ArticlePDF Available

Abstract and Figures

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.
Content may be subject to copyright.
Contents lists available at ScienceDirect
Acta Psychologica
journal homepage:
Anisotropy of perceived numerosity: Evidence for a horizontalvertical
numerosity illusion
Alessandra Pecunioso
, 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
Numerosity illusions
Non-symbolic numerical abilities
Numberspace interaction
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 identied 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 horizontalvertical
illusions with inverted T and L congurations. 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 reects the perceptual biases
underlying horizontalvertical illusions. In addition, we identied an enhanced illusory eect 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 rst evidence that numerical estimation
diers 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.
1. Introduction
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,
Pier, & 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
tness 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 eld 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, Pier,
Bisazza, & Butterworth, 2015;Halberda, Mazzocco, & Feigenson, 2008;
Cantlon & Brannon, 2007). Stimuli in this research eld 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 hemields (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 dierential 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 horizontalvertical (HV) illusion oers 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: (C. Agrillo).
evidence of this phenomenon. In its classical conguration, the illusory
pattern is an inverted T gure 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
eld. The binocular visual eld is a horizontally oriented ellipse with a
horizontal to vertical aspect ratio of 1.53, whereas the monocular visual
eld 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 eld. 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 eect). However, the horizontal line is far from the
boundary, which leads to a contrast eect. 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 aected 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 eect was approximately 1620%, than
in the L version, in which the magnitude of the eect was 36% (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 eect could be
identied in non-symbolic numerical estimation. The fact that a sig-
nicant 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,
Woldor, & Brannon, 2016 for a dierent 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 aect spatial judgments may also
inuence 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 aected 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 eect in the
presence of the inverted T pattern compared to the L pattern due to
length bisection bias.
2. Methods
2.1. Participants
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 dierence 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 dierent 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-
normal vision.
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.
2.2. Apparatus
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.
2.3. Stimuli
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; 1218 white dots
were used in the experiment (1812 black dots). In particular, we
classied 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. $FWD3V\FKRORJLFD
2.4. Procedure
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 xation 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 reects 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 coecient 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 reected
by a stable coecient of variation across numerosities (Izard &
Dehaene, 2008;Revkin et al., 2008;Whalen, Gallistel, & Gelman,
1999). Thus, this coecient provides a measure of the overall precision
of the underlying numerical representation (Izard & Dehaene, 2008).
The lower the value of the coecient of variation, the more precise the
numerical estimation.
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. Results
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 eect 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 eect
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. $FWD3V\FKRORJLFD
3.2. Test trials
We performed a 2 ×2 repeated measures ANOVA with spatial
conguration (15 white dots vertically vs. horizontally located) and
shape (L or T) as within-subjects factors. We found a main eect of
spatial conguration, F(11, 39) = 14.531, p< 0.001, η
= 0.271,
meaning that white dots in the vertical space are less underestimated.
We found no main eect of shape, F(1, 39) = 1.208, p= 0.317,
= 0.026. However, we identied a signicant 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
(Fig. 4).
We conrmed 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
coecient of variation as dependent variable, and Spatial
Conguration (15 white dots vertically, horizontally, or randomly lo-
cated) and Shape (L or T) as within-subjects factors. We found a main
eect of Spatial Conguration, F(2, 78) = 17.730, p< 0.001,
= 0.313, meaning that the spatial arrangement of white dots af-
fected the coecient of variation. No eect 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-
nicant dierence 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 xation 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 reect an under-
estimation of the target items, whereas positive
values indicate an overestimation of the target
items. Participants signicantly underestimated
the value of all target numbers. Red dots re-
present individual data points. (For interpreta-
tion of the references to colour in this gure
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 eect, however, was greater in the T shape. Red dots re-
present individual data points. (For interpretation of the references to colour in
this gure legend, the reader is referred to the web version of this article.)
