Is It a Cube? Common Visual Perception of Cuboid Drawings

Abstract

A cube is one of the most fundamental shapes we can draw and can observe from a drawing. The two visualization methods most commonly applied in mathematics textbooks and education are the axonometric and the perspective representations. However, what we see in the drawing is really a cube or only a general cuboid (i.e., a polyhedron with different edge lengths). In this experimental study, 153 first-year ( 19–20-year-old) students, two-thirds of them being female, were asked to interactively adjust a cuboid figure until they believe what they see is really a cube. We were interested in how coherently people, who are actually students of arts studies and engineering with advanced spatial perception skills in most cases, evaluate these drawings.

What we have experienced is that for most people there is a common visual understanding of seeing a cube (and not a general cuboid). Moreover, this common sense is surprisingly close to the conventions applied in axonometric drawings, and to the theoretical, geometric solution in the case of three-point perspective drawings, which is the most realistic visualization method.

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Article
Is It a Cube? Common Visual Perception of Cuboid Drawings
Miklós Hoffmann 1,* and László Németh 2,*


Citation: Hoffmann, M.; Németh, L.
Is It a Cube? Common Visual
Perception of Cuboid Drawings. Educ.
Sci. 2021,11, 577. https://doi.org/
10.3390/educsci11100577
Academic Editor: Liudmila Liutsko
Received: 19 August 2021
Accepted: 18 September 2021
Published: 24 September 2021
Publisher’s Note: MDPI stays neutral
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Licensee MDPI, Basel, Switzerland.
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conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
1Faculty of Informatics, Institute of Mathematics and Computer Science, Eszterházy Károly University,
Leányka Str. 4, 3300 Eger, Hungary
2Faculty of Wood Engineering and Creative Industries, Institute of Informatics and Mathematics,
University of Sopron, Bajcsy Zs. Str. 4, 9400 Sopron, Hungary
*Correspondence: hoffmann.miklos@uni-eszterhazy.hu (M.H.); nemeth.laszlo@uni-sopron.hu (L.N.)
Abstract:
A cube is one of the most fundamental shapes we can draw and can observe from a drawing.
The two visualization methods most commonly applied in mathematics textbooks and education
are the axonometric and the perspective representations. However, what we see in the drawing
is really a cube or only a general cuboid (i.e., a polyhedron with different edge lengths). In this
experimental study, 153 first-year ( 19–20-year-old) students, two-thirds of them being female, were
asked to interactively adjust a cuboid figure until they believe what they see is really a cube. We were
interested in how coherently people, who are actually students of arts studies and engineering with
advanced spatial perception skills in most cases, evaluate these drawings. What we have experienced
is that for most people there is a common visual understanding of seeing a cube (and not a general
cuboid). Moreover, this common sense is surprisingly close to the conventions applied in axonometric
drawings, and to the theoretical, geometric solution in the case of three-point perspective drawings,
which is the most realistic visualization method.
Keywords:
axonometric drawing; perspective drawing; visualization in education; visual perception;
common understanding of drawings; cube
MSC: 97C30; 51N05
1. Introduction
Most people with average spatial abilities can draw a cuboid, and, vice versa, can ob-
serve a cuboid from a line drawing. Of course, depending on the personal drawing abilities
and cultural background, this drawing as well as the perception and understanding of this
drawing can be diverse (see, e.g., [
1
]). Having that said, in spatial line drawings of artistic
pieces, design materials and mathematics illustrations, we can distinguish fundamentally
two basic approaches: the axonometric (or oblique) type, when the parallel sides of the
cube will also be parallel in the drawing, and the perspective type, when—due to the
classical rules of perspectivity—images of lines of parallel edges meet at one point (the
so-called vanishing point) in some or all of the three directions.
It is by no means trivial whether a drawing is well understood by the observer
and if this is a correct drawing to provide the specific mathematic knowledge or artistic
message [
2
–
4
]. This holds already in the case of a simple cube [
5
,
6
]. Accurate drawing and
understanding of a drawing is of utmost importance to effectively support mathematical
performance [
7
], especially in geometry [
8
], but its interdisciplinary impact is also crucial.
Understanding and representing spatial relationships are fundamental skills in many fields,
and the development of these skills can have a mutual impact on various disciplines from
pure mathematics to arts [9].
Our central question in this study is whether a correct line drawing depicts a general
cuboid (i.e., a box-like polyhedron with six rectangular sides but different edge lengths
in the three dimensions) or indeed a cube, that is, a cuboid with three spatial dimensions
Educ. Sci. 2021,11, 577. https://doi.org/10.3390/educsci11100577 https://www.mdpi.com/journal/education

