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Anisotropy of perceived numerosity: Evidence for a horizontal–vertical
, Maria Elena Miletto Petrazzini
, Christian Agrillo
Department of General Psychology, University of Padova, Italy
School of Biological and Chemical Science, Queen Mary University of London, UK
Padua Neuroscience Center, University of Padova, Italy
Non-symbolic numerical abilities
Many studies have investigated whether numerical and spatial abilities share similar cognitive systems. A novel
approach to this issue consists of investigating whether the same perceptual biases underlying size illusions can
be identiﬁed in numerical estimation tasks. In this study, we required adult participants to estimate the number
of white dots in arrays made of white and black dots displayed in such a way as to generate horizontal–vertical
illusions with inverted T and L conﬁgurations. In agreement with previous literature, we found that participants
tended to underestimate the target numbers. However, in the presence of the illusory patterns, participants were
less inclined to underestimate the number of vertically aligned white dots. This reﬂects the perceptual biases
underlying horizontal–vertical illusions. In addition, we identiﬁed an enhanced illusory eﬀect when participants
observed vertically aligned white dots in the T shape compared to the L shape, a result that resembles the length
bisection bias reported in the spatial domain. Overall, we found the ﬁrst evidence that numerical estimation
diﬀers as a function of the vertical or horizontal displacement of the stimuli. In addition, the involvement of the
same perceptual biases observed in spatial tasks supports the idea that spatial and numerical abilities share
similar cognitive processes.
Mathematical abilities represent some of human beings' highest
cognitive skills. Symbolic representation of numbers and mathematical
operations has facilitated technological progress throughout human
history. However, cultural (Butterworth, Reeve, & Reynolds, 2011;
Pica, Lemer, Izard, & Dehaene, 2004), cognitive (Krause, Bekkering, &
Lindemann, 2013;Price, Palmer, Battista, & Ansari, 2012;Revkin,
Piazza, Izard, Cohen, & Dehaene, 2008), developmental (Izard, Sann,
Spelke, & Streri, 2009;Xu & Spelke, 2000), and comparative (Agrillo &
Bisazza, 2014;Beran, 2006) psychology provide evidence for the ex-
istence of rudimentary numerical abilities that are independent from
educational factors and predate the emergence of language. Such
abilities –often referred to as non-symbolic numerical abilities (Agrillo,
Piﬀer, & Adriano, 2013;Gilmore, McCarthy, & Spelke, 2010;Park,
Bermudez, Roberts, & Brannon, 2016)–seem to be evolutionarily an-
cient, given the unquestionable advantages they provide in terms of
ﬁtness and survival in the natural environment. Indeed, the capacity to
estimate whether a group is larger or smaller allows organisms to make
optimal decisions, such as selecting a larger number of food items, as
well as selecting a larger group of social companions or sexual partners
(reviewed in Agrillo & Bisazza, 2018).
Studies that investigate non-symbolic numerical abilities commonly
present stimuli in the same portion of the visual ﬁeld to control for
spatial biases and focus on the independent variables under investiga-
tion (e.g., the role of numerical ratio in the performance; Agrillo, Piﬀer,
Bisazza, & Butterworth, 2015;Halberda, Mazzocco, & Feigenson, 2008;
Cantlon & Brannon, 2007). Stimuli in this research ﬁeld are presented
either sequentially in the centre of the screen (e.g., one group of dots
followed by another group of dots; Shuman & Kanwisher, 2004;Ansari,
Lyons, van Eimeren, & Xu, 2007;Agrillo et al., 2015) or simultaneously
in the centre of the two hemiﬁelds (e.g., one group of dots on the left
and another group of dots on the right; Hurewitz, Gelman, & Schnitzer,
2006;Piazza et al., 2010;Agrillo et al., 2015). As far as we are aware,
no study has investigated whether humans' capacity to estimate nu-
merosities is identical in vertical and horizontal spaces.
Many researchers have reported diﬀerential perception of the hor-
izontal and vertical axes in spatial tasks (Higashiyawa, 1992;Loomis &
Philbeck, 1999). This is commonly known as ‘anisotropy of the per-
ceived space’. The horizontal–vertical (HV) illusion oﬀers compelling
Received 17 May 2019; Received in revised form 20 January 2020; Accepted 27 February 2020
Corresponding author at: University of Padova, Department of General Psychology, Via Venezia 8, 35131 Padova, Italy.
