Effect of egocentric and allocentric reference frames on spatial-numerical associations

Abstract

From an embodied view of cognition, sensorimotor mechanisms are strongly involved in abstract processing, such as Arabic number meanings. For example, spatial cognition can influence number processing. These spatial-numerical associations (SNAs) have been deeply explored since the seminal SNAs of response code (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively). Although these SNAs along the transverse plane (left-to-right axis) have been extensively studied in cognitive sciences, no systematic assessment of other planes of the tridimensional space has been afforded. Moreover, there is no evidence of how SNAs organise themselves throughout the changes in spatial body-reference frames (egocentric and allocentric). Hence, this study aimed to explore how SNAs organise themselves along the transverse and sagittal planes when egocentric and allocentric changes are processed during body displacements in the environment. In the first experiment, the results revealed that, when the participants used an egocentric reference, SNAs were observed only along the sagittal plane. In a second experiment that used an allocentric reference, the reversed pattern of results was observed: SNAs were present only along the transverse plane of the body. Overall, these findings suggest that, depending on the spatial reference frames of the body, SNAs are strongly flexible.

Keywords

Allocentric egocentric numerical cognition sagittal and transverse planes SNARC effect SNAs

Introduction

The ability to represent spatial information is fundamental in human cognition (Previc, 1998). Some interactions with the environment require a distance estimation between our own body and the localisation of the goal location. For example, during forest hiking, it is essential to be able to situate our body orientation according to the cardinal points and to make landmarks on the sagittal and transverse planes of the environment. These landmarks are deeply rooted in space navigation for humans and rest on quantity representation, which establishes the basis for any common representation of topographic references and relies on closely related brain networks (for a review, see Hubbard et al., 2005). Furthermore, this quantity representation provides anticipation and adjustment mechanisms for the trajectory to compute the distance and the time for the required action.

From a theoretical point of view, Walsh (2003; see also Anobile et al., 2021; Cona et al., 2021; Hawes et al., 2019) suggested that quantity representation could be based on a generalised magnitude system for time, quantity, and space. This view was supported by a recent meta-analysis of neuroimaging studies showing a cortical gradient organisation from time to space in the core magnitude neural system (i.e., bilateral insula, presupplementary motor area, right frontal operculum, and intraparietal sulci; Cona et al., 2021). For example, the right supramarginal gyrus (i.e., parietal area) is mainly involved during the subjective experience of time processing (Prete et al., 2021). Behavioural studies have also shown, for instance, that the physical characteristics of objects such as weight (Dalmaso & Vicovaro, 2019), size (Reike & Schwarz, 2017), visual luminance (Cohen Kadosh & Henik, 2006), or even loudness of sounds (Hartmann & Mast, 2017) are processed in this generalised magnitude system (Leibovich et al., 2017). Importantly, this system is also engaged in the processing of symbols such as Arabic numbers and their associated meanings (Nourouzi Mehlabani et al., 2020). As defined in the theory of magnitude (ATOM), action should play a key role in the link between all these quantity dimensions (Walsh, 2003). Thus, if this system shares mutual resources for action, then sensorimotor processing should influence the processing of these quantity dimensions. For example, Badets and Pesenti (2010) showed that the observation of biological stimuli such as hand actions can influence number processing. Specifically, after seeing two Arabic digits on a screen, participants needed to recall a digit according to parity depending on the type of movement (i.e., for hand closing and opening, it was necessary to recall the odd and even digits, respectively). The results showed that participants recalled small and large digits faster after seeing closing and opening hands, respectively (for similar findings, see Andres et al., 2004; Chiou et al., 2012; Lindemann et al., 2007; Namdar & Ganel, 2018). Badets and Pesenti suggested that these findings are due to a compatibility effect between the magnitudes of the numbers and the size covered by the hand movement. There is an automatic coding system that shares a magnitude dimension between numbers, finger movements, and, implicitly, objects. This could be attributed to the finding that fingers and numbers are interconnected through actions that are devoted to object processing. In their experiments, when fake hands were used as stimuli, this compatibility effect disappeared. Consequently, the embodied account can support the number−hand interaction found for biological stimuli. In this embodied theoretical perspective, the cognitive system uses the environment and sensory-motor information of the body to construct the internal representations of the mind (Barsalou, 2010; Meteyard et al., 2012; Wilson, 2002). Accordingly, the environment and the body serve as external support for quantity representations. This interpretation through embodiment has been strengthened by studies showing overlapping parietal brain areas for processing number, spatial, and hand action representations (Andres et al., 2012; Hubbard et al., 2005).

