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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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Zeki et al. Neural correlates of mathematical beauty

Non-mathematical subjects

We also tried to gauge the reaction of 12 non-mathematical

subjects to viewing the same equations. This was, generally,

an unsatisfactory exercise because many had had some, usu-

ally elementary, mathematical experience [up to GCSE (General

Certiﬁcate of Secondary Education) level, commonly taken at

ages 14–16]. Reﬂecting this, the majority indicated that they

had no understanding of what the equations signiﬁed, rating

them 0, although some gave positive beauty ratings to a minor-

ity of the equations. Overall, of the 720 equations distributed

over 12 non-mathematical subjects, 645 (89.6%) were given a 0

rating (no understanding), 49 (6.8%) were given a rating of 1

(vague understanding) and the remainder were rated as 2 (good

understanding) or 3 (profound comprehension). To the question

“When you consider a particularly beautiful equation, do you expe-

rience an emotional response?,” the majority (9 out of 12) gave a

negative response. Given this, we hypothesized that, when such

non-mathematical subjects gave a positive beauty rating to the

equations, they were doing so on a formal basis, that is to say

on how attractive the form of the equations was to them. This

hypothesis receives support from the contrast to reveal the para-

metric relationship between brain activity and understanding in

mathematicians (see Results).

BRAIN ACTIVATIONS

Parametrically related activity in mOFC with beauty ratings,

independently of understanding

Results should that activity that was parametrically related to the

declared intensity of the experience of mathematical beauty was

conﬁned to ﬁeld A1 of mOFC (Ishizu and Zeki, 2011), where

there was a signiﬁcant difference in the BOLD signal when view-

ing equations rated as beautiful on the one hand and as neutral

and ugly on the other (Figure 3). Even though our subjects were

experts, with an understanding of the truths that the equations

depict, we had nevertheless asked them to rate, post-scan, how

well they had understood the formulae, on a scale of 0 (no

comprehension) to 3 (profound understanding) (Data Sheet 2:

UnderstandingForm.pdf), as a way of dissociating understanding

from the experience of beauty. There was a good but imperfect

correlation between their understanding and the beauty ratings

given, in that some formulae that had been understood were

not rated as beautiful (Pearson’s correlation coefﬁcient ranging

from −0.0362 to 0.6826). As described above, separate paramet-

ric modulators were used for understanding and beauty ratings in

the SPM analysis. This allowed us to model both understanding

and beauty effects and examine responses to one that could not

be explained by the other, thus separating out the two faculties

in neural terms. Hence, cr ucially, the parametri cally related activ-

ity in mOFC (Figure 3) was speciﬁcally driven by beauty ratings,

after accounting for the effects of understanding.

The mOFC was the only brain region that showed a BOLD

signal that was parametrically related to beauty ratings, sig-

niﬁcant at cluster level (see Tabl e 1A and Figure 3). Previous

studies have nevertheless shown a number of areas that are active

when subjects undertake mathematical tasks (see Arsalidou and

Taylor, 2011 for a meta-analysis) and we observed some activ-

ity (which did not reach signiﬁcance) in three regions which

previous studies of mathematical cognition had reported to be

active (see Ta ble 1 B). One of these is located in the left angu-

largyrus,oneinthemiddletemporalgyrusandoneinthe

FIGURE 3 | Parametric “Activations” with Beauty. (A) Second level

parametric analysis derived from 15 subjects, to show parametric

modulation by scan-time beauty rating (after orthogonalization to

understanding rating). One-sample t-test (df = 14) thresholded at

unc

< 0.001 with an extent threshold of 10 voxels, revealed a cluster of

95 voxels in medial orbito-frontal cortex (mOFC) with hot-spots at (−6,

56, −2) and (0, 35, −14), signiﬁcant at cluster level, with familywise error

correction over the whole brain volume. The location and extent of the

cluster is indicated by sections along the three principal axes through the

two hot-spots, pinpointed with blue crosshairs and superimposed on an

anatomical image which was averaged over all 15 subjects. (Note:

“activation” in this case relates to a positive parametric relationship; i.e.,

an increase in activity with an increase in scan-time beauty rating. Overall,

as can be seen in (B), these locations were deactivated relative to

baseline, i.e., there was a relative activation in a deactivated region. (B) A

separate categorical analysis, based on scan-time beauty ratings alone,

was used to generate contrast estimates for the three categories Ugly,

Neutral, and Beautiful vs. Baseline at each of the locations in (A).

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Zeki et al. Neural correlates of mathematical beauty

Table 1 | Activation for the contrast parametric beauty.

Location xyzk

Clust-FEW

FWE

(A) ACTIVATIONS—PARAMETRIC BEAUTY —SIGNIFICANT AT

CLUSTER LEVEL

mOFC −656−2 95 0.047 5.14 0.619

mOFC 0 35 −14 4.90 0.733

(B) ACTIVATIONS—PARAMETRIC BEAUTY —NOT REACHING

CLUSTER LEVEL SIGNIFICANCE

R caudate 12 8 19 17 0.703 5.72 0.369

L angular gyrus −36 −55 22 28 0.489 4.96 0.705

L middle temporal

gyrus

−63 −4 −17 16 0.725 4.48 0.895

SPM 2nd level analysis (15 subjects). One sample t-test (df = 14). (A) Cluster

level signiﬁcant activation with familywise error correction over the whole brain

volume (P

Clust-FWE

< 0.05, background threshold P

unc.

