Olgeir Sigmarsson
email: olgeir.sigmarsson@uca.fr

Séverine Moune

Pierre-Jean Gauthier

Fractional degassing of S, Cl and F from basalt magma in the Bárðarbunga rift zone, Iceland

Keywords: Rayleigh distillation, volcanic gas, partition coefficients, Holuhraun lava, Kilauea, Etna

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Degassing of basaltic magma is known to impact the environment by emission of several different gaseous species but the physics of the degassing process is less well understood. At depth, where CO2 is the dominant volatile species, gas segregation from basaltic or lowviscosity magma is thought to follow a cyclic behaviour, which may lead to alternating explosive and effusive activity [START_REF] Sparks | Dynamics of magma degassing[END_REF]. The gas phase and the rising melt can evolve from a state of thermodynamic equilibrium to that of a progressive open-system fractional distillation, especially during the waning stage of an eruption [START_REF] Aiuppa | Degassing of halogens from basaltic volcanism: insights from volcanic gas observations[END_REF]. Fractional degassing at shallow depth has been inferred from changing gas composition during single eruptions from craters at different altitude and during different types of volcanic activity, as in the case of the persistently degassing Mount Etna, Sicily (e.g. [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF][START_REF] Burton | FTIR remote sensing of fractional magma degassing at Mount Etna[END_REF][START_REF] Allard | Spectroscopic evidence for a lava fountain driven by previously accumulated magmatic gas[END_REF]. In addition, the syn-eruptive gas phase of a basaltic fissure eruption is expected to be different from the residual gas phase of a cooling lava during emplacement and crystallization [START_REF] Swanson | Loss of volatiles during fountaining and flowage of basaltic lava at Kīlauea Volcano, Hawaii[END_REF]. A well-known example of a basaltic fissure eruption is that of Laki in Iceland which may have released up to 20% of the total emitted SO2 and significant proportion of the halogens during lava field emplacement and crystallisation [START_REF] Thordarson | Sulfur, chlorine, and fluorine degassing and atmospheric loading by the 1783-1784 AD Laki (Skaftár Fires) eruption in Iceland[END_REF]. The final degassing stage may thus lead to local hazes with different gas composition than the principal gas phase. Fluorine, for instance, with its relatively high solubility in basaltic melt compared to Cl and S [START_REF] Carroll | Solubilities of sulfur, noble gases, nitrogen, chlorine, and fluorine in magmas[END_REF], will principally enter the final gas phase. Such degassing of F can lead to fluorosis in grazing animals, a well-known effect that caused the death of approximately two thirds of the Icelandic livestock during and soon after the Laki eruption [START_REF] Thorarinsson | The Lakagigar eruption of 1783[END_REF][START_REF] Pétursson | Um eituráhrif af völdum Skaftárelda (On the poisoning effect of the Skaftá Fires (Laki eruption))[END_REF][START_REF] Thordarson | Atmospheric and environmental effects of the 1783-1784 Laki eruption: A review and reassessment[END_REF].

Variations in the gas composition at basaltic volcanoes are also related to the global tectonic context, with hotspot volcanoes generally emitting gas of higher S/halogens than those in subduction-zone settings, despite several common factors as reviewed by [START_REF] Aiuppa | Degassing of halogens from basaltic volcanism: insights from volcanic gas observations[END_REF]. At Kilauea, Hawaii, gas of different composition is released at summit craters than is emitted in flank eruptions in rift zones (e.g. gas type I and II of [START_REF] Gerlach | Volatile budget of Kilauea volcano[END_REF][START_REF] Greenland | Gases from the 1983-84 east-rift eruption[END_REF][START_REF] Symonds | Volcanic-gas studies: Methods, results, and applications[END_REF][START_REF] Edmonds | Halogen degassing during ascent and eruption of water-poor basaltic magma[END_REF]. Likewise, during the 2001 eruption of Etna, [START_REF] Burton | FTIR remote sensing of fractional magma degassing at Mount Etna[END_REF] observed remarkably similar S/halogens in gas from the Northeast crater (NEC) and the Central craters as well as in the bulk gas plume, whereas gas from the Southeast crater (SEC) had considerably lower S/halogens and that of the flank crater, which fed the 2001 summer lava flow, had even lower ratios. Their results confirmed observations by [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF], which in addition measured decreasing Cl/F in gas from the flank crater. These studies suggest a primary gas phase from single magma body degassing beneath NEC and Central craters, whereas SEC and the southern flank crater emitted a secondary gas phase. [START_REF] Aiuppa | Degassing of halogens from basaltic volcanism: insights from volcanic gas observations[END_REF] extended the open system Rayleigh-type degassing model for Etna to other basaltic volcanoes, a model that assumes uniform vapour-melt partition coefficients for S, Cl and F. A similar method was applied by [START_REF] Mather | Halogens and trace metal emissions from the on-going 2008 summit eruption of Kīlauea volcano, Hawai'i[END_REF] when discussing degassing mechanism at Kilauea. However, neither Cl nor F volatility and their partition coefficients between vapour and melt phases are well understood. A comparison of the primary gas phase released during basaltic fissure eruption to that produced during the final degassing stages at the end of an eruption should provide better constraints on the mechanism, volatile solubility, and therefore the volatile partition coefficients between the silicate melt and the gas phase, needed for better understanding of the degassing processes.

In this article syn-and post-eruptive gas phases of the Holuhraun eruption (29 August 2014 to 27 February 2015) within the rift zone of the Bárðarbunga volcanic system in Iceland are compared. The eruption site was located approximately 40 km N of the Bárðarbunga central caldera, which is positioned above the presumed centre of the Iceland mantle plume. The eruptive fissure produced approximately 1.5 km 3 of basalt of uniform composition (ol-tholeiite) and liberated close to 10 Tg of SO2 into the atmosphere (Sigmundsson et al., 2014;[START_REF] Gíslason | Environmental pressure from the 2014-15 eruption of Bárðarbunga volcano, Iceland[END_REF][START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF][START_REF] Bali | Melt inclusion constraints on volatile systematics and degassing history of the 2014-2015 Holuhraun eruption, Iceland[END_REF][START_REF] Halldórsson | Petrology and geochemistry of the 2014-2015 Holuhraun eruption, central Iceland: compositional and mineralogical characteristics, temporal variability and magma storage[END_REF][START_REF] Pfeffer | Ground-Based Measurements of the 2014-2015 Holuhraun Volcanic Cloud (Iceland)[END_REF].

The large amount of gas emission from Holuhraun caused repeated air pollution events characterized by atmospheric SO2 concentrations exceeding air quality standards, as well as high concentrations of fine-grained volcanogenic aerosols [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF][START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF]. Snow precipitated around the lava field, rain and the surrounding rivers were monitored during the eruption and had elevated F concentrations [START_REF] Gíslason | Environmental pressure from the 2014-15 eruption of Bárðarbunga volcano, Iceland[END_REF][START_REF] Galeczka | Pollution from the 2014-15 Bárðarbunga eruption monitored by snow cores from the Vatnajökull glacier, Iceland[END_REF][START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF] despite its very low abundance in the main gas plume [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. Understanding the origin and fate of fluorine during degassing processes is thus of importance in order to assess its release to the atmosphere and subsequent environmental impact. Here, results are presented on S and halogen (Cl and F) concentrations in the post-eruptive gas phase released at Holuhraun in order to verify whether or not fluorine is ultimately degassed and if a simple Rayleigh distillation process occurring during the progressive basalt degassing explains the observed gas compositions. Halogen partition coefficients D vapour/melt are estimated for Cl and F from S/Cl and S/F ratios and their variability towards the end of the eruption is assessed. The results are finally compared with those from Etna and Kilauea volcano (e.g., [START_REF] Gerlach | Volatile budget of Kilauea volcano[END_REF][START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF][START_REF] Burton | FTIR remote sensing of fractional magma degassing at Mount Etna[END_REF][START_REF] Allard | Spectroscopic evidence for a lava fountain driven by previously accumulated magmatic gas[END_REF][START_REF] Edmonds | Halogen degassing during ascent and eruption of water-poor basaltic magma[END_REF][START_REF] Mather | Halogens and trace metal emissions from the on-going 2008 summit eruption of Kīlauea volcano, Hawai'i[END_REF] and show that the gas phase composition reflects the intrinsic parameters of the fractional degassing mechanism.

Sample collection and analysis

High-quality syn-eruptive gas samples are not easily collected during intense effusive activity such as the Holuhraun eruption. Two syn-eruptive gas plume samples could be collected in still weather on 2 October 2014, when the gas plume was grounded for a few hours because of a short-lived temperature inversion of the atmosphere [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]Fig. 1). A few months later, 21 January 2015, a third sample of the gas plume was collected from a helicopter [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF]. This latter sample has an order of magnitude lower SO2 concentration compared to the earlier samples, possibly reflecting greater air dilution on the plume's edge, or increasing distance from the vent.

Recently published results [START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF][START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF] on ground-collected syn-eruptive gas composition can be compared with those given by [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. Out of four filter-pack gas samples collected during the eruption by [START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF], three have very similar S/Cl mass ratios with an averaged value of 41±7, characteristic of the syneruptive gas plume [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. In marked contrast, their samples exhibit much higher fluorine concentrations than reported by [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF], leading to lower Cl/F ratios, somewhat intermediate between primary vapour compositions [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] and residual vapour compositions (this work, Table 1). Marked enrichments in fluorine may arise though from other sources than the syn-eruptive gas phase, such as secondary vapour from the cooling lava field. The same holds true for ground-collected gas samples taken by [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF], who also suggested that extreme variability in their own samples may be explained by a gas phase "partially sourced from the degassing lava". In addition to the fortunate two samples from [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF], we therefore consider only their helicopter-aided airborne plume sample as representative of the principal syn-eruptive gas phase [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF].

