1. Introduction

One of the most salient properties of an object is its axis. An axis in general terms is a line that passes through an object and which minimizes the distances between opposing boundary contours. Axes capture important information. They can specify an object’s orientation and the relationship between its parts, both of which are crucial for judging orientation and for recognition [1, 2] (Boutsen, 2001; Sekuler, 1996). Axes for two-dimensional shapes can be classified into several basic types. An elongation axis is the single longest line that can fit through an object, sometimes also referred to as a principle axis. A symmetry axis specifies the line about which the whole or part of an object is reflected. A medial axis or shape skeleton results from a grassfire transformation and is formed from the boundaries where “fire” travelling inward from outer contours meets [3] (Blum, 1973). It is possible for one or more of these axes to coincide. For instance, the medial can also be the symmetry axis.

Recent research shows that shape axes have a neural basis and are thus functionally significant. Hung (2012) used In an adaptive shape sampling technique neurons in the inferotemporal cortex (area IT) of the macaque monkey wer found to code for both medial axis and surface characteristics [4]. Neurons in this part of the mammalian brain can thus represent both internal object structure and surface features, both of which are necessary for object perception. Lescroart and Biederman (2013) In another study line drawings of novel geometrical shapes were shown to human observers [5]. Using an fMRI-based voxel categorization method they found evidence of medial axis structure representation in area V3 of the visual system. They propose that medial axes are used to specify the relative position of an object’s parts.

Other psychophysical evidence points to the importance of axes in shape perception. Kovacs and Julesz (1994) Researchers measured contrast sensitivity for a target in different areas enclosed by a border [6]. Local sensitivity was enhanced within a boundary even when the distance between these areas and the border was large. In other work using a reverse mapping procedure they found that maxima in contrast sensitivity maps for bounded shapes fell along the medial axis [7] (Kovacs, Feher, & Julesz, 1998). These researchers propose that the visual system extracts shape “skeletons” as an intermediate-level representation of an object. This representation forms a structurally simple shape description that can be used to specify object location. The skeleton could then be operated upon by higher-level perceptual operations, for example to group parts or to match the object against a stored memory representation.

Firestone and Scholl (2014) Investigators in one study presented shapes to participants on a touch-sensitive tablet and asked them to tap the shape anywhere they wanted [8]. The tap locations across subjects formed the shape’s medial-axis skeleton. Predicted changes conforming to an axis interpretation held across different shape variations, when the shape borders were perturbed and under conditions of amodal completion. They argue that shape skeletons allow for shape constancy across alterations in orientation and perspective and note that this solution as been used in computer vision models to accomplish the same ends.

Despite the research cited above there has been very little work investigating the role that axes might play in perceived shape aesthetics. Friedenber (2012) The current author had observers rate right triangles at different axis lengths defined by the ratio of the two sides forming the right angle [9]. He found that an increase in this ratio produced a decrease in judgments of perceived beauty and concluded that compactness, not elongation was preferred. It was hypothesized that compact shapes are preferred because they are perceptually stable, i.e., less likely to bend or break. Right triangles however are typically seen in more compact forms and so familiarity may explain these results.

To determine if this is the case one would need to use random polygons that observers cannot interpret as meaningful. In the current study we utilize random polygons rather than familiar geometric shapes. The number of sides for a given condition is kept constant. In experiment 1 the shapes we use have eight sides. In experiment 2 they have more. But in each case the orientation of the shapes and the location of the vertices is determined randomly. By using a rectangular template of different lengths we can vary how long these shapes are, effectively increasing their elongation.

In addition to compactness there is an alternate and competing prediction for the judged beauty of elongated shape. According to the hedonic processing fluency account, stimuli that are more easily processed are judged as more pleasing and beautiful [10] (Winkielman et. al., 2003). Fluency here refers to the relative speed or ease of mental operations, either at the perceptual or cognitive level. Fluency can occur through repeated exposure to a stimulus, what is deemed the mere exposure effect [11] (Bornstein, 1989). Since the visual system is receptive to processing shape axes and has experience in doing so (most shapes both natural and manmade have an axis), it follows that shapes with an axis might be judged more beautiful than those without. It is unknown, however if there is a limit to this. Preference may peak for shapes at some maximum length and then drop off after this.