Other empirical evidence linking space representation and number processing is well documented in the literature (see Hawes et al., 2019; Winter et al., 2015 for a review). Specifically, there is an interaction between spatial and numerical processing, called spatial-numerical associations (SNAs). The SNA of response code (SNARC; Dehaene et al., 1993) effect is the most reliable evidence of SNAs, especially on the transverse plane according to body orientation (i.e., the left to right axis; Cipora et al., 2022). In these studies, participants had to judge the parity of digits by pressing a button on the left or right side. The results revealed faster reaction times for responses with the left and right hands after the processing of small and large numbers, respectively (Dehaene et al., 1993; see Cipora et al., 2022 for a review). Studies suggest that there is a compatibility effect between spatial information and the magnitude of numbers that rest on neuronal networks located in parietal areas (Hawes et al., 2019; Hubbard et al., 2005; Toomarian & Hubbard, 2018).

This effect appears to be malleable and largely shaped by cultural factors such as the direction of reading and writing (Toomarian & Hubbard, 2018). For right-to-left readers, the SNARC tends to disappear when similar response times are found for the left and right hands after the processing of small and large numbers, respectively (Dehaene et al., 1993; Experiment 7). Moreover, some studies have revealed a reversed effect (Hung et al., 2008; Shaki & Fischer, 2008, 2012; Shaki et al., 2009; Zebian, 2005). For example, Shaki et al. (2009) reported an inverted SNARC with Palestinian participants who read Arabic words and Arabic–Indic numbers from right to left; their responses were faster for small and large numbers when pressing right- or left-side buttons, respectively (see also Shaki et al., 2012). All of these findings indicate that SNAs are influenced by ontogenic factors such as the direction of writing and reading that rely on sensorimotor experiences. However, these left-to-right SNAs also appear to follow a universal organisation shared by different species (de Hevia et al., 2017; Rugani et al., 2010, 2015). For instance, a study recently observed in honey bees an organisation of numerosity according to magnitude with a small association to the left and a large association to the right side (Giurfa et al., 2022).

Nevertheless, most studies on SNAs have focused on the transversal plane (i.e., the left to right axis) with no systematic assessment of other planes (i.e., the frontal plane with bottom to top axis and the sagittal plane with near to far axis) of the tridimensional representation (Previc, 1998). For the ATOM (2003), the interaction between space representation and Arabic number processing should also influence the three different planes and the related axis. Accordingly, SNAs have also been observed along the frontal plane in the vertical axis (Hartmann et al., 2014). When participants judged the parity of a number, they answered faster when pressing a button located in the down and up parts of space when processing small and large numbers, respectively.

This finding along the vertical axis can also be interpreted through the embodied cognition approach (Barsalou, 2008, 2010, 2016; Wilson, 2002). The vertical distance appears to be constantly congruent with the physical characteristics of the natural world, which are defined by quantitative facts (Fischer & Brugger, 2011). In this way, abstract concepts and representations (e.g., number magnitude) are derived from sensorimotor experiences during interactions with the universal properties of the world. Accordingly, it has recently been argued that horizontal SNAs are derived from the experience of finger counting (Fischer & Brugger, 2011 but see also Prete & Tommasi, 2020 for no concrete link between finger counting and the SNARC).

To the best of our knowledge, very few studies have examined the organisation of SNAs along the third plane, namely, the sagittal plane (Aleotti et al., 2020; Chen et al., 2015; Cooney et al., 2021; Santens & Gevers, 2008). Importantly, these studies have explored only the effect of number processing on action and not the reverse relationship (i.e., the effect of the processing of action on number processing). Participants usually needed to judge the magnitude of numbers and then answer with a motor response (i.e., by pressing a button) with their left or right hands if a viewed number was, for instance, smaller or larger than five (Cooney et al., 2021; Santens & Gevers, 2008) or according to parity (Aleotti et al., 2020; Chen et al., 2015). Therefore, all of these experimental paradigms focused only on hand action, and none of them investigated the processing of whole-body displacement. According to the ATOM, body displacements should also influence number processing. Because studies on SNAs were mainly hand-centred, they only explored the number−space association according to an egocentric reference frame (for a review, see Winter et al., 2015). Therefore, for the transverse or sagittal planes, there is no evidence about SNAs from an allocentric perspective. From a more ecological approach, it seems necessary to specify how SNAs organise themselves when egocentric and allocentric changes are present during displacements in the environment. According to ATOM, SNAs should adapt to changes in these spatial reference frames.

