Seeing and Perceiving 23 (2010) 263–271 brill.nl/sp
Fechner’s Aesthetics Revisited ∗
Flip Phillips 1,∗∗, J. Farley Norman 2,∗∗ and Amanda M. Beers 2
1 Department of Psychology and Neuroscience Program, Skidmore College, 815 North Broadway,
Saratoga Springs, NY 12866, USA
2 Department of Psychology, Western Kentucky University, USA
Received 6 September 2009; accepted 5 May 2010
Abstract
Gustav Fechner is widely respected as a founding father of experimental psychology and psychophysics
but fewer know of his interests and work in empirical aesthetics. In the later 1800s, toward the end of his
career, Fechner performed experiments to empirically evaluate the beauty of rectangles, hypothesizing that
the preferred shape would closely match that of the so-called ‘golden rectangle’. His findings confirmed
his suspicions, but in the intervening decades there has been significant evidence pointing away from that
finding. Regardless of the results of this one study, Fechner ushered in the notion of using a metric to evaluate
beauty in a psychophysical way. In this paper, we recreate the experiment using more naturalistic stimuli.
We evaluate subjects’ preferences against models that use various types of object complexity as metrics. Our
findings that subjects prefer either very simple or very complex objects runs contrary to the hypothesized
results, but are systematic none the less. We conclude that there are likely to be useful measures of aesthetic
preference but they are likely to be complicated by the difficulty in defining some of their constituent parts.
© Koninklijke Brill NV, Leiden, 2010
Keywords
Fechner, history, aesthetics, beauty, shape, form
1. Introduction
Widely respected as a founding father of experimental psychology and psy-
chophysics, toward the end of his career, Fechner also made a rather significant
contribution to the field of empirical aesthetics. His curiosity with the oft-worshiped
golden section (Fechner, 1865, 1871, 1876) and the authenticity questions sur-
rounding the Holbein Madonna (Fechner, 1876) was the beginning of a significant
movement whose goal was to empirically investigate art and aesthetics. Of course,
structured inquiry into the questions of beauty and aesthetics can be found as far
* This article is part of the 2010 Fechner collection, guest-edited by J. Solomon.
** To whom correspondence should be addressed. E-mail: flip@skidmore.edu or Farley.Norman@wku.edu
© Koninklijke Brill NV, Leiden, 2010 DOI:10.1163/187847510X516412

264 F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271
back as the works of Plato and Aristotle. Fechner was aware of other efforts to de-
fine beauty and its elements, such as Hogarth’s ‘line of beauty’ (Hogarth, 1873),
as seen in Fig. 1 (Barasch, 2000). Nevertheless, Fechner’s was the first effort to
empirically and psychophysically parameterize the elements of beauty.
One of the ongoing fundamental problems in empirical aesthetics is the defi-
nition and discovery of appropriate metrics that can be used to evaluate artwork.
McWhinnie (1965, 1968, 1971) provides a somewhat older review of the topic and
Gibson (1975), Gibson and Pickford (1976a, b) provide an interesting discussion
of the pros and cons of such an endeavor. To make empirical aesthetics a truly psy-
chophysical enterprise there must be a meaningful way to compare the artwork with
subjects’ ratings, preferences, and evaluations.
The golden ratio (sometimes called the ‘golden section’ or the ‘divine propor-
tion’) is a well-known geometrically-oriented metric that, for thousands of years,
has been thought to possess special aesthetic properties. The ratio,
1 +
√
5
2
≈ 1.61803, (1)
typically denoted by the symbol ϕ, appears in artistic artifacts that date back as
early as 440 BC with the construction of the Parthenon (Hemenway, 2005) as well
as various locations in nature, such as nautilus shells and sunflower seeds (Ghyka,
1977).
In his rectangle study, Fechner found peak pleasingness with rectangles possess-
ing the proportions of the golden section. Alas, there has been a terribly mixed bag
Figure 1. Hogarth’s line of beauty, an ‘s’-shaped curve. Here we have scaled it to have the proportions
of the golden ratio.