Despite being strongly diluted compared to the gas plume samples of [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF], the helicopter-aided sample from [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF] has identical syn-eruptive S/Cl but significantly lower S/F and Cl/F. Ten days after the lava extrusion ended, the residual gas phase released from the cooling magma was collected. Three post-eruptive filter-pack samples were obtained in calm weather on 9

March 2015 inside the main crater, approximately 50 m above the cooling lava lake (Fig. 1).

Intensive degassing was observed from the lava-filled crater with visually estimated maximum gas emission from cracks inside the crater walls, one of which was selected for gas sampling.

The same filter-pack technique as presented in [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] was employed. Each filter pack was connected to a pump operating at a flow rate of 11 l/min for 23-72 min, collecting three gas and aerosol samples (HH1, HH2 and HH3; Table 1) from a total air volume of 0.25-0.79 m 3 . For comparison, the atmospheric composition free of volcanic gas (blank) was collected approximately 500 m west of the lava field over one hour. Filters were subsequently analysed for S, Cl and F concentrations using ion chromatography at ICCF-Sigma, Université Clermont Auvergne, with analytical uncertainty of approximately 5%. For the consolidated and thus outgassed lava, S concentration was measured with a CHNS/O analyser at LMV-Université Clermont Auvergne, but those of Cl and F using absorption spectrophotometry and selective electrodes, respectively, at CRPG-Nancy, France. The 2σ errors for S, Cl and F concentrations in the lava are less than 10%.

Results

The post-eruptive gas phase has S, Cl and F concentrations two to three orders of magnitude higher than the atmospheric background, which is therefore neglected. The gas samples are characterised by S, Cl and F concentrations ranging from 49-164, 53-104 and 27-46 (in mg/m 3 ), respectively, and their composition significantly differs from the syn-eruptive gas plume. The S concentrations in the post-eruptive gas phase are lower than those emitted during the primary eruptive phase by a factor of 3 on average. In marked contrast, both Cl and F are in much higher concentrations in the post-eruptive gas: 11 times more Cl and 62 times more F. The S/Cl mass ratio is close to unity (0.92 -1.58), substantially lower than the 40-52 of the primary gas phase [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. So is the Cl/F mass ratio, which is on average 2 in the post-eruptive gas phase against 11 ± 4 in the syn-eruptive plume. Consequently, the S/F mass ratio (1. 85 -3.53) is two orders of magnitude lower in the final gas phase and its decline is already observed in the late January 2015 gas (Table 1). Although S remains the principal minor gas component, these observations suggest a much stronger halogen contribution to the vapour phase at the end of the eruption.

Discussion

Arguments for almost complete S degassing

The sulphur-rich gas plume from the Holuhraun eruption affected people over most part of Iceland because of the high volatility of S from the olivine tholeiitic basalt. Sulphur is known to degas from basaltic melt until melt quenching or solidification, which for example explains higher concentrations in water-vs air-quenched tephra, and tephra vs crystallized lava [START_REF] Thordarson | Sulfur, chlorine, and fluorine degassing and atmospheric loading by the 1783-1784 AD Laki (Skaftár Fires) eruption in Iceland[END_REF][START_REF] Óladóttir | Did the Mýrdalsjökull ice-cap survive the Holocene Thermal Maximum? Evidence from sulfur contents in Katla tephra layers (Iceland) from the last ~8400 years[END_REF][START_REF] Haddadi | Magma storage beneath Grímsvötn volcano, Iceland, constrained by clinopyroxene-melt thermobarometry and volatiles in melt inclusions and groundmass glass[END_REF]. The very low S concentration in the degassed lava (97 ppm; [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] compared to the initial amount preserved in melt inclusions (up to 1644 ppm S; [START_REF] Bali | Melt inclusion constraints on volatile systematics and degassing history of the 2014-2015 Holuhraun eruption, Iceland[END_REF] suggests that 94% of sulphur was lost during magma ascent and evolution. Sulphide minerals could play a role in the sulphur budget of the Holuhraun products since a few globules were observed in the early lava and tephra in September 2014. However, most of the sulphides are present as irregular, partially dissolved particles and exhibit variable S isotope composition, whereas other isotope ratios remain mostly constant [START_REF] Halldórsson | Petrology and geochemistry of the 2014-2015 Holuhraun eruption, central Iceland: compositional and mineralogical characteristics, temporal variability and magma storage[END_REF]. These sulphides are partially xenocrystic and most likely picked up during magma ascent. Smaller euhedral sulphide spheres are crystallized immiscible melt in the Holuhraun magma itself. However, they are scarce enough to ensure that their crystallization (acting as a S sink) and their destabilization (acting as a S source) does not drastically change the S budget at Holuhraun. It thus appears that most of the sulphur escaped the magma as a volatile species before the crystallisation of the lava, mostly as gaseous SO2 since H2S does not exceed a few percent in the gas phase [START_REF] Gíslason | Environmental pressure from the 2014-15 eruption of Bárðarbunga volcano, Iceland[END_REF]. With SO2 averaging ca 30% in the gas phase (e.g. [START_REF] Bali | Melt inclusion constraints on volatile systematics and degassing history of the 2014-2015 Holuhraun eruption, Iceland[END_REF], the S liquid-gas partition coefficient (D l/g (S)) can be estimated at 97 . 10 -6 / 0.15 = 6 . 10 -4 . The figure must be considered as a maximum value for D l/g (S) (note that D l/g is simply the reciprocal of D V/M ) because the final S concentration in the most degassed eruptive products could be lower than 97 ppm. Whichever the exact value, it appears reasonable to assume D l/g (S) close to nil. Finally, even if some S remains in quenched groundmass glass or in the rare sulphides, the bearing on its partition coefficient between vapour and melt (D V/M ) is none, since that coefficient only measures the vapour-melt partition but neither vapour-glass nor melt-solid fractionation.

Estimation of vapour/melt partition coefficients of Cl and F

In contrast to S, which is severely depleted in the final solid product, Cl and F concentrations in the crystallized lava reach 110 and 190 ppm, respectively, which is within the range observed in melt inclusions and the lava groundmass [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF][START_REF] Bali | Melt inclusion constraints on volatile systematics and degassing history of the 2014-2015 Holuhraun eruption, Iceland[END_REF]. Moreover, given their higher solubility in basaltic melt [START_REF] Carroll | Solubilities of sulfur, noble gases, nitrogen, chlorine, and fluorine in magmas[END_REF], both Cl and F are expected to degas principally at or very close to the surface. The important decreases in S/Cl, S/F and Cl/F from the syn-eruptive gas plume to the post-eruptive gas phase clearly indicate decreasing availability of S for the gas phase in the crystallizing basalt magma and increasing relative concentrations of Cl and especially F in the final gas phase. This concentration pattern strongly suggests a Rayleigh distillation process, with most of the S extracted from the melt.

The Rayleigh law, C = C0 x f D-1 , where C denotes the measured concentration of a given gas species, C0 its initial gas concentration, f the fraction of gas remaining in the magma (vapour plus melt) and D the concentration in liquid divided by that of the gas (i.e. the partition coefficient, D l/g ) can be linearized as:

log 𝐶𝑙 = log 𝐶𝑙 0 + (𝐷 𝐶𝑙 𝑙/𝑔 -1) 𝑥 log 𝑓 log 𝑆 = log 𝑆 0 + (𝐷 𝑆 𝑙/𝑔 -1) 𝑥 log 𝑓 When D l/g (S)
is close to zero and can be neglected, the equations can be combined to eliminate f, which leads to:

log 𝐶𝑙 = (1 -𝐷 𝐶𝑙 𝑙/𝑔 ) 𝑥 log 𝑆 -[(1 -𝐷 𝐶𝑙 𝑙/𝑔 ) 𝑥 log 𝑆 0 ] + log 𝐶𝑙 0 .
The D l/g (Cl) can thus be determined from the slope of the linear relationship between log Cl and log S. The vapour/melt partition coefficient (D V/M ) for Cl, being simply the reciprocal of D l/g (Cl), thus can readily be calculated. The D V/M for F is obtained in the same way. Note that if D l/g (S) were slightly higher than 0, these calculated D V/M for halogens would be the maximum possible values.

The results are plotted in Figure 2 where Cl and S concentrations in gas samples collected during the eruption form a strong linear correlation (R 2 = 0.990). From the slope a surprisingly low D l/g (Cl) is calculated that leads to unusually high value for D V/M (= 1/ D l/g = 85), meaning ). Given the difficulties in determining D V/M for Cl, such as the presence or absence of a third phase (i.e. brine) in the experimental products [START_REF] Webster | Chloride and water solubility in basalt and andesite melts and implications for magmatic degassing[END_REF], we consider the value of 13 to 85 realistic, albeit possibly a maximum value. Chlorine most likely formed gas species and outgassed at very low pressure, close to or at the surface, aided by extensive water and sulphur degassing. In fact, evidence for sulphur-aided degassing of Cl does exist. Sulphur chloride species such as S2Cl can form in a gas phase, depending on the H/C and oxygen fugacity [START_REF] Zolotov | Chemical models for volcanic gases on Venus[END_REF], and complex ligands such as S2Cl2 4--, S2C1 3-, SCl2 3-, and S . Cl2 3-have been proposed for volatile transport of platinum-group elements [START_REF] Fleet | Volatile transport of platinum-group elements in sulfide-chloride assemblages at 1000°C[END_REF]. Clearly experimental results are needed to improve understanding of sulphur degassing effect on the volatility of Cl in mafic magma. The slope of the trend line for the three posteruptive gas samples (R 2 = 0.84) is significantly lower leading to an order of magnitude smaller D V/M for Cl, or only 2.2. The low D V/M in the post-eruptive gas phase could reflect the decreased role of a carrier gas for Cl such as H2O or SO2, which concentrations can be assumed as relatively low in the post-eruptive gas phase. Interestingly, similar D V/M has been determined in several experimental studies on Cl solubility in basalt without much sulphur present [START_REF] Webster | Chloride and water solubility in basalt and andesite melts and implications for magmatic degassing[END_REF][START_REF] Stelling | Solubility of H2O-and chlorinebearing fluids in basaltic melt of Mount Etna at T= 1050-1250 C and P= 200 MPa[END_REF][START_REF] Alletti | Chlorine partitioning between a basaltic melt and H2O-CO2 fluids at Mount Etna[END_REF]. Such a low D V/M for Cl may thus be characteristic for residual degassing of basalts with a gas phase with relatively low S/Cl as observed at Holuhraun.