E-mail address: firstname.lastname@example.org (C. Agrillo).
evidence of this phenomenon. In its classical conﬁguration, the illusory
pattern is an inverted T ﬁgure in which the horizontal and vertical lines
are identical in size. In the presence of this pattern, most human ob-
servers estimate the vertical line as longer than the horizontal line
(Avery & Day, 1969).
HV illusion seems to be related to the shape of the human visual
ﬁeld. The binocular visual ﬁeld is a horizontally oriented ellipse with a
horizontal to vertical aspect ratio of 1.53, whereas the monocular visual
ﬁeld has a ratio of 1.23 (Harrington, 1981;Künnapas, 1955). Thus, the
ends of the vertical line in this illusory pattern are nearer to the
boundary in the binocular visual ﬁeld. Because lines appear longer
when they are close to a surrounding frame (Künnapas, 1955, 1957;
Prinzmetal & Gettleman, 1993), humans probably experience percep-
tual overestimation because the vertical lines extend to the boundary
edges (assimilation eﬀect). However, the horizontal line is far from the
boundary, which leads to a contrast eﬀect. An alternative explanation is
related to the ‘inappropriate size-scaling hypothesis’(Girgus & Coren,
1975;Gregory, 1963, 1997;Pecunioso & Agrillo, 2020a), according to
which an observer perceives the vertical line as receding into the dis-
tance but perceives the horizontal line as lying on the same plane.
Because the two lines are the same size in the retina, the observer in-
terprets the former as longer (Girgus & Coren, 1975).
Interestingly, the HV illusion is aﬀected by another factor unrelated
to the anisotropy of the perceived space, the ‘length bisection bias.’If a
line is bisected by a second line, the former appears to be shorter than
the latter (Finger & Spelt, 1947;Mamassian & de Montalembert, 2010).
One way to test the role of length bisection bias consists in presenting
an L version of the HV illusion, in which vertical and horizontal axes are
still involved but no line is bisected. When comparing the performance
of participants in the presence of the T and L versions of the HV illusion,
researchers reported a greater misperception of length in the T version,
in which the magnitude of the eﬀect was approximately 16–20%, than
in the L version, in which the magnitude of the eﬀect was 3–6% (Avery
& Day, 1969;Mamassian & de Montalembert, 2010).
Given the strong asymmetry in the perception of vertical and hor-
izontal sizes, it is reasonable to ask whether a similar eﬀect could be
identiﬁed in non-symbolic numerical estimation. The fact that a sig-
niﬁcant body of literature indicates a link between spatial and nu-
merical abilities (e.g., de Hevia, Vallar, & Girelli, 2008;Krause,
Bekkering, Pratt, & Lindemann, 2017) also legitimates such a question.
Many studies suggest that non-symbolic numerical estimation is af-
fected by continuous quantities, such as the cumulative surface area
(sum of areas) or convex hull (overall space occupied by the most lat-
eral items) of the stimuli (Gebuis & Reynvoet, 2012a, 2012b;Leibovich,
Katzin, Harel, & Henik, 2017; but see DeWind, Adams, Platt, &
Brannon, 2015;Cicchini, Anobile, & Burr, 2016;Park, Dewind,
Woldorﬀ, & Brannon, 2016 for a diﬀerent perspective). The mental
representation of numbers is often described as oriented from left to
right on the so-called mental number line (Galton, 1880;Izard &
Dehaene, 2008;Zorzi, Priftis, & Umiltà, 2002). Walsh (2003) also ad-
vanced the ATOM theory of magnitude, which holds that time, space,
and number are processed by a common neural circuit located in the
parietal lobe. If spatial and numerical abilities share a common cogni-
tive metric, perceptual biases that aﬀect spatial judgments may also
inﬂuence numerosity judgments.
