People verbally overestimate the slant of visually perceived geographical, virtual, and manmade surfaces, viewed from either the base or the top, by between 5° and 25° (Bhalla & Proffitt, 1999; Durgin & Li, 2011; Hajnal, Abdul-Malak, & Durgin, 2011; Li & Durgin, 2009, 2010; Proffitt, Bhalla, Gossweiler, & Midgett, 1995; Shaffer & Flint, 2011). People’s verbal overestimates of these slanted surfaces are explained nicely by a scale expansion theory, whereby people expand the scale by which slanted surfaces are overestimated by a factor of ~ 1.5 (Durgin & Li, 2011; Li & Durgin, 2010). Less work has looked at slanted surfaces that are commonly thought of as being sloped downward starting from a vertical position. This is partly because much of the aforementioned work grew out of work focusing on geographical slant (e.g., Proffitt et al., 1995). However, as common as it is to encounter slopes (and thus, as important as it is to focus work on these slopes), which we view as tilted upward from horizontal, we commonly think of many objects conceptually as leaning down (or over) from a vertical position rather than upward from a horizontal position. This is the case with people leaning themselves, chairs, pictures and paintings, drywall, ladders, broom and mop handles, bulletin boards, bicycles, mirrors, or window screens against walls, as well as people orienting computer monitors away from a straight up-and-down position. Here people seem to be very good at finding fairly stable angular orientations at which to place these objects against a vertical reference frame. This begs the questions—how do we assess the slopes of objects typically tilted away from vertical, as compared with those slanted from horizontal; likewise, since such objects need to be leaned against something in order not to fall over, can this tell us something about assessing the angles of objects that have a vertical reference frame; and finally, does this process differ from making assessments of surfaces without a vertical reference frame?

Previous work has investigated whether sloped surfaces would be estimated differently from a vertical than from a horizontal reference. For instance, Durgin, Li, and Hajnal (2010) had participants estimate one of 18 irregularly shaped wooden surfaces at oriented at 16 different angles from 0° to 90° within a hemispheric enclosure of black felt that served as the background for the surface. Durgin, Li, and Hajnal told some of the participants to make their estimates relative to horizontal (0° to 90°), and other participants to make their estimates relative to vertical (90° to 0°). In both groups, the participants made overestimates relative to horizontal, exhibiting what the authors called a “vertical tendency.” That is, the participants gave estimates that were farther away from the horizontal than from the vertical.

In the present work, we examined the perception of sloped surfaces that are typically thought of as being slanted downward from vertical, not upward from horizontal. Although the results from Durgin, Li, and Hajnal (2010) demonstrated a vertical tendency, their participants may have conceptually used a horizontal reference frame to make their judgments even when they were told to make their estimates relative to vertical, perhaps because it is easier to imagine a horizontal than to imagine a vertical frontal surface. As we sought to investigate the perception of sloped surfaces typically thought of as being slanted downward from vertical, we incorporated a vertical reference frame away from which we slanted our apparatus. This procedure differed from those employed in previous work examining slant perception, in that the previous work had used backgrounds that purposely did not allow participants to use a vertical reference frame. To examine this issue, we wanted to investigate whether people are able to understand at what angle an object is most stable when it leans against a vertical reference frame, by having participants (1) actually position a ladder in what they perceived to be its most stable position against a wall, (2) draw a line on a piece of paper at an angle that represented a ladder’s most stable position and then to verbally estimate at what angle the ladder/drawn line was oriented, or (3) verbally estimate the angle at which an object was most stable when leaning against a vertical reference frame. We chose to use a ladder because ladders best represent a stimulus that people have a lot of “hands-on” experience with leaning against a vertical reference frame. Doing this also ensured that we would invoke the representation and/or memory of something that was conceived of as being specifically tilted downward from above against a vertical reference frame, and not upward from a horizontal reference frame. Next, we systematically manipulated the orientation of a ladder placed against a wall, at angles from 50° to 86°, and had participants verbally estimate the ladder’s orientation.