Fluorine concentrations in all syn-eruptive samples show more scatter, possibly due to contributions from different sources as suggested by [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF]. Therefore, we again calculate best fit only for the two plume samples of [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] and the helicopter-aided sample of [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF]. The resulting D V/M for F in the syn-eruptive gas phase is close to that obtained for the three post-eruptive gas samples: 1.9 vs 1.7, respectively. The much lower D V/M for F compared to Cl during the eruption illustrates its higher solubility in the basaltic melt, whereas in the residual gas phase similar volatility is obtained for both halogens. Knowledge of D V/M for F is limited and experimental results are needed to test the accuracy of our approach. The halogen concentration in the residual gas phase is thus approximately twice that of the final crystallizing melt.

Comparison with Etnean gas

Halogen degassing has been extensively studied at Mount Etna, where persistent open-conduit degassing presents ample opportunities for gas sampling. Fractional degassing was proposed for the 2000 eruption at the SE crater where variations of S/Cl correlated with eruptive behaviour [START_REF] Allard | Spectroscopic evidence for a lava fountain driven by previously accumulated magmatic gas[END_REF]. Moreover, temporal decline of S/Cl and Cl/F was observed from the central craters to the flank lava-forming eruption in 2001 [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF][START_REF] Burton | FTIR remote sensing of fractional magma degassing at Mount Etna[END_REF]. Despite the obvious difference between the subduction zone-related Etna stratovolcano, where gas was sampled from several craters at different altitude, and Holuhraun in the Bárðarbunga rift zone, where both syn-and post-eruptive gas was sampled from the same fissure crater, an interestingly similar degassing patterns can be observed. In Figure 2, the S, Cl and F gas concentrations from [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF] are plotted and the slope of the best-fit lines used to infer the vapour-melt partition coefficients for Cl and F during the 2001 flank eruption.

Severe S depletion relative to the halogens was observed and the calculated D V/M for Cl is 2.3, and 1.8 for F, indistinguishable from that of the Holuhraun post-eruption gas phase. Flank eruptions at Etna thus bear similarities with fractional degassing at Holuhraun and, as suggested by [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF][START_REF] Aiuppa | Degassing of halogens from basaltic volcanism: insights from volcanic gas observations[END_REF], with that of Kilauea volcano, Hawaii. However, Etna produces gas especially enriched in Cl, as expected from its subduction-zone settings. The much higher gaseous S concentrations and S/Cl at the hotspot-related Bárðarbunga and Kilauea is worth further discussion.

Comparison with Kilauea

In Figure 3, we compare the S/Cl and S/F mass ratios measured in syn-and post-eruptive gas phases of Holuhraun, as well as in the degassed lava. The decreasing trend observed from the syn-eruptive gas plume through the residual gas phase to the degassed lava is consistent with a simple Rayleigh fractionation. The simplest explanation for the observed behaviour is progressive distillation of S from the rift-zone magma. When the eruption ceases, lava cools down and residual volatiles form a gas phase causing the melt to crystallise. The flux of carrier gas (mostly H2O and SO2) is still important enough to flush halogens out of the cooling lava but the low S concentration in the residual melt leads to a drastic decrease in Cl volatility, and/or increase in Cl solubility. Because fraction f of volatiles remaining in the melt decreases as the outgassing progresses, the Rayleigh equation implies increasing halogens, and notably fluorine, concentrations, not only in gas collected close to the degassing lava but also in the nearenvironment. The volcanic haze of basaltic fissure eruptions consequently comprises, in addition to water and sulphur dioxide, chlorine and fluorine gas species that may carry many metal and metalloid species out of the final melt of crystallizing basaltic lava. Accordingly, plume samples collected close to advancing lava field, like those sampled by [START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF] and [START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF] are prone to be mixed with F-enriched vapour, as exemplified in Figure 3 where their samples lie in between the two end members. [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF], and more recently [START_REF] Edmonds | A distinct metal fingerprint in arc volcanic emissions[END_REF], showed that hotspot-related volcanoes exhibit common trace-element degassing patterns. One of the best-studied hotspot basaltic volcanoes is Kilauea, Hawaii. Although the geodynamics of Bárðarbunga volcano is under the influence of both the Iceland mantle plume and extensional tectonics, while Kilauea is merely related to the Hawaii hotspot, their volcano-tectonic context is surprisingly comparable. Type II SO2-rich gas is characteristic of the Kilauea east rift zone [START_REF] Gerlach | Volatile budget of Kilauea volcano[END_REF], and also of the gas plume released from the Holuhraun craters in the Bárðarbunga fissure swarm (= rift zone). In addition, both volcanoes share common physical parameters such as a magma velocity of ~1 m.s -1 [START_REF] Swanson | Chronological narrative of the 1969-71 Mauna Ulu eruption of Kilauea volcano, Hawaii[END_REF][START_REF] Eaton | Cycling magma between the summit reservoir and Kilauea Iki lava lake during the 1959 eruption of Kilauea volcano[END_REF][START_REF] Parfitt | The 1983-86 Pu' u 'O'o eruption of Kilauea volcano, Hawaii: a study of dike geometry and eruption mechanisms for a long-lived eruption[END_REF][START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. Therefore, the comparison of S/Cl and S/F mass ratios at both volcanoes is relevant for better understanding degassing processes at basaltic volcanoes. It can be seen in Figure 3 that syn-eruptive gas samples from the rift zone of Kilauea and the Bárðarbunga fissure swarm (Holuhraun) have similar S/Cl and S/F ratios. The same holds true for residual gases as well as the most degassed products collected at Kilauea. In fact, a continuous trend of decreasing S concentrations is observed from Type II gas of the Kilauea rift zone [START_REF] Symonds | Volcanic-gas studies: Methods, results, and applications[END_REF] towards the Pele's tears of one of the rift-zone craters [START_REF] Edmonds | Halogen degassing during ascent and eruption of water-poor basaltic magma[END_REF]. Gas burst and lava spatter data extend this trend to even lower S/halogens. The simplest explanation is a progressive distillation of S from the rift-zone magma leading to decreasing solubility and increasing volatility of the halogens.

Within this framework, it might seem surprising not to detect at Bárðarbunga-Holuhraun the equivalent of CO2-enriched Type I gas released from the Halema`uma`u crater inside the Kilauea caldera [START_REF] Gerlach | Volatile budget of Kilauea volcano[END_REF]. Indeed, the gas from Kilauea's summit crater exhibits the highest recorded S/Cl and S/F ratios [START_REF] Mather | Halogens and trace metal emissions from the on-going 2008 summit eruption of Kīlauea volcano, Hawai'i[END_REF] and it likely represents the very first step of the Rayleigh distillation process (Figure 3). Whether such primary gas is absent or hidden within the Bárðarbunga volcanic system is worth considering. [START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] observed that 210 Po, a short-lived isotope from the highly volatile element polonium, was at unusually low level in the main gas plume released from the Holuhraun craters. Their preferred interpretation was that earlier degassing of Po, and of CO2 which was also depleted in the main gas plume, took place somewhere under the glacier-covered Bárðarbunga caldera and the magma pathway towards the Holuhraun fissure craters. Because of the ice cover of Bárðarbunga, such a first-stage CO2-enriched gas phase remains obscure and hard to detect.

Nevertheless, ice-cauldrons formed 1-5 km SE of Bárðarbunga during the rifting event in the second-half of August 2014 [START_REF] Reynolds | Changes in geothermal activity at Bárdarbunga, Iceland, following the 2014-15 caldera collapse, investigated using geothermal system modelling[END_REF], and shortly before the eruption started, could be the surface manifestations of CO2-rich gas burst during magma ascent. Alternatively, the chimney of seismicity detected 12 km SE of Bárðarbunga caldera and interpreted to be caused by vertical melt migration from 24 km depth towards the brittle-ductile crustal transition at 9 km could be another candidate for the site of early CO2 degassing [START_REF] Hudson | Deep crustal melt plumbing of Bárðarbunga volcano, Iceland[END_REF].

Conclusion

The different compositions of the syn-and post-eruptive gas emitted during the Holuhraun 2014-2015 eruption is explained by fractional degassing following the Rayleigh distillation law.

Since S is extremely volatile, its partition into the gas phase can be assumed close to quantitative. In that case, the vapour-melt partition coefficient D V/M can be calculated for both

Cl and F from their correlation with S concentrations. Elevated D V/M for Cl during the eruption may reflect the effect of sulphur degassing on Cl concentration, but experimental verification is needed to test this proposition. In the post-eruptive gas phase, D V/M (Cl) is significantly lower, most likely because S is strongly depleted at the end of the Rayleigh process, and Cl and F reach similar vapour-melt partition coefficients. Improved knowledge of D V/M for minor volatile species is required for the use of halogens in gas monitoring at active basaltic volcanoes (e.g., [START_REF] Aiuppa | Degassing of halogens from basaltic volcanism: insights from volcanic gas observations[END_REF]) but our study suggests a complex behaviour, partly dependent on sulphur degassing. Further constraints on D V/M (Cl) will also aid interpretation of Cl isotopes, which are known to fractionate during degassing [START_REF] Sharp | An experimental determination of chlorine isotope fractionation in acid systems and applications to volcanic fumaroles[END_REF]).