F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271 265
of results when attempting to replicate these findings (see Höge, 1997, for an exten-
sive review). The most recent findings seem to suggest that Fechner’s results were
anomalous and/or due primarily to methodology (e.g., Green, 1995; Höge, 1997).
Of course, this has not put the question to rest (e.g., Dio et al., 2007).
Furthermore, we would contend that it is somewhat difficult to assign a precise
aesthetic valuation to a simple rectangle based solely on its proportions, and in iso-
lation from all other contextual information. It is rare to observe a debate over the
appeal of a simple four-sided shape, yet, as is well known by experimental psychol-
ogists, the nature of the demand characteristic is such that subjects will report some
preference in these experiments. Though not reported by Fechner, we suspect that,
even if the effect of the golden section is real, its effect size is somewhat limited by
other factors.
There are certainly other metrics that can be used as a hypothetical model for
aesthetic choice. One frequently posited metric is that of complexity (see Berlyne,
1968, 1971, 1974; Birkhoff, 1933; Boselie and Leeuwenberg, 1985; Leeuwenberg
and Helm, 1991, for some examples).
Berlyne suggested that the aesthetic value and pleasurableness of a stimulus
starts at a relatively indifferent level, then increases as a function of complexity
up to a certain level, then decreases and becomes more unpleasant as complex-
ity increases. This theory is clearly inspired by the ideas of Wundt (Wundt, 1874)
with respect to stimulus intensity rather than complexity (see Fig. 2). This rela-
tionship has gained further currency within the field of robotics, concerning the
desired realism of humanoid robots, known commonly as The Uncanny Valley
(Mori, 1970/2005; Pollick (in press); Tondu and Bardou, 2009).
The mathematician George Birkhoff extended the notion of simple complexity
to include another factor, order:
M =
O
C
. (2)
In this scenario, complexity, C is thought of as attentional ‘effort’ while order, O ,
takes on a role as the amount of ‘harmony’ or symmetry in the object, their ratio,
Figure 2. The Wundt curve, an inspiration to Berlyne and others with respect to complexity and
preference.

266 F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271
yielding the aesthetic measure, M . There is a certain appeal to this extension —
the idea that complexity and the potential for complexity are somehow in balance.
Still, as with complexity, ‘order’ has its own definitional difficulties.
What follows is a simple experiment in the spirit of the Fechner’s rectangle
study — using three-dimensional objects with greater visual and structural com-
plexity.
2. Experimental
2.1. Method
2.1.1. Stimuli
Fifty randomly-shaped, smoothly-curved virtual objects (5 sets of 10 objects) with
varying levels of complexity were created using procedures developed by Norman
et al. (1995). To produce each object, a sphere was sinusoidally modulated in depth
(i.e., along the z-axis). The object was then rotated randomly about each of the
three Cartesian coordinate axes and modulated in depth (sinusoidally) again. This
process was repeated a number of times to create the final object. We manipulated
the complexity of the objects by varying the number of iterations, which ranged
from 2 (producing the least complex object in each set, object 1) to 31 (producing
the most complex object in the set, object 10). The magnitude of the modulation in
depth was constant at each iteration for all objects. The surface of each object was
defined by the positions and orientations of 8192 triangular polygons (there were
4098 unique vertices). One set of 10 stimulus objects is shown in Fig. 3.
We used the variability in vertex distance (across all 4098 vertices) from the cen-
ter of the objects as a measure of complexity (see Fig. 2 of Norman and Raines,
2002). Consider a sphere, the simplest globally-convex object. For a sphere, there
is no (i.e., zero) variation in vertex distance from its center; the standard deviation
of vertex distances is zero. Next consider a potato: its shape is more ‘complex’ in
that there is more variability in the distances from its center to each surface location
(i.e., there is a larger standard deviation of surface distances). In comparison, the
shape of a bell-pepper is even more complex, in that there is even more variation in
the surface distances from its center. All of our objects have the same overall size
(the mean vertex distance was approximately 6.7 cm for all objects). However, the
complexity of our objects varied widely. For each of our five sets of ten objects, the
least complex object (object 1) possessed a standard deviation (of vertex distances)
of 0.2 cm, while the most complex object (object 10) possessed a standard devia-
tion of 1.1 cm. Intermediate surface complexities possessed intermediate standard
deviations.
The objects were rendered with texture and Lambertian shading. The texture
resembled red granite and the shading was produced using OpenGL’s standard re-
flectance model, with an ambient component of 0.3 and a diffuse component of
0.7; the specular component was set to zero, simulating matte surfaces. The ob-

F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271 267
Figure 3. An example stimulus set, ordered by complexity. This figure is published in colour online,
see http://www.brill.nl/sp
ject surfaces were illuminated by a simulated point light source at infinity that was
coincident with the observer’s line of sight.
The stimulus displays were rendered by a dual-processor Apple Power Macin-
tosh G4 computer using OpenGL. Four random spatial arrangements of the ten
objects for each set were created and printed, in color, on white paper. There were
thus a total of 20 stimulus sheets, each depicting a different spatial arrangement of
one of the five sets of ten objects.
2.1.2. Procedure
Each observer was shown one of the five sets of ten stimulus objects. One of the
four spatial arrangements of that object set was chosen at random. We used Fechn-
er’s method of choice (Fechner, 1876/1997). The observers were asked to indicate
which of the objects possessed the most attractive (i.e., most aesthetically pleasing)
3-D shape. The observers were given as much time as they needed to make their
judgment.
2.1.3. Subjects
Two-hundred observers (101 males and 99 females) participated in the experiment.
All had normal, or corrected-to-normal vision.