Experiment 1

In Condition 1 of Experiment 1, we first tested how accurate people are at performing an action-based task. To do this, we asked people to place a ladder at the most stable position against the wall and told them we would ask them to climb the ladder after doing so. We then asked people to estimate the orientation of the same ladder that they had placed against the wall. We did this to evaluate people’s intuitive beliefs about stability and compare their verbal estimates to their positioning of the ladder. It is known that action-based tasks or verbal estimations of the orientations of objects within reach are fairly accurate for orientations less than 45° (Durgin, Hajnal, Li, Tonge, & Stigliani, 2010). We wanted to test whether the same was true for an action-based task with orientations closer to 90°. We expected that placing the ladder against the wall (the action-based task) would result in closer estimates than would verbal estimation, as had been shown in the past (Durgin, Hajnal, et al., 2010). In Condition 2, we asked another group of participants to verbally estimate the orientation of a ladder’s most stable position against a wall. In Condition 3, we asked a separate group of participants to draw a line representing a ladder and its most stable orientation. We expected that people would verbally overestimate and draw the orientation of the ladder away from vertical, as has been shown with a clear vertical reference frame (Dick & Hochstein, 1989), in much the same way that people overestimate slopes tilted upward from the ground away from the horizontal (reference frame).

Method

Condition 1

Participants

Seventy undergraduates from the Ohio State University at Mansfield (37 female, 33 male), mean age = 18.73 years (SD = 3.11), participated in fulfillment of an introductory psychology requirement.

Procedure

Participants placed a free-standing ladder against a wall at what they thought was the most stable position to climb. The ladder used in all conditions of Experiment 1 and in Experiment 2 is shown in Fig. 1. Participants were told that they would be asked to climb the ladder a few steps, although later we never asked them to do this. They were then asked to estimate at what angle the ladder was oriented. They were told that 0° was horizontal—lying flat on the ground—and 90° was vertical, or straight up and down. We randomly assigned (1) the initial ladder placement prior to a participant’s picking it up and orienting it against the wall—either on the ground or straight up and down against an adjacent wall—and (2) the participant’s distance from the ladder when making verbal estimates—either standing at the base of the ladder or 1 m away.

Fig. 1
figure 1

Picture of the 6-foot aluminum sectional ladder used in Experiments 1 and 2.

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Condition 2

Participants

Seventy-eight undergraduates from the Ohio State University at Mansfield independent of those participating in Condition 1 (45 female, 33 male), mean age = 18.88 years (SD = 1.55), participated in fulfillment of an introductory psychology requirement.

Procedure

We asked participants to give a verbal (conceptual) estimate of the most stable orientation of a ladder against a wall to support a climber. Prior to giving any estimates, all participants were given a brief review of standard geometry, to clarify that they understood different angular positions. They were told that 0° was horizontal—lying flat on the ground—and 90° was vertical, or straight up and down. We did not proceed until all participants had clearly understood the instructions.

Condition 3

Participants

Forty-six undergraduates (independent of those participating in Condition 1 or 2) from the Ohio State University at Mansfield (26 female, 20 male), mean age = 18.8 (SD = 1.94) participated in fulfillment of an introductory psychology requirement.

Procedure

Participants made an estimate of the most stable position at which to place a ladder against a wall in order to support a climber, by drawing a single line on a piece of paper to indicate the most stable orientation. We had a vertical line drawn on each piece of paper to indicate a vertical wall, against which they would draw the ladder. We then asked participants to verbally estimate the orientation of the drawn line. Prior to producing any estimates, all participants were given a brief review of standard geometry in order to clarify that they understood the different angular positions. They were told that 0° was horizontal—lying flat on the ground—and 90° was vertical, or straight up and down. We did not proceed until all participants had clearly understood the instructions.