An open-system Rayleigh-type outgassing of the lava field explains elevated F concentrations measured in the environmental samples close to the lava field. Sustained halogen, and notably fluorine, degassing from cooling lavas may play a major role in the transportation of volatile trace elements. The results of this study underline the importance of integrating post-eruptive degassing processes when quantifying the entire volcanic gas emissions and their impact on the environment.

The mass ratios of sulphur over halogens (S/Cl and S/F) in the gas phase illustrate a strong analogy between degassing processes operating at both Kilauea and Bárðarbunga volcanoes. In contrast to Kilauea, CO2-enriched primary gases were not released into the atmosphere during the Holuhraun eruption. However, we speculate that the ice cauldrons generated before and during lateral magma transfer beneath the Vatnajökull ice-sheet resulted from CO2 gas burst rather than small sub-glacial eruptions. [START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF].

  that 85 times more mass of Cl went into the gas phase compared to what remained in the melt phase. The results from Stefánsson et al. (2017) fall on the lower concentration end of the correlation. Taken together with the present results, a somewhat lower regression coefficient (R 2 = 0.987) is obtained and a slope that yields less extreme D V/M of 13. At first glance, our values might seem extreme, but in fact, they are on the same order of magnitude as the highest experimentally determined D V/M by Stelling et al. (2008) and Alletti et al. (2009; D V/M for Cl as high as 24

Fig. 1

 1 Fig. 1 Grounded gas plume at Holuhraun on 2 October 2014 caused by atmospheric temperature inversion. Top left and right insets show, respectively, sampling conditions within the syn-eruptive gas plume and of the post-eruptive gas phase inside the eruptive fissure on 9 March 2015.

Fig. 2

 2 Fig. 2 Halogen vs sulphur concentrations in syn-and post-eruptive gas (plume and residual gas) from Holuhraun. Circles and squares denote Cl and F concentrations, respectively. The post-eruptive gas phase data are from the present study. Filled symbols for the syneruptive gas plume data are from Gauthier et al. (2016) and Ilyinskaya et al. (2017), whereas results from Stefánsson et al. (2017; ref AS on plot) are shown with open symbols. The squared correlation coefficient, R 2 , ranges from 0.837 for post-eruptive Cl vs S to 0.999 for syn-eruptive F vs S concentrations. Also plotted are results from Aiuppa et al. (2002) for the 2001 flank eruption on the south slope of Mt. Etna, Sicily. The vapourmelt partition coefficient (D V/M ) shown is simply the reciprocal of D l/g calculated from the slope of the linear best-fit and by assuming D l/g for S = 0. See text for further discussion.

Fig. 3

 3 Fig.3Mass ratios of S/Cl versus S/F illustrating the decreasing sulphur concentration from the syn-eruptive gas plume[START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF] to the post-eruptive residual gas phase at Holuhraun, Bárðarbunga volcanic system, Iceland. Also displayed are syn-eruptive results from[START_REF] Ilyinskaya | Understanding the environmental impacts of large fissure eruptions: Aerosol and gas emissions from the 2014-2015 Holuhraun eruption (Iceland)[END_REF] and[START_REF] Stefánsson | Major impact of volcanic gases on the chemical composition of precipitation in Iceland during the 2014-2015 Holuhraun eruption[END_REF]. The Kilauea summit, rift zone and Pu'u O'o gas and Pele's tears compositions are respectively from[START_REF] Mather | Halogens and trace metal emissions from the on-going 2008 summit eruption of Kīlauea volcano, Hawai'i[END_REF],[START_REF] Symonds | Volcanic-gas studies: Methods, results, and applications[END_REF],Edmonds and Gerlach (2006) and[START_REF] Edmonds | Halogen degassing during ascent and eruption of water-poor basaltic magma[END_REF]. Kilauea rift zone and Pu'u O'o gas are of similar composition as that of Holuhraun, whereas the Pele's tears record the residual gas phase comparable to that of the post-eruptive gas and outgassed lava of Holuhraun. Gas collected during the Etna 2001 eruption displays similar exhaustion of S during fractional degassing at significantly lower S/halogens[START_REF] Aiuppa | S, Cl and F degassing as an indicator of volcanic dynamics: The 2001 eruption of Mount Etna[END_REF].

Table 1 .

 1 Volatile species concentrations and element mass ratios in post-(prefix HH) and syneruptive (prefix BARB) gas samples from the 2014-2015 Holuhraun eruption, Bárðarbunga volcanic system, Iceland. 'Blank' refers to the local atmosphere. Samples BARB-A and -B are taken from[START_REF] Gauthier | Elevated gas flux and trace metal degassing from the 2014-15 Holuhraun eruption (Bárðarbunga volcanic system, Iceland)[END_REF]. Residual concentration in whole-rock sample of the degassed Holuhraun lava is given as g/g.

Acknowledgment

The field mission was financed by the French Laboratory of Excellence Program "ClerVolc" and Institute of Earth Science, University of Iceland through Icelandic Governmental Civil Protection Grant. Sveinbjörn Steinþórsson and Ármann Höskuldsson are acknowledged for field assistance and photos taken during the gas sampling. Magnús T. Guðmundsson and Guðmundur H. Guðfinnsson are thanked for discussion on ice-cauldron formations and correcting the English. Comments from two anonymous reviewers were appreciated as well as the editorial handling by the three editors: Patrick Allard, Tobias Fischer and Jacopo Taddeucci. This is ClerVolc contribution number 411.


Moira R Dillon

Marianne Duyck

Stanislas Dehaene

Marie Amalric

Véronique Izard

Journal of Experimental Psychology: Human Perception and Performance

Keywords: categories, shape discrimination, spatial cognition, angle, geometry, parallel, perpendicular

At the scale in which we live, space is continuous. Nevertheless, our perception and cognition parse the world into categories, whether physical, like scene or object, or abstract, like infinitesimal point or seven. The present study focuses on two categories of special angles in planar geometry, parallels and perpendiculars, and we evaluate how these categories might be reflected in adults' basic angle discrimination. In the first experiment, participants were most precise when detecting two parallel or perpendicular lines among other pairs of lines at different relative orientations. Detection was also enhanced for two connected lines whose angle approached 90°, with precision peaking at 90°. These patterns emerged despite large variations in the scales and orientations of the angle exemplars. In the second experiment, the enhanced detection of perpendiculars persisted when stimuli were rotated in depth, indicating a capacity to discriminate shapes based on perpendicularity in three dimensions despite large variation in angles' two-dimensional projections. The results suggest that two categorical concepts which lie at the foundation of Euclidean geometry, parallelism and perpendicularity, are reflected in our discrimination of simple visual forms, and they pave the way for future studies exploring the developmental and evolutionary origins of these cognitive categories.

Introduction

Categories of all kinds pervade human cognition and organize our physical and mental worlds. Some categories are present early in human development and are rooted in perception.

For example, very young infants show more precise discrimination (indicated by an increase in sucking frequency) of two synthetic speech sounds separated by a fixed distance in voice onset time when that distance distinguishes voiced and unvoiced stop consonants (e.g. /b/ and /p/) compared to when it does not [START_REF] Eimas | Speech perception in infants[END_REF][START_REF] Liberman | The discrimination of relative onset-time of the components of certain speech and nonspeech patterns[END_REF]. Moreover, even early brain responses are modulated by categories of visual stimuli: Localized regions of the visual cortex of infants as young as four months respond preferentially to exemplar pictures belonging to categories like faces and scenes [START_REF] Deen | Organization of high-level visual cortex in human infants[END_REF]. Nevertheless, such early emerging categorical processing is reshaped by experience. For example, in the first year of life, infants' auditory discrimination becomes specialized to the phonemic categories present in their native language [START_REF] Kuhl | Linguistic experience alters phonetic perception in infants by 6 months of age[END_REF][START_REF] Werker | Cross-language speech perception: Evidence for perceptual reorganization during the first year of life[END_REF], and children's explicit discrimination of phonemes develops reciprocally with culturally constructed reading and writing systems (see [START_REF] Anthony | Development of phonological awareness[END_REF] for a review). Such complex developmental stories raise intriguing questions about the origins of our explicit categorical knowledge in adulthood: To what extent is such knowledge innate, rooted in our perceptual experiences, or acquired by explicit education or immersion in a specific language and culture? And what of those categories with no apparent perceptual origins or constraints, like the concept of irrational number or other such abstract concepts often found in formal mathematics? In the present study, we investigate whether two categories of special angles that lie at the foundation of formal, Euclidean geometry, parallelism and perpendicularity (Euclid, 1990/300 B.C.E.), are reflected in adults' basic angle discrimination. We do so by evaluating whether adults' discrimination acuity is enhanced around these category boundaries in a variety of perceptual contexts.

At the turn of the 20 th century, Wilhelm Wundt and Hermann von Helmholtz independently suggested that angle discrimination may not be a wholly continuous process. They observed that small angles are judged to be somewhat bigger than their actual size and that big angles are judged to be somewhat smaller [START_REF] Von Helmholtz | Outlines of Psychology[END_REF]Wundt, 1897). More recent work on angle discrimination has aimed to explain and quantify this observation by suggesting, for example, that errors in discrimination reflect orientation selectivity in the visual cortex that leads to orientation distortions [START_REF] Carpenter | Interactions between orientations in human vision[END_REF] or that the errors are rooted in an inference about how an angle's appearance reflects its real-world size [START_REF] Howe | Natural-scene geometry predicts the perception of angles and line orientation[END_REF][START_REF] Nundy | Why are angles misperceived[END_REF].