Educ. Sci. 2021,11, 577 2 of 17
of the same length. The proportions of the three edge-lengths of the image of a cuboid,
including the foreshortening or depth ratio, can be chosen in the drawing in many different
ways (see Figures 1and 2); therefore, there is no unique correct way of drawing a cube.
However, does (and if yes, when) a specific drawing really depict a cube and not a
general cuboid? And can we observe this difference? In this paper, we look for an answer
to whether there is a common visual sense in this observation, whether there is a specific
ratio of width, height and depth of the figure along the three dimensions when people
generally feel that they are really watching a cube.
Figure 1.
Five different axonometric images of a cuboid from the same direction of view, with fixed edge lengths in
two directions (“height” and “width”), but an alternated foreshortening in the third direction (“depth”)—which one best
resembles a cube?
Figure 2.
Five different perspective-like images of a cuboid from the same viewpoint—which one best resembles a cube?
Which one is the real perspective image of a cube?
To do this, we conducted an experiment with an interactive computer model in which
the width and height of the cuboid drawing are fixed, but the foreshortening in the third
direction (the “depth”) can be changed by the user with simple means [
10
]. This is an
innovative investigation because although students are taught to see and draw the represen-
tation of the cube, no previous investigation has been carried out on the visual perception
of these drawings. We were interested in how coherent people, who were actually first-year
students with advanced spatial perception skills (c.f. [
11
]), but without specific knowledge
of descriptive geometry principles, judge this situation, and to what extent the set foreshort-
enings coincide or differ for different students. In the case of axonometric drawing, we have
also compared the selected depth of the cube to the foreshortening conventionally used
in engineering drawings and figures in educational materials. In the case of perspective
drawings, we have examined how these positions are coherent to the exact solution of this
problem in the geometric sense (what we mean by the exact solution here will be discussed
in detail in Section 2).
It is essential to understand what people will perceive from a geometric spatial
drawing on a blackboard or a canvas, which is related to the question of perceiving extents
and distance (for an overview of this question, see [12]).
The main outcome of this study is that there is a somewhat common sense of observing
a cube (and not a cuboid), but there is a small number of people who have a significantly
different view of these drawings. Moreover, while this common sense highly coincides
with the conventional and geometrically correct drawing and selection of foreshortenings,
in some particular cases, there is a significant discrepancy between perception and con-

Educ. Sci. 2021,11, 577 3 of 17
vention. This all must be taken into account when a figure, an illustration or an artistic
drawing is used to explain spatial relationships.
In the following section, we briefly review the geometric background of the two types
of representation. In Section 3, we describe the data collection and analysis. In Section 4,
results of the experiment are presented in detail. Conclusions close the paper in Section 5.
2. Geometric Background
Consider first the so-called axonometric representation, which plays a crucial role
in many applications in knowledge production (for a detailed discussion of this method,
see, e.g., [
13
]). An axonometric image of a cube or any general cuboid is fairly easy to
produce. For simplicity, imagine that the three extensions of the cuboid are in the
x
,
y
,
and
z
axes mutually perpendicular to each other, with one vertex at the origin. Let us fix
a point, the image of the origin, in the drawing plane and three half-lines starting from
it, the images of the three axes. Taking an additional vertex on each of these half-lines,
separately, the image of the cuboid is already uniquely defined, we only need to draw
parallel segments from the corresponding points to finalize the figure. If the spatial cube is
considered to be of unit edge-length, then it is evident that in the drawing, not all edges of
the image will necessarily be of unit length, but the parallel edges will be the same length
in each direction. The ratio of the original edge length to the edge length shown in the
drawing is characterized by the foreshortening values
qx
,
qy
, and
qz
in the three directions,
separately. These values can freely be chosen by the person who prepares the drawing.
The axonometric representation is closely related to parallel projection. While in the
former, as we have seen above, one can draw the image with a great deal of freedom after
observing the spatial shape (cuboid), and in parallel projection, the image of the spatial
cuboid is prepared through a projection onto the plane with rays parallel to a predefined
direction. One of the central questions in axonometry is that if we draw a cuboid with
arbitrary axial directions and arbitrary foreshortenings
qx
,
qy
, and
qz
in the above way, then
there exists a spatial cube in some position whose image through a parallel projection with
a well-chosen direction will exactly coincide with the original drawing. Pohlke’s famous
theorem gives a positive answer to the question: no matter how we draw the axonometric
image of the cuboid, it can be not only the image of a general cuboid but also a parallel
projection of a cube, well-positioned in space [14].
Thus, according to Pohlke’s theorem, any axonometric image of a cuboid can be
considered a cube. However, it is clear that in the case of a very elongated image, most
observers associate it with a general cuboid rather than a cube. In our experiment, we
examined whether there was any “common sense” about the axonometric image of a
cuboid being a cube.
Persons were given an axonometric drawing depicting a cuboid where the foreshort-
ening was fixed in two directions (the edge lengths of width and height were predefined in
the image), but in the third direction, persons were able to freely adjust the edge length
in an interactive way. Our request was to adjust this third-direction foreshortening until
they themselves observe the image as a cube, and not as a cuboid with different side
lengths. Here, we emphasize once again that in the case of an axonometric mapping, there
is no single solution, no specific “perfect” drawing according to Pohlke’s theorem. That
is, in principle, any foreshortening is equally good from a geometric point of view, in all
cases the drawing can be considered as an image of a cube. However, as our results show,
the foreshortenings set by testing persons, with a few exceptions, were clearly culminated
around a single value. All this means is that for each setting, there is some kind of common
sense as to whether this image represents a cube.
In the case of perspective images, the story is a bit different. The essence of the per-
spective representation is that the images of lines of parallel edges in space converge at a
point (the so-called vanishing point) in the drawing. This method of graphical represen-
tation results in a much more realistic figure than the axonometric image and has deeply
influenced art and science since the Renaissance.