Dormal, Larigaldie, Lefevre, Pesenti, and Andres (2018) recently
adapted a size illusion to test non-symbolic numerical abilities. The
authors presented two linear dot arrays surrounded by outward or in-
ward arrows, a pattern that resembled the Müller-Lyer illusion. In one
experiment, participants decided whether the presented test stimuli
contained more or fewer dots than a reference stimulus. In another
experiment, they provided verbal estimates of the number of dots
contained in the array. The results indicated that when arrows pointed
inward, participants overestimated the number of dots compared to
when the arrows pointed outward. In this sense, the spatial arrange-
ment of the stimuli aﬀected numerosity estimation.
In the present study, we used the approach adopted by Dormal et al.
(2018) to investigate whether non-symbolic numerical estimation
varies as a function of the vertical or horizontal presentation of the
stimuli, a fact that would also suggest the existence of anisotropy of
perceived numerosity. To achieve this goal, we used a visual pattern
made up of white and black dots arranged in the inverted T and L
patterns to generate a HV illusion. In control trials, black and white dots
were randomly presented in the array; in test trials, white dots were
arranged in a line (e.g., vertical line), and black dots were arranged in
the other line (the horizontal one). We required participants to verbally
estimate the number of white dots. We make two predictions: If nu-
merical and spatial abilities rely on a common magnitude system, in the
test trials we should expect (1) an overestimation of white dots when
presented on the vertical axis and (2) a greater illusory eﬀect in the
presence of the inverted T pattern compared to the L pattern due to
length bisection bias.
We assessed sample size preliminarily using Fisher's exact test,
choosing α= 0.05 and a desired power of 0.80. Assuming an error rate
value of 7.747 as minimal detectable diﬀerence in means and a stan-
dard deviation of 11.30 (data taken from a previous experiment at our
laboratory that used the same methodology to test a diﬀerent hypoth-
esis on non-symbolic numerical estimation; Pecunioso & Agrillo,
2020b), we found a total of 36 participants appropriate to test our
hypothesis. We accordingly tested a total of 40 volunteers (21 males
between 19 and 25 years old; mean age 21.90 years). We sampled and
tested the participants at the Department of General Psychology at the
University of Padova. All participants had normal or corrected-to-
The study was approved by the ethics committee (prot. N. 2576) of
the Department of General Psychology (University of Padova, Italy).
Before the experiment began, all participants gave their informed
consent in accordance with the Declaration of Helsinki.
We conducted the task in a dimly lit room. Our testing setup was
identical to that of Agrillo, Parrish, and Beran (2016) and Pecunioso
and Agrillo (2020b). It included a 17-in. LCD colour monitor, a personal
computer, and a keyboard.
Stimuli appeared in the centre of the screen and comprised white
(255 R, 255 G, 255 B) and black (0 R, 0 G, 0 B) dots (diameter: 1 cm) on
a grey (127 R, 127 G, 127 B) background. Dots were arranged in either
an inverted T shape (a vertical line bisected a horizontal line at the 8th
dot) or a L shape (no bisection). In both arrays, dots appeared in spatial
continuity to one another (inter-item distance: 0 cm; Fig. 1). The hor-
izontal and vertical lines contained 15 dots each (line length: 15 cm).
The number of white and black dots varied as a function of the ex-
perimental phase (see below). However, the overall number of dots was
always 30. For example, if an array contained 18 black dots, it also
contained 12 white dots.
Ten, 11, 19 and 20 white dots (20, 19, 11, and 10 black dots, re-
spectively) were used in the initial training phase; 12–18 white dots
were used in the experiment (18–12 black dots). In particular, we
classiﬁed participants' estimations of 12, 13, 14, 15, 16, 17, and 18
white items –randomly presented in the L-shape and T-shape –as
control trials. We considered estimations of 15 items arranged in the HV
illusory pattern (all white dots appeared on one axis, and all black dots
appeared on the other) test trials.
A. Pecunioso, et al. $FWD3V\FKRORJLFD
Participants remained seated 60 cm from the monitor. Initially, we
presented a short training phase with a total of 16 trials, 4 for each
target number. To familiarise participants with the spatial arrange-
ments under investigation, we arranged half of the stimuli for each
target number in the L shape and half in the T shape. During the
training phase, white and black dots appeared randomly on the hor-
izontal and vertical axes.
After the presentation of a ﬁxation cross (250 ms), we presented a
single array comprising black and white dots (200 ms). We required
participants to report aloud the number of white dots they saw (Fig. 2).