In this study, we aimed to highlight SNAs along the sagittal plane of space to assess their organisation from different spatial reference frames (egocentric vs. allocentric). This issue has been investigated with a random number generation (RNG) task (Loetscher & Brugger, 2007) using a street view. During this task, participants had to generate a random number ranging from 1 to 30 just after the observation of the avatar displacement. The displacement of the avatar targeted the near-far (i.e., the sagittal plane) and left-right (i.e., the transverse plane) spaces. Our analyses offer the opportunity to explore the relative implications of these two kinds of SNAs (i.e., sagittal and transverse).

Two hypotheses were considered throughout two experiments. In a first experiment, participants used an egocentric perspective (i.e., the avatar was oriented in the same direction as the participant). From the ATOM, we expected a relationship between the distance of displacements on the sagittal plane and the side on the transverse plane with the numerical magnitude. Specifically, displacements in near space should be associated with the production of small numbers, whereas displacements in far space should be associated with large numbers (sagittal SNARC). Furthermore, displacements to the left should be associated with small numbers, whereas those to the right should be associated with large numbers (transversal SNARC). In a second experiment, we assessed whether these space−number associations were transferable throughout changes in the subject’s spatial reference frames (i.e., the avatar was oriented in front of the participant with a rotation of 180° in the screen). In this allocentric perspective, we expected the same findings for the space−number association with a rotation of 180°.

Experiment 1

Methods

Participants

A total of 28 participants were recruited (19 females, M = 24.14, SD = 3.89, ranging from 18 to 31). The scores of the Edinburgh test revealed 25 right-handers (mean handedness score of 82.39 ± 23.54) and 3 left-handers (−75.76 ± 41.98). For this test, a score of −100 or +100 reports to a complete left or right preference, respectively. For both experiments, G Power software (Faul et al., 2007) was used to make the a priori calculation of our sample size. The calculation relied on a repeated-measures within-interaction analysis of variance (ANOVA) from the results obtained in a previous work (Hartmann et al., 2019). We referred to the ANOVA on the mean of generated numbers following a 2 × 2 plane with a partial eta-squared of 0.09, correlations between repeated measures of .5, alpha level set to .05, and statistical power of 1 − β at 0.85. The results indicated that 24 participants per group would be sufficient. Consequently, we recruited 28 participants to ensure a sufficient sample size. All participants had normal or corrected-to-normal vision and no neurodevelopmental or neurological and perceptual disorders. Informed consent was obtained before participation, and participants were naïve to the purpose of the study. The study conformed to the ethical standards of the Declaration of Helsinki and to the institutional review board of the Centre National de la Recherche Scientifique (approval number ID-RCB: 2018-A02426-49).

Apparatus and stimuli

The apparatus was a response microphone connected to a Dell laptop computer with an LCD screen 38.1 cm in diagonal and a spatial resolution of 1,920 × 1,080 pixels. For manual responses, participants used the trackpad of the computer. A customised E-Prime programme controlled the presentation of stimuli and recorded the verbal responses (Schneider et al., 2002). Participants sat in front of the microphone and 60 cm from the computer with the keyboard hidden in a dimly lit, quiet room.

For the stimuli, a drawing of a shopping street with an avatar at its extremity comprising two types of food stores (i.e., a bakery and a cheese shop) was used (Figure 1). The participants heard two recordings in French (i.e., tarte [pie] and camembert [cheese]). These two words were matched in intensity (the same decibel level) and sample size (the same duration of recording) to avoid any confounding factors due to the effects of continuous magnitudes (Leibovich et al., 2017). In the same vein, the words were chosen because they referred to foodstuffs that coincided in weight and size (area and convex hull). The location of food stores was randomly defined across trials on four positions, near left, near right, far left, and far right, to avoid any favour towards the transversal or sagittal part of space.

Figure 1.

Sequence of events in Experiment 1. After hearing the name of one foodstuff, participants had to say the name of the associate food store. Then, depending on the colour, they had to generate a random even or odd number (here the rule requires to associate the yellow to odd numbers).