Educ. Sci. 2021,11, 577 4 of 17
Imagine a cuboid again with three edges in the directions of
x
,
y
and
z
axes perpendic-
ular to each other and one vertex in the origin. Let us define a point (the image of the origin)
in the drawing and the three half-lines starting from it; hence, we have fixed the image in
the three directions. Contrary to the axonometric drawing, however, on these half-lines we
must define not only one additional vertex (the image of the unit point) but also, along the
directions, we must determine the points at which the lines parallel to the given direction
intersect each other. These latter points are called vanishing points. If we select such a
point in each of the three directions, the image will be called a three-point perspective
(see points
Vx
,
Vy
,
Vz
in perspective figures, but if we define these points in only one or
two directions, while lines in the other directions remain parallel, we obtain a one-point
and two-point perspective drawing, separately. The perspective image of the cuboid thus
becomes uniquely defined by selecting the origin, the axes (half lines), the unit points,
and the vanishing points: if these are given, the image of the cuboid is uniquely determined
and can be easily drawn.
The perspective representation is closely related to central projection. If a spatial
cuboid is projected from a center to the image plane, then the result is a perspective
image of the cuboid. A question similar to the axonometric case can be asked: if one
prepares a perspective drawing of a cube specifying the origin, the three arbitrary directions,
the unit points and vanishing points along these directions, is there a cube in space and
an appropriate center of projection, from which we obtain the given image by projecting
the cube from that center onto the plane? Whilst the answer was always positive in
the axonometric case for any drawing, the answer is negative in almost every case of a
perspective drawing. The one-point perspective can always be considered as an image of a
centrally projected cube. However, given a two- or three-point perspective drawing, there
are a number of conditions that must be met in order to find a spatial cube and a projection
center from which the cube can be projected to obtain the given image. For an overview of
these conditions, see, e.g., [15–17].
In other words, for most perspective drawings, there is no appropriate spatial structure
from which we can receive an analogous visual perception. If we fix the origin, the three
directions, the (two or three) vanishing points and two out of three unit points, as it
happened in our experiment, then the mathematical conditions mentioned above allow us
to determine the only position of the third unit point, for which such a central projection
exists. In all the other cases, the perspective drawing can be correct in itself, but cannot be
the result of a central projection.
We have conducted the experiment related to perspective drawings similarly to the
case of axonometric drawings. Persons were given a perspective image depicting a cuboid
where the vanishing points were given in all three directions, the side length was also fixed
in two directions, but in the third direction, persons were able to adjust the side length at
will. Our request was to adjust this third direction length (“depth” of the cuboid) until they
themselves feel they see the image of a cube and not a cuboid with different side lengths.
As we have discussed above, unlike axonometry, not all positions here are correct in a
geometric sense, i.e., there was practically one single position in each figure that actually
resulted in an image that could be the central projection of a cube. This single correct
position of the third unit point has been calculated and visualized as a green unit point in
each of the perspective figures with two or three vanishing points.
In our experiment, the foreshortenings set by students in most of the cases showed
a surprising coincidence in the case of perspective drawing as well. Moreover, with a
good approximation, the adjusted vertex was positioned close to the only geometrically
correct position. All this means, surprisingly, is that there is common sense in terms of the
perception of perspective images as well as whether this image represents a cube.
3. Method of the Survey
For our investigation, we created a twenty-drawing-questions online test. Test figures
were created by GeoGebra. Given a slider tool, students were able to find the best position

