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ORIGINAL RESEARCH ARTICLE

published: 13 February 2014

doi: 10.3389/fnhum.2014.00068

The experience of mathematical beauty and its neural

correlates

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,

USA

Reviewed by:

Miriam Rosenberg-Lee, Stanford

University, USA

Marie Arsalidou, The Hospital for

Sick Children, Canada

*Correspondence:

Semir Zeki, Wellcome Department

of Neurobiology, University College

London, Gower Street, London,

WC1E 6BT, UK

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

sources.

Keywords: mathematics, neuroesthetics, fMRI, beauty, mofc

INTRODUCTION

“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

“understanding.”

MATERIALS AND METHODS

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.

EXPERIMENTAL PROCEDURE

To allow a direct comparison between this study and previous

ones in which we explored brain activity that correlates with

Frontiers in Human Neuroscience www.frontiersin.org February 2014 | Volume 8 | Article 68

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HUMAN NEUROSCIENCE

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:

BehavioralData.xls.

STIMULI

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.

SCANNING

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

−2

;the

width of equations varied from 4

◦

to 24

◦

visual angle and the

height varied between 1

◦

and 5

◦

ANALYSIS

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

unc.

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

FWE

< 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).

RESULTS

BEHAVIORAL DATA

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

iπ

= 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 Ramanujan’s inﬁnite s eries for 1/π,

√

9801

∞



k =0

(

)

(

1103 + 26390k

)

(

)

396

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—

Equations2,5,and54).Formulaecommonlyratedasneutral

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 Riemann’s functional equation, the smallest

number expressible as the sum of two cubes in two different

ways,andanexampleofanexactsequencewheretheimage

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

understanding).

An excel ﬁle containing raw behavioral data is provided as Data

Sheet 3: B ehavioralData.xlsx., which gives the following eight

tables:

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 (Pearson’s 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 (Pearson’s 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.

(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)

experiences

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?,”9gaveanunqualiﬁed“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, Newton’s 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

beautiful.

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