### TL;DR This paper explores the fascinating connection betwee...
> ***"The beauty of mathematical formulations lies in abstracting, ...
You can find the Data Sheet with all of the 60 mathematical equatio...
$$1+e^{i\pi}=0$$ Euler's Identity is a special case of Euler's F...
The results show a strong positive correlation between the level of...
Expert mathematicians had an emotional response described as an "ae...
Art and mathematics are sometimes perceived as polar opposites. The...
> ***"Perhaps one of the most awkward, and at the same time challen...
published: 13 February 2014
doi: 10.3389/fnhum.2014.00068
The experience of mathematical beauty and its neural
Semir Zeki
, John Paul Romaya
, Dionigi M. T. Benincasa
and Michael F. Atiyah
Wellcome Laboratory of Neurobiology, University College London, London, UK
Department of Physics, Imperial College London, London, UK
School of Mathematics, University of Edinburgh, Edinburgh, UK
Edited by:
Josef Parvizi, Stanford University,
Reviewed by:
Miriam Rosenberg-Lee, Stanford
University, USA
Marie Arsalidou, The Hospital for
Sick Children, Canada
Semir Zeki, Wellcome Department
of Neurobiology, University College
London, Gower Street, London,
e-mail: s.zeki@ucl.ac.uk
Many have written of the experience of mathematical beauty as being comparable to that
derived from the greatest art. This makes it interesting to learn whether the experience
of beauty derived from such a highly intellectual and abstract source as mathematics
correlates with activity in the same part of the emotional brain as that derived from more
sensory, perceptually based, sources. To determine this, we used functional magnetic
resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians
when they viewed mathematical formulae which they had individually rated as beautiful,
indifferent or ugly. Results showed that the experience of mathematical beauty correlates
parametrically with activity in the same part of the emotional brain, namely field A1 of
the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other
Keywords: mathematics, neuroesthetics, fMRI, beauty, mofc
“Mathematics, rightly viewed, possesses not only truth, but
supreme beauty”
Bertrand Russell, Mysticism and Logic (1919).
The beauty of mathematical formulations lies in abstracting, in
simple equations, truths that have universal validity. Many—
among them the mathematicians Bertrand Russell (1919) and
Hermann Weyl (Dyson, 1956; Atiyah, 2002), the physicist Paul
Dirac (1939) and the art critic Clive Bell (1914)—have written of
the importance of beauty in mathematical formulations and have
compared the experience of mathematical beauty to that derived
from the greatest art (Atiyah, 1973). Their descriptions suggest
that the experience of mathematical beauty has much in common
with that derived from other sources, even though mathematical
beauty has a much deeper intellectual source than visual or musi-
cal beauty, which are more “sensible and perceptually based. Past
brain imaging studies exploring the neurobiology of beauty have
shown that the experience of visual (Kawabata and Zeki, 2004),
musical (Blood et al., 1999; Ishizu and Zeki, 2011), and moral
(Tsukiura and Cabeza, 2011) beauty all correlate with activity in a
specific part of the emotional brain, field A1 of the medial orbito-
frontal cortex, which probably includes segments of Brodmann
Areas (BA) 10, 12 and 32 (see Ishizu and Zeki, 2011 for a review).
Our hypothesis in this study was that the experience of beauty
derived from so abstract an intellectual source as mathematics will
correlate with activity in the same part of the emotional brain as
that of beauty derived from other sources.
Plato (1929) thought that “nothing without understanding
would ever be more beauteous than with understanding, mak-
ing mathematical beauty, for him, the highest form of beauty.
The premium thus placed on the faculty of understanding when
experiencing beauty creates both a problem and an opportunity
for studying the neurobiology of beauty. Unlike our previous
studies of the neurobiology of musical or visual beauty, in which
participating subjects were neither experts nor trained in these
domains, in the present study we had, of necessity, to recruit
subjects with a fairly advanced knowledge of mathematics and
a comprehension of the formulae that they viewed and rated.
It is relatively easy to separ ate out the faculty of understanding
from the experience of beauty in mathematics, but much more
difficult to do so for the experience of visual or musical beauty;
hence a study of the neurobiology of mathematical beauty carried
with it the promise of addressing a broader issue with implica-
tions for future studies of the neurobiology of beauty, namely
the extent to which the experience of beauty is bound to that of
Sixteen mathematicians (3 females, age range = 22–32 years, 1
left-handed) at postgraduate or postdoctoral level, all recruited
from colleges in London, took part in the study. All gave
written informed consent and the study was approved by the
Ethics Committee of University College London. All had nor-
mal or corrected to normal vision. One subject was eliminated
from the study after it transpired that he suffered from atten-
tion deficit hyperactivity disorder and had been on medication,
although his exclusion did not affect the overall results. We
also recruited 12 non-mathematicians who completed the ques-
tionnaires described below but were not scanned, for reasons
explained below.