Educ. Sci. 2021,11, 577 5 of 17
of the adjustable side length of the cube in each of the 20 drawings. The students received
a wide range of various axonometric and perspective drawings of cuboids, 5-5 tasks of
axonometric, 1-point perspective, 2-point perspective and 3-point perspective drawings
(all of these drawings can be seen and tested on our website [10]).
We asked first-year (19–20-year-old) bachelor students of Arts and Engineering from
two Hungarian universities, University of Sopron and Eszterházy Károly University, Eger.
We received 153 filled tests from 107 students of Arts (20 males and 87 females) and 46
students of Engineering (34 males and 12 females) in May of 2020. None of the students
had been reported to have any vision deficiencies, and they all had normal or corrected to
normal vision.
They had to adjust the depth of various axonometric and perspective drawings of
a cuboid until they felt they could see a cube. By moving the slider, students fixed the
depth between the two extremal positions [
10
], which was transformed to a value from 0
to 100 in every case. They had to send back these 20 numbers. These scores can easily be
transformed to foreshortening values in axonometric and perspective drawings as well
(see the next section).
We collected the foreshortening values in a table and analyzed them by the software
Statistica [
18
]. We used its base statistics, normal distribution fitting, box and whisker plots
and cluster analysis modules. We calculated the mean, standard deviation, median, modus,
and interquartile range for each case, and we applied a cluster analysis for the participants.
4. Results and Discussion
In this section, we discuss our results, and we present most of the drawings of a cube
with an additional box-plot of the selected foreshortenings.
4.1. Axonometry Test Results
In the following axonometric figures, the lengths of the edges of the cuboid in direction
y
and
z
(width and height) are set to be 1, while in direction
x
(depth), the length of the
edge can be freely adjusted by the participants. Thus, the foreshortenings of these figures
are qy=1, and qz=1, while qxis the free parameter.
We provided a relatively wide range for
qx
to be adjusted, so that
qx∈[
0.2, 1.2
]
in the axonometric figures, where
qx=
0.2 means an extremely “thin” cuboid, while
qx=1.2 yields
a very “thick”, elongated solid. To avoid confusion with various foreshort-
enings, these values have been transformed to the unified parameter
s
from the
[
0, 100
]
interval in each test figure. From GeoGebra, we received the value
s
of the chosen length of
the edge in direction x, where
s=100 qx−20, s={0, 1, 2, . . . , 99, 100}
from which the inverse function is
qx=s+20
100
. The relationship between the received value
sand the foreshortening qxcan also be seen in Table 1.
Table 1. Scores and foreshortenings.
s0 10 20 30 40 50 60 70 80 90 100
qx0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
In our survey, we provided five different axonometric images (Axo01–05) to the students.
Note that these figures represent classical axonometric views, frequently applied
in various scientific fields, mostly in engineering. In these applications, there is a usual
standard value of foreshortening, but this is purely a convention of these fields. In other,
less frequently applied cases, there is no such convention. The participants were not aware
of these engineering conventions.
In the first axonometric task, we provided a cuboid in the so-called Cavalier-axonome-
try: the edges of the cuboid in the
y
and
z
directions are perpendicular in the image.

Educ. Sci. 2021,11, 577 6 of 17
Students were able to adjust the foreshortening in the third,
x
direction. After the evaluation,
we found that the mean of the chosen foreshortenings is
qx=
0.54, the median value of the
answers is
qx=
0.52. In Figure 3, the cube is shown with this latter foreshortening—this is
the drawing for which most students believe they see a cube (and not a general cuboid).
The engineering convention is 0.5.
On the right side of the figure, the box-plot of the most important foreshortening test
data has been displayed as well: 75% of the responses are in the range
[
0.48, 0.6
]
, which
is its interquartile range or IQR (green rectangle parallel to axis
x
). This is a remarkably
narrow interval. The so-called non-outlier range is
[
0.3, 0.75
]
. This range contains most
of the participants. The remaining few results are the extremes, which will be discussed
in Section 4.3.
0.52
0.48
0.6
0.3
0.75
x
y
z
Figure 3.
The first axonometric task (Axo01) with a box plot of the test results. The cube is visualized
with the mean value
qx=
0.54 of selected depths. Non-outliers minimum:
qx=
0.3, non-outliers
maximum: qx=0.75, median: qx=0.52, IQR between qx=0.48 and qx=0.6 (see also Table 2).
We have also evaluated the distribution of the variable of foreshortening, and we
found that it follows a normal distribution (see Figure 4). This is typical in all the other
cases as well.
0.24 0.30 0.36 0.42 0.48 0.54 0.60 0.66 0.72 0.78 0.84 0.90 0.96 1.02 1.08 1.14 1.20
Figure 4. Distribution fitting dialog for Axo01.
In the following four figures (Figures 5–8), we provide analogous data and views of
further axonometric tasks. Data are also summarised for the axonometric tasks in Table 2,
where the conventional foreshortenings are also presented in the first row.