Procedure

The experiment comprised two phases: a familiarisation phase followed by an experimental phase. For the familiarisation phase, the events of a trial were as follows: a street drawing was displayed on the screen with an avatar located at its extremity, and after 1,500 ms, the name of a foodstuff was heard. Participants were then required to say the name of the food store where the foodstuff was available. Once the microphone detected the vocal response, the avatar moved. This sequence was performed for all locations (near left, near right, far left, and far right) and included eight trials in total. Subsequently, to ensure that the participants integrated the avatar’s spatial references, they had to click on the right and left sides of the street from the point of view of the avatar. For the experimental phase, for each trial, the participants completed the following sequence (Figure 1): a black fixation cross was displayed (96-point Arial font) on a white background for 500 ms. Next, a street drawing was displayed on the screen with an avatar located at its extremity. From an egocentric perspective, participants observed the street drawing with the avatar located from behind at the bottom of the screen. To maintain attention on this initial location, the avatar was first displayed in black for 500 ms, then in grey for 500 ms, and then returned to black for 500 ms. Next, the drawing remained displayed, and the name of a foodstuff was heard (e.g., pie). Participants were then required to say the name of the food store where the foodstuff was available (e.g., bakery). As soon as the verbal response was detected by the microphone, the sign of the corresponding store was coloured yellow or blue in association with the corresponding avatar’s movement. After the avatar displacement, the participants had to randomly say an odd or even number between 1 and 30 if the colour sign was yellow or blue, respectively. The view of the shopping street remained on the screen for 2,000 ms. The parity instructions for colours were used to ensure that participants accurately processed the spatial location of the food store and to be sure that the rule not fostered any plane. Thus, the participants explicitly processed spatial features that have recently been proven to be a necessary condition for the establishment of SNAs (Shaki & Fischer, 2018). The parity instructions were counterbalanced between the 2 trial blocks (each block consisting of 48 trials), and the order of blocks was counterbalanced between participants. For each block, the near and far conditions were equally distributed throughout the right and left sides (i.e., 12 trials for the near-right side; 12 trials for the far-right side; 12 trials for the near-left side; 12 trials for the far-left side). Finally, the instructions emphasised that the participants should take their time responding to focus on the randomisation constraint to avoid parity errors.

Analyses

Two participants were excluded of the analyses because they did not follow task instructions (they generated simply non-random sequences such as “12, 13, 15, 16”). The data concerning the other participant could not be included due to a programme crash. Consequently, we have added three other participants to balance the sample size between Experiments. Trials with parity-rule errors (3.42% of the total data) were not included in the analyses. For the main dependent variable, we calculated the mean of generated numbers between 1 and 30 (Hartmann et al., 2019; Shaki & Fischer, 2014). This dependent variable was then submitted to a repeated-measures ANOVA with the factors of distance (near and far spaces) and side (left and right spaces) as within-subject variables. To exclude any confounding interpretation with the manual laterality of the participants, the analysis with the factor laterality quotient as a covariate has been performed.

The median response times recorded for the vocal responses were submitted to a repeated-measures ANOVA with the factors of distance (near and far spaces) and side (left and right spaces) as within-subject variables.

Finally, to control whether our experimental protocol was comparable with the usual protocol used in the literature, we used an additional analysis to assess the classical small number bias (Loetscher & Brugger, 2007). Specifically, we computed the mean of generated numbers regardless of the distance or the side and submitted it to a one-sample t-test with 15 reference values (i.e., the median of the number range used in the RNG task).

Results

The one-sample t-test revealed a tendency to the typical small number bias with an average RNG of 14.4 ± 1.58 smaller than 15, t(27) = −1.90, p = .068.

The repeated-measures ANOVA revealed a main effect of distance, F(1, 27) = 5.185, p = .031, $η_{p}^{2} = . 161$ (Figure 2), indicating participants generated the lowest mean numbers for the near space condition compared with the far space condition. The analysis revealed no effect of side, F(1, 27) = 0.552, p = .4640, $η_{p}^{2} = . 020$ . Finally, the Distance × Side interaction was not significant, F(1, 27) = 7.50e-6, p = .9978, $η_{p}^{2} = . 000$ . Table 1 presents the mean generated numbers as function of the distance and the side.

Figure 2.

The mean of generated numbers on the sagittal plane according to the distance of the displacements from Experiment 1.

Table 1.

Mean generated number as function of the distance and the side in Experiment 1 (means that differ significantly are shown in bold, and standard deviations are in brackets).