To allow a direct comparison between this study and previous
ones in which we explored brain activity that correlates with
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Zeki et al. Neural correlates of mathematical beauty
the experience of visual and musical beaut y (Kawabata and Zeki,
2004; Ishizu and Zeki, 2011), we used similar experimental pro-
cedures to these previous studies. About 2–3 weeks before the
scanning experiment, each subject was given 60 mathematical
formulae (Data Sheet 1: EquationsForm.pdf) to study at leisure
and rate on a scale of 5 (ugly) to +5 (beautiful) according
to how beautiful they experienced them to be. Two weeks later,
they participated in a brain scanning experiment, using func-
tional magnetic resonance imaging (fMRI), during which they
were asked to re-rate the same equations while viewing them
in a Siemens scanner, on an abridged scale of ugly—neutral—
beautiful. The pre-scan ratings were used to balance the sequence
of stimuli for each subject to achieve an even distribution of pre-
ferred and non-preferred equations throughout the experiment.
A few days after scanning, each subject received a questionnaire
(Data Sheet 2: UnderstandingForm.pdf) asking them to (a) report
their level of understanding of each equation on a numerical
scale, from 0 (no understanding) to 3 (profound understanding)
and (b) to report their subjective feelings (including emotional
reaction) when viewing the equations. The data from these ques-
tionnaires (pre-scan beauty ratings, scan-time beauty ratings,
and post-scan understanding ratings) is given in Data Sheet 3:
Stimuli consisting of equations were generated using Cogent 2000
(http://www.vislab.ucl.ac.uk/Cogent2000) and displayed by an
Epson EH-TW5910 LCD projector at a resolution of 1600 × 1200
with a refresh rate of 60 Hz. The display was back-projected onto
a translucent screen (290 × 180 mm, 27.2
× 18.1
visual angle),
which was viewed by subjects using an angled mirror.
Subjects viewed the formulae during four functional scanning
sessions, with breaks between sessions which gave them an oppor-
tunity to take a rest if required and us to correct any anoma-
lies, for example to correct rare omissions in rating a stimulus.
Scans were acquired using a 3-T Siemens Magnetom Trio MRI
scanner fitted with a 32-channel head volume coil (Siemens,
Erlangen, Germany). A B0 fieldmap was acquired using a double-
echo FLASH (GRE) sequence (duration 2
). An echo-planar
imaging (EPI) sequence was applied for functional scans, measur-
ing BOLD (Blood Oxygen Level Dependent) signals (echo time
TE = 30 ms, TR = 68 ms, volume time = 3.264 s). Each brain
image was acquired in an ascending sequence comprising 48
axial slices, each 2 mm thick, with an interstitial gap of 1 mm
and a voxel resolution of 3 × 3 × 3 mm, covering nearly the
whole brain. After functional scanning had been completed a
T1 MDEFT (modified driven equilibrium fourier transform)
anatomical scan was a cquired in the saggital plane to obtain
high resolution structural images (176 slices per volume, constant
isotropic resolution 1 mm, TE = 2.48 ms, TR = 7.92 ms).
Fifteen equations were displayed during each session
(Figure 1A), so that each of the 60 equations appeared once over
the four sessions. Each session started with a blank gray screen
for 19.5 s, followed by 15 trials, each of 16 s, interspersed with
four blanks, each of 16–17 s, to acquire baseline signal. A plain
gray blank screen was used, of an equivalent overall brightness
to the equation screens. Pre-scan beauty ratings were used to
divide the 60 equations into three groups; 20 “low” rated, 20
“medium rated, and 20 high rated equations. The sequence
of equations viewed by each subject in the scanner was then
organized so that 5 low-, 5 medium- and 5 highly-rated equations
appeared in each session and the pseudo-randomized sequence
was organized so that a low-rated equation was never followed by
another low-rated equation and the same held for medium and
highly rated equations. The session ended with a blank screen of
duration 30 s.