A. Pecunioso, et al. $FWD3V\FKRORJLFD
4. Discussion
The present study aimed to investigate whether perceptual biases
aecting 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 diered 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
eect 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 linemay play an important role. Indeed,
humans seem to reproduce numbers logarithmically (visualizing, for
instance, the number 10 near the midpoint of a 1100 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 (1218), 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 occupancymodel (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 eect 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 0.42 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 eect of spatial conguration was
found, supported our rst 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
signicant dierence between the horizontal and vertical condition
showed with both dependent variables (mean error rates and coecient
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 nding). 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. Coecient of variation. Participants showed to be
less precise when 15 dots were randomly located. In ad-
dition, a signicant dierence 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. $FWD3V\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 signicant dierence in accuracy when partici-
pants had to estimate items presented vertically rather than horizon-
tally. However, the analysis of the coecient of variation suggests a
more parsimonious explanation of the signicant dierence between
orderedand randomconditions: 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 conrmed our second prediction in terms of the T pattern's
enhanced illusory eect (Finger & Spelt, 1947;Mamassian & de
Montalembert, 2010). Although the lack of main eect 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 cong-
uration types, the signicant 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 nd a main eect of target number, meaning that
participants' accuracy did not signicantly vary as a function of the
number to be estimated. The limited range of target numbers used in
our study (6-unit dierence from the smallest number to the largest)
may have obscured this eect. The other unexpected result is the main
eect 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 inuenced 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 rst evidence of a dierential per-
ception of numerosity in vertical and horizontal spaces. This result has
important methodological implications in this eld. Because various
studies identied conicting 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 ne
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-
nicantly dier 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 aect 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 horizontalvertical
numerosity illusionto dene 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
reects 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.
Funding information
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
or behavioural).
Declaration of competing interest
The authors report no potential conict of interest with respect to
the research, authorship, and publication of this article.
Agrillo, C., & Bisazza, A. (2014). Spontaneous versus trained numerical abilities: A
comparison between the two main tools to study numerical competence in non-
human animals. Journal of Neuroscience Methods, 234, 8291.
Agrillo, C., & Bisazza, A. (2018). Understanding the origin of number sense: A review of
sh studies. Philosophical Transactions of the Royal Society of London B, 373,
Agrillo, C., Parrish, A. E., & Beran, M. J. (2016). How illusory is the solitaire illusion?
Assessing the degree of misperception of numerosity in adult humans. Frontiers in
Psychology, 7, 1663.
Agrillo, C., Pier, L., & Adriano, A. (2013). Individual dierences in non-symbolic nu-
merical abilities predict mathematical achievements but contradict ATOM. Behavioral
and Brain Functions, 9, 26.
Agrillo, C., Pier, L., Bisazza, A., & Butterworth, B. (2015). Ratio dependence in small
number discrimination is aected by the experimental procedure. Frontiers in
Psychology, 6, 1649.
Allik, J., & Tuulmets, T. (1991). Occupancy model of perceived numerosity. Perception &
Psychophysics, 49(4), 303314.
Anobile, G., Cicchini, G. M., Pomè, A., & Burr, D. C. (2017). Connecting visual objects
reduces perceived numerosity and density for sparse but not dense patterns. Journal
of Numerical Cognition, 3(2), 133146.
Ansari, D., Lyons, I. M., van Eimeren, L., & Xu, F. (2007). Linking visual attention and
number processing in the brain: The role of the temporo-parietal junction in small
and large symbolic and nonsymbolic number comparison. Journal of Cognitive
Neuroscience, 19(11), 18451853.
A. Pecunioso, et al. $FWD3V\FKRORJLFD
Apthorp, D., & Bell, J. (2015). Symmetry is less than meets the eye. Current Biology, 25(7),
Avery, G. C., & Day, R. H. (1969). Basis of the horizontal-vertical illusion. Journal of
Experimental Psychology, 81(2), 376.
Beran, M. J. (2006). Quantity perception by adult humans (Homo sapiens), chimpanzees
(Pan troglodytes) and rhesus macaques (Macaca mulatta) as a function of stimulus
organization. International Journal of Comparative Psychology, 19, 386397.
Butterworth, B., Reeve, R., & Reynolds, F. (2011). Using mental representations of space
when words are unavailable: Studies of enumeration and arithmetic in indigenous
Australia. Journal of Cross Cultural Psychology, 42, 630638.
Cantlon, J. F., & Brannon, E. M. (2007). Basic math in monkeys and college students. PLoS
Biology, 5, 328.
Cicchini, G. M., Anobile, G., & Burr, D. C. (2014). Compressive mapping of number to
space reects dynamic encoding mechanisms, not static logarithmic transform.
Proceedings of the National Academy of Sciences, 111(21), 78677872.
Cicchini, G. M., Anobile, G., & Burr, D. C. (2016). Spontaneous perception of numerosity
in humans. Nature Communications, 7, 12536.
Cleland, A. A., & Bull, R. (2015). The role of numerical and non-numerical cues in non-
symbolic number processing: Evidence from the line bisection task. Quarterly Journal
of Experimental Psychology, 68, 18441859.
Crollen, V., Castronovo, J., & Seron, X. (2011). Under- and over-estimation: A bi-direc-
tional mapping process between symbolic and non-symbolic representations of
number? Experimental Psychology, 58(10), 3949.
de Hevia, M. D., Vallar, G., & Girelli, L. (2008). Visualizing numbers in the minds eye:
The role of visuo-spatial processing in numerical abilities. Neuroscience &
Biobehavioral Reviews, 32, 13611372.