If compactness determines perceived beauty then we would expect to replicate the results obtained with right triangles and find a decrease in preference as shapes get longer. If processing fluency determines liking then we may see the opposite effect in which shapes that are longer would be more preferred. Should this be the case, the ease with which the axis is identified should be positively correlated with beauty judgments. The axes on more elongated shapes should be more easily identified and therefore more beautiful. However there may be a limit to this effect, with beauty ratings dropping at extremely high or low axis lengths.

2. Experiment 1

2.1. Method

2.1.1. Participants

Sixteen undergraduates participated to fulfill a course requirement. There were ten females and six males in the group. All vision was normal or corrected to normal. Average age was approximately 20 years.

2.1.2. Stimuli

To eliminate familiarity we used random polygons at different axis lengths. We employed five axis lengths running from 70-190 mm in 30 mm increments (70 mm, 100 mm, 130 mm, 160 mm, 190 mm). These values were chosen because they spanned a wide range of values, the smallest being easily visible, the longest being about as long as could be presented on the computer monitor without getting close to the edge of the screen. Viewing distance was on average about 45 cm. At this distance the lengths in increasing order corresponded to the following visual angles: 76°, 96°, 111°, 121° and 129°.

The polygons were created using rectangular templates whose longer sides equaled the five lengths. Locations parallel to the elongation axis were randomly determined for each longer side. Next, locations perpendicular to the axis at these sites were randomly determined. These positions were the x and y coordinate points for the polygon’s longer sides. Following this end points were determined by randomly selecting sites along the smaller sides of the rectangular template. In a final step, each of these points was connected with a straight line to form the polygon’s vertices.

The smaller side of each rectangle was determined to be 70 mm so that the smallest rectangle formed a square. This was the limiting case in which there was no elongation. This smaller side of the rectangle was also made just long enough so that an internal space could be inserted preventing the two longer sides from touching. The minor axis of the polygons will at its shortest be equal to 10 mm, the width of the internal space, or at its longest be equal to 70 mm, the length of the smaller side of the rectangular template.

Figure 1 shows an example of how a polygon for the 100 mm case was constructed. To generate the upper contour three points between 1 and 100 were randomly determined. Next, three values between 1 and 30 were randomly determined. These constituted the paired x and y values for the upper vertices. This same procedure was then applied to the bottom contour with a different set of random values. The left and right endpoints in this example were each chosen as independent random values between 1 and 70. All of the resulting points were then connected with straight lines to form the vertices and the rectangular frame was deleted. The original shapes were horizontal when constructed. Orientations were then randomized during presentation.

Figure 1. The dimensions of the rectangular template used to construct a 100 mm elongated polygon

The construction produced octagons. We decided to use eight-sided figures because they are a relatively simple shape and because they allow comparison with other work with octagons [12] (Friedenberg & Bertamini, 2015). All of the polygons were centered in the middle of the computer screen. They appeared as black lines against a white background. Figure 2 shows examples for each of the five length conditions.

Figure 2. Examples of polygons from each of the elongation conditions in experiment 1

2.1.3. Procedure

Ten polygons were generated for each of the five axis conditions yielding a total of 50 unique patterns. These constituted a single block of trials. There were four blocks in the experiment for a total of 200 trials. Presentation order of patterns within each block was randomized. Each polygon was presented at a different random orientation on every trial and appeared on the screen for as long as a participant needed to respond.

A 1-7 rating scale was used with a 1 labeled as “Very Ugly” and a 7 labeled as “Very Beautiful”. Participants were instructed that they could use any number in between these two extremes to indicate their response, including a 4 that corresponded to neutral. They were additionally told that there was no right or wrong answer and to judge the beauty of the patterns in their own way. Reaction times were also recorded. Following the experiment participants completed a debrief form and answered several questions regarding themselves and the stimuli.

2.1.4. Results and Discussion

The data were first screened for outliers. Any trials more than three standard deviations away from the mean, those longer than 16 seconds, were discarded. These constituted .007 percent of the data. A simple regression was next performed with axis length as the predictor and the ratings response as the dependent variable. The regression was run on the data averaged across subjects. Each data point in the scatterplot thus corresponds to this average. There was a significant fit F(4, 78) = 6.45, p < .01 with an R-square value of .076 and a Pearson correlation coefficient of .28. The slope for this analysis was .27 with an intercept value of 2.78. Effect size based on Cohen’s method was .082. Beauty ratings increased linearly with an increase in axis length as shown in Figure 3. The means and standard deviations for beauty ratings for axis length are provided in Table 1.