	Distance
Side	Near	Far	(Mean)
Left	14 (2.41)	15 (2.15)	14.5 (2.32)
Right	13.8 (2.25)	14.8 (1.99)	14.3 (2.17)
(Mean)	13.9 (2.32)	14.9 (2.06)

The analysis with the covariate factor (laterality quotient) revealed a main effect of distance, F(1, 26) = 5.143, p = .0310, $η_{p}^{2} = . 165$ . There is no interaction between the distance and the laterality quotient, F(1, 26) = 1.100, p = .3040, $η_{p}^{2} = . 041$ . No main effect for the factor side was revealed, F(1, 26) = 0.312, p = .5810, $η_{p}^{2} = . 012$ , and there is no interaction between the factors side and the laterality quotient, F(1, 26) = 0.014, p = .9063, $η_{p}^{2} = . 001$ . Consequently, the laterality quotient was not relevant for the main finding on the mean generated numbers.

The ANOVA performed for the vocal responses revealed no significant effect for the factors: distance (median in the near-left condition: 2,353.91 ms ± 808.04, near right: 2,330.62 ± 878.11, far left: 2,310.96 ± 616.03, and far right: 2,266.34 ± 717.80; F(1, 27) = 1.061, p = .3122, $η_{p}^{2} = . 038$ ), side, F(1, 27) = 0.5682, p = .4575, $η_{p}^{2} = . 021$ , or interaction between the two factors, F(1, 27) = 0.0449, p = .8339, $η_{p}^{2} = . 002$ .

Discussion

This first experiment revealed an effect of congruency between the numerical (i.e., the generated number) and physical (i.e., the physical distance browsed by the avatar) magnitudes in the sagittal plane. Specifically, participants generated more small and large numbers when the avatar spanned short and long distances, respectively. From a theoretical point of view, these results are in line with the generalised magnitude system described in the ATOM. They revealed an interaction between space representation and Arabic number processing along the sagittal plane. Unexpectedly, this congruency effect was not observed along the transverse plane. We can assume that, in this egocentric condition, SNAs along the sagittal plane were prioritised and trumped the classical SNARC effect, which is in line with previous findings (Greenacre et al., 2022; Holmes & Lourenco, 2011). This interpretation will be addressed in more detail in the general discussion.

From an embodied approach, we could argue that, if the spatial reference frames of participants switch from an egocentric to an allocentric perspective with a twist of 180°, we should observe the same effect following the perspective of the avatar. Specifically, in the second experiment, we expected to observe similar SNAs with the generation of small and large numbers when the avatar covered short and long distances, respectively.

Experiment 2

Methods

Participants

A total of 28 new participants were recruited (18 females, M = 22.82, SD = 2.93, ranging from 19 to 31; 25 right-handers with a mean handedness score of 84.47 ± 16.44 and 3 left-handers with a mean handedness score of −73.33 ± 46.19). All participants had normal or corrected-to-normal vision and no neurodevelopmental or neurological and perceptual disorders. Informed consent was obtained before participation, and the participants were naïve to the purpose of the study. The study conformed to the ethical standards of the Declaration of Helsinki.

Procedure

The procedure was identical to the first experiment except that participants performed the task from an allocentric perspective. Specifically, the participants observed a field shot with the avatar located from the front at the top of the screen (see Figure 3). In this condition, the left side of the avatar was the right side of the participants and vice versa.

Figure 3.

Street drawing used in the second experiment with an avatar located at its extremity, requiring participants to use an allocentric spatial reference frame.

Analyses

Trials with parity-rule errors (3.81% of the total) were not included in the analyses.

Results

In line with the tendency observed in Experiment 1, the one-sample t-test revealed a significant small number bias with an average RNG of 14.6 ± 0. 93 significantly smaller than 15, t(27) = −2.31 p = .029.

The repeated-measures ANOVA revealed a main effect of side, F(1, 27) = 21.883, p < .0001, $η_{p}^{2} = . 448$ (Figure 4), indicating that participants generated the lowest mean numbers for the left space condition compared with the right space condition (from the avatar’s point of view). The analysis did not reveal a main effect of distance, F(1, 27) = 0.351, p = .556, $η_{p}^{2} = . 013$ , without a significant difference in the mean generated numbers for near space in comparison to far space. Finally, the Distance × Side interaction was not significant, F(1, 27) = 2.657, p = .115, $η_{p}^{2} = . 09$ . Table 2 presents the mean generated numbers as function of the distance and the side.

Figure 4.

The mean of generated numbers on the transverse plane according to the side of the displacements from Experiment 2.

Table 2.

Mean generated number as function of the distance and the side in Experiment 2 (means that differ significantly are shown in bold, and standard deviations are in brackets).