Educ. Sci. 2021,11, 577 7 of 17
0.7
0.65
0.74
0.53
0.86
Figure 5.
The axonometric task Axo02 with mean
qx=
0.7. Non-outliers minimum:
qx=
0.53,
non-outliers maximum: qx=0.84, median: qx=0.7, IQR= [0.65, 0.74](see Table 2).
Figure 6.
The axonometric task Axo03 with mean
qx=
1. Non-outliers minimum:
qx=
0.88,
non-outliers maximum: qx=1.15, median: qx=1.02, IQR= [0.98, 1.05](see Table 2).
Figure 7.
The axonometric task Axo04 with mean
qx=
0.84. Non-outliers minimum:
qx=
0.71,
non-outliers maximum: qx=0.95, median: qx=0.83, IQR= [0.79, 0.88](see Table 2).
Figure 8.
The axonometric task Axo05 with mean
qx=
0.49. Non-outliers minimum:
qx=
0.27,
non-outliers maximum: qx=0.7, median: qx=0.5, IQR= [0.43, 0.55](see Table 2).

Educ. Sci. 2021,11, 577 8 of 17
Table 2. Mean and interquartile range (IQR) of selected foreshortenings of axonometric tasks.
Foreshortening (qx) Axo01 Axo02 Axo03 Axo04 Axo05
usual/convention 0.5 0.5 1 0.8 0.5
mean 0.54 0.7 1.0 0.84 0.49
standard deviation 0.1 0.07 0.12 0.06 0.09
minimum 0.26 0.5 0.2 0.71 0.27
non-outlier minimum 0.3 0.53 0.88 0.71 0.27
25% 0.48 0.65 0.98 0.79 0.43
median 0.52 0.7 1.02 0.83 0.5
modus 0.5 0.7 1 0.79 0.5
75% 0.6 0.74 1.05 0.88 0.55
non-outlier maximum 0.75 0.86 1.15 0.95 0.7
maximum 0.8 0.95 1.2 1.0 0.8
Box-plots of the axonometric test figures are also provided separately in Figure 9. Here
we highlighted the mild outlier and extreme outlier values as well. For example, in the
case of the third axonometric task, there were two upper outliers and one under the outlier
(small circles). They are farther from the boundaries of IQR than 150% of the length of IQR
but do not go farther than 300%. Moreover, we found that few students provided extreme
values, which will be discussed in Section 4.3.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Figure 9.
Box plots of axonometric test results. Median, interquartile range (IQR), non-outlier range,
outliers and extremes are shown.
4.2. Perspective Test Results
Students received 5-5 tasks of one-point, two-point, and three-point perspective
drawing, separately. Let
Ex
,
Ey
,
Ez
be the unit points,
Vx
,
Vy
,
Vz
be the vanishing points
of
x
,
y
and
z
axes, respectively, while the origin is denoted by
O
.
Vx
always exists (i.e., it
is in finite position); therefore, we can compute the so-called depth ratio (briefly depth)
rx=OEx
OVx
,
rx∈[
0, 1
]
of the perspective image cube along the
x
-axis by the ratio of the edge
length of direction
x
of the cube and the distance of the origin and the vanishing point
Vx
.
In the case of two- and three-point perspectives, one can analogously calculate the depth
ratio along the other two axes by
ry=OEy
OVy
and, in the case of the three-point perspective,
rz=OEz
OVz.
In our survey, the only point that the students were able to adjust was
Ex
, thus
automatically changing the depth ratio
rx
. We restricted the value of
rx
to the range

Educ. Sci. 2021,11, 577 9 of 17
rx∈[
0.05, 0.75
]
(only in the case of Pers10 to
rx∈[
0.01, 0.26
]
). Just as in the case of
axonometric tasks, we transformed this depth ratio to a parameter
s∈[
0, 100
]
, and from
GeoGebra we received the value sof the chosen length of the edge in the following form:
s=100 ·100
70 (rx−0.05),s={0, 1, 2, . . . , 99, 100}.
The inverse function is
rx= (
7
/
1000
)s+
0.05. (In the case of Pers10, we have
s=400(rx−0.01)and rx=s/400 +0.01.)
4.2.1. One-Point Perspective Test
In the five tasks of one-point perspective drawings, students were able to adjust point
Ex
and thus the depth ratio along the
x
axis. As we have discussed above, the crucial
difference between the one-point perspective and other perspectives is that here there is
not any convention of depth ratio nor a single correct value from the geometric point of
view. This method has been frequently applied from the early Renaissance in classical
paintings, such as The Cestello Annunciation by Botticelli, The School of Athens by Raphael,
or The Last Supper from Leonardo, just to name a few, but with various shortenings. As a
consequence, in Table 3, where the outcomes of the test of one-point perspectives are
summarised, we did not provide any conventional or expected value.
Table 3.
Mean and interquartile range (IQR) of the in depth ratio in the case of one-point perspec-
tive tasks.
Ratio (rx) Pers6 Pers7 Pers8 Pers9 Pers10
mean 0.23 0.21 0.33 0.21 0.1
standard deviation 0.05 0.05 0.07 0.05 0.02
non-outlier minimum 0.11 0.12 0.16 0.11 0.06
25% 0.19 0.19 0.29 0.18 0.09
median 0.23 0.21 0.33 0.22 0.1
75% 0.26 0.24 0.4 0.25 0.11
non-outlier maximum 0.33 0.31 0.5 0.33 0.14
Perhaps this is also the reason why the interval between the non-outlier minimum
and maximum is mostly larger in these tasks than in the case of two-point and three-
point perspectives, with the only exception of the task “Perspective 10” (fifth one-point
perspective), where the only vanishing point
Vx
is very far from the origin, consequently
the drawing is very close to an axonometric image. The first five plots in Figure 10 show
the box-plots of all the one-point perspective tasks, and the cubes with all the data can be
seen in Figures 11–15.