DeWind, N. K., Adams, G. K., Platt, M. L., & Brannon, E. M. (2015). Modeling the ap-
proximate number system to quantify the contribution of visual stimulus features.
Cognition, 142, 247265.
Dormal, V., Larigaldie, N., Lefevre, N., Pesenti, M., & Andres, M. (2018). Eect of per-
ceived length on numerosity estimation: Evidence from the Müller-Lyer illusion.
Quarterly Journal of Experimental Psychology, 71(10), 21422151.
Finger, F. W., & Spelt, D. K. (1947). The illustration of the horizontal-vertical illusion.
Journal of Experimental Psychology, 37(3), 243.
Fornaciai, M., Cicchini, G. M., & Burr, D. C. (2016). Adaptation to number operates on
perceived rather than physical numerosity. Cognition, 151, 6367.
Fornaciai, M., & Park, J. (2018). Early numerosity encoding in visual cortex is not suf-
cient for the representation of numerical magnitude. Journal of Cognitive
Neuroscience, 30(12), 17881802.
Franconeri, S. L., Bemis, D. K., & Alvarez, G. A. (2009). Number estimation relies on a set
of segmented objects. Cognition, 113(1), 113.
Frith, C. D., & Frith, U. (1972). The solitaire illusion: An illusion of numerosity. Perception
& Psychophysics, 11, 409410.
Galton, F. (1880). Visualised numerals. Nature, 21, 252256.
Gebuis, T., & Reynvoet, B. (2012a). The interplay between visual cues and non-symbolic
number. Journal of Experimental Psychology: General, 141, 642648.
Gebuis, T., & Reynvoet, B. (2012b). The role of visual information in numerosity esti-
mation. PLoS One, 7, e37426.
Gebuis, T., & van der Smagt, M. J. (2011). False approximations of the approximate
number system? PLoS One, 6(10), e25405.
Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2010). Non-symbolic arithmetic abilities
and mathematics achievement in the rst year of formal schooling. Cognition, 115,
Ginsburg, N. (1976). Eect of item arrangement on perceived numerosity: Randomness vs
regularity. Perceptual and Motor Skills, 43, 663668.
Ginsburg, N., & Nicholls, A. (1988). Perceived numerosity as a function of item size.
Perceptual & Motor Skills, 67, 656658.
Girgus, J. S., & Coren, S. (1975). Depth cues and constancy scaling in the horizontal-
vertical illusion: The bisection error. Canadian Journal of Psychology, 29(1), 59.
Gregory, R. L. (1963). Distortion of visual space as inappropriate constancy scaling.
Nature, 199(4894), 678680.
Gregory, R. L. (1997). Knowledge in perception and illusion. Philosophical Transactions of
the Royal Society of London B, 352, 11211127.
Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008). Individual dierences in non-
verbal number acuity correlate with maths achievement. Nature, 455, 665668.
Harrington, D. O. (1981). The visual elds. Anatomy of visual pathway.
He, L., Zhang, J., Zhou, T., & Chen, L. (2009). Connectedness aects dot numerosity
judgment: Implications for congural processing. Psychonomic Bulletin & Review,
16(3), 509517.
Higashiyawa, A. (1992). Anisotropic perception of visual angle: Implications for the
horizontal-vertical illusion, overconstancy of size, and the moon illusion. Perception &
Psychophysics, 51, 218230.
Hurewitz, F., Gelman, R., & Schnitzer, B. (2006). Sometimes area counts more than
number. Proceedings of the National Academy of Sciences of the United States of America,
103, 1959919604.
Izard, V., & Dehaene, S. (2008). Calibrating the mental number line. Cognition, 106,
Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract
numbers. Proceedings of the National Academy of Sciences of the United States of
America, 106(25), 1038210385.
Jazayeri, M., & Shadlen, M. N. (2010). Temporal context calibrates interval timing. Nature
Neuroscience, 13, 10201026.
Kramer, P., Di Bono, M. G., & Zorzi, M. (2011). Numerosity estimation in visual stimuli in
the absence of luminance-based cues. PLos ONE 6e17378.
Kramer, P., & Bressan, P. (2017). Commentary: From sense of numberto sense of
magnitude’–The role of continuous magnitudes in numerical cognition. Frontiers in
Psychology, 7, 2032.
Krause, F., Bekkering, H., & Lindemann, O. (2013). A feeling for numbers: Shared metric
for symbolic and tactile numerosities. Frontiers in Psychology, 4(7).
Krause, F., Bekkering, H., Pratt, J., & Lindemann, O. (2017). Interaction between numbers
and size during visual search. Psychological Research, 81(3), 664677.