An experimenter in the room recorded their responses. We provided no
feedback throughout the experiment. The participants pressed the space
bar to begin the next trial. As soon as the training phase ended, the
experiment began. It consisted of two equal blocks (108 trials per block,
for a total of 216 trials; 168 control trials, 48 test trials), each of which
began when participants pressed the space bar. For each target number,
12 trials presented the L shape, and 12 trials presented the T shape.
We calculated the mean error rate as a dependent variable with the
following formula: [(participant's response −target number)/target
number] ×100. A positive error rate reﬂects an overestimation of
numerosity, a negative error rate indicates an underestimation, and an
error rate of zero indicates a correct estimation (Dormal et al., 2018).
We also calculated the coeﬃcient of variation (standard deviation of
response divided by mean response) for test and control trials in which
we presented same number of white dots (15). Weber's law is reﬂected
by a stable coeﬃcient of variation across numerosities (Izard &
Dehaene, 2008;Revkin et al., 2008;Whalen, Gallistel, & Gelman,
1999). Thus, this coeﬃcient provides a measure of the overall precision
of the underlying numerical representation (Izard & Dehaene, 2008).
The lower the value of the coeﬃcient of variation, the more precise the
As the data were normally distributed, we used parametric statistics
(Kolmogorov-Smirnov, average performance of control trials,
p= 0.200, test trials, p= 0.200).
3.1. Control trials
We performed a 7 ×2 repeated measures ANOVA with the mean
error rate as dependent variable, and target number (12/13/14/15/16/
17/18; all arrays composed of randomly arranged white and black dots)
and shape (L or T) as within-subjects factors. We found a main eﬀect of
shape, F(1, 39) = 6.899, p= 0.012, η
= 0.150, meaning that parti-
cipants were overall more accurate with the L shape. We found no eﬀect
of target number, Greenhouse-Geisser correction F(2.775,
180.206) = 1.477, p= 0.227, η
= 0.036, or interaction, F(6,
234) = 1.788, p= 0.102, η
= 0.044 (Fig. 3). The participants un-
derestimated all target numbers (one-sample t-tests with 0 as chance
level, p< 0.001).
Fig. 1. Examples of stimuli used. The black and white dots appeared on a grey background. We arranged half of the stimuli in an inverted T shape (left column) and
arranged the other half in an L shape (right column). In control trials (top row), we arranged white and black dots randomly on the two axes; in test trials (bottom
row), we arranged the white dots in a line on one axis and arranged black dots in a line on the other.
A. Pecunioso, et al. $FWD3V\FKRORJLFD
3.2. Test trials
We performed a 2 ×2 repeated measures ANOVA with spatial
conﬁguration (15 white dots vertically vs. horizontally located) and
shape (L or T) as within-subjects factors. We found a main eﬀect of
spatial conﬁguration, F(11, 39) = 14.531, p< 0.001, η
meaning that white dots in the vertical space are less underestimated.
We found no main eﬀect of shape, F(1, 39) = 1.208, p= 0.317,
= 0.026. However, we identiﬁed a signiﬁcant interaction, F(1,
39) = 8.237, p= 0.007, η
= 0.174, meaning that when white dots
were arranged in a vertical line, participants exhibited a lower tendency
to underestimate the number of dots with the T shape than the L shape
We conﬁrmed this asymmetrical estimation in the vertical and
horizontal space with a comparison of the performance when white
dots were arranged in the vertical or horizontal line separately for the
two shapes (T shape: t(39) = 3.925, p< 0.001, Cohen's d= 0.621; L
shape: t(39) = 2.968, p< 0.001, d= 0.469).
Lastly, we performed a 3 ×2 repeated measures ANOVA with the
coeﬃcient of variation as dependent variable, and Spatial
Conﬁguration (15 white dots vertically, horizontally, or randomly lo-
cated) and Shape (L or T) as within-subjects factors. We found a main
eﬀect of Spatial Conﬁguration, F(2, 78) = 17.730, p< 0.001,
= 0.313, meaning that the spatial arrangement of white dots af-
fected the coeﬃcient of variation. No eﬀect of Shape, F(1, 39) = 0.337,
p= 0.565, η
= 0.009, nor interaction F(2, 78) = 0.370, p= 0.692,
= 0.009 was found (Fig. 5). Post hoc tests (Tukey) revealed a sig-
niﬁcant diﬀerence between vertically oriented dots and randomly or-
iented dots (p= 0.001), between horizontally oriented dots and ran-
domly oriented dots (p= 0.001), and between vertically oriented dots
and horizontally oriented dots (p= 0.019).