Educ. Sci. 2021,11, 577 10 of 17
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 10.
Box plots of perspective tasks with median, interquartile range (IQR), non-outlier range,
outliers and extremes.
x
x
E
O
V
y
E
z
E
Figure 11.
The 1-point perspective task Pers6 with mean value
rx=
0.23 of selected depth ratio.
Non-outliers minimum:
rx=
0.11, non-outliers maximum:
rx=
0.33, median:
rx=
0.23, IQR
between rx=0.19 and rx=0.26 (see Table 3).
Figure 12.
The 1-point perspective task Pers7 with mean
rx=
0.21. Non-outliers minimum:
rx=
0.12,
non-outliers maximum:
rx=
0.31, median:
rx=
0.21, IQR between
rx=
0.19 and
rx=
0.24 (see
Table 3).
Figure 13.
The 1-point perspective task Pers8 with mean
rx=
0.33. Non-outliers minimum:
rx=
0.16,
non-outliers maximum:
rx=
0.5, median:
rx=
0.33, IQR between
rx=
0.29 and
rx=
0.4 (see Table 3).

Educ. Sci. 2021,11, 577 11 of 17
Figure 14.
The 1-point perspective task Pers9 with mean
rx=
0.21. Non-outliers minimum:
rx=
0.11,
non-outliers maximum:
rx=
0.33, median:
rx=
0.22, IQR between
rx=
0.18 and
rx=
0.25
(see Table 3).
Figure 15.
The 1-point perspective task Pers10 with mean
rx=
0.1. Non-outliers min.:
rx=
0.06,
non-outliers max.: rx=0.14, median: rx=0.1, IQR= [0.09, 0.11](see Table 3).
4.2.2. Two-Point Perspective Test
We have provided five different two-point perspective tasks, where, again, students
were able to adjust the depth ratio along the
x
axis by moving one single point,
Ex
. In the
case of the two-point perspective, although all figures can be technically correct as a
perspective drawing, there is one single solution where the drawing can be a result of a
central projection, due to Stiefel [
19
]. This single solution is presented in the figures as
point
G
(green), and the associated depth ratio is listed in the first row of Table 4, as the
expected value.
The two-point perspective tasks with all the data and box-plots next to the cubes can
be seen in Figures 16–20. Box plots can also be seen separately in Figure 10 (second 5 plots).
It is worth noting that in this perspective method, the expected (optimal) value and
the mean value of the depth ratio are fairly different in three out of five tasks with very
small standard deviation. This means that the depth ratio seems to tendentiously be over-
or underestimated in some cases by most of the particpants (see the position of the green
dot in the figures).
y
x
E
G
y
E
O
V
z
E
x
V
Figure 16.
The 2-point perspective task Pers11 with mean value
rx=
0.41 of selected depth ratios.
The expected depth ratio at
G=Ex
is
rx=
0.31. Non-outliers minimum:
rx=
0.36, non-outliers
maximum: rx=0.47, median: rx=0.42, IQR between rx=0.14 and rx=0.43 (see Table 4).