A number of studies have focused on comparing the threshold at which we are able to discriminate various angles, but the results have been mixed. In addition, such studies often incompletely controlled for a variety of visual cues in the angle stimuli, such as orientation or size, and so conclusions about angle discrimination specifically have been difficult to make [START_REF] Werkhoven | Visual size invariance does not apply to geometric angle and speed of rotation[END_REF][START_REF] Wenderoth | The effects of angle-arm length on judgments of angle magnitude and orientation contrast[END_REF]Snippe & Koenderink 1994;Regan & Hamstra 1992). [START_REF] Chen | Angle judgment: Is the whole the sum of its parts?[END_REF] suggested, for example, that there are different detection thresholds for angles of specific sizes, reporting more fine-grained discrimination of 90º angles in a discrimination space that otherwise follows Weber's Law. While such thresholds were measured for 12 different reference angles (ranging from 15º -180º), the angles were presented at only two different orientations, vertical or oblique, so it remains unclear whether the thresholds were generalizable to more variable orientations. [START_REF] Heeley | Mechanisms specialized for the perception of image geometry[END_REF] found a similar pattern of discrimination thresholds with angles presented at random orientations, but they did not simultaneously vary the lengths of the lines that formed the angles, allowing for other, global shape cues to drive participants' performance. [START_REF] Regan | Evidence for a neural mechanism that encodes angles[END_REF], in contrast, found fairly constant thresholds of angle discrimination between 20º and 160º when angles were presented at a wide range of orientations. The authors suggest, however, that their discrepant findings might have been due to their unique methodology: Participants compared each stimulus to a reference angle size internalized through many practice trials, rather than to a physical display. The existing literature thus suggests that discrimination may be most precise around 90º angles, but it remains inconclusive.

In the present study, we measured the discrimination thresholds for many different angles using an intruder task (after [START_REF] Dehaene | Core knowledge of geometry in an Amazonian indigene group[END_REF]. We evaluated whether participants could locate an intruder that differed in angle size among five other angle exemplars of the same angle size as each other. The lines forming the angles varied considerably in their lengths and orientations to ensure that responses were made on the basis of angle alone within an individual trial. In addition, no individual lines were oriented within 10° of the horizontal or vertical to avoid any specialized angle discrimination that might occur when individual lines are orientated at the horizontal or vertical [START_REF] Xu | The human visual system estimates angle features in an internal reference frame: A computational and psychophysical study[END_REF].

We hypothesized that angle discrimination may reflect categories of parallelism and perpendicularity in two ways. First, detection thresholds may be more precise when parallels or perpendiculars serve as reference angles, compared to intruder angles of other sizes. Second, the detection of angle intruders may be asymmetric such that, for reference angles near 0°/180° and 90°, intruders whose angle size moves towards versus away from the parallel or perpendicular category boundaries will be easier to detect. This second prediction should hold when intruders cross the category boundary and also, perhaps, when intruders approach but do not cross the boundary. In Experiment 1, we find evidence for such categories. In Experiment 2, we demonstrate the robustness of these categories by imposing an additional rotation in depth in the experimental displays, dissociating 3D perpendicularity from 2D angle.

Experiment 1

Methods

Participants

Eight adults (four women; Mage = 25 years; range 19-28 years) participated in this experiment.

The sample size was set in advance based on the maximum sample size (8 participants) used in several other studies investigating angle discrimination and presenting large numbers of trials to individual participants [START_REF] Chen | Angle judgment: Is the whole the sum of its parts?[END_REF][START_REF] Regan | Evidence for a neural mechanism that encodes angles[END_REF][START_REF] Heeley | Mechanisms specialized for the perception of image geometry[END_REF]Snippe & Koenderink 1994;Regan & Hamstra 1992[START_REF] Xu | The human visual system estimates angle features in an internal reference frame: A computational and psychophysical study[END_REF]. In addition, stimuli were piloted in advance on three of the study's authors, and effects were robust enough to emerge in each pilot participant. As such, we also illustrate individual participant results in the figures and SM. All participants had normal or corrected-to-normal vision and had completed high school; most of them had also received a college or advanced degree. None were informed of the purpose and hypotheses of the study until after it was completed. This study was part of a research program approved by the Paris Descartes Ethics Committee (Conseil d'Évaluation Éthique pour les Recherches En Santé, CERES) and each participant provided informed consent prior to the experiment. Participants were paid 10€ per hour plus an additional sum, which depended on their performance (see below). On average, participants earned 77.38€ (range 75.60€-80.07€).

Design, Apparatus, Procedure, and Stimuli

For each trial, participants were presented with six angles, five of which were identical in their angle size (hereafter referred to as the "reference" angle) and one of which differed in its angle size (hereafter referred to as the "deviant" angle). Participants were asked to identify the deviant angle. In the connected lines condition (Figure 1A), angles were composed of two lines that met at one end. Participants saw trials in seven different blocks, each block with a different reference angle. The angle measures of the stimuli are summarized Table 1. The reference angles were chosen to evaluate discrimination thresholds symmetrically around 90°, and the angle differences between the reference angles and each deviant were chosen to capture potentially large differences in discrimination thresholds across the different references (if, e.g., discrimination followed Weber's Law). To probe the detection of 90° deviants specifically, +/-10° were presented in the 80° and 100° reference blocks. In the non-connected lines condition (Figure 1B), two lines were oriented relative to one another, but did not meet, and participants saw eight different reference angles in separate blocks (Table 1). We again added 90° deviants in the 80° and 100° blocks. We also added 0° (parallel) deviants in the 10° block and removed from that block the -12° and -18° deviants since these values were not geometrically possible (i.e., they would have resulted in -2° and -8° deviants, which would be equivalent to 2° and 8° angles).

Each type of deviant was presented 18 times in a random order in a block, three times at each of six possible target locations. The number of trials per block thus ranged from 126 to 162. Finally, despite the addition of 0° and 90° deviants in some reference blocks, those angles were not overrepresented in the experiment. For example, in the connected lines condition, a total of 756 90° exemplars were presented, compared to 810 exemplars of 80° and 100° angles and 720 exemplars of the angles in the other reference blocks ( Participants were seated in a lit room (62 cd.m -2 ) at eye level and 48 cm away from the center of an LCD monitor (60 Hz) subtending 44 x 32 degrees of visual angle (dva). On every trial, six angles appeared at the same time in one of the six white circular placeholders (124 cd.m -2 , 5 dva radius). These placeholders were equally distributed around a 11 dva radius circle centered on a central black fixation dot (0.15 cd.m -2 , 0.4 dva radius) on a light grey screen (92 cd.m -2 ). While maintaining their head position with a chin rest, participants were given 5 s to look at all of the angles before the angles disappeared. Participants could respond 250 ms after the presentation onset or up to 30 s after the angles disappeared. At the beginning of each trial, the cursor was positioned at the center of the screen. Participants clicked on the location of the deviant angle and received informative auditory feedback. The next trial started 500 ms after the response. After every quarter of a block, the percentage of correct responding was displayed as well as the sum earned during that block. Participants earned 0.014€ per correct response.

In both conditions, the lines that formed each angle were the same length, but these lengths varied across angles in the same display (chosen randomly from a uniform distribution between 2 and 4.4 dva). Lines were never within 10° of the vertical or horizontal axes of the screen. In the non-connected lines condition, one of the lines was displaced relative to the other line both along it (up to its midpoint) and orthogonally (0.6-2.5 dva) to it. Finally, to ensure that all angles in each display were presented at sufficiently different orientations, each was initially assigned to either 0°, 60°, 120°, 180°, 240°, and 300° relative to the screen and then each was jittered randomly between +/-30°. Each angle was then translated to a random location within 0.4 dva of its placeholder's edge.

The experiment was divided into two sessions corresponding to the two conditions, which took place on different days within a two-week period. The order of conditions was counterbalanced across participants. Each session started with a short training phase consisting of two trials per reference angle displaying +/-24° angle deviants. These trials were first presented with unlimited viewing time and then with the 5 s viewing time used in the actual experiment. Each block started with an introductory screen instructing participants to click on the shape that had a different angle size from the rest. This introductory screen also displayed one example of the reference angle for that block, oriented such that a vertical line would bisect the angle. The order of the blocks was random for each participant. Each session lasted approximately 3 hours, with a 15-30 min break after the fourth block in each condition.

Analyses

Participants' performance for each reference angle and for both the connected and nonconnected line conditions was fit with down-pointing Gaussian curves constrained to chance performance (chance = 0.167) at a difference of zero degrees and perfect performance at a difference of infinity. The fitted Gaussians were used to estimate individual participants' thresholds. These thresholds corresponded to the difference in degrees between the reference and the deviant such that the participant could detect the deviant on half of the trials and was guessing on the other half (i.e., performance of 0.583, halfway between chance and perfect performance). To capture possible asymmetries in participants' responses, performance was fit separately for smaller and larger deviants. Overall thresholds for each reference were obtained by averaging these two values. Because we did not present the same number and measure of angle deviants across reference angles (e.g., we included 10° deviants in the 80° block, but not in the 55° block), such thresholds were more appropriate than raw accuracies to compare performance across reference angles. This measure was decided on after pilot testing and in advance of any data collection.

In the connected lines condition, all eight participants performed above chance on all seven reference angle conditions with both smaller and larger deviants (binomial tests, twotailed, all ps < 0.05), yielding 112 data sets with above-chance performance. In the nonconnected lines condition, however, 17 of the 128 data sets did not significantly differ from chance, thus yielding unreliable detection thresholds (see thresholds' relative magnitudes. As confirmation of our results for this condition using this method, we also analyzed participants' accuracy using parametric tests, and we obtained the same results (see SM, Table S1, Figure S2).

First, we evaluated participants' performance in the connected lines condition. Using planned one-tailed t-tests, we examined whether participants' detection thresholds were more precise for the perpendicular (90°) reference angles compared to the 80° and 100° reference angles (the closest references to 90°, a 10° difference) and to the 55° and 125°reference angles (references further from 90°, a 35° difference). In those same reference blocks, we then examined whether deviant detection was more precise when deviants approached or crossed the 90° boundary compared to when they did not. For example, we evaluated whether thresholds in detecting larger deviants in 80° reference blocks were more precise than detecting smaller deviants in 80° reference blocks. Since all four comparisons tested whether angle discrimination was influenced by the category of right angles, p-values were adjusted for multiple comparisons using Holm's method.

