, in the last years, thanks to the
advent of digital technology, they show curiosity, if nothing
else in the evocative power of images, whether realistic or
symbolic.
What is the difference between the drawing of an archi-
tect and the drawing commonly understood as that of an
artist or a painter?
To respond effectively (to children) it was decided to com-
pare two representations of the same object. If we look at
Pablo Picasso’s Guitar and Violin painting we might think, at
first glance, that the painter was not good at drawing or
did not really like musical instruments [8]. Naturally, neither
of the two statements are true. Picasso was a great drawer
and was a great lover of music. The other figure shows the
technical drawing of a violin by Antonio Stradivari (fig.1).
The difference between the two drawings, that is between
that of a painter and that of an architect (or a designer),
is that the former interprets the form in a subjective way
and transmits this emotion to the observer for empathy,
while the latter, the technical drawing, measures the form
and transmits it as an objective datum, aiming to remove
any margin of ambiguity. This drawing must be transmis-
sible and have a bijective relationship with the reality it
represents: in other words, given a plan representation and
elevation, to that correspond, in reality, the object repre-
sented and, vice versa, given a real object, it is possible to
draw a plan and an elevation that match it.
Likewise, the fundamental characteristic of architectural
drawing is to incorporate the code that allows us to move
from reality to model and vice versa.
But can an architect use the free sketch to express his ideas
as a painter? Yes, even the architect can use the drawing in a
freer way, but only to follow with his mind, as with a pencil
or computer, the representation of that space that can and
must be measured and constructed.
Therefore, when designing a house, the method of dou-
ble orthogonal projections is generally used to invent and
measure space. It is also possible to accompany this study
with axonometric representations to well define the vo-
lumes, the relationships between the parts and to better
analyse the space. Furthermore, it is often advisable to
construct a physical model, to check volumes and propor-
tions in scale.
The perspective, instead, is used to study the perception of
space, or to understand how that space will be seen and
experienced, even from an emotional point of view.
The children seem to have appreciated the scientific part
of the class in which the projective principles of drawing
are described.
The operations that define drawing are the projection
and the section. To get a drawing, we have to imagine that
the sheet of paper is the picture plane, like the screen
of the cinema or the computer, and there is a projection
centre out of the picture plane. The image is obtained by
projecting the object through the centre and dissecting the
line star that derives from the picture. There may be two
Fig. 4. The size of our mirror image is always equal to the half of our real height.
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Fig. 5. Rendering of the constrained view of the Ames room digital model.
cases: in the first one the projection centre is a point that
has a finite distance from the picture; in the second case,
the projection centre has an infinite distance from the pic-
ture, i.e. it is very far : it is the direction of the projecting
straight lines.
In the first case the drawing is a perspective, otherwise
called the central projection. In the second case, i.e. the
parallel projection, the drawing can be a plan, a prospect
or an axonometry.
In conclusion, the former case serves to study the world
as it appears and the latter serves to study the world as it
is. A good architect is able to manage both situations well.
However, the perspective view can be misleading, because
sometimes the world is not exactly what it looks like!
What is the perspective machine we use every day?
The first answer that comes to mind might be the mobile
phone camera. Well no, there is a perspective machine that
we use every day from before the advent of mobile pho-
ne; this machine is the mirror (fig. 2). The mirror recreates
a parallel world beyond the glass, still three-dimensional,
which is exactly symmetrical of the real world. In the mor-
ning, when we wash or get dressed, we all usually mirror
each other. Have you ever wondered how big your mirror
image is?
Well, this image is no smaller when we move away and
neither is it bigger when we approach the mirror. Our
reflected image is always the same size and measures a
specific quantity. To understand the problem we can ima-
gine to observe our alter ego in front of the mirror and
reconstruct the virtual world that is created beyond the
mirror (fig. 3). There are two worlds: in the real one there
are us and in the virtual one there is our twin. The only
difference is that if we are right-handed our twin will be
left-handed or vice-versa. In other words, in the symmetry
of the mirror the right and the left exchange roles. If we
look at the figure, it is easy to see that the distance betwe-
en us and the mirror, and between this and our twin is the
same and does not change with our distance from the
mirror : if we approach the mirror, even our twin will ap-
proach the mirror in equal measure. If we now look at the
projective triangle that forms our image, it is equally easy
to see that the height of our image does not change with
the change of the distance from the mirror : this height is a
constant and is always half of the real one. To conclude, the
measurement of our mirror image is always equal to half
our height (fig. 4). To test this surprising truth, all we need
to do is measure our own image on a mirror : two small si-
gns are enough, above and below, to prove that this image
will always be half our height whether we move away or
approach the mirror.