Educ. Sci. 2021,11, 577 12 of 17
Table 4.
Mean and interquartile range (IQR) in foreshortening in the case of 2-point perspective tasks.
Ratio (rx) Pers11 Pers12 Pers13 Pers14 Pers15
expected 0.31 0.28 0.47 0.33 0.34
mean 0.41 0.34 0.44 0.31 0.32
standard deviation 0.05 0.02 0.05 0.03 0.03
non-outlier minimum 0.36 0.31 0.33 0.25 0.27
25% 0.4 0.32 0.41 0.3 0.31
median 0.42 0.33 0.44 0.31 0.33
75% 0.43 0.34 0.47 0.33 0.34
non-outlier maximum 0.47 0.35 0.56 0.36 0.37
Figure 17.
The 2-point perspective task Pers12 with mean
rx=
0.34. The expected depth ratio:
rx=
0.28. Non-outliers minimum:
rx=
0.31, non-outliers maximum:
rx=
0.35, median:
rx=
0.33,
IQR between rx=0.32 and rx=0.34 (see Table 4).
Figure 18.
The 2-point perspective task Pers13 with mean
rx=
0.44. The expected depth ratio:
rx=
0.47. Non-outliers minimum:
rx=
0.33, non-outliers maximum:
rx=
0.56, median:
rx=
0.44,
IQR between rx=0.32 and rx=0.34 (see Table 4).
Figure 19.
The 2-point perspective task Pers14 with mean
rx=
0.31. The expected depth ratio:
rx=
0.33. Non-outliers minimum:
rx=
0.25, non-outliers maximum:
rx=
0.36, median:
rx=
0.31,
IQR between rx=0.3 and rx=0.33 (see Table 4).
Figure 20.
The 2-point perspective task Pers15 with mean
rx=
0.32. The expected depth ratio:
rx=
0.34. Non-outliers minimum:
rx=
0.27, non-outliers maximum:
rx=
0.37, median:
rx=
0.33,
IQR between rx=0.31 and rx=0.37 (see Table 4).

Educ. Sci. 2021,11, 577 13 of 17
4.2.3. Three-Point Perspective
Analogously to the previous task groups, in this part of the test, five different three-
point perspective drawings were provided, and the students were able to adjust the depth
ratio along the
x
axis by moving point
Ex
. In the three-point perspective drawing, similarly
to the two-point case, there is a single position of
Ex
(and single value of depth ratio),
where the image can be considered as a central projection of a cube. This somewhat optimal
position of
Ex
is denoted in the figures by a green dot. Data of these tasks can be seen in
Table 5, with that single depth ratio as the expected value.
Note that—contrary to what we have observed in the two-point case—in the three-
point perspective tasks, there is a remarkable coincidence between the optimal (expected)
value of depth ratio and the mean value of the test results. This is also obvious from
Figures 21–25
where the three-point perspective tasks can be seen with the optimal position
of the green point. Box plots can also be seen in these figures as well as in Figure 10 (third
group of 5 plots).
x
E
O
y
E
z
E
x
z
V
V
y
V
Figure 21.
The 3-point perspective task Pers16 with mean value
rx=
0.4 of depth ratio. The expected
value is
rx=
0.42. Non-outliers minimum:
rx=
0.36, non-outliers maximum:
rx=
0.46, median:
rx=0.4, IQR between rx=0.39 and rx=0.42 (see Table 5).
Table 5.
Mean and interquartile range (IQR) in foreshortening in the case of the 3-point perspective.
Ratio (rx) Pers16 Pers17 Pers18 Pers19 Pers20
expected 0.42 0.41 0.43 0.38 0.37
mean 0.4 0.42 0.41 0.37 0.35
standard deviation 0.03 0.03 0.04 0.04 0.04
non-outlier minimum 0.36 0.36 0.3 0.32 0.28
25% 0.39 0.41 0.38 0.36 0.34
median 0.4 0.42 0.4 0.38 0.36
75% 0.42 0.44 0.44 0.39 0.38
non-outlier maximum 0.46 0.48 0.51 0.44 0.44

Educ. Sci. 2021,11, 577 14 of 17
Figure 22.
The 3-point perspective task Pers17with mean
rx=
0.42. The expected value:
rx=
0.41,
non-outliers minimum:
rx=
0.36, non-outliers maximum:
rx=
0.48, median:
rx=
0.42, IQR between
rx=0.41 and rx=0.44 (see Table 5).
Figure 23.
The 3-point perspective task Pers18 with mean
rx=
0.41. The expected value:
rx=
0.43,
non-outliers minimum:
rx=
0.3, non-outliers maximum:
rx=
0.51, median:
rx=
0.4, IQR between
rx=0.38 and rx=0.44 (see Table 5).
Figure 24.
The 3-point perspective task Pers19 with mean
rx=
0.37. The expected value:
rx=
0.38,
non-outliers minimum:
rx=
0.32, non-outliers maximum:
rx=
0.44, median:
rx=
0.38, IQR between
rx=0.36 and rx=0.39 (see Table 5).
Figure 25.
The 3-point perspective task Pers20 with mean
rx=
0.35. The expected value:
rx=
0.37,
non-outliers minimum:
rx=
0.28, non-outliers maximum:
rx=
0.44, median:
rx=
0.36, IQR between
rx=0.34 and rx=0.38 (see Table 5).
4.3. Cluster Analysis
Five-five tasks of axonometric, one-point perspective, two-point perspective and
three-point perspective drawings in this order have been provided to the test participants
(students of arts and engineering) to find the common sense, if it exists, of seeing a cube
(and not a general cuboid). Most students found their best view and foreshortening
in a remarkably narrow interval, with small deviation, confirming that there is such a
common view. However, different additional lessons learnt from each type of task were
provided. We sorted the students into four clusters by the software Statistica, and we