Each trial (Figure 1B) began w ith an equation which was dis-
played for 16 s followed by a blank lasting 1 s. The response screen
appeared for 3 s, during which the subject selected interactively
a scan-time beauty rating (Beautiful, Neutral, or Ugly) for each
equation by pressing keypad buttons. A second blank lasting 1–2 s
ended each trial. Equations were all drawn in the same sized font
in white (CIE 1931 XYZ: 755, 761, 637) and the same gr ay back-
ground was used throughout (CIE 1931 XYZ: 236, 228, 200). The
overall screen brightness varied between 280 and 324 cd m
width of equations varied from 4
to 24
visual angle and the
height varied between 1
and 5
SPM8 (Statistical Parametric Mapping, Friston et al., 2006)was
used to analyze the results, as in our previous studies (Zeki and
Romaya, 2010; Ishizu and Zeki, 2011). At the single-subject level
the understanding rating (0–3) and the scan-time beauty rating
(coded as 1 for “Ugly, 0 for “Neutral, and 1 for “Beautiful”)
for each equation were included as first and second paramet-
ric modulators, respectively, of a boxcar function which modeled
the appearance of each equation [in fact, the beauty and under-
standing ratings correlated but imperfectly (see behavioral data
below)]. There were fewer “Ugly” rated equations than “Neutral”
or “Beautiful” (see behavioral data below). Indeed, in 2 of the
60 functional sessions there was no “Ugly” rating. This imbal-
ance does not bias the estimation of, or inference about, the
effects of beauty—it only reduces the efficiency with which the
effects can be estimated (Friston et al., 2000). Happily, this reduc-
tion was not severe, because we were able to identify significant
effects. As a result of SPM orthogonalization, the beauty rating
parametric modulator can only capture variance that cannot be
explained by the understanding rating, thus allowing us to dis-
tinguish activations that correlate with beauty alone. Contrast
images for each of the 15 subjects for the parametric beauty rat-
ing and for All equations vs. Baseline were taken to a 2nd-level
(random effects) analysis. We used a conjunction-null analysis
(Nichols et al., 2005), to determine whether there was an overlap,
within mOFC, in regions of parametric activation with beauty
and the general de-activation produced by viewing mathemati-
cal formulae. In order to examine the activity of Ugly, Neutral,
and Beautiful stimuli relative to baseline at locations identified in
the parametric beauty analysis we also carried out a categorical
analysis of the beauty rating alone, by coding contrasts for Ugly,
Neutral, and Beautiful stimuli vs. Baseline for each subject at the
first level and taking these to a 2nd level, random effects, analysis
as before.
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Zeki et al. Neural correlates of mathematical beauty
FIGURE 1 | Experimental paradigm and stimulus presentation. There
were four functional scanning sessions, with a break between sessions.
(A) 15 equations were displayed during each session, with each of the 60
equations appearing only once over the four sessions. The pre-scan beauty
ratings were used to sequence the equations for each subject so that 5
low-rated (L), 5 medium-rated (M), and 5 highly-rated equations (H)
appeared in each session and their sequence within a session organized
so that a low-rated equation was never followed by another low-rated
equation (similarly for medium and highly-rated equations). During each
session four blank periods, each varying in duration from 16 to 17 s, were
inserted to collect baseline signal. Each session commenced with a 19.5 s
blank period to allow T1 equilibration effects to subside and ended with a
30 s blank period, giving a tot al duration of about 7 min 20 s per functional
session. (B) Each equation trial began with an equation displayed for 16 s,
followed by a response period during which subjects indicated their
scan-time beauty rating by pressing one of three keypad buttons. This
started with a 1 s blank period (B1) followed by 3 s during which subjects
pressed keypad buttons and their selection (either “Ugly, “Neutral, or
“Beautiful”) was interactively displayed to them. The response was
followed by an inter-trial interval which was randomly varied between 1
and 2 s during which a blank screen appeared (B2). Blank periods displayed
a uniform mid-gray background.
In a similar way, we undertook another parametric analysis
with the scan-time beauty rating as the first parametric mod-
ulator and the understanding rating as the second one. This
time the understanding modulator can only capture variance that
cannot be explained by the beauty rating, thus allowing us to
distinguish activations that are accounted for by understanding
alone. Contrast images for the 15 subjects were taken to the sec-
ond level, as before. To supplement this we also undertook a
categorical analysis of the four understanding categories (0–3) in
order to obtain parameter estimates for the four understanding
categories at locations identified as significant in the parametric
understanding analysis (see Figure 5B).