Krueger, L. E. (1982). Single judgments of numerosity. Perception & Psychophysics, 31(2),
Krueger, L. E. (1984). Perceived numerosity: A comparison of magnitude production,
magnitude estimation, and discrimination judgments. Perception & Psychophysics,
35(6), 536542.
Künnapas, T. M. (1955). An analysis of the vertical-horizontal illusion.Journal of
Experimental Psychology, 49(2), 134.
Künnapas, T. M. (1957). The vertical-horizontal illusion and the visual eld. Journal of
Experimental Psychology, 53(6), 405.
Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2017). From sense of numberto sense
of magnitude”—The role of continuous magnitudes in numerical cognition.
Behavioral and Brain Sciences, 31, 647648.
Loomis, J. M., & Philbeck, J. W. (1999). Is the anisotropy of perceived 3-D shape invariant
across scale? Perception & Psychophysics, 61, 397402.
Mamassian, P., & de Montalembert, M. (2010). A simple model of the verticalhorizontal
illusion. Vision Research, 50(10), 956962.
Mejias, S., & Schiltz, C. (2013). Estimation abilities of large numerosities in kindergart-
ners. Frontiers in Psychology, 4, 518.
Park, J., Bermudez, V., Roberts, R. C., & Brannon, E. M. (2016). Non-symbolic approx-
imate arithmetic training improves math performance in preschoolers. Journal of
Experimental Child Psychology, 152, 278293.
Park, J., Dewind, N. K., Woldor, M. G., & Brannon, E. M. (2016). Rapid and direct
encoding of numerosity in the visual stream. Cerebral Cortex, 26(2), 748763.
Parrish, A. E., Beran, M. J., & Agrillo, C. (2019). Linear numerosity illusions in capuchin
monkeys (Sapajus apella), rhesus macaques (Macaca mulatta), and humans (Homo
sapiens). Animal Cognition, 22(5), 883889.
Pecunioso, A., & Agrillo, C. (2020a). Is the horizontal-vertical illusion mainly a by-pro-
duct of Petters rule? Symmetry, 12,6.
Pecunioso, A., & Agrillo, C. (2020b). Do musicians perceive numerosity illusions dier-
ently? Psychology of Music. online rst.
Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., et al.
(2010). Developmental trajectory of number acuity reveals a severe impairment in
developmental dyscalculia. Cognition, 116, 3341.
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in
an Amazonian indigene group. Science, 306, 499503.
Price, G. R., Palmer, D., Battista, C., & Ansari, D. (2012). Nonsymbolic numerical mag-
nitude comparison: Reliability and validity of dierent task variants and outcome
measures, and their relationship to arithmetic achievement in adults. Acta
Psychologica, 140(1), 5057.
Prinzmetal, W., & Gettleman, L. (1993). Vertical-horizontal illusion: One eye is better
than two. Perception & Psychophysics, 53(1), 8188.
Revkin, S. K., Piazza, M., Izard, V., Cohen, L., & Dehaene, S. (2008). Does subitizing
reect numerical estimation? Psychological Science, 19(6), 607614.
Shuman, M., & Kanwisher, N. (2004). Numerical magnitude in the human parietal lobe:
Tests of representational generality and domain specicity. Neuron, 44, 557569.
Walsh, V. A. (2003). A theory of magnitude: Common cortical metrics of time, space and
quantity. Trends in Cognitive Sciences, 7, 483488.
Whalen, J., Gallistel, C. R., & Gelman, R. (1999). Nonverbal counting in humans: The
psychophysics of number representation. Psychological Science, 10, 130137.
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants.
Cognition, 74, B1B11.
Zhang, Y., & Okamoto, Y. (2017). Encoding 10nessimproves rst-graders' estimation of
numerical magnitudes. Journal of Numerical Cognition, 2(3), 190201.
Zorzi, M., Priftis, K., & Umiltà, C. (2002). Brain damage: Neglect disrupts the mental
number line. Nature, 417, 138139.
A. Pecunioso, et al. $FWD3V\FKRORJLFD
... Non-symbolic numerosity in sets with illusory contours exploits a contextsensitive, but contrast-insensitive, visual boundary formation process. & Agrillo, 2020). For instance, the so-called connectedness illusion has been adopted to manipulate the perceived segmentation (or grouping strength) of the items in the set, keeping constant the low-level features across connectedness levels (Franconeri et al., 2009;He et al., 2009). ...