Educ. Sci. 2021,11, 577 15 of 17
found that the results of the students in three clusters differ slightly, while the fourth
cluster is spectacularly and significantly different from the other three (see Figure 26).
Cluster 2 contains only three engineering students out of 153. The values sent by them
are extremes, as perhaps they did not understand the task or they have slightly different
spatial abilities than the majority. The numbers of the members of the other three clusters,
Cluster 1, Cluster 3 and Cluster 4, are 61, 45 and 44, respectively. The ratios of the arts
and engineering students in the clusters, respectively, are 75–25%, 87–13% and 48–52%,
while this ratio in the case of all the students is 70–30%. We found that there is only a
slight difference considering Cluster 1, Cluster 3 and Cluster 4 in one-point perspectives,
but Cluster 2 has a relevant difference from the others in most of the tasks.
Figure 26.
Four clusters of the participants. Students in Cluster 2 (red) produced results significantly
different from the others.
5. Conclusions
In the case of axonometry, every geometrically correct drawing can be considered as a
projected image of a cube; therefore, there is no absolute, single right solution. However,
in many axonometric situations, there is a kind of convention for the value of foreshortening,
inherited mostly from engineering applications. The results of our survey, where students
were not aware of these conventions, show that the mean value of the test results is
very close to these conventions in most cases, somewhat confirming their existence. One
important exception is the task Axo02, a frequently used axonometric framework, where
the usual foreshortening is 0.5, but the mean and median of the responses are 0.7. This fact
may raise the possibility of changing the convention in education in this specific case.
In the case of one-point perspective drawing tasks, analogously to axonometry, every
geometrically correct drawing can be considered as a projected image of a cube, but in
this case, there is no such convention. Among the perspective drawings, these one-point
perspective tasks resulted in the largest interquartile and full range, showing a kind of
uncertainty about this view.
In the case of two-point perspective drawing tasks, the depth ratio has been chosen
from a much smaller interval but with a definite over- or underestimation compared to
the single correct geometric solution where these drawings can be considered as a central
projected image of a cube.
Finally, in the case of the three-point perspective drawing tasks, having received the
same small interquartile range as in the previous case, the mean was very close to the single
correct geometric solution. It seems to us that this view, which is the one closest to our
everyday life experience, provided the most obvious outcome in terms of common view.

Educ. Sci. 2021,11, 577 16 of 17
Our experience can be concluded as follows: the perspective and axonometric drawing
of cuboids in math textbooks and lessons yield a deeply congruent common understanding
in most cases. This especially holds for two-point and three-point perspective represen-
tations of which students evidently have a greater everyday experience because they are
used to seeing representations of cubes mostly with those dimensions. However, potential
reasons of congruent views as well as deviations and discrepancies need further elaboration.
All of this means that when we intend to draw a cube (and not a cuboid), we must
take into account this common sense, otherwise students will have a feeling of seeing a
general cuboid, and not a cube.
However, as we can observe from the box plots and statistical data of the test results,
there is a small group of students, whose results are significantly and consistently different
from the others. These students evidently have a very different view of space and suffer
from a serious lack of spatial abilities. The study of potential reasons for this difference
goes beyond the scope of this paper but can be the subject of future investigations. In a
future extended research, we are planning to study whether there is any difference in the
perception of the “common sense of cube” in terms of movement type, spatial abilities,
age or gender, analogously to what has been reported in perception and spatial abilities
in [20–24].
Author Contributions:
Conceptualization, M.H. and L.N.; methodology, M.H. and L.N.; software,
L.N.; validation, M.H. and L.N.; formal analysis, M.H. and L.N.; investigation, M.H. and L.N.;
resources, M.H. and L.N.; data curation, M.H. and L.N.; writing—original draft preparation, M.H.;
writing—review and editing, M.H. and L.N.; visualization, L.N. All authors have read and agreed to
the published version of the manuscript.
Funding:
This research was partially funded by the MTA KOZOKT2021-44 Public Education Devel-
opment Research Program of the Hungarian Academy of Sciences.
Institutional Review Board Statement:
The study was conducted according to the guidelines of the
Declaration of Helsinki.
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement:
The data will be available upon reasonable request to the correspond-
ing authors.
Conflicts of Interest: The authors declare no conflict of interest.
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