A categorical analysis differs from the corresponding paramet-
ric analysis in two respects: first, the parametric analysis looks
for a relationship between BOLD signal and differences in the
rated quantity (beauty or understanding) on an individual ses-
sion basis, while a categorical analysis will average the BOLD
signal for each category of the rated quantity over all sessions
and subjects; second, when using two parametric modulators we
can isolate beauty effects from understanding and vice versa by
using orthogonalization, but this is not available in a categori-
cal analysis. When we use orthogonalization of two parametric
regressors to partition the variance in the BOLD signal into a
“beauty only” and an “understanding only” component, there
remains a common portion which cannot be directly attributed
to either component.
For the main contrasts of interest, parametric, and cate-
gorical beauty, we report activation at cluster-level significance
Clust -FWE
< 0.05) with familywise error correction over the
whole br ain volume as reported by SPM8 based on random field
theory (Friston et al., 1994). A statistical threshold of P
0.001 and an extent threshold of 10 voxels was used to define
clusters. There were also some activations which did not reach sig-
nificance (clearly noted) which we nevertheless report since they
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Zeki et al. Neural correlates of mathematical beauty
suggest areas which may be contributing to the main activation.
Whether they actually do will be left to future studies.
For other contrasts vs. baseline which are not of principal
interest to this study we report activations that survive a peak
voxel threshold of P
< 0.05, with familywise error correction
over the whole brain volume.
Co-ordinates in millimeters are given in Montreal
Neurological Institute (MNI) space (Evans et al., 1993).
Beauty ratings
The formula most consistently rated as beautiful (average rat-
ing of 0.8667), both before and during the scans, was Leonhard
Euler’s identity
1 + e
= 0
which links 5 fundamental mathematical constants with three
basic arithmetic operations, each occurring once; the one most
consistently rated as ugly (average rating of 0.7333) was
Srinivasa Ramanujans infinite s eries for 1/π,
k =0
1103 + 26390k
which expresses the reciprocal of π as an infinite sum.
Other highly rated equations included the Pythagorean iden-
tity, the identity between exponential and trigonometric func-
tions derivable from Euler’s formula for complex analysis, and the
Cauchy-Riemann equations (Data Sheet 1: EquationsForm.pdf—
included Euler’s formula for polyhedral triangulation, the Gauss
Bonnet theorem and a formulation of the Spectral theorem (Data
Sheet 1: EquationsForm.pdf—Equations 3, 4, and 52). Low rated
equations included Riemanns functional equation, the smallest
number expressible as the sum of two cubes in two different
of one morphism equals the kernel of the next (Data Sheet 1:
EquationsForm.pdf—Equations 15, 45, and 59).
Pre-, post-, and scan-time ratings
In pre-scan beauty ratings, each subject rated each of the 60 equa-
tions according to beauty on a scale of 5(Ugly)through0
(Neutral) to +5 (Beautiful) while during scan time ratings, sub-
jects rated each equation into the three categories of Ugly, Neutral
or Beautiful.
Post-scan understanding ratings. After scanning, subjects rated
each equation according to their comprehension of the equa-
tion, from 0 (no comprehension whatsoever) to 3 (Profound
An excel file containing raw behavioral data is provided as Data
Sheet 3: B ehavioralData.xlsx., which gives the following eight
Table 1: Pre-scan beauty ratings for each equation by subject
Table 2: Scan-time equation numbers by subject, session
and trial
Table 3: Scan-time beauty ratings for each equation by subject
Table 4: Scan-time beauty rating s by subject, session, and trial
Table 5: Scan-time beauty ratings by subject—Session and
experiment totals
Table 6: Post-scan understanding ratings by subject
Table 7: Post-scan understanding ratings by subject, session,
and trial
Table 8: Post-scan understanding ratings by subject—Session
and experiment totals.