... Only recently, we assisted to a continuously increasing number of studies focused on perceptual aspects of numerosity processing using visual illusions and grouping rules. It follows that future (new) models of numerosity processing should account for the effects of several visual factors that may affect how individual local items are segmented, such as item-proximity (e.g., Allik & Raidvee, 2021;Chakravarthi & Bertamini, 2020), homogeneity (Redden & Hoch, 2009;DeWind et al., 2020), physical connectedness (Franconeri et al., 2009;He et al., 2009;He et al., 2015), illusory connectedness (Kirjakovski & Matsumoto, 2016) and several other spatial factors (Dormal et al., 2018;Pecunioso et al., 2020;Picon et al., 2019). In short, we suggest that the Number sense model should be extended integrating also the knowledge gained in visual perception and Gestalt psychology. ...
... Chapter adapted from:.Attention, Perception, & Psychophysics,[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Recently, visual illusions have been successfully employed to understand which visual features represent the building block of numerosity perception, because they can be used to selectively manipulate particular features of numerical sets without altering other physical features in the image (e.g.,Dormal, Larigaldie, Lefèvre, Pesenti, & Andres, 2018;Picon, Dramkin, & Odic, 2019;Pecunioso, Petrazzini, ...
Full-text available
The natural environment in which animals are forced to survive shapes their brain and the way in which they behave to adapt and overcome natural pressures. These selective pressures may have determined the emergence of an evolutionary ancient neural system suited to rapidly extract abstract information from collections, such as their numerosity, to take informed decisions pivotal for survivance and adaptation. The “Number Sense” theory represents the most influential neural model accounting for neuropsychological and psychophysical evidence in humans and animals. However, this model is still largely debated because of the methodological difficulties in isolating neural signals related to “discrete” (i.e., the real number of objects in a collection) abstract numerosity processing from those related to other features correlated or confounded with numerosity in the raw sensory input (e.g., visual area, density, spatial frequency, etc.). The present thesis aimed to investigate which mechanisms might be at the basis of visual numerosity representations, overcoming the difficulties in isolating discrete from continuous features. After reviewing the main theoretical models and findings from the literature (Chapter 1 and 2), in the Chapter 3 we presented a psychophysical paradigm in which Kanizsa-like illusory contours (ICs) lines were used to manipulate the connectedness (e.g., grouping strength) of the items in the set, controlling all the continuous features across connectedness levels. We showed that numerosity was underestimated when connections increased, suggesting that numerosity relies on segmented perceptual objects rather than on raw low-level features. In Chapter 4, we controlled for illusory brightness confounds accompanying ICs. Exploiting perceptual properties of the reverse-contrast Kanizsa illusion, we found that underestimation was insensitive to inducer contrast direction, suggesting that the effect was specifically induced by a sign invariant boundary grouping and not due to perceived brightness confounds. In Chapter 5, we concurrently manipulated grouping with ICs lines and the perceived size of the collections using classic size illusions (Ponzo Illusion). By using a combination of visual illusions, we showed that numerosity perception is not based on perceived continuous cues, despite continuous cue might affect numerical perception. In Chapter 6 we tackled the issue with a direct physical approach: using Fourier analysis to equalize spatial frequency (SF) in the stimuli, we showed that stimulus energy is not involved in numerosity representation. Rather segmentation of the items and perceptual organization explained our main findings. In Chapter 7 we also showed that the ratio effect, an important hallmark of Weber-like encoding of numerical perception, is not primarily explained by stimulus energy or SF. Finally, in Chapter 8, we also provided the first empirical evidence that non-symbolic numerosity are represented spatially regardless of the physical SF content of the stimuli. Overall, this thesis strongly supports the view that numerosity processing is not merely based on low-level features, and rather clearly suggests that discrete information is at the core of the Number Sense.
... Despite this it is the inner group that is perceived as more numerous. Note that the convex hull measures objective size; it is also known that perceived (subjective) size can affect numerosity, as illustrated in the context of the Ponzo illusion (Ponzo, 1928), the horizontal vertical illusion (Pecunioso et al., 2020) or in terms of changes of perceived size after adaptation (Zimmermann & Fink, 2016). ...
Full-text available
In four experiments we investigated the Solitaire illusion. In this illusion, most observers see as more numerous a set of dots that forms a single central group, compared to dots on the outside of that group. We confirmed and extended the effect to configurations with much higher numerosity than the original and of various colours. Contrary to prediction, separating the two groups, so that they are presented side by side, reduced but did not abolish or reverse the illusion. In this illusion, therefore, neither total size of the region (area), not average distance of the elements has the expected effect. In Experiments 3 and 4 we eliminated the regularity of the pattern, by sampling 50% (Exp 3) or only a 10% (Exp 4) of the elements. These produces quasi‐random configurations. For these configurations the bias for the inner groups was still present, and it was only eliminated when the groups were shown as separate. However, the effect never reversed (no bias for the outer group, despite its larger area). We conclude that the Solitaire illusion is evidence of a strong bias in favour of centrally located elements, a bias that can overcome other factors.