Fig. 2. Experimental setup. After presenting a ﬁxation cross, we presented an array of white and black dots on the monitor; participants verbally reported the number
of white dots in the array.
Fig. 3. Control trial results. The participants'
estimation as a function of the actual numer-
osity. Negative values reﬂect an under-
estimation of the target items, whereas positive
values indicate an overestimation of the target
items. Participants signiﬁcantly underestimated
the value of all target numbers. Red dots re-
present individual data points. (For interpreta-
tion of the references to colour in this ﬁgure
legend, the reader is referred to the web version
of this article.)
Fig. 4. Test trial results. We observed a reduced tendency to underestimate the
number of white dots when they were arranged vertically in a line in both the L
and T shapes. This eﬀect, however, was greater in the T shape. Red dots re-
present individual data points. (For interpretation of the references to colour in
this ﬁgure legend, the reader is referred to the web version of this article.)
A. Pecunioso, et al. $FWD3V\FKRORJLFD
The present study aimed to investigate whether perceptual biases
aﬀecting spatial estimation also impact numerical estimation, a fact
that would support the existence of similar cognitive processes related
to space and number in the brain. To achieve this goal, given the well-
known anisotropy of perceived space that characterises the HV illusion,
we investigated whether the participants' accuracy when estimating the
number of dots diﬀered depending on whether the dots were presented
on the vertical or horizontal axis. If participants demonstrated the same
perceptual biases for the two cognitive domains, we predicted (1) an
overestimation of vertically presented dots and (2) a greater illusory
eﬀect in the presence of the inverted T.
Before discussing our results with respect to the predictions men-
tioned above, the overall results merit a preliminary discussion. In the
control and test trials, participants underestimated all target numbers.
This result is not totally unexpected because several studies found that
humans tend to underestimate large numbers in numerical estimation
tasks (Zhang & Okamoto, 2017;Izard & Dehaene, 2008;Krueger, 1982,
1984;Kramer & Bressan, 2017;Kramer, Di Bono, & Zorzi, 2011;
Crollen, Castronovo, & Seron, 2011). On the contrary, small numbers
are often overestimated (e.g., Cicchini, Anobile, & Burr, 2014). To ex-
plain this phenomenon, it has been argued that the mental re-
presentation of the ‘number line’may play an important role. Indeed,
humans seem to reproduce numbers logarithmically (visualizing, for
instance, the number 10 near the midpoint of a 1–100 interval). Also, in
a numerical estimation task, participants' responses may be biased to-
ward the mean of the stimulus distribution, a phenomenon known as
‘regression to the mean’(Jazayeri & Shadlen, 2010). The possibility
exists that, because of the logarithmic representation of numbers and
the small range of numbers used in our study (12–18), the mean of the
stimulus distribution was wrongly biased on the left of the mental
number line (average estimation of control trials: 11.5 dots, instead of
15), leading to a general underestimation of all target values.
Alternatively, the tendency to underestimate large numbers, as well
as the tendency to underestimate the number of large dots, might be
explained by the ‘occupancy’model (Allik & Tuulmets, 1991). Ac-
cording to this model, numerical estimations should be linearly related
to the total occupied area (occupancy) by concentric virtual disks that
cover each dot. When dots are close to one other, the virtual disks
overlap, leading to an underestimation of the dots; when the dots are
further apart, the overall space occupied by these disks is larger, leading
to an overestimation of the dots. Large dots' virtual disks are more likely
to overlap (Cleland & Bull, 2015;Gebuis & van der Smagt, 2011;
Ginsburg & Nicholls, 1988). In addition, our dots were adjacent one to
another in vertical and horizontal lines with no inter-item distance. This
might have led to a general underestimation of the target dots, given
the larger degree of virtual disk overlap. Such a strong eﬀect was not
observed in the study by Dormal et al. (2018), as the arrays the authors
presented contained dots that were not adjacent to each other. Instead,
their study's inter-dot spacing varied from 0.4–2 cm. Related to this
issue, it was found that arrays of connected dots can be underestimated
by up to 20% (Anobile, Cicchini, Pomè, & Burr, 2017;Fornaciai,
Cicchini, & Burr, 2016;Fornaciai & Park, 2018;Franconeri, Bemis, &
Alvarez, 2009;He, Zhang, Zhou, & Chen, 2009).