	Distance
Side	Near	Far	(Mean)
Left	14.2 (1.66)	13.9 (1.88)	14.1 (1.76)
Right	14.8 (1.55)	15.5 (1.53)	15.1 (1.57)
(Mean)	14.5 (1.61)	14.7 (1.88)

The analysis with the covariate factor (laterality quotient) revealed a main effect of side, F(1, 26) = 7.830, p = .0096, $η_{p}^{2} = . 231$ . As for Experiment 1, there is no interaction between the side and the laterality quotient, F(1, 26) = 0.001, p = .9784, $η_{p}^{2} = . 000$ . No main effect for the factor distance was revealed, F(1, 26) = 0.076, p = .7854, $η_{p}^{2} = . 003$ , and there is no interaction between the factors distance and the laterality quotient, F(1, 26) = 0.651, p = .4271, $η_{p}^{2} = . 024$ . As for Experiment 1, the laterality quotient was not relevant for the main finding on the mean generated numbers.

The ANOVA performed for the vocal responses revealed no significant effect for the factors: distance (median in the near-left condition: 2,349.14 ms ± 558.77, near right: 2,318.14 ± 540.58, far left: 2,240.95 ± 570.57, and far right: 2,265.91 ± 578.82; F(1, 27) = 3.597, p = .0686, $η_{p}^{2} = . 118$ ), side, F(1, 27) = 0.0043, p = .9482, $η_{p}^{2} = . 000$ , or interaction between the two factors, F(1, 27) = 0.665, p = .4219, $η_{p}^{2} = . 024$ .

The results of this second experiment showed an effect of congruency between the numerical magnitude and the side of the space in the transverse plane. Specifically, participants generated more small numbers when their avatars moved to the left and large numbers when they moved to the right regardless of the position on the sagittal plane. It is noteworthy that this SNA followed the mental rotation of the participants. In other words, the displacement of the avatar to the right part of the screen was equivalent to moving to the left and was associated with small numbers and vice versa for large numbers. These results show that SNAs on the transverse plane are flexible and can adapt according to the spatial reference frames used by participants.

Unexpectedly, we did not observe SNA along the sagittal plane as during Experiment 1. Most likely, the additional cognitive mechanisms of the mental rotation necessary to switch from an egocentric to allocentric perspective engaged participants towards the classical number−space association, which is deeply influenced by cultural factors (i.e., the transversal classical SNARC effect organising from the left to right axis in Western countries; Toomarian & Hubbard, 2018). Thus, sagittal SNAs that were previously observed and that correlated more with universal features of the natural world (e.g., the distance in the present task) seemed to be trumped. The general discussion will return to these points and seek to explain the differences observed between the first and second experiments.

Comparison across Experiments

To confirm that both experiments differed for the factors “distance” and “side,” the mean of generated numbers was submitted to a repeated-measures ANOVA with the factors of distance (near and far spaces) and side (left and right spaces) as within-subject variables and the factor of spatial reference frames (egocentric and allocentric; i.e., Experiments 1 and 2, respectively) as between-subject factors. All results are given according to the perspective of the participants (as a reminder, in the allocentric group, the left and the right of the avatar corresponded to the right and left side of space for the participants).

This analysis did not revealed a main effect of distance, F(1, 54) = 2.047, p = .1582, $η_{p}^{2} = . 037$ . Importantly, the analysis revealed a significant interaction between the distance and the spatial reference frames, F(1, 54) = 4.558, p = .0373, $η_{p}^{2} = . 078$ . Duncan’s post hoc analysis indicated that participants generated the lowest mean numbers for the near space condition compared with the far space condition only in the egocentric group (p = .0232). The analysis also revealed a main effect of side, F(1, 54) = 11.420, p = .0014, $η_{p}^{2} = . 175$ , and most importantly, a significant interaction between the side and the spatial reference frames, F(1, 54) = 5.005, p = .0294, $η_{p}^{2} = . 085$ . Duncan’s post hoc analysis revealed that only the participants of the allocentric group generated the lowest mean numbers after avatar movements to the right space of participants (i.e., the left side of the avatar) compared with the left space (i.e., the right side of the avatar) (p = .0004).

Finally, we have performed a t-test between the laterality quotients for the two samples. This analysis revealed no difference between experiments, t(27) = −0.139, p = .8905.