... This study aims to investigate whether non-symbolic numerical estimation varies as a function of the perceived area size, adopting a similar approach to the abovementioned studies (Dormal et al. 2018;Pecunioso et al. 2020;Picon et al. 2019) not only in humans but also in non-human animals. We were interested in understanding, from a behavioural point of view, whether the same perceptual biases that influence spatial estimation also affect numerical estimation and whether this occurs similarly in humans and fish. ...
Full-text available
Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.
... Recently, visual illusions have been successfully employed to understand which visual features represent the building blocks of numerosity perception, because they can be used to selectively manipulate particular features of numerical sets without altering other physical features in the image (e.g., Dormal et al., 2018;Pecunioso et al., 2020;Picon et al., 2019). For instance, the so-called connectedness illusion has been adopted to manipulate the perceived segmentation (or grouping strength) of the items in the set, keeping constant the low-level features across connectedness levels (Franconeri et al., 2009;He et al., 2009). ...
Full-text available
The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by Illusory Contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule-out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2 or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs-lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs-lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.
... subjects estimate spatially spread out magnitudes like line length 22 and numerosity 23 to be larger along the vertical axis. We anticipate this incidental observation to be of interest for vision-and numerical cognition scientists using similar stimuli. ...
Full-text available
Previous work has shown bidirectional crosstalk between Working Memory (WM) and perception such that the contents of WM can alter concurrent percepts and vice versa. Here, we examine WM-perception interactions in a new task setting. Participants judged the proportion of colored dots in a stream of visual displays while concurrently holding location- and color information in memory. Spatiotemporally resolved psychometrics disclosed a modulation of perceptual sensitivity consistent with a bias of visual spatial attention towards the memorized location. However, this effect was short-lived, suggesting that the visuospatial WM information was rapidly deprioritized during processing of new perceptual information. Independently, we observed robust bidirectional biases of categorical color judgments, in that perceptual decisions and mnemonic reports were attracted to each other. These biases occurred without reductions in overall perceptual sensitivity compared to control conditions without a concurrent WM load. The results conceptually replicate and extend previous findings in visual search and suggest that crosstalk between WM and perception can arise at multiple levels, from sensory-perceptual to decisional processing.
... Assimilation and contrast effects seem to operate also in the perception of the HV illusion: the ends of the vertical line are supposed to be closer to the boundary of the visual field, leading to an assimilation of its length to the boundary edges of the visual field. Instead, the ends of the horizontal line are supposed to be far from the boundary of the visual field, eliciting a contrast effect [28,29]. A reduced susceptibility to contrast effects, found in the Delboeuf illusion study [19], may have impacted the behaviour exhibited with the L-shape, thereby reducing underestimation of the horizontal line and nulling the illusory effect. ...
Full-text available
The horizontal-vertical (HV) illusion is characterized by a tendency to overestimate the length of vertically-arranged objects. Comparative research is primarily confined to primates, a range of species that, although arboreal, often explore their environment moving along the horizontal axis. Such behaviour may have led to the development of asymmetrical perceptual mechanisms to make relative size judgments of objects placed vertically and horizontally. We observed the susceptibility to the HV illusion in fish, whose ability to swim along the horizontal and vertical plane permits them to scan objects’ size equally on both axes. Guppies (Poecilia reticulata) were trained to select the longer orange line to receive a food reward. In the test phase, two arrays, containing two same-sized lines were presented, one horizontally and the other vertically. Black lines were also included in each pattern to generate the perception of an inverted T-shape (where a horizontal line is bisected by a vertical one) or an L-shape (no bisection). No bias was observed in the L-shape, which supports the idea of differential perceptual mechanisms for primates and fish. In the inverted T-shape, guppies estimated the bisected line as shorter, providing the first evidence of a length bisection bias in a fish species.
Full-text available
The horizontal-vertical (HV) illusion is a classical example of an asymmetrical perception of size in the vertical and horizontal axes, also known as ‘anisotropy of the perceived space’. Several authors argued that the horizontally-oriented ellipse of the binocular visual field might play an important role in the emergence of this illusion. Alternatively, a length bisection bias and size-constancy mechanisms have been advocated to account for the asymmetrical perception in the two dimensions. To investigate this phenomenon, participants are commonly required to estimate the length of two separate lines, one vertical and one horizontal, often arranged in an inverted-T pattern. Here we suggest that this type of stimulus may introduce physical and subjective biases that prevent a fine investigation. In particular, we believe that Petter’s rule, that applies to two-dimensional patterns formed by two overlapping surfaces, may play a critical role that will not support an interpretation based on the shape of the binocular visual field nor a length bisection bias.