Provided that the participants underestimated all target numbers,
we can draw important conclusions with respect to our predictions in
terms of a higher or lower tendency to underestimate the target num-
bers as a function of the spatial arrangements of white dots. The results
of the test trials, in which a main eﬀect of spatial conﬁguration was
found, supported our ﬁrst prediction. Participants' accuracy varied if
the target dots were presented entirely on the vertical axis, entirely on
the horizontal axis, or randomly on the two axes. In particular, the
signiﬁcant diﬀerence between the horizontal and vertical condition
showed with both dependent variables (mean error rates and coeﬃcient
of variation) indicates that items on the vertical axis are less under-
estimated than those on the horizontal axis. This result fully aligns with
the anisotropy of the vertical space reported in the HV illusion
(Harrington, 1981;Künnapas, 1955).
When items were arranged in an orderly fashion (regardless of
whether they were vertically or horizontally oriented), participants
were less inclined to underestimate target numbers compared to when
the same number of items was randomly arranged on the two axes. This
might be due to the presence of two concomitant perceptual biases that
operate in the same direction. First, it was found that humans tend to
overestimate the number of items in regularly arranged stimulus sets
compared to randomly arranged sets. This phenomenon is known as the
regular random numerosity illusion (RRNI; Beran, 2006;Ginsburg,
1976; but see Apthorp & Bell, 2015 for an opposite ﬁnding). When the
white dots formed a line, they represented a more regular pattern
compared to the pattern in which white dots were randomly arranged,
leading to a reduced underestimation of linearly arranged stimuli. The
other perceptual bias deals with item clustering. Observers tend to
overestimate items that form a more coherent Gestalt compared to
items that form small, separate clusters. This phenomenon describes a
numerosity illusion known as the solitaire illusion (Agrillo et al., 2016;
Fig. 5. Coeﬃcient of variation. Participants showed to be
less precise when 15 dots were randomly located. In ad-
dition, a signiﬁcant diﬀerence in their precision was re-
ported when 15 white dots were located either horizon-
tally or vertically, with participants less incline to
underestimate dots' numerosity if they were presented in
the vertical axis.
A. Pecunioso, et al. $FWD3V\FKRORJLFD
Frith & Frith, 1972;Parrish, Beran, & Agrillo, 2019). When the white
dots were linearly arranged, they formed a more coherent Gestalt than
when they were randomly presented in small, separate clusters on the
horizontal and vertical axes. In any case, the potential compresence of
the perceptual biases underlying the RRNI and solitaire illusions did not
prevent us to notice a signiﬁcant diﬀerence in accuracy when partici-
pants had to estimate items presented vertically rather than horizon-
tally. However, the analysis of the coeﬃcient of variation suggests a
more parsimonious explanation of the signiﬁcant diﬀerence between
‘ordered’and ‘random’conditions: Participants might have been simply
more precise in assessing the numerosity of ordered dots because this
represented an easier condition compared to control trials with random
We also conﬁrmed our second prediction in terms of the T pattern's
enhanced illusory eﬀect (Finger & Spelt, 1947;Mamassian & de
Montalembert, 2010). Although the lack of main eﬀect of shape (T vs.
L) indicates that the accuracy of numerical estimation of 15 items ar-
ranged in a T or L shape is similar when collapsing the spatial conﬁg-
uration types, the signiﬁcant interaction indicates that when white dots
were arranged in a vertical line, participants underestimated the dots
less in the T shape than in the L shape. Thus, the T shape seems to
increase the overestimation of vertically arranged dots.