Figure 3. The scatterplot and fitted regression line for mean beauty ratings and axis length in experiment 1

Table 1. Means and Standard Deviations for Beauty Ratings by Axis Length in Experiment 1

It is surprising that the results we obtain here are the opposite of those for elongated right triangles where there was a preference for less elongated shapes [9] (Friedenberg 2012). The explanation might be due to familiarity. Right triangles are more often presented as compact versions, with elongation not usually exceeding an aspect ratio of 3:1. When familiarity is removed as we did in this study, preference changes and longer shapes are preferred. It is unclear whether compactness is still useful as an explanatory construct. It may useful as a predictor for other types of shapes taking familiarity into account.

3. Experiment 2

Previous research shows that observers prefer random octagons with greater perimeter lengths and with a greater number of concavities, suggesting beauty judgments may be driven by object complexity [12] (Friedenberg & Bertamini, 2015). The results of experiment 1 also suggest a complexity preference because elongated polygons on average have longer contours and hence may be considered more complex. In the next experiment we create elongated polygons with an increased number of sides. If viewers prefer complex shapes then ratings will be higher for those with more sides.

3.1. Method

3.1.1. Participants

Twenty-three undergraduate college students participated to satisfy a course requirement. There were four males and 19 females in total. Vision was normal or corrected to normal. Average age of the participants was 19.6 years.

3.1.2. Stimuli

Method of construction was the same as in the previous experiment. There were three side number conditions corresponding to polygons with 16, 24 and 32 sides respectively. Shapes with 16 sides had 16 randomly generated locations parallel to the axis, those with 24 sides had 24 locations, and so on. We connected the endpoints on the short sides of each rectangle to close the figure. The dimensions of the template were the same as before. Figure 4 shows examples for each condition.

Figure 4. Examples of polygons from each of the elongation and number of side conditions in experiment 2

3.1.3. Procedure

Five examples were created for each of the five axis length conditions and for each of the three side number conditions yielding a total of 75 unique patterns. These constituted a single block of trials. There were four blocks in the experiment with a total of 300 trials. Presentation order of polygons within each block was randomized. On each trial the shapes were presented at a different randomly determined orientation. As before, the polygons were centered in the middle of the computer screen. They appeared as black lines against a white background. The template and construction lines were removed prior to the experiment. Viewing distance was about 45 cm.

Each pattern appeared on the screen for as long as a participant needed to respond. The 1-7 rating scale from experiment 1 was used again. Participants were not given any special instructions regarding responding and were told that they should judge the aesthetics of the forms in their own way. Response times were gathered. At the end of the experiment participants completed a debrief form containing a series of questions about themselves and the stimuli.

3.1.4. Results and Discussion

We considered responses more than three standard deviations from the mean (those that took longer than 17 seconds) to be outliers and removed them from any subsequent analysis. The amount of removed data was .006 percent of the total. We performed a linear models regression for the response data with axis length cast as a continuous variable, number of sides as a categorical variable and their interaction as the three factors. The regression analysis was run on the data averaged across subjects. There was a significant main effect for axis length, F(4, 88) = 38.36, p < .01 and number of sides, F(2, 66) = 81.98, p < .01. Their interaction did not attain significance. Tables 2 and 3 show the means and standard deviations for these effects, Table 4 shows the intercepts and slopes. As in the previous experiment ratings increased linearly with an increase in axis length, replicating this finding. The main effects means for number of sides showed that beauty judgments were highest for the 16-sided shapes, second highest for 24-sided shapes and lowest for 32-sided shapes. Figure 5 shows the axis length scatterplot with the corresponding regression lines.

Table 2. Means and Standard Deviations for Beauty Ratings by Axis Length in Experiment 2

Table 3. Means and Standard Deviations for Beauty Ratings by Number of Sides in Experiment 2

Table 4. Intercept and Slope Values for Beauty Ratings in the Axis Length Regression for Number of Sides in Experiment 2

Figure 5. Scatterplot and fit regression lines for mean beauty ratings split by number of sides in experiment 2

We replicated our previous finding that observers prefer more elongated polygons. However, the result for number of sides runs contrary to a complexity prediction. Shapes with more sides were considered less beautiful. The results thus show there is a limit to preference for contour complexity. The Friedenberg and Bertamini (2015), Our previous work showed a complexity preference used eight-sided figures that were radial and lacked any main axis, these polygons being much simpler [12]. In the current experiment we used up to four times this many sides with a salient main axis.