Full-text available
Numerosity illusions emerge when the stimuli in one set are overestimated or underestimated relative to the number (or quantity) of stimuli in another set. In the case of multi-item arrays, individual items that form a better Gestalt are more readily grouped, leading to overestimation by human adults and children. As an example, the Solitaire illusion emerges when dots forming a central cluster (cross-pattern) are overestimated relative to the same number of dots on the periphery of the array. Although this illusion is robustly experienced by human adults, previous studies have produced weaker illusory results for young children, chimpanzees, rhesus macaques, capuchin monkeys, and guppies. In the current study, we presented nonhuman primates with other linear arrangements of stimuli from Frith and Frith’s (Percept Psychoph 11:409–410, 1972) original paper with human participants that included the Solitaire illusion. Capuchin monkeys, rhesus macaques, and human adults learned to quantify black and white dots that were presented within intermingled arrays, responding on the basis of the more numerous dot colors. Humans perceived the various illusions similar to the original findings of Frith and Frith (1972), validating the current comparative design; however, there was no evidence of illusory susceptibility in either species of monkey. These results are considered in light of illusion susceptibility among primates as well as considering the role of numerical discrimination abilities and perceptual processing mode on illusion emergence.
Full-text available
Recent studies have demonstrated that the numerosity of visually presented dot arrays is represented in low-level visual cortex extremely early in latency. However, whether or not such an early neural signature reflects the perceptual encoding of numerosity remains unknown. Alternatively, such a signature may indicate the raw sensory representation of the dot-array stimulus before becoming the perceived representation of numerosity. Here, we addressed this question by using the connectedness illusion, whereby arrays with pairwise connected dots are perceived to be less numerous compared with arrays containing isolated dots. Using EEG and fMRI in two independent experiments, we measured neural responses to dot-array stimuli comprising 16 or 32 dots, either isolated or pairwise connected. The effect of connectedness, which reflects the segmentation of the visual stimulus into perceptual units, was observed in the neural activity after 150 msec post stimulus onset in the EEG experiment and in area V3 in the fMRI experiment using a multivariate pattern analysis. In contrast, earlier neural activity before 100 msec and in area V2 was strictly modulated by numerosity regardless of connectedness, suggesting that this early activity reflects the sensory representation of a dot array before perceptual segmentation. Our findings thus demonstrate that the neural representation for numerosity in early visual cortex is not sufficient for visual number perception and suggest that the perceptual encoding of numerosity occurs at or after the segmentation process that takes place later in area V3.
Full-text available
The ability to use quantitative information is thought to be adaptive in a wide range of ecological contexts. For nearly a century, the numerical abilities of mammals and birds have been extensively studied using a variety of approaches. However, in the last two decades, there has been increasing interest in investigating the numerical abilities of teleosts (i.e. a large group of ray-finned fish), mainly due to the practical advantages of using fish species as models in laboratory research. Here, we review the current state of the art in this field. In the first part, we highlight some potential ecological functions of numerical abilities in fish and summarize the existing literature that demonstrates numerical abilities in different fish species. In many cases, surprising similarities have been reported among the numerical performance of mammals, birds and fish, raising the question as to whether vertebrates' numerical systems have been inherited from a common ancestor. In the second part, we will focus on what we still need to investigate, specifically the research fields in which the use of fish would be particularly beneficial, such as the genetic bases of numerical abilities, the development of these abilities and the evolutionary foundation of vertebrate number sense. This article is part of a discussion meeting issue ‘The origins of numerical abilities’.
Full-text available
Previous studies showed that the magnitude information conveyed by sensory cues, such as length or surface, influences the ability to compare the numerosity of sets of objects. However, the perceptual nature of this representation and how it interacts with the processes involved in numerical judgements remain unclear. This study aims to address these issues by studying the interference of length on numerosity under different perceptual and response conditions. The first experiment shows that the influence of length does not depend on the actual length but on subjective values reflecting the way length is perceived in a given visual context. The Müller-Lyer illusion was used to manipulate the perceived length of two dot arrays independently of their actual length. When the length of two dot arrays was equal but perceived as different due to the illusion, participants erroneously reported differences in the number of dots contained in each array, evidencing a similar effect of Müller-Lyer illusion on length and numerosity comparison. This finding was replicated in a second experiment where participants had to give a verbal estimate of the number of dots contained in a given array, thereby eliminating the choice between a small or large response. Compared with a neutral condition, estimations were systematically larger than the actual number of dots as the illusory length increased. These results demonstrate that the illusory-induced experience of length influences numerosity estimation over and beyond objective cues and that this influence is not a response selection bias.