The pre-scan beauty ratings were used to assemble the equa-
tions into three groups, one containing 20 low-rated, another
20 medium-rated, and a third 20 high-rated equations, indi-
vidually for each subject. These three allocations were used to
organize the sequence of equations viewed during each of the
four scanning sessions so that each session contained 5 low-
rated, 5 medium-rated, and 5 high-rated equations. Each subject
then re-rated the equations during the scan as Ugly, Neutral,
or Beautiful. In an ideal case, each subject would identify 5
Ugly, 5 Neutral, and 5 Beautiful equations in each session. In
fact, this did not happen. Figure 2A shows the frequency dis-
tribution of pre-scan beauty ratings for all 15 subjects; it is
positively skewed, indicating that more equations were rated as
beautiful than ugly. This is reflected in the frequency distr ibu-
tion of the scan-time beauty ratings (Figure 2B)which,again,
shows a bias for beautiful equations. Figure 2C shows the rela-
tionship between pre-scan and scan-time beauty ratings. There
was a highly significant positive correlation (Pearsons r = 0.612
for 898 values, p < 0.001) but there were departures; for exam-
ple, one equation received a pre-scan rating of 4butwas
classed as Beautiful at scan-time and three equations with a
pre-scan rating of 5 were subsequently classed as Ugly. These
infrequent departures are not of great concern providing there
was still a reasonable ratio of Ugly: Neutral: Beautiful scan-time
designations for each session, which was the case. Ideally this
ratio would always be 5:5:5 but, due to the predominance of
Beautiful over Ugly scan-time ratings, we twice recorded 0:7:8
and 1:5:9 for particular sessions (see Table 5 in Data Sheet 3:
BehavioralData.xlsx). Other sessions in general showed more
equable ratios and, even with an extreme ratio such as 0:7:8, a
relationship between Neutral and B eautiful equations could still
be established.
The frequency distribution of post-scan understanding ratings
is given in Figure 2D, which shows that more of the equations
were well understood, as would be expected from a group of
expert mathematicians. Figure 2E shows that there was a highly
significant positive correlation (Pearsons r = 0.413 for 898 val-
ues, p < 0.001) between understanding and scan-time beauty
ratings. In this case, departures from a fully correlated relation-
ship allow us to separate out effects of beauty from those of
understanding, so that, for example, i n the well understood cate-
gor y (3) the ratio of Ugly: Neutral: Beautiful is 31:97:205. In order
to analyze scanning data with regard to understanding ratings we
would ideally have equal ratios of the four understanding ratios
(0, 1, 2, and 3) in each scanning session. These ratios are recorded
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Zeki et al. Neural correlates of mathematical beauty
FIGURE 2 | Summary of Behavioral data. Behavioral data scores
summated over all 15 subjects. (A) Frequency distribution of pre-scan
beauty ratings. (B) Frequency distribution of scan-time beauty ratings.
(C) Pre-scan beauty ratings plotted against scan-time beauty ratings.
(D) Frequency distribution of post-scan understanding ratings.
(E) Post-scan understanding ratings plotted against scan-time beauty
ratings. Numbers in brackets give the count for each group. Area of each
circle is proportional to the count for that group.
in Table 8 in Data Sheet 3: BehavioralData.xlsx. We occasionally
find missing categories in some sessions (such as 0:2:4:9) but we
could still establish a relationship w hen one category is missing in
a particular session.
Post-scan questionnaires regarding subjective (emotional)
Mathematical subjects were as well given four questions to
answer, post-scan. One subject did not respond to this part of the
questionnaire, leaving us with 14 subjects. To the question: When
you consider a particularly beautiful equation, do you experience
an emotional response?,”9gaveanunqualied“Yes,”1reported
a “shiver of appreciation, 1 reported being “ a bit excited, 1
reported The same kind of response as when hearing a beauti-
ful piece of music, or seeing a particularly appealing painting, 1
reported that “the feeling is visceral” and 1 was “unsure To the
question: Do you derive pleasure, happiness or satisfact ion from
a beautiful equation?” 14 subjects answered affirmatively; all 14
also gave a positive response to the question: Is there any math-
ematical equation which, in the past, you have found particularly
beautiful and, if so, was it among the list of equations which we
gave you?” but some regretted that variations of the equations
were not on the list [e.g., the Einstein field equations, related to
equation 60 (contracted Bianchi identity), and Cauchy’s integral
formula for the special case where n = 1 (Equation 29)]; three
regretted that the following equations were not on the list: the
analytical solution of the Abel integral equation, Noether’s the-
orem, the Euler- Lagrange and the Liouville, Navier-Stokes and
Hamilton equations, Newtons Second Law (F = ma), and the
relativistic Dirac equation. Finally, variable answers were given to
the question:“Do you experience a heightened state of conscious-
ness when you contemplate a beautiful equation?” In summary,
our subjects had an emotional experience when viewing equa-
tions which they had rated as beautiful (the “aesthetic emotion”),
and which they also qualified as satisfying or pleasurable. They
also showed a very sophisticated knowledge of mathematics, by
specifying equations that they considered particularly beautiful
(which the y had known), as well as by the regret expressed at
not finding, in our list, equations that they consider especially
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