The second experience, which I want to propose, to test
the illusory power of the perspective vision is Ames room.
Fig. 6. Fig. 6. Projective construction of the Ames room model.
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Fig. 7. The mathematical model of the Ames room.
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Adalbert Ames Jnr. was an American ophthalmologist
expert in optics. He is known for his experiments on visual
perception which explained some fundamental principles
of the visual perception. Demonstrations began in Hanno-
ver in 1938 and were carried out with the University of
Princeton. These experiments are still reproduced in many
departments of psychology and museums worldwide.
Among these experiments, the best known and particularly
peculiar is the Ames room.
The existence of this space was theorized for the first
time by Fiermann von Helmholtz in 1866. He realized that
objects of a multitude of different shapes and sizes can re-
turn to the eye the same image , and that a distorted room,
constructed to return to the eye the same image of a
rectangular room, may result in the perspective view iden-
tical to a regular room. The merit of Ames was to have built
this distorted room and to have included two subjects in
the room, studying its effects also on a group of volunteers.
Observing the space of the room from the special hole,
one gets the impression of being in front of a perfectly
regular room. But if we put two subjects inside the room
or facing the two windows at the back, we realize that so-
mething is wrong (fig. 5). One seems to be much bigger
than the other or, conversely, if they change place, the they
also change size.
We are so used to perceiving size and space in a certain
way, that at first we cannot see that space is deformed and
that we are not facing a regular space but a trapezoidal
room. This space is specially constructed according to the
projection centre which is positioned exactly in the centre
of the hole. The illusion of finding ourselves in front of a
perfectly regular space is disorienting; we just need to ob-
serve the space on the opposite side to immediately realize
the trick.
But the most unusual feature of this experience is that even
if the trick is known, the illusion does not lose its effecti-
veness at all: we cannot see the distortion of the room
and we continue to perceive the two subjects one much
smaller than the other or, vice versa, one much larger than
the other. Ames is convinced that there is a memory of
perception that conditions human perception, that is, the
habit of living in regular spaces influences our vision and
our perception. There are other theories and explanations
in this regard but until now a conclusive and convincing the-
ory that can explain this phenomenon well has not been
formulated [9].
As far as we are concerned, these two experiences tell
how important it is for an architect to know how to obser-
ve and represent shapes in space, both in their real form
and in their appearance.
Fig. 8. Images of the constrained view of the polystyrene model of the Ames room.
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The Construction of the Ames Room
Compared to the many Ames room models built in mu-
seums, this Room was designed in parts and made of a
single material. This choice, as mentioned, was dictated by
the need to test the use of polystyrene and make the mo-
del safe for children.
To model the Ames room, the following procedure must
be followed.
You design a regular room as you like, that is, a rectangular
prismatic space that has all the walls perpendicular to each
other. To make the illusion more effective you can design
two regular openings on the front wall of the room. On
the opposite side, on the anterior wall, chose the position
of the optical hole and projection centre of the transfor-
mation. It would be a good idea to choose the height of
the projection centre at the height of a man’s point of view.
In the case study examined this height was calculated at
the height of a child.
In order to construct the transformed prism, or Ames
room, it is necessary to respect the planarity of all the
faces. This geometric characteristic is respected if the four
faces perpendicular to the front plane, once the prism has
been transformed, belong to a pyramid which has the apex
on the axis perpendicular to the front wall passing through
the projection centre (fig. 6). In fact, only respecting this
geometric condition the four faces will all be flat: otherwi-
se, one or more of the faces will turn into hyperbolic
paraboloids.
Ultimately, in order to easily control the transformation,
you can draw the pyramid axis first. The vertex of the
pyramid can be chosen on this axis: the more the ver-
tex will be close to the anterior wall, the more impor-
tant the projective transformation will be and vice versa.
You choose a vertical edge of the front wall to maintain
a fixed position. You select an upper (or lower) vertex of
the other vertical edge and project it (from the projection
centre) until you meet the straight line passing through the
apex of the pyramid and the upper (or lower) vertex of
the anterior wall. This sets the position of the transformed
vertical corner. The remaining side faces must all belong to
the apex of the pyramid, or pass through that point; while
the anterior and front faces are sections, always flat, of the
pyramid. To construct the windows of the wall placed in
front of the observer it is possible to project (from the
centre of projection) on the transformed wall itself the
vertices of the edges. All edges parallel to the axis of the
regular prism are transformed into lines that still belong to
the apex of the pyramid.