Full-text available
How is numerosity encoded by the visual system? – directly, or derived indirectly from texture density? We recently suggested that the numerosity of sparse patterns is encoded directly by dedicated mechanisms (which have been described as the “Approximate Number System” ANS). However, at high dot densities, where items become “crowded” and difficult to segregate, “texture-density ” mechanisms come into play. Here we tested the importance of item segmentation on numerosity and density perception at various stimulus densities, by measuring the effect of connecting visual objects with thin lines. The results confirmed many previous studies showing that connecting items robustly reduces the apparent numerosity of patterns of moderate density. We further showed that the apparent density of moderate-density patterns is also reduced by connecting the dots. Crucially, we found that both these effects are strongly reduced at higher numerosities. Indeed for density judgments, the effect reverses, so connecting dots in dense patterns increases the apparent density (as expected from the physical characteristics). The results provide clear support for the three-regime framework of number perception, and suggest that for moderately sparse stimuli, numerosity – but not texture-density – is perceived directly.
Full-text available
Understanding numerical magnitudes is a foundational skill that significantly impacts later learning of mathematics concepts. The current study tested the idea that encoding of “10ness” is crucial to improving children’s estimation of two-digit number magnitudes. We used commercially available base-10 blocks for this purpose. The children in the experimental condition were asked to construct two-digit numbers by laying down the precise combinations of 10- and 1-blocks horizontally (e.g., three 10-blocks and seven 1-blocks for 37). Two control conditions were also included. In one control condition, children used 1-blocks only. In another control condition, children used one 10-block and as many 1-blocks as necessary. After working with the experimenter for only 15 minutes twice, the children in the experimental condition were significantly more accurate on the estimation task than those in the control conditions. The findings confirmed the importance of encoding 10ness as a unit in making accurate estimates of two-digit number magnitudes. The importance of encoding other units in the base-10 system is discussed.
Full-text available
The Solitaire illusion occurs when the spatial arrangement of items influences the subjective estimation of their quantity. Unlike other illusory phenomena frequently reported in humans and often also in non-human animals, evidence of the Solitaire illusion in species other than humans remains weak. However, before concluding that this perceptual bias affects quantity judgments differently in human and non-human animals, further investigations on the strength of the Solitaire illusion is required. To date, no study has assessed the exact misperception of numerosity generated by the Solitaire arrangement, and the possibility exists that the numerical effects generated by the illusion are too subtle to be detected by non-human animals. The present study investigated the strength of this illusion in adult humans. In a relative numerosity task, participants were required to select which array contained more blue items in the presence of two arrays made of identical blue and yellow items. Participants perceived the Solitaire illusion as predicted, overestimating the Solitaire array with centrally clustered blue items as more numerous than the Solitaire array with blue items on the perimeter. Their performance in the presence of the Solitaire array was similar to that observed in control trials with numerical ratios larger than 0.67, suggesting that the illusory array produces a substantial overestimation of the number of blue items in one array relative to the other. This aspect was more directly investigated in a numerosity identification task in which participants were required to estimate the number of blue items when single arrays were presented one at a time. In the presence of the Solitaire array, participants slightly overestimated the number of items when they were centrally located while they underestimated the number of items when those items were located on the perimeter. Items located on the perimeter were perceived to be 76% as numerous as centrally located items. The magnitude of misperception of numerosity reported here may represent a useful tool to help to understand whether non-human animals have different perceptual mechanisms or, instead, do not display adequate numerical abilities to spot the illusory difference generated in the Solitaire array.
A large body of experimental evidence suggests that long-term musical training has profound consequences on the functional organization of the brain, leading to an improvement of cognitive abilities that are non-primarily involved in music. Here, we tested the hypothesis stating that long-term musical training has effects in the perceptual laws underlying vision. To achieve our goal, we compared the susceptibility of musicians and non-musicians to the Solitaire illusion, an illusion of numerosity based on the Gestalt law of proximity and good continuation. Both groups were observed in a relative (Experiment 1) and an absolute (Experiment 2) numerosity task: the former required an estimation of which array contained more blue dots; the latter required an estimation of the number of blue dots presented. In both experiments, the illusory pattern was presented as well. In control trials, no difference was found between musicians and non-musicians in the overall performance. The two groups were susceptible to the illusion in both experiments, although the musicians in Experiment 2 varied in their susceptibility to the numerosity misperception, perceiving a smaller illusory ratio compared with non-musicians. Based on these results, we suggest that prolonged music training may alter the perceptual laws in visual modality.