It is possible participants prefer contours of a moderate level of complexity and that the stimuli in this experiment exceeded that amount. There is a well-known preference across many types of pattern for moderate complexity with an inverted U-shaped preference function [13] (Nadal, 2007). It is interesting to note that in our data there was no interaction between axis length and number of sides. The slopes for these functions were quite similar. This suggest that the effect of sides is additive to elongation, the differences between them being characterized by intercepts, not slopes.

4. General Discussion

We tested aesthetic preference for elongated shape and in two experiments found observers judge longer shapes to be more beautiful. In experiment 1 this was found using octagons and in experiment 2 it was replicated using shapes with 16-, 24- and 32-sides. To provide the reader with a more concrete sense of what shapes were preferred we show the most beautiful and least beautiful polygons for each experiment in Figure 6. For experiment 2 these are shown by number of sides.

Figure 6. Examples of the least and most beautiful shapes in both experiments

Most shapes that we encounter have an axis. The visual system seems to be set up to process axes because they are useful in recognizing objects and in guiding action toward them [14] (Kimia, 2003). Shapes with a prominent axis appear to automatically activate neural receptors in area IT in monkey and area V3 in human cortex [4, 5] (Hung, 2012; Lescroart & Biederman, 2013). This automatic activation may then produce an associated hedonic response as explained by processing fluency theory [10] (Winkielman et. al., 2003). In this view shapes with longer axes would produce even more neural activation and so elevate aesthetic judgment further.

Processing fluency may also account for our second major finding that shapes with more sides were less preferred. A shape with more sides contains more contour and is therefore more complex. More complex shapes may take longer to process, reducing processing fluency and the corresponding hedonic response. The results of the second experiment might therefore demonstrate a sort of “streamlining” effect where longer objects are preferred but only if they are relatively simple, without too many external details. It is speculative at this point, but shapes with increased aerodynamic or hydrodynamic properties may be judged beautiful because of their perceived dynamism or ability to move through a surrounding medium. However, it is possible to construct more streamlined shapes with a greater number of sides, depending upon how they are angled, so the current data do not address this conjecture.

We do not claim that processing fluency is the single or only theoretical account of our data. It is merely plausible. In this study we seek to demonstrate the effect, not provide a definitive explanation. If axis-based and aesthetic brain areas were concurrently activated in response to presentation of our stimuli, it would provide additional support for this theoretical account. Future work could explore whether this axis effect extends to familiar objects and whether it correlates with other perceptual processes. Lawson (2004) for example, found that extending the axis of familiar objects aided orientation-based judgments but not object recognition.

A complexity interpretation is supported by the experiment 2 ratings data where we see an overall drop in perceived beauty with an increase in number of sides. Participants preferred 16-sided polygons to 24-sided ones and 24-sided polygons to 32-sided ones. More complex shapes could be more aesthetically ambiguous and require greater visual exploration prior to passing judgment.

Friedenberg and Bertamini (2015), Our former work showed a complexity preference for octagonal shapes with increased contour [12]. The results of experiment 1 can be explained in the same way, with increased liking for more elongated shapes that contain more contour. However, there seems to be a limit to contour complexity because in experiment 2 the pattern reverses, with participants preferring fewer sided shapes containing less overall contour. The shift occurs for our polygons somewhere between 8 sides and 16 sides. So like the proverbial tale of Goldilocks, we appear to like contours that are complex, but not too complex.

5. Conclusions

An elongated axis is a prominent property of most objects. Recent evidence suggests that the extraction of an axis may be one of the steps necessary to form a structural description of a shape so that it can be recognized. However, no studies to date have investigated the aesthetic properties of elongation. If fast automatic processes like axis extraction are judged aesthetic as might be predicted according to processing fluency theory, then objects with prominent axes ought to be considered more beautiful.

This hypothesis was tested in two experiments. In experiment 1, polygons with random contours at five increasing elongations were presented to undergraduates. Longer shapes were consistently judged as more beautiful, with a strong linear effect of elongation length, thus supporting the processing fluency account. In experiment 2 these lengths were used again but with a varying number of sides. Shapes with longer axes have greater contour so if increased boundary contour is responsible for the effect then increasing the number of sides ought to enhance beauty judgments.

Surprisingly the results showed the opposite of this. Shapes with a greater number of sides were judged less rather than more beautiful. It is not clear why we obtained this effect. Complexity may play a role. Shapes with a large number of sides may be considered too complex. Shapes with more sides have more boundary contour but the orientation of the lines may detract from the overall orientation of the shape reducing how streamlined it appears, which perhaps may also detract from its appearance. This idea couldbe tested in future work.