Results

Condition 1

Positioning of the ladder at the most stable position

We compared where the participants oriented the ladder against the wall to 75.5°—or an ~ 4:1 ratio of pitch, which is the most stable position at which one can place a free-standing ladder (American National Standards Institute, 2007a, b; Occupational Safety & Health Administration, US Department of Labor, 2017; Young & Wogalter, 2000). An independent-samples t test showed no difference in positioning due to ladder starting position (horizontal or vertical), t(68) = 0.3, p = .765. Therefore, we collapsed the data for the next analysis. A one-sample t test indicated that people tipped the ladder away from vertical statistically more than the ideal stable position of 75.5° (90° – 75.5°), t(69) = 6.02, p < .001, M = 71.91°, SD = 4.99°. This statistical difference resulted from only a small difference between means (MDiff = 3.59°).

Verbal estimates of ladder orientation

Neither ladder starting position nor distance from the ladder affected the verbal estimates of ladder position, t(68) = 0.38, p = .705, and t(68) = 0.79, p = .432, respectively. Therefore, we collapsed the data for the remaining analyses. A paired samples t test showed that participants underestimated the slant of the ladder position relative to its actual position at which they oriented it against the wall, t(69) = 6.26, p < .001, MDiff = 11.37°, SDDiff = 15.2°. The mean setting of the ladder was 71.91°, and the mean verbal estimate of where the ladder was placed was 60.64°.

Condition 2

Verbal (conceptual) estimates of ladder orientation

The mean verbal (conceptual) estimate of the most stable orientation of a ladder to be placed against a wall was significantly shallower than 75.5°, t(77) = – 14.17, p < .001, M = 50.31°, SD = 15.7°. This was a large effect, Cohen’s d = 1.6.

Condition 3

Drawing of the ladder at the most stable orientation

A one-sample t test indicated that people drew the orientation of the ladder at a statistically shallower orientation than the ideal stable position of 75.5°, t(45) = – 7.68, p < .001, M = 58.04°, SD = 15.41°. This was also a large effect, Cohen’s d = 1.13.

Verbal estimates of drawn ladder orientation

A paired-samples t test showed that participants estimated the slant of the ladder position similarly to the orientation at which they drew it, t(45) = 1.27, p = .21, MDiff = 4.24°, SDDiff = 22.67°. The mean drawn orientation of the ladder was 58.04°, and the mean verbal estimate of the drawn orientation of the ladder was 53.8°.Footnote 1

The perceived most stable ladder placement positions and estimated ladder orientations, along with drawn lines representing the most stable orientations and the estimated drawn orientations, are shown in Fig. 2.

Fig. 2
figure 2

(Left) Mean orientation of the ladder that participants positioned against the wall at what they thought was the most stable position, and the mean estimated angle at which they believed the ladder to be oriented (with standard errors). (Right) Mean slope of drawn lines that a different group of participants drew representing the orientation of a ladder at the most stable position, and the mean estimated orientation of the line that they drew (with standard errors).

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Although the difference between the drawn and estimated drawn orientations (right panel) was not statistically different, both panels show a pattern of the mean verbal estimates being shallower than either the positioned ladder or the drawn ladder orientation.

Comparison of estimates across conditions

To compare estimates made across the different conditions, we performed a one-way analysis of variance (ANOVA) on the orientations in the three conditions: the ladder placed physically against the wall, verbal estimates of the most stable position at which a ladder should be placed against a wall, and a drawn line representing the most stable position at which a ladder should be placed against a wall. The one-way ANOVA indicated a significant difference among the conditions, F(2, 191) = 53.03, p < .001. This was a large effect, η2 = .36. Tukey post-hoc tests indicated that all three sets of estimates were significantly different from one another (ps < .05) and formed the pattern shown in Fig. 3.

Fig. 3
figure 3

Mean placed, drawn, and verbal estimates of the most stable position for a ladder, with standard errors.