The model was designed with ashlars so that it can be
assembled and disassembled (like a dry-stone wall) in a
short time.
To construct the room, a mathematical model was first
created and from this model the measurements of the in-
dividual ashlars were obtained (fig. 7). The scale of the mo-
del was dictated by two factors. The first was to make the
children protagonists of the experience and to do so the
model of the room had to be large enough. The second
was dictated by reasons of external space, i.e. the model
had to be small enough to easily enter inside the entrance
space of the main hall of the Psychological Faculty of the
University of Bologna. The final model is a room of about
two meters by three, inside of which it is not possible to
walk (as in the original Ames room) but it is possible to
look out of the windows and observe the space directly
(fig. 8).
Another fundamental choice was to build the model
entirely in polystyrene in order to experiment with the
construction of the ashlars using a wire cutting machine;
moreover, as I already said, polystyrene is a light and safe
material for children [10].
Each ashlar has been designed and modelled conside-
ring the projection centre: observing the space from the
projection centre , the ashlars appear to divide the space
in a regular way according to the conventional horizontal
and vertical directions. In reality, the ashlars are all skewed
and the faces that form such parts are not perpendicular
to each other.
Initially we tried to realize the skewed segments making
only two cuts. This solution, however, was immediately dis-
carded after the first attempt. In fact, to obtain this result,
the machine for processing the oblique cut had to move
the two motors that carry the wire independently; doing
this, however, the wire stretched too much, until it breaks.
Consequently, we decided to let the motors work to-
gether and in parallel so as to avoid breaking or loosening
the wire.
To obtain this result, we had to calculate the angle betwe-
en the planes that form the various individual ashlars and
the exact size of the regular volume that enclose each pie-
ce (fig. 9).
Starting with a piece of regular polystyrene, equal to the
overall volume of the single ashlar, the various cuts were
made separately, each time placing a worked piece in the
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machine chamber according to the calculated angles. The
cuts were also designed to fit the various dry-ashlars to-
gether.
The final model is dry-mounted in about twenty minutes
and can be dismantled just as easily to transport it (fig. 10).
The floor has been designed as a checkerboard to accen-
tuate the illusion of a regular space.
Conclusions
Through the game and the direct experience of space, we
illustrated the illusory power of perspective and experien-
ced the stereometric construction of a small Ames room.
The children’s response was positive, that is they seemed
to have understood and appreciated the experiments on
perspective. The lesson on the world as it is and the world as
it appears will be repeated next year for the new edition
of Unijunior 2018.
In the future, the idea is to be able to design and
construct other models that can stimulate the study of
visual perception and space. Regarding the question of
perspective and vision, there are still open questions that
would be interesting to investigate. Perspective continues
to be a stimulating and mysterious theme: each time it is
dealt with, it reveals its elusive and profound nature that
has ancient roots. Today, living in the digital age, we have
the opportunity to simulate the construction of various
models and can study their potential. However, the need
to physically build models that allow direct experience of
those deceptions is even more surprising. Perhaps one
day we will be able to definitively unravel the question of
perspective artificialis and naturalis. Or the case will simply
remain a matter of perspective points of view. However, we
cannot forget that the experiences and research described
here have been realized thanks to the geometric theory of
perspective. The fact that the two experiments, the twins
and the Ames room, are effective seems to bring a further
element in favour of the existence of a single perspective
that corresponds to the human sensible vision.
Fig. 9. Schematic illustration of the cutting stages for the construction of a
polystyrene ashlar.
Fig. 10. Building of the polystyrene model at the entrance hall of the Aula
Magna of Psicology faculty, University of Bologna.
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Acknowledgments
I would like to thank the architect and technical manager Davide Giaffreda
and the collaborator Marika Mangano for the indispensable help in the de-
sign and construction of Ames Room’s polystyrene model. The model was
built entirely with the tools of the Department of Architecture; in particular,
the model was realized within the Laboratory Lamo of the Department
of Architecture of Cesena, scientific director Francesco Gulinello. Thanks
to Fabrizio Ivan Apollonio and Riccardo Foschi for the publication of the
photos. Thanks also to Valentina Orioli who has supported the initiative.