For the non-connected lines condition, we investigated whether detection thresholds were more precise for the perpendicular (90°) and parallel (0°) reference angles using planned Holmcorrected, one-tailed Wilcoxon signed-ranks tests. As in the connected lines condition, we compared the thresholds of detecting deviants with a 90° reference both to those with an 80°/100° reference and also to those with a 55°/125° reference. In addition, we also compared the thresholds of detecting deviants with a 0° reference to the two closest reference blocks (10° and 25°). Finally, we tested for asymmetries in detection thresholds around these categories by examining whether deviant detection was more precise when deviants approached or crossed the 90° or 0° boundaries compared to when they did not.

Results

Connected Lines Condition

Figure 3A displays the individual and average detection thresholds for each angle reference in the connected lines condition. All participants had similarly shaped curves, in which detection thresholds were smaller as reference angles approached 0°/180°. Strikingly, all participants also showed steep drops in their detection thresholds as angle references approached 90°. Group-wise analyses, summarized in 2. Group-wise mean (for the connected lines condition) or median (for the non-connected lines) detection thresholds for perpendicular and parallel reference angle blocks compared to other reference blocks and detection asymmetries. Planned, Holm-corrected one-tailed t-tests (for the connected lines condition) or Wilcoxon signed-ranks tests (for the non-connected lines condition) compare the 90° or 0° reference blocks to the other reference blocks and the magnitude of the asymmetry effects. All angle measures are in degrees. *p < .05, ** p < .01, ***p < .001.

Non-Connected Lines Condition

Figure 4A displays the individual and median detection thresholds for each reference angle in the non-connected lines condition. All participants had similarly shaped curves, in which detection thresholds were smallest at 0° compared to other reference angles (see Figure S1 for individual participant accuracy curves). Group-wise analyses corroborated these results (see Table 2), finding more precise detection thresholds for 0° reference angles compared to 10° and 25° reference angles (10°: Z = 2.15, p = .016, 8/8 participants showed the effect; 25°: Z = 2.15, p = .016, 8/8 participants showed the effect). Moreover, participants performed better with angles deviating towards versus away from 0° with both 10° and 25° reference angles (10°: Z = 2.15, p = .016, 8/8 participants showed the effect; 25°: Z = 2.15, p = .016, 8/8 participants showed the effect; see Figure 4B) suggesting that parallel lines might serve as an anchor for judgments of discrimination.

Detection thresholds were quite variable across participants between 10° and 80°, though they exhibit a similar shape, first increasing at smaller reference angles and then decreasing at larger reference angles. Evidently, acute angles are difficult to differentiate with non-connected lines. Nevertheless, as in the connected lines condition, participants' precision increased around 90° compared to 80°/100° reference angles (Z = 2.15, p = .016; 8/8 participants showed the effect) and to 55°/125° reference angles (Z = 2.15, p = .016; 8/8 participants showed the effect; see Table 2). We also observed significant asymmetries in detection thresholds, with better discrimination as deviants approached or crossed the 90° boundary in the 80°/100° reference blocks (Z = 2.15 p = .016, 8/8 participants showed the effect), but not in the 55°/125° reference blocks (Z = 0.07, p = .473, 4/8 participants showed the effect; see Fig. 4B).

Discussion

This experiment presents two main findings. First, acuity in angle discrimination varies massively across angles. In particular, thresholds are smallest for parallels and perpendiculars.

When angles are acute or obtuse and far from these two categories, thresholds of discrimination are high, though when angles deviate from such acute or obtuse angles and approach the parallel or perpendicular categories, discrimination becomes more precise. These effects persist over variations in the orientation and scale of the angle exemplars. While other studies reviewed above have found similar categorical responses to parallels and perpendiculars under some conditions, our results suggest that angle discrimination in general and the more precise discrimination of these angle categories in particular are invariant to scale and orientation.

How robust, then, are the categories of parallelism and perpendicularity in the face of more drastic variation in the angle stimuli, such as rotations in 3D depth, which are prevalent in everyday viewing conditions? In Experiment 1, where stimuli were presented in the frontoparallel plane, the higher precision for parallel and perpendicular lines could have arisen from a specific sensitivity at the retinal or retinotopic level. If, however, the effect arises at a more abstract representational level, i.e., that parallelism and perpendicularity are properties that apply to the arrangement of lines in 3D space, then the effect should remain even if the stimuli are rotated in 3D depth, off the frontoparallel plane. This manipulation applies specifically to perpendicular lines: While parallel lines remain parallel even with rotation in depth (save a few, "accidental" viewpoints where their projections coincide: Amir, [START_REF] Amir | The neural basis for shape preferences[END_REF][START_REF] Lupyan | Linguistically modulated perception and cognition: the label-feedback hypothesis[END_REF][START_REF] Biederman | Representation of shape in individuals from a culture with minimal exposure to regular, simple artifacts: Sensitivity to nonaccidental versus metric properties[END_REF], perpendicular lines vary greatly in their projected angle measure when the lines are rotated in depth [START_REF] Nundy | Why are angles misperceived[END_REF]. Experiment 2 was therefore conducted both to evaluate how well the detection of perpendicular lines is preserved under viewing conditions that include additional rotations in 3D depth and to examine whether perpendicularity as a category persists under these conditions.

Experiment 2

In this experiment, we presented new participants with stimuli derived from the connected lines condition of Experiment 1. In the slanted screen condition (Figure 5A), participants viewed the same stimuli as Experiment 1, but on a screen that was rotated in depth by 45°. In the normal viewing condition (Figure 5B), participants were presented with the same stimuli as Experiment 1 with an un-rotated, frontoparallel screen (i.e., a direct replication of Experiment 1). By comparing participants' performance in these two conditions, we could evaluate whether both angle discrimination in general and the perpendicular category in particular persist with the rotation of our visual stimuli in depth. In the projected stimuli condition (Figure 5C), participants viewed the projection of the stimuli presented in the slanted screen condition on a frontoparallel screen. With this condition, we could evaluate whether the 3D context, presenting consistent cues to the reference frame transformation (like would be present in our everyday object recognition, see [START_REF] Nakayama | Visual surface representation: A critical link between lower-level and higher-level vision[END_REF] or some general property of the stimuli's projection (which has a less clear connection to our everyday perceptual experience) might explain participants' responses in the slanted screen condition. If participants showed the same pattern of performance on the projected stimuli condition and the slanted screen condition, then there may be some geometric information at the level of the retinal projection that is guiding responses on the slanted screen condition. The lower panel illustrates the same trial (100° referent angle and 124° deviant) as it would be seen on the screen in each condition. The angles in the slanted screen condition are identical to the angles in the projected stimuli condition when projected on to the frontoparallel plane, as illustrated here. For illustration purposes, the correct response is circled in red.

Methods

Participants

Twelve adults (10 women; Mage = 23 years; range 19-38 years) participated in this experiment. The sample size was set in advance of data collection. With the sample size and smallest effect size from Experiment 1 (N = 8, Cohen's d = 1.94), our power to detect groupwise categorical effects within condition would be .999. We decided to increase our sample size by four participants compared to Experiment 1 since we also planned to compare results across conditions in this experiment and investigate effects by condition. All participants had normal or corrected-to-normal vision and had completed high school; most of them had also received a college or advanced degree. None were informed of the purpose and hypotheses of the study until after it was completed. Each provided informed consent prior to the experiment and were paid 10€ per hour plus an additional sum, which depended on their performance. On average, participants earned 72.95€ (range 70.47€-75.56€).

Design, Apparatus, Procedure, and Stimuli

The distance of the participant to the center of the screen, the lighting conditions, the placeholder sizes, the screen luminance, the presentation duration, and the stimuli parameters were identical in all conditions of Experiment 2 to the connected lines condition of Experiment 1, except that the lines that formed each angle were slightly shorter (uniform distribution from 1.6 to 3.5 dva). In the projected stimuli condition, the angles were transformed by reducing all horizontal coordinates of each angle by a factor of , while keeping all vertical coordinates constant. This transformation was applied to the angles only so that the external reference frame in the slanted and projected stimuli conditions (placeholders' shapes and screen frame) provided cues to enforce the perception of a slanted or frontoparallel presentation of the stimuli. As in Experiment 1, trials were blocked by reference angle but fewer reference angles were presented to focus on the comparisons that were most relevant to exploring the categorical effect (see Table 1). Each type of deviant was presented 18 times in a random order in a block, three times at each of six possible target locations. The total number of trials per block thus ranged from 144 to 162.

The experiment was divided into three sessions, one for each of the three experimental conditions. The sessions took place on different days, one to five days apart, and the order of the condition presented at each session was counterbalanced across participants. Sessions started with a short training phase consisting of two trials per reference angle displaying +/-24° angle deviants. Five blocks corresponding to the five reference angles followed the training phase. The order of the blocks within a session was chosen randomly for each participant. Each session lasted approximately 1.5 hours. As in Experiment 1, at the beginning of each block, participants were given a slide with instructions, showing an example of the reference angle in the vertical orientation. This slide was identical in the slanted screen, normal viewing, and projected stimuli conditions.

Analyses

As in Experiment 1, we fit performance with asymmetric down-pointing Gaussian curves. However, in several cases participants responded at chance (across participants: 10/120 blocks in the slanted screen condition; 3/120 blocks in the normal viewing condition; and 45/120 blocks in the projected condition; see Figure S3 for individual accuracy curves). Such at-chance performance resulted in very high and unreliable estimated threshold values. We thus again used non-parametric rank tests to minimize the impact of these extreme values in our comparisons, and we also conducted parametric analyses on accuracy data, which yielded identical results except where indicated (see SM, Table S2).