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Discussion

The findings from Experiment 1 showed that people are pretty accurate at placing a ladder at the most stable position, erring on the side of placing the ladder slightly too shallow, as had been found in previous work (Häkkinen, Pesonen, & Rajamäki, 1988; Irvine & Vejvoda, 1977). This also fits the results of palm board and forearm tasks for near surfaces (Durgin, Hajnal, et al., 2010), in which gains are ~ 1, and also with manmade and geographical slants when there is no anchoring (e.g., Shaffer, McManama, & Durgin, 2015; Shaffer, McManama, Swank, Williams, & Durgin, 2014; Shaffer & Taylor, 2017). However, whereas the gains of free-hand or forearm measures are typically 1 for near surfaces and greater than 1 but less than the gains of verbal (over)estimates for geographical and ramp surfaces (i.e., steeper than the surface itself), the ladder placement was close to but shallower than the most stable position. Additionally, across the conditions of Experiment 1, verbal estimations of positioned and drawn orientations of the most stable position for a ladder to be leaned against a wall and verbal (conceptual) estimates of the most stable position at which to position a ladder were all shallower than the actual orientation of positioned or drawn ladders (i.e., lines representing ladder orientation). This is counter to most previous work showing that there is a perceptual scale expansion of geographic and manmade hills, roads, ramps, and people’s own orientation, leading to steeper verbal estimates (Durgin & Li, 2011; Li & Durgin, 2010; Proffitt et al., 1995; Shaffer & McManama, 2015; Shaffer & Taylor, 2017; Shaffer, Taylor, McManama, et al., 2016; Shaffer, Taylor, Thomas, et al., 2016).

This finding is also contrary to what Durgin, Li, and Hajnal (2010) found. They told some participants that vertical was 0°, and others that horizontal was 0°. In both groups, their participants made overestimates relative to horizontal, exhibiting what the authors called a “vertical tendency.” There are two differences between the present work and theirs. The first is a methodological difference. In the Durgin, Li, and Hajnal study, no vertical reference frame (planar wall) was visible. However, in our Experiment 1, a clear reference frame was evident to all of the participants. This vertical wall gave participants a clear angular deviation from vertical information that was unavailable to the participants in Durgin, Li, and Hajnal’s work. The second difference was that even though some of the Durgin, Li, and Hajnal participants were told to make their estimations from vertical, the surface that they used, like the surfaces that had most been used in previous studies, was one that was probably conceived of as being tilted up from horizontal. Therefore, participants may have used the intrinsic horizontal reference frame to perform the task, even though they were asked to judge orientation from the vertical, because it is easier to imagine a horizontal than a vertical reference frame. In the present work, there was a clear vertical reference frame to which to compare the ladder orientation, and ladders are generally conceived of as being tilted away from straight up-and-down, and not upward from horizontal.

Interestingly, people’s verbal estimates (conceptual idea, if you will) of the most stable orientation at which to place a ladder against a wall were significantly shallower than the actual orientation, but also were very close to an estimate of 45°, or halfway between vertical and horizontal. In fact, 52.6% of the 78 participants verbally estimated the most stable position to be 45°. This may be because the ladder is about as far away from its tipping point backward at this position (although it is at a position where the possibility of it slipping out is greater). It may be that people rely on this approximate angle as a conceptual anchor for making estimates of orientations close to 45°. Although 45° is not the most stable position for a ladder in terms of supporting a climber, it is an orientation that, with the ladder by itself, is highly unlikely to tip over backward. Support for the idea that people may use 45° as an anchor came from the results of Experiment 1, Condition 1. When estimating at what angle the ladder was oriented after they had positioned it against the wall, 14.3% of the participants estimated it to be at 45°. This was in spite of the fact that the median ladder orientation for the participants who estimated the ladder to be positioned at 45° was 71.25°. Using 45° as a perceptual anchor has been found in other work, as well. For instance, it has been shown that when making verbal and manual estimates of different angles close to 45°, people’s estimates tend to converge on 45°, or people first use 45° as an anchor from which to make their estimates (Feresin, Agostini, & Negrin-Saviolo, 1998; Haun, Allen, & Wedell, 2005).

In Experiment 2, we sought to systematically investigate people’s perception of different ladder orientations.