Notes
[1] The internet website of the Unijunior association is: (accessed 2018, February 20).
[2] The digital methods are basically two: the method of mathematical re-
presentation and the method of numerical representation. The mathema-
tical method describes continuously and accurately the geometric shapes
in space. NURBS mathematics is the most widely implemented to describe
curves and surfaces in mathematical modeling programs. On the other
hand, the numerical or polygonal method describes the shapes in space
in a discrete and approximate way. Polygonal shapes or mesh shapes are
used to describe curves and surfaces in polygon modeling programs. Of
course the two methods have advantages and disadvantages that make
them suitable for some purposes. Mathematical modeling is generally used
in the design phase and to accurately construct and measure shapes in
space. In this sense we can say that mathematical representation is the
equivalent of the two parallel projections in classical methods, namely the
representation in plan and elevation and axonometry. While numerical re-
presentation is generally used to visualize and formally study shapes in spa-
ce, ie to construct static and dynamic perspective and static and dynamic
rendering. In this sense we can say that numerical representation is the
equivalent of perspective in classical methods. Today we speak commonly
of BIM or of the generative parametric representation (for example the
use of visual programming languages like Grasshopper). The latter digital
representations can be considered as digital representation techniques and
not real methods. They do not change the geometric nature of the objects
described; which can be mathematical, polygonal and hybrid. Furthermore,
both techniques can be used to obtain accurate or approximate models.
Unfortunately, at the state of the art there is no univocal consensus on
the classification of digital methods at national and international level. The
reason is naturally due to the novelty of these methods and techniques and
the rapid development that these techniques are having over the years.
[3] In this context ‘perceiving’ has the meaning proposed by Italian Dictio-
nary Zinagarelli 2018, that is: ‘grasp the data of reality through the senses’.
To avoid misunderstandings in this article I will use the terms ‘to see’ and
‘perceive’ strictly with the first meaning reported in the Italian language
dictionary. It is natural that man is able to imagine and see space also in
axonometry (and in double orthogonal projections). For some cultures
such as Asian, in particular the Chinese and Japanese, the parallel projection
method has been the main method of representing the surrounding world.
And perhaps it is no coincidence that when man designs and analyses
space, he is naturally led to use and prefer parallel projections.
[4] There is an episode of a British television series, Black Mirror, released
in 2011 in which a situation is described that recalls the experiment men-
tioned. The third and final episode of the first season, entitled Hazardous
Memories, is set in an alternative reality, where most people have a grain
implanted behind the ear, which records everything that is done, seen or
heard. This allows memories to be played in front of the owner’s eyes or
on a screen through a process known as ‘re-do’, just like videos. It seems
that this grain is implanted since newborns, but that a person can decide
to have it removed.
[5] Here the verb ‘to see’ is to be understood in a broader sense.
[6] In this sense it is sufficient to think about how natural it is for man to
imagine and read space in axonometry. Beau Lotto, in his essay [Lotto
2017], while not making any direct reference to the perspective, describes
numerous examples that demonstrate how man reconstructs in his mind
what he sees. In conclusion, even the latest theories of perception do not
seem to help us on the question of perspective. Nevertheless, the scientific
theses supporting the existence of a naturalis perspective different from
the artificialis one are not conclusive [Gioseffi 1957].
[7] To deepen the notions and the history of Gaspard Monge and descrip-
tive geometry refer to Cardone [Cardone 2017]. With regard to descripti-
ve geometry and the ‘scuola romana’ refer to Migliari [Migliari 2010].
[8] Picasso’s painting is from 1912. While the technical drawing of the violin
refers to one of the instruments construct by the well-known Italian luthier
Antonio Stradivari (1644-1737).
[9] Gregory says that with the Ames room it is possible to put in place an
experiment that is perhaps even more disturbing and it is able to challenge
a fundamental law of physics. Simply take two objects, like two balls, and
drop them. We will then see the two spheres falling at different times
defying the gravitational law. Even in this case, at first glance, the impression
is to be in front of objects that do not respect the same physical laws and
we can not perceive that the height from which the two objects were
dropped were different [Gregory 1994].
[10] The machine of the Laboratory Lamo (Laboratorio modelli di Ar-
chiettura) of University of Bologna is the model 120P Box of Nettuno
Sistemi; (accessed 2018,
February 20).
Author
Federico Fallavollita, Department of Architecture, University of Bologna, federico.fallavolllita@unibo.it
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