In our study, we also presented control trials with various target
numbers ranging from 12 to 18. We found two unexpected results. First,
accuracy in numerical estimation tasks typically decreases when target
numbers increase (Mejias & Schiltz, 2013;Revkin et al., 2008). How-
ever, we did not ﬁnd a main eﬀect of target number, meaning that
participants' accuracy did not signiﬁcantly vary as a function of the
number to be estimated. The limited range of target numbers used in
our study (6-unit diﬀerence from the smallest number to the largest)
may have obscured this eﬀect. The other unexpected result is the main
eﬀect of shape, as participants were overall more accurate in the pre-
sence of the L shape. Unfortunately, we can only speculate on why this
occurred. It is possible that, when white dots are randomly distributed
in the two shapes, estimation of white dots is easier if no line is bi-
sected. Because the bisected line is perceived as shorter than the other
line, participants might have underestimated the white dots included in
the subjectively shorter line, leading to an overall greater under-
estimation of the target number compared to what we observed with
the L shape.
In non-symbolic numerical tasks, several physical attributes of the
stimuli co-vary with numerosity and are commonly referred to as
continuous quantities (see Leibovich et al., 2017). These include cu-
mulative surface area, density, inter-item distance, and convex hull.
Because we used same-sized dots, the larger number of dots also pre-
sented a larger cumulative surface area and was more likely to be
denser. One may argue that the participants' estimations might have
been partly driven by these continuous quantities instead of genuine
numerical processing. However, even assuming that continuous quan-
tities might have inﬂuenced the control trial results, such criticism
cannot be ascribed to the most important result of the study, the
varying performance reported in test trials with illusory arrays. In those
arrays, the same number of white dots had the same cumulative surface
area, density, and convex hull whether on the vertical or the horizontal
In conclusion, we provided the ﬁrst evidence of a diﬀerential per-
ception of numerosity in vertical and horizontal spaces. This result has
important methodological implications in this ﬁeld. Because various
studies identiﬁed conﬂicting results in terms of participants' numerical
acuity (e.g., Agrillo et al., 2015;Revkin et al., 2008), part of the
variability reported in the literature might result from a lack of a ﬁne
control of the vertical and horizontal dimensions of the stimuli. For
instance, when assessing the threshold of numerical discrimination,
researchers might inappropriately present dots asymmetrically in the
horizontal and vertical space (especially if a reduced number of sti-
mulus pairs is presented). If nine dots are more distributed in the
vertical space, and ten dots are more distributed in the horizontal space,
the possibility exists that participants' performance does not sig-
niﬁcantly diﬀer from chance not because of their lack of numerical
acuity, but because their capacity to detect the larger number would be
contrasted by the perceptual bias in overestimating vertically located
dots (more frequent in the smaller array).
Our study also suggests that the perceptual biases that aﬀect spatial
estimation play an important role in numerical estimation, particularly
with regards to the anisotropy of the perceived space reported in the HV
illusion. This encourages us to introduce the term ‘horizontal–vertical
numerosity illusion’to deﬁne the visual pattern presented here and
strengthen the idea that similar cognitive systems govern spatial and
numerical estimation. Future studies investigating the neuroanatomical
correlates of such perceptual biases are welcome to assess whether the
similar performance reported in the two cognitive domains actually
reﬂects the activation of the same neural circuits.
CrediT authorship contribution statement
Alessandra Pecunioso: Conceptualization, Investigation, Writing -
original draft. Maria Elena Miletto Petrazzini: Formal analysis,
Writing - original draft. Christian Agrillo: Conceptualization, Formal
analysis, Writing - original draft.
The authors wish to thank Amos Segato and Sebastiano Cinetto for
their help in testing participants.
This research was funded by PRIN 2015 (grant number
2015FFATB7) from “Ministero dell'Istruzione, Università e Ricerca”
(MIUR, Italy) and by STARS@unipd (acronym: ANIM_ILLUS) from the
University of Padova (Italy) to C. Agrillo. We performed the present
work within the scope of the research grant ‘Dipartimenti di Eccellenza’
entitled ‘Innovative methods or technologies for assessment, interven-
tion, or enhancement of psychological functions (cognitive, emotional
Declaration of competing interest
The authors report no potential conﬂict of interest with respect to
the research, authorship, and publication of this article.
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