Experiment 2

In Experiment 1 we found initial evidence of verbally overestimating the amount that a ladder is tilted downward from a vertical position, similar to the findings from previous work showing that people verbally overestimate the amount that manmade and geographical surfaces are slanted upward. However, in Experiment 1 we only used one angle of the ladder, and the ladder was put in that orientation by the participant. To investigate whether there is a systematic pattern of verbal estimates of objects typically thought of as tilted downward from the vertical with a salient vertical reference frame present, we randomly placed the ladder against the wall at seven different angles, all in the range 45° < α < 90°, and then had participants estimate the ladder’s orientation.

Method

Participants

The seventy-eight undergraduates used in Condition 2 of Experiment 1 also participated in Experiment 2. All of the participants took part in the present experiment after they had participated in Condition 2 of Experiment 1.

Procedure

We placed a 1.83-m ladder at one of the following seven angles against a vertical wall: 50°, 56°, 62°, 68°, 74°, 80°, and 86°. These angles correspond to angles measured from the ground up. Each participant stood directly in front of the ladder 1 m from its base and made one verbal estimate for only the one orientation of the ladder. Prior to giving any estimates, all participants were given a brief review of standard geometry, to clarify that they understood different angular positions. They were told that 0° was horizontal—lying flat on the ground—and 90° was vertical, or straight up and down. We did not proceed until all participants had clearly understood the instructions.

Results

Verbal estimates of ladder orientation

Means and standard errors are shown for each ladder orientation in Fig. 4, and means and standard deviations are shown in Table 1.

Fig. 4
figure 4

Mean estimated orientations of the ladder, plotted against the actual orientations of the ladder, with standard errors.

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Table 1 Means and standard deviations are shown for actual and estimated orientations of the ladder
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The use of 45° as a perceptual anchor was found in Experiment 1 and was supported by previous work (Feresin et al., 1998; Haun et al., 2005). Therefore, we tested whether participants were using 45° as an anchor for the angles closest to 45°. We first compared the angle estimations of 50°, 56°, and 62° ladder placements to 45° as it seems like those three values are close enough to 45° so that participants might use that as an anchor to make their estimates. As is shown in Fig. 4, the mean estimates for those ladder placements are all about equal, as well. We used a Bonferroni correction alpha level of .017 (.05/3) to correct for the multiple one-sample t tests. Three one-sample t tests verified that the mean estimates for 50° and 56° were not statistically different from 45°, whereas the mean estimate for 62° was just barely statistically different from 45°: 50°, t(12) = 1.7, p = .113; 56°, t(12) = 2.14, p = .052; 62°, t(11) = 3.1, p = .01.

We then compared verbal estimates of the remaining ladder placements to the actual ladder placements, to test whether the verbal estimates were significantly shallower than the actual ladder orientations. We did not use a Bonferroni correction for multiple one-sample t tests here, as participants were independent of one another and the values we were using to compare the verbal estimates were different between groups. The verbal estimates for every ladder placement were statistically shallower than the actual ladder placement: 62°, t(11) = – 3.92, p = .002; 68°, t(13) = – 2.57, p = .023; 74°, t(8) = – 7.26, p < .001; 80°, t(8) = – 3, p = .017; 86°, t(6) = – 3.16, p = .020.

Another pattern that is evident from Fig. 4 is that, starting at ~ 68°, the difference between the verbal estimates of ladder orientations and the actual ladder orientation increases, from an average difference between the verbal estimates and actual placements of 3.49°, for ladder placements of 50°, 56°, and 62°, to 13.36°, 19.48°, 14.43°, and 6.73° for ladder placements of 68°–86°.