General discussion

This study aimed to highlight SNAs along the sagittal plane of space to assess their organisation from different spatial reference frames (egocentric vs. allocentric). As expected, during Experiment 1 under the egocentric perspective, we observed SNAs along the sagittal plane with small and large numbers produced after the observation of short and long displacements of the avatar, respectively. These findings confirmed our first hypothesis based on the ATOM and were in line with previous studies showing similar space−number associations along the sagittal plane (Aleotti et al., 2020; Chen et al., 2015; Cooney et al., 2021; Santens & Gevers, 2008). Specifically, the congruency effect between the physical and numerical magnitudes during displacements along the sagittal plane confirmed that space and quantity are processed inside a generalised magnitude system (Cona et al., 2021; Hawes et al., 2019; Walsh, 2003). The goal of Experiment 2 was to assess whether SNAs were flexible when the participants used allocentric spatial reference frames. In this view, a rotation of 180° was applied to the avatar. The findings showed a compatibility effect between the displacement along the transverse plane and numbers, whereas SNAs along the sagittal plane disappeared. Consequently, we can assume that SNAs, according to the plane of reference, do not have the same stability under allocentric or egocentric perspectives.

Experiment 1 revealed an SNA for the sagittal plane and did not show the same effect along the transverse plane (i.e., SNARC effect; Dehaene et al., 1993). At first glance, our finding contradicts data from a study that highlighted simultaneous associations along the transverse, vertical, and sagittal planes in the prehension space during a parity task (Aleotti et al., 2020). However, we suggest that, in our present experiment, SNAs along the sagittal plane may have trumped SNAs along the transverse plane. Accordingly, Holmes and Lourenco (2011) found that SNAs processing along a specific plane can trump another plane during a number processing task. Specifically, the authors asked participants to judge the parity of numbers by pressing keys along the transversal and sagittal planes on a numerical keypad. In the congruent condition, participants responded faster by pressing near-left and far-right keys when they processed small and large numbers, respectively. In this condition, SNAs for the transverse and sagittal planes were observed. However, for the incongruent condition, the compatibility effect was observed only between the near-right and far-left keys after the processing of small and large numbers, respectively (Experiment 1B, Holmes & Lourenco, 2011). Therefore, these results revealed that only the near and far parts of the space were processed during this last condition. Our present findings are in accordance with Holmes and Lourenco’s study in showing a compatibility effect only for the sagittal plane (see also Greenacre et al., 2022 for a similar effect with vertical SNA dominance against horizontal dominance). From a theoretical point of view, Holmes and Lourenco suggested that the generalised magnitude system (Walsh, 2003) may be involved when the task refers directly to physical magnitudes (i.e., distance in our study). In Experiment 1, near and far conditions may have emulated the distance between the two, which corresponded with analogous differences in the spatial dimension and consequently produced a “direct magnitude-on-magnitude mapping” (Holmes & Lourenco, 2011). However, left-to-right mapping of the classical SNARC effect appears to be more dependent on cultural conventions from teaching, such as reading and writing behaviours (Göbel et al., 2011). In this view, this left-to-right mapping also implies an ordinal organisation that depends on the storage of sequences in memory, well known as the serial order effect (Abrahamse et al., 2016; Rasoulzadeh et al., 2021; van Dijck et al., 2020; van Dijck & Fias, 2011). This ordinality effect has been suggested to explain SNAs from left-to-right mapping (Sixtus et al., 2023). In the same vein, an alternative account argues that these effects are mainly due to decisional process rather than magnitude processing per see (Van Opstal & Verguts, 2013). Thus, we suggest that the findings of Experiment 1 are due to the magnitude processing of the distance rather than a spatial order or cultural conventions. In these conditions, the universal organisation shared by different species illustrating by SNAs in the transverse plane (see de Hevia et al., 2017; Rugani et al., 2010, 2015, for this phylogenetic account) appears to be trumped by the sagittal plane. Our results reveal that, when the context induces genuinely the processing of the physical distance with a “direct magnitude-on-magnitude mapping,” this phylogenetic influence seems to be trumped by the ontogenetic one (i.e., the processing of magnitude related to the distance). More studies are needed to test and disentangle both mechanisms.

It is worth noting that the compatibility effect of Experiment 1 was observed, whereas the size of the shops and the avatar decreased with increasing distance. This finding eliminates any confounding interpretation about congruency effects between the selected numbers and the size of the surrounding visual elements. Indeed, when the avatar moved to far space, the size of the shop and the avatar were smaller than in near space, and participants generated large numbers. The congruity effect between object size and the spatial location of the response is well known in the literature (for a review and a meta-analysis, see Macnamara et al., 2018; Seegelke et al., 2022). Consequently, we could argue that our present experimental task privileged an embodiment of the avatar’s spatial reference frames, which in turn facilitated the processing of magnitudes according to the distance, especially during the displacement in the road.