268 F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271
Figure 4. Each bar in the histogram indicates the number of times a given stimulus was selected as
the most attractive/aesthetically pleasing. Object complexity increases along the x-axis.
2.1.4. Results
The results are shown in Fig. 4. Each bar in the histogram indicates the number
of times a given stimulus was selected as the most attractive/aesthetically pleasing.
Object complexity increases along the abscissa. As can be easily seen, performance
was decidedly not random, χ2 = 134.7 (p < 0.0001), with the two most complex
objects garnering the greatest preference, followed by the simplest object. There
was no effect of object set or gender on preference (both p > 0.1).
This result runs counter to the simple complexity-based predictions and is, in
fact, the inverse of such a prediction. Factoring in an ‘ordering’ parameter that em-
phasized symmetry (as one possible factor) would only affect the results slightly.
The geometric symmetry decreases with object complexity with these object sets.
We constructed one such measure, based on a method proposed by Beran (1968)
for testing the uniformity of spherical distributions. The Gauss map (see Note 1)
of a symmetrical spherical object will be uniformly distributed. As the spherical
object becomes more asymmetric (especially in the way that our stimulus genera-
tion procedure distorts them) the Gauss map will exhibit clusters of normals and
regions relatively devoid of them. The calculation of this hypothetical M is shown
in Fig. 5 for our stimuli. Comparing this predicted result to the actual results shown
in Fig. 4 accounts for the preference of the most simple stimuli but not for the most
complicated.
One possible explanation for the discrepancy of our results with those predicted
by the Berlyne/Wundt hypothesis might be that our range of stimuli did not feature
objects of sufficient complexity to drive preference toward the middle. While this
is certainly possible, we would suggest that, since the entire range of stimuli were

F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271 269
Figure 5. Calculations of M ala Birkhoff, using an ordering calculation for O based on the distribution
of each stimulus’ surface normals relative to those of a sphere, and a complexity C based on the
average displacement of the surface from that of a sphere. Comparing this predicted result to the actual
results shown in Fig. 4 accounts for the preference of the most simple stimuli but not for the most
complicated.
presented simultaneously, the subjects were able to easily calibrate their judgement
against the full range of available complexities. Additionally, there are very few nat-
ural objects that possess a complexity greater than our maximally complex object.
3. General Discussion
We believe that much of the problem with applying a metric to beauty comes from
the simultaneous interplay of the denotative and, more significantly the connotative
properties of artwork. While the denotative properties are usually clear cut and eas-
ily discernible by the viewer, the connotation is a more private event that may be
vulnerable to very large individual differences.
For example, there are numerous examples in the art world of the notorious
‘blank painting’ but we contend the aesthetic appeal of such works lies more in
their connotative value — information that is not completely conveyed by the work
itself but is suggested in addition to its literal interpretation. For example, the shape
of the ‘C3’ series Corvette, manufactured from 1968 to 1982, is sometimes said to
be implicative of a reclining nude woman (see Note 2). Clearly the car was not a
literal expression of this shape, but rather, only suggestive of it. For this study we
restricted ourselves to the denotative or literal values of the artwork under exami-
nation, but the effect of connotation should not be underestimated. Indeed, several
subjects responded that the objects reminded them of other, real objects (hearts,

270 F. Phillips et al. / Seeing and Perceiving 23 (2010) 263–271
various parts of human anatomy, faces, rocks). Any emotional attachment to such
denotation may sway an observer toward or away from a particular stimulus.
In our study, the most complex objects were preferred, followed by the most
simple. This runs counter to several existing theories of aesthetic choice. Of course,
the notion of complexity is a difficult one. By some definitions, such as description
length, our objects possess the same or similar complexity, whereas by others, such
as the variability-based metric used, they fall into a naturally-appearing progression.
Adding other factors to normalize the complexity, such as order, further complicates
the matter. Our symmetry measure helped to explain the observers’ preference for
the simpler objects, but is totally contrary to our findings of preference for the most
complex objects.
Lastly, demand characteristics clearly come into play. It is interesting to note
that, across 200 subjects, not a single one refused to pick a preferred object, no
one complained that they couldn’t do the task. In fact, some subjects ruminated
significantly, trying to reduce their choice to a single object.
Along with others who seek empirical explanations for aesthetic phenomena, we
believe that the problem is tractable. Clearly, our results and the results of others
are not random — there is structure to many of the findings. We simply must carry
on Fechner’s quest to find the right metrics to explain it.
Notes
1. This is essentially, the mapping of an object’s surface normals onto a sphere.
2. It should be noted that the Corvette is manufactured in Bowling Green Ken-
tucky — home of Western Kentucky University and two of the authors.
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