As in Experiment 1, for all conditions of Experiment 2, we examined whether angle detection was more precise for the 90° reference angle compared to the other reference angles that were close to and far from 90° (i.e., 80°/100° reference angles and 55°/125° reference angles). We also evaluated whether deviant detection was more precise when deviants approached or crossed the 90° boundary compared to when they did not (all using Holmcorrected, one-tailed Wilcoxon signed-ranks tests). We only evaluated the asymmetries in the 80°/100° reference angles since the asymmetries in the 55°/125° reference angles were not significant in Experiment 1. By comparing the findings across the three conditions, first we asked whether categorical effects were present in the slanted screen condition, as in the normal viewing condition (we compared the thresholds and asymmetries in these two conditions using Scheirer-Ray-Hare [SRH] tests, a non-parametric equivalent of a two-way repeated measures ANOVA). Second, we asked whether the effects observed with the slanted screen were due to information present in the stimuli's projection rather than based on shape information invariant to 3D rotations (by comparing the slanted screen condition to the projected stimuli condition using a second SRH test).

Results

Figure 6A shows the median detection thresholds for each reference angle in all three conditions, and Table 3 shows tests for within-condition categorical effects. First, in the slanted screen condition, we observed more precise detection thresholds for 90° reference angles compared to both the 80°/100° reference angles (Z = 3.18, p < .001, 12/12 participants showed the effect), and also the 55°/125° reference angles (Z = 3.18, p < .001, 12/12 participants showed the effect). Second, this categorical effect did not differ across the slanted screen and normal viewing conditions. An SRH test comparing these two conditions revealed a significant main effect of reference angle (H = 8.37, p = .004), a significant main effect of condition (H = 5.87, p = .015), but, crucially, no interaction between reference angle and condition (H = 0.20, p = .653). As such, our normal viewing condition also replicated the results of Experiment 1, finding smaller detection thresholds with the 90° reference angle compared to the other reference angles (80°/100°: Z = 2.47, p = .007, 11/12 participants showed the effect; 55°/125°: Z = 2.98, p = .001, 11/12 participants showed the effect). Does the increased performance with the 90° reference angle in the slanted screen condition result from an analysis of the stimuli as projected or from an invariance of the detection of perpendicularity over rotations in depth? If the former, then the same categorical effects should be present in the projected stimuli condition. An SRH test comparing the slanted screen and projected stimuli conditions revealed a significant main effect of reference angle (H = 5.89, p = .015), a significant main effect of condition (H = 11.67, p < .001), and an interaction between reference angle and condition (H = 8.56, p = .003). Indeed, contrary to the slanted screen condition, there was no sign of categorical effects in the projected stimuli condition, as performance with the 90° reference angle was no different from the other reference angles (ps = 1.00).

Next, we assessed whether performance for reference angles 80°/100° was better when the deviant approached or crossed the 90° boundary (see Figure 6B). Performance was significantly asymmetric in the slanted screen condition (Z = 1.85, p = 0.032, 9/12 participants showed the effect). An SRH test comparing the thresholds towards and away from 90° in the slanted screen and normal viewing conditions revealed a significant main effect of deviant direction (H = 5.00, p = .025), condition (H = 7.29, p = .007), but no interaction between these two factors (H = 0.02, p = .896). In the normal viewing condition too, there was a significant asymmetry in detection thresholds (Z = 1.85, p = .032, 8/12 participants showed the effect).

Thus, the asymmetry effects were comparable across the slanted screen and normal viewing conditions and were comparable to Experiment 1.

Finally, an SRH test comparing the slanted screen and projected stimuli conditions on the asymmetry of their deviant detection thresholds revealed a main effect of deviant direction (H = 6.88, p = .009), condition (H = 11.38, p < .001), and no interaction between the two factors (H = 0.09, p = .770). Indeed, asymmetries followed the same pattern in the projected stimuli condition as in the other conditions, with slightly better performance for angles deviating towards 90° versus away from 90°, though this difference was not significant (Z = 1.20, p = .116, 8/12 participants showed the effect; see SM, Table S2). These weak asymmetry effects in the projected stimuli condition may have been due to a residual property of the angle transformation, rather than to participants' categorical discrimination of projected 90° angles. In particular, when the stimuli were transformed, the range of angle sizes presented on the screen differed based on the reference angle, with the greatest range occurring at 90°. Moreover, projected deviant angles fell outside the range of projected reference angles around 90° more often for deviants towards 90° than for deviants away from 90°, making deviants towards 90° easier to detect.

Discussion

This experiment builds on the findings of Experiment 1 by suggesting that both angle discrimination in general, and the perpendicular category in particular, are largely invariant to the rotation of shape stimuli in depth. A 45° rotation of the screen in the slanted screen condition degraded discrimination performance uniformly, yet left the relative discriminability of angles of different sizes intact, including preserving peak discriminability at 90°. Because the 90° angles in that condition were defined only in three dimensions with widely varying 2D projections, these results suggest that participants detected the angle intruder by monitoring the 3D angle sizes rather than their retinotopic, 2D projections. This conclusion is further supported by comparisons between the slanted screen condition and the projected stimuli condition. Here, the 2D angles that were projected on the retina were identical between conditions, yet accuracy varied greatly, suggesting that the preserved shape discrimination in the slanted screen condition was not due to properties of the 2D projection of the stimuli. Indeed, participants performed so poorly in the projected stimuli condition that the data were difficult to model, and it is thus hard to make specific conclusions about the characteristics of our shape discrimination with stimuli transformed in this way. By contrast, rotating shapes in depth, with all of the real-world depth cues intact, led to a largely preserved ability to discriminate angles.

General Discussion

Across two experiments presenting angle-intruder detection tasks, we demonstrated that the discriminability of both connected and non-connected lines forming different angles varies greatly depending on the size of the angles being discriminated. Most notably, thresholds of discrimination were significantly smaller when lines formed parallels and perpendiculars. When angles were acute or obtuse and far from these two special angle categories, thresholds of discrimination were high, though when angles deviated from nearer acute or obtuse angles to approach the parallel or perpendicular categories, discrimination became more precise. This pattern of results persisted not only over variations in the orientations and scale of the angle exemplars, but also over their rotations in depth. In particular, when the very same angle stimuli were presented on a screen rotated 45° in depth, the relative discriminability of angles of different sizes persisted. Most remarkably, the significantly smaller threshold for detecting perpendicular angles persisted, despite the variability in the 2D projections of these angles, and this result was not due to properties of the stimuli's 2D projection. Our findings thus suggest that angle discrimination in general and the special angle categories of parallelism and perpendicularity in particular affect participants' shape judgments by their real-world angle information.

The prior literature had outlined different models of angle discrimination to which we can compare our results. In two cases, more precise discrimination for horizontal and vertical lines was invoked to explain better acuity for 90° and 0/180° angles: [START_REF] Chen | Angle judgment: Is the whole the sum of its parts?[END_REF] suggested an orientation-independent, Weber-like discrimination space except at 90°, where discrimination is heightened and rooted in the presence of vertical and horizontal lines; [START_REF] Xu | The human visual system estimates angle features in an internal reference frame: A computational and psychophysical study[END_REF] proposed that angles which include either a vertical or horizonal line are discriminated most precisely. In contrast, Heely & Buchanan- [START_REF] Heeley | Mechanisms specialized for the perception of image geometry[END_REF] proposed that, unlike the discrimination of individual line orientations, angle discrimination operates over a reference frame that is object-centered, like the discrimination of other, more complex objects. Our findings thus support and extend this last model: Not only did our stimuli specifically avoid using lines at or near the horizontal or vertical, but also more precise discrimination persisted when the angles were rotated in 3D depth, suggesting that angle discrimination unfolds at the level of the 3D angle size, as it might for the shape of complex 3D objects.

What then is the status of the geometric categories of parallelism and perpendicularity? Some research has suggested that sensitivity to parallelism, at least, arises early in child development. Even four-year-old children indicate that a pair of parallel lines is the "most different" from five other pairs of lines that present continuous differences in angle at varied absolute orientations and scales (Izard et al., 2011a;[START_REF] Izard | Flexible intuitions of Euclidean geometry in an Amazonian indigene group[END_REF]). Children's knowledge of the word "parallel," moreover, has no relation to this choice (Izard et al., 2011a;[START_REF] Izard | Flexible intuitions of Euclidean geometry in an Amazonian indigene group[END_REF]. As such, children's judgments of parallelism may be rooted in the recognition of more basic shape properties, for example, that two lines have the same orientation or that they maintain a constant distance from each otherand perhaps these very properties contribute to the categorical effects for parallels documented here. Future research should explore whether children display the same categorical effects around the discrimination of parallels as adults and whether their specialized treatment of parallel lines changes with more visual experience or with explicit learning of formal geometry in school.

What about perpendicularity, whose angle measure changes greatly with rotations in 3D depth? Unlike their performance with parallel lines, not until age seven or older do children pick out a pair of perpendicular lines from other pairs of lines that vary continuously in angle, and individual children's knowledge of the lexical terms "right angle" and "perpendicular" correlates with this choice (Izard et al., 2011a). The present task with adults did not rule out the possibility that explicitly learned conceptual representations were supporting participants' performance. All the adults in this study had benefitted from a formal education at least through high-school, which likely included a geometry class where the concepts of parallelism and perpendicularity were taught. In particular, since participants were introduced to each block of trials with an image that depicted the reference angle, it was possible that, for the parallel and perpendicular categories, participants labeled the reference and used that label to access a stored representation of an exemplar from that category from memory, making its recognition more accurate [START_REF] Firestone | Cognition does not affect perception: Evaluating the evidence for "top-down" effects[END_REF]. Like other studies examining advantages in discrimination for complex spatial stimuli, defined, e.g., by topological relations, it is unknown the extent to which the present findings reflect verbal coding or pure visual processing [START_REF] Lovett | Topological relations between objects are categorically coded[END_REF]. Indeed, a suite of studies using search tasks and deviant detection tasks show enhanced detection of visual stimuli including colors, simple shapes, and more complex objects when category boundaries line up with linguistic labels (see [START_REF] Goldstone | Categorical perception[END_REF][START_REF] Lupyan | Linguistically modulated perception and cognition: the label-feedback hypothesis[END_REF].