During Experiment 2, participants were required to reverse their spatial reference frames from their own referential frame to the avatar’s frame. The finding revealed an effect only for the transversal plane. Consequently, we can assume that transversal SNAs can adapt to the position of the avatar embodied by the participant. For this allocentric perspective, when the avatar moved to the left and right sides of the space, these movements corresponded to the right and left sides of the participant’s space, respectively. The results revealed that, when the avatar moved to the right of the participant, small numbers were produced. When the avatar moved to the left of the participant, large numbers were produced. This mapping corresponded to the classical SNARC effect following the perspective of the avatar embodied by the participant. This adaptation of the SNAs is in accordance with the joint numerical cognition effect (Hartmann et al., 2019). In this study, a dyad-participant performed an RNG task, and results revealed that the participants located to the left of the other participant generated more small numbers. This study and our present finding indicate that SNAs can flexibly adapt to the presence of another person.

However, no effect was observed for the sagittal plane, as we found during Experiment 1. From the congruity effect between object size and the spatial location of the response (Macnamara et al., 2018; Seegelke et al., 2022), we could expect that the size of the surrounding visual elements highlights the appearance of SNAs on the sagittal plane. Indeed, when the avatar browsed a small distance and reached near space, the size of elements was being small (opposite effect for large distance and far space). The present finding revealed no effect of such likelihood on the sagittal plane. Consequently, the results of Experiment 2 suggest that, when the participants were required to use an allocentric perspective and then to focus on the mental rotation of 180°, a spatial order account was likely prioritised. As previously mentioned, left-to-right SNAs appeared to follow the sense of reading, writing, or finger counting (Göbel et al., 2011; Toomarian & Hubbard, 2018) and to reflect ordinality (Sixtus et al., 2023). From an allocentric perspective, these ordinality and cultural factors seem to take primacy over alternative planes, and magnitude processing per se is likely not processed as spatial information.

Finally, according to the embodied view of cognition, the representation of a concept and, more generally, human knowledge requires the reactivation of the sensorimotor mechanisms that were activated during the initial experience of what that knowledge refers to (Barsalou, 2008; Wilson, 2002). From this perspective, cognition is based on simulations involving a reconstruction of perceptual, motor, and introspective states acquired during experiences with the world, body and mind (Barsalou, 2016). Recently, this view has been extended to numerical cognition (Anobile et al., 2021; Sixtus et al., 2023). It has been suggested that the meaning of numbers arises from how they are grounded in diverse sensorimotor experiences such as perception and action (Sixtus et al., 2023). From this perspective, several studies have shown that a “sensorimotor numerosity system” can process nonnumerical quantitative information such as space (Anobile et al., 2021). For this study, during Experiment 1, participants were required to embody the position of the avatar in the street. This embodiment required simulating the perceptual experience of the avatar. This simulation involved mechanisms of action representation depending on the spatial features of the environment. In this context, when participants needed to randomly select a number, the concept of magnitude was therefore jointly recruited with the process of distance browsing by the avatar. This direct magnitude-on-magnitude mapping between distance and numbers caused the effect of congruency observed along the sagittal plane (a phenomenon analogous to the effect of congruency between the magnitude of numbers and other continuous magnitudes such as physical size; see Henik & Tzelgov, 1982; Leibovich et al., 2017). In Experiment 2, this effort of embodiment, increased by the mental rotation of 180°, led to favouring the observation of cultural transversal SNAs (i.e., the classical SNARC effect).

To conclude, our findings provide new evidence of the presence of specific SNAs that are modulated in function to the investigated plane and the spatial reference frames used by the participant. In this way, SNAs along the transverse and sagittal planes seem to be mutually exclusive in regard to the spatial features of the task and the required spatial reference frames. Theoretically, our interpretation of numerical information is informed by embodied factors such as perceptions and actions, which are linked to various aspects of daily experiences, such as the processing of space and other magnitudes. In a dynamic spatial-semantic relationship, the flexibility mechanisms of SNAs need to be further investigated to understand how numerical and spatial cognitions influence each other under the weight of ecological constraints specific to the spatial orientation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Arnaud Badets

Data accessibility statement

The data and materials from the present experiment are publicly available at: https://doi.org/10.5281/zenodo.10017153, https://doi.org/10.5281/zenodo.10017371, https://doi.org/10.5281/zenodo.10017401, https://doi.org/10.5281/zenodo.10017304, and

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