Discussion

In Experiment 2 we first found that participants verbally estimated ladder orientations of 50°, 56°, and 62° statistically no differently from or very close to 45°. In fact, overall, 14.1% of the participants gave estimates of exactly 45°. This was in spite of the fact that the overall mean ladder orientation was 62°. Using 45° as a perceptual anchor had been found in Experiment 1 and was supported by previous work, as well (Feresin et al., 1998; Haun et al., 2005). Next, we found that participants gave verbal estimates that were significantly shallower than the actual ladder placements for orientations from 68° to 86°. This is contrary to the results of Durgin, Li, and Hajnal (2010, Experiment 1), who found verbal estimates that were steeper than the sloped surface oriented in 6° increments from 60° to 90°. This may have been because the surface Durgin, Li, and Hajnal used is typically thought of as being sloped upward from horizontal, and even though they had 15 out of their 27 participants make estimates from vertical, these participants may have used an intrinsic horizontal reference frame to make their estimates. This may also have occurred because, in the present work, the participants had a very clear vertical reference frame to use when making their judgments. However, in spite of the shallower estimates for steeper orientations found in the present work, we did find a similar scale compression pattern in verbal responses to the five ladder orientations above 60°, similar to what Durgin, Li, and Hajnal found for angles from 60° to 90°.

Previous work supported the idea that a clear reference frame affects judgments from vertical in a similar way to that found in the present work, and it also supported the idea that differently oriented stimuli without a clear reference frame affect judgments in a way similar to that found in Durgin, Li, and Hajnal (2010; Dick & Hochstein, 1989). In Dick and Hochstein’s work, the authors found that participants made systematic errors in their estimates of tilted bar stimuli, showing that the estimates were closer to the vertical axis than were the stimuli (Experiment 1). They also found that participants made systematic errors when estimating stimuli in what the authors called “an experimental set appropriate to the minute hand of a clock” (p. 231). One minute on a clock is equivalent to a separation of 6°. They found that the estimates of times between 1 and 6 min before or after the hour were estimated as being farther from the hour. Their data (see the bottom panel of their Fig. 4) also show a scale compression for angles greater than 45°, similar to angles from 60° to 90° shown in Experiment 2 and in the Durgin, Li, and Hajnal data.

General discussion

When asked to perform an action-based task by placing the ladder against the wall in the most stable position to support a climber, people positioned a ladder against a wall at a shallower than ideal orientation. However, people do pretty well, only slightly exaggerating the proper orientation by a little under 4°. This is consistent with previous work showing that free-hand measures of slant are well calibrated to near surfaces (Durgin, Hajnal, et al., 2010). Our previous work showed that placing the ladder at this orientation probably feels to participants like it is at a steeper orientation, consistent with the proprioception calibration hypothesis, which proposes that similar biases exist in both verbal estimates of visually perceived slant and proprioceptively perceived haptic orientation (Shaffer, Taylor, McManama, et al., 2016).

Additionally, our participants also verbally estimated a self-positioned, systematically positioned, and drawn line representing a ladder as shallower than actual. This finding is contrary to Durgin, Li, and Hajnal (2010), who found steeper verbal estimates than the actual slope of a surface when it was oriented at angles greater than 45°. However, although this difference may have been due to Durgin, Li, and Hajnal’s participants who were told to estimate different orientations from vertical instead using an intrinsic horizontal reference frame to make their estimates, the difference also seems to be reflective of the clear vertical reference frame (Dick & Hochstein, 1989).

The present work shows that using a vertical reference frame and a device (a ladder) typically seen and thought of as being tipped away from the vertical, the ladder is perceived to be oriented at shallower than the actual angle from horizontal, as compared to surfaces shown in previous work that do not have a clear reference frame but are sloped at orientations between 45° and 90°, similar to those in the present work. One of the translations of the findings of this work is to people leaning objects against walls. When people do this, they will perceive the leaned object as leaning more away from them than it actually is. This perceived exaggeration of the orientation of an object typically viewed as upright or starting from vertical, then, is similar to the perceived exaggeration of the orientation of hills and ramps, which are typically viewed as slanting upward from a horizontal reference frame. This may point to a generic perceived exaggeration of slant whose direction depends on the conceptual or actual reference frame being used.

Author note

We thank Meghan Burkhardt, Ky Mattingly, Brandon Short, Allyson Thomas, Ally Taylor, and Eric McManama for their help in collecting and coding data.