Future studies using the same angle stimuli with adults from other cultures, children, and nonhuman animals might shed further light on whether the effects observed in this study were perceptual or cognitive.

Nevertheless, the detection of perpendicular angles may indeed be universal. A group of adults from the Mundurucu tribe in the Amazon, who, unlike the adults in the present sample, have no specialized geometric training or vocabulary, pick out a pair of perpendicular lines as the "most different" from other pairs of lines that vary continuously in angle (Izard et al., 2011a).

While categorization of perpendiculars (as in this study with the Mundurucu adults, Izard et al., 2011a) and categorical effects in discrimination (as measured with the adults in this paper) may dissociate, it is possible that a perceptually based perpendicular category arises spontaneously in development and that education in formal geometry refines it or heightens attention to it (see [START_REF] Piazza | Education enhances the acuity of the nonverbal approximate number system[END_REF] for further exploration of this suggestion in the numerical domain). If this is the case, it remains a challenge to identify a perceptual learning mechanism that might operate over such a long period of development and create such a robust category.

Might visual experience through development bolster our recognition of 3D shape information from which a heightened sensitivity to perpendicular angles could emerge? Previous work has shown that high-level perceptual learning can occur over extended periods of time, for example, to create or refine specific object categories like letters and faces [START_REF] Dehaene | The neural code for written words: a proposal[END_REF][START_REF] Maurer | The many faces of configural processing[END_REF]. This kind of perceptual learning is nevertheless limited, in that categorical effects disappear when, for example, face stimuli are inverted [START_REF] Maurer | The many faces of configural processing[END_REF]. Such limits may be adaptive to the context: While face recognition is not invariant to large 2D orientation changes, it is invariant over large changes in viewpoint (i.e., in full view or in profile), expression, and lighting [START_REF] Anselmi | Representation learning in sensory cortex: a theory[END_REF].

The perceptual learning associated with 3D object recognition may accumulate 2D and 3D rotational invariance through development since objects are often seen rotated in 2D or 3D space. While newborns differentiate among 2D angle shapes given long periods of habituation [START_REF] Slater | Form perception at birth: Revisited[END_REF], seven-month-old infants fail to differentiate shapes differing in angle during brief exposures to 2D forms with simultaneous variations in scale, direction, and orientation (Dillon, Izard, & Spelke, in preparation). Research with toddlers has suggested that 3D object recognition undergoes protracted development, with a spurt in the ability to recognize objects by their 3D geometric shapes between the ages of 18-24 months [START_REF] Augustine | Parts and relations in young children's shapebased object recognition[END_REF][START_REF] Smith | From fragments to geometric shape: Changes in visual object recognition between 18 and 24 months[END_REF], and studies with older children have shown that the period during which we become better able to recognize unfamiliar viewpoints of 3D objects captured in 2D drawings extends to adulthood [START_REF] Jüttner | A developmental dissociation of view-dependent and view-invariant object recognition in adolescence[END_REF][START_REF] Jüttner | Developmental trajectories of part-based and configural object recognition in adolescence[END_REF][START_REF] Landau | Object recognition with severe spatial deficits in Williams syndrome: sparing and breakdown[END_REF]. Further developmental work, investigating invariances in angle detection may begin to shed light on how such shape representations become more robust to every-day 3D viewing conditions.

Even if angle detection in general becomes more robust to rotations in 3D depth through development, such development alone does not explain the emergence of a perpendicular category. A rotationally invariant representation of perpendicularity could be singled out because a perpendicular line is the most symmetrical position for a line relative to another line (i.e., splitting a line in half) or to a plane, and both infants and young children show some sensitivity to symmetry [START_REF] Bornstein | Perception of symmetry in infancy[END_REF][START_REF] Huang | The aesthetic preference for symmetry dissociates from early-emerging attention to symmetry[END_REF]. A rotationally invariant representation of perpendicularity could also derive from representations of the absolute vertical and horizontal. These orientations are robustly represented early in development and across animal species [START_REF] Appelle | Perception and discrimination as a function of stimulus orientation: the" oblique effect" in man and animals[END_REF]. For example, 5month-old infants take longer to look towards a deviant oblique line among other oblique lines versus among vertically oriented lines the same angle measure away [START_REF] Franklin | Hemispheric asymmetries in categorical perception of orientation in infants and adults[END_REF].

Finally, the present study raises questions about the relations between our perception of angles and our conception of formal Euclidean geometry. Euclid famously did not explain what he means by "equal" angles when, at the beginning of his Elements (Book 1, Definition 10), he defines a perpendicular as a line with equal angles to either side as it stands up from a baseline. We are only just beginning to probe how the human mind navigates the perceptual and conceptual worlds of geometry.

Figure 2

 2 for the accuracy curves of a representative participant and Figure S1 for all individual accuracy curves). To analyze the results of the non-connected lines condition, we thus used non-parametric, rank-order tests instead of parametric tests, which replaced estimated threshold values with ranks based on

Figure 2 .

 2 Figure 2. Model-fit curves for one participant (S2) for all reference angles in the connected lines condition of Experiment 1. Individual data points are additionally represented for the 80°, 90°, and 100° reference blocks. These curves illustrate lower thresholds in the 90° reference block (i.e., a more narrow curve, in yellow) and lower thresholds for deviants towards versus away from 90° (i.e., an asymmetry in the curves around 0° of deviation) in the 80° (in green) and 100° (in orange) reference blocks.

Figure 3 .

 3 Figure 3. A. Estimated detection thresholds for each participant (grey lines) and on average (black line) at different reference angles for the connected lines condition. All curves are characterized by threshold drops towards 0°/180° as well as a sharp drop at 90°. The dashed curve corresponds to the participant shown in Fig. 2. B. Asymmetries in the thresholds for detecting smaller versus larger deviants at different reference angles. Positive values on the yaxis indicate greater success in detecting larger versus smaller deviants, and negative values indicate the opposite. The dashed curve corresponds to the participant shown in Fig.2. For 80° and 100° references, acuity is better (smaller thresholds) when the deviant is in the direction of 90°. The asymmetry in detection thresholds crosses zero at almost exactly 90°, and there appears to be an approximately identical advantage on both sides of 90°.

Figure 4 .

 4 Figure 4. A. Estimated detection thresholds for each participant (grey lines) and the group medians (black line) at different reference angles for the non-connected lines condition. Thresholds at some reference angles for some individuals exceed the graph limit. All curves are characterized by low thresholds for the 0° reference angle as well as a sharp drop at the 90° reference angle. B. Asymmetries in the detection thresholds for larger versus smaller deviants at

Figure 5 .

 5 Figure 5. The top panel illustrates on overhead perspective of the setup for the screen and participant for the three conditions: (A) slanted screen; (B) normal viewing; (C) projected stimuli.The lower panel illustrates the same trial (100° referent angle and 124° deviant) as it would be seen on the screen in each condition. The angles in the slanted screen condition are identical to the angles in the projected stimuli condition when projected on to the frontoparallel plane, as illustrated here. For illustration purposes, the correct response is circled in red.

  Figure 6. A. Median thresholds for different reference angles in the three conditions of Experiment 2. While the slanted screen and normal viewing conditions show a sharp drop at the 90° reference angle, the projected stimuli condition does not. B. Asymmetries in the threshold of detection for larger versus smaller deviants at different reference angles.

  This definition is strikingly perceptual: We see what he means. Do we need to see the abstractions of geometry to conceive of them? Are they driven by mental imagery of specific spatial exemplars, abstract idealized concepts, or linguistic representations? Indeed, what are the mental representations guiding our geometric reasoning? Individuals differ in their ability to discriminate angles: Might individual differences in our angle discrimination predict our ability to reason about the properties of points, lines, and figures on the Euclidean plane? Could training in angle discrimination cause short-or long-term benefits to geometric reasoning or judgment?

Table 1 ).

 1

	Reference Angles

Table 1 .

 1 Summary of the stimuli presented in Experiments 1 & 2. The 25° and 155° reference 161 angles were presented only in Experiment 1 and the 0° and 10° reference angles were presented 162 only in the non-connected lines condition of Experiment 1. All angle measures are in degrees.

163 164 165 166 Figure 1. A. Exemplar trials from the connected lines condition. Participants showed

	167	smaller distinguished non-parallel lines among parallel lines (top) but had difficulty with larger deviants when reference angles were 10° (bottom). Here, deviants differ from the reference angles by 24° (the difference used during training trials). For illustration purposes, the correct responses are discrimination thresholds with 90° reference angles (top) than with 80° reference angles (bottom). B. Exemplar trials from the non-connected lines condition. Participants easily circled in red.

Table 2

 2

	, corroborated these results, finding

Cohen's d = 0.07). Thus, as deviants crossed 90°, their discrimination became more precise, again signaling an influence of the perpendicular category on performance (see Figure

3B

).

Acknowledgements

This work was supported by a Starting Grant of the European Research Council to V.I. (FP7 Project MathConstruction 263179), by a grant from the National Science Foundation to M.R.D. (DGE-1144152), and by the Norman Henry Anderson Graduate Psychology Fund to M.R.D.

Author Contributions

All authors conceived of and designed the study. M. Duyck programmed the stimuli. M. R. Dillon, M. Duyck, and V. Izard collected the data. M. R. Dillon, M. Duyck, and V. Izard performed the data analysis. M. R. Dillon, M. Duyck, and V. Izard drafted the manuscript, and all authors provided critical revisions. All authors approved the final version of the manuscript.

Authors' Note

The full data set and analysis code are available on the Open Science Framework at: https://osf.io/wa64q/. Code for the generation of the experimental stimuli are available upon request.