< 0.001, extent thresh-

old 10 voxels). Co-ordinates of the two hotspots in the cluster are given in MNI

space. (B) In addition to the signiﬁcant cluster activation shown in Figure 3 and

tabulated above in (A) there were some clusters which failed to reach signiﬁ-

cance but which are listed here as possible contributors to the main cluster of

activation. For completeness we also give the peak signiﬁcance at each location,

FWE

, with familywise error correction over the whole brain volume. The peak

activation in each cluster is shown in bold with up to two other peaks in the

same cluster listed below.

caudate nucleus. Although they did not attain signiﬁcance, we

nevertheless document them here and leave it to future studies

to ascertain their possible role in the experience of mathematical

beauty.

Beauty—categorical analyses

A categorical analysis of Beauty ratings vs. Baseline is less sophis-

ticated than a parametric analysis in two respects, for reasons

given in the methods section. Nevertheless, we thought it useful

to employ such an analysis to examine the parameter estimates for

Ugly, Neutral and Beautiful vs. Baseline at the locations in mOFC

identiﬁed as signiﬁcant in the parametric study. Figure 3B shows

the parameter estimates for the three beauty categories vs. base-

line at the two locations in mOFC identiﬁed in the parametric

analysis: at (−6, 56, −2) and (0, 35, −14). It is evident that, at

both, overall activit y within this area of deactivation was greater

for Beautiful than for Neutral or Ugly stimuli. In neither case is a

linear relationship particularly evident, probably due to the infe-

rior sensitivity of the categorical analysis, for the reasons given

above.

Tab le 2 tabulates the categorical contrast Beauty > Neutral.

There are cluster-level signiﬁcant activations in mOFC, the left

angular gyrus and the left superior temporal sulcus. The relative

activation within mOFC occurs within a region of de-activation

relative to baseline (see section Cortical de-activations when

viewing mathematical formulae). As with the previous study

of Kawabata and Zeki (2004), parameter estimates show that

it is a change in relative activity within a de-activated mOFC

that correlates with the experience of mathematical beauty. Data

Sheet 4: BeauNeutUglyBase.pdf tabulates activations and deacti-

vations of the Beautiful, Neutral, and Ugly categories relative to

baseline.

Table 2 | Activations for the categorical contrast beautiful > neutral.

Location xyz k

Clust-FWE

FWE

ACTIVATIONS—BEAUTIFUL > NEUTRAL

L angular gyrus −48 −67 22 143 0.015 6.32 0.180

mOFC −12 53 4 285 <0.001 5.52 0.416

mOFC −950−5 5.16 0.576

mOFC −938−5 4.70 0.790

L superior temporal gyrus −12 29 61 122 0.026 4.98 0.633

L superior temporal gyrus −3 17 67 4.65 0.809

L superior temporal gyrus −18 38 46 4.52 0.858

SPM 2nd level analysis (15 subjects). One sample t-test (df = 14). Cluster level

signiﬁcant activations corrected for familywise error over the whole brain volume

Clust-FWE

< 0.05, background threshold P

unc.

< 0.001, extent threshold 10 vox-

els). Co-ordinates of hotspots within each cluster are given in MNI space. For

completeness we also give the peak signiﬁcance at each location, P

FWE

,with

familywise error correction over the whole brain volume. The peak activation in

each cluster is shown in bold with up to two other peaks in the same cluster

listed below.

Activity unrelated to beauty ratings

While the experience of mathematical beauty correlated para-

metrically with activity in mOFC, the contrast All equations >

Baseline showed that many sites were generally active when

subjects viewed the equations (Tabl e 3). These included sites

implicated in a variety of relatively simple arithmetic calcula-

tions and problem solving (Dehaene et al., 1999; Fias et al., 2003;

Anderson et al., 2011; Arsalidou and Taylor, 2011; Wintermute

et al., 2012 for a meta-analysis), and symbol processing (Price

and Ansar i, 2011) as well as activity in three sites in the cere-

bellum, generally ignored in past studies of the mathematical

brain: one of these cerebellar sites, located in Crus I, may be

involved in working memory (Stoodley, 2012), another one,

also located in Crus I, in grouping according to numbers (Zeki

and Stutters, 2013) and the processing of abstract information

(Balsters and Ramnani, 2008) while the third one, located in

the para-ﬂocculus, is involved in smooth pursuit eye move-

ments (Ilg and Their, 2008). That these areas should have been

active when subjects view more complex formulae suggests that

they are also recruited in tasks that go beyond relatively simple

arithmetic calculations and involve more complex mathematical

formulations.

Cortical de-activations when viewing mathematical formulae

As well, the contrast All equations < Baseline revealed widespread

cortical de-activations (Ta bl e 3 ), many in areas not speciﬁcally

related to mathematical tasks or aesthetic ratings; their distri-

bution corresponds closely to areas active in the resting state

and de-activated during complex cognitive tasks (Shulman et al.,

1997; Binder et al., 1999), including arithmetic ones (Feng et al.,

2007). The most interesting of these is in mOFC. A Conjunction-

Null analysis (Nichols et al., 2005) of the contrasts “Paramet ric

beauty rating” and “De-activations with Equations,” both thresh-

olded at P

unc

< 0.001, showed that this de-activation overlaps the

mOFC activation which correlates parametrically with the expe-

rience of mathematical beauty (Figure 4); unlike other areas of

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Zeki et al. Neural correlates of mathematical beauty

Table 3 | Activations and de-activations for the contrast all equations

vs. baseline.

Location xyzk

FWE

ACTIVATIONS—ALL EQUATIONS > BASELINE

L fusiform gyrus −33 −82 −8 188 15.20 <0.001

L fusiform gyrus −30 −91 −514.96 <0.001

R fusiform gyrus 42 −79 −11 377 14.98 <0.001

R cerebellum Crus I 48 −70 −29 13.02 <0.001

R fusiform gyrus 39 −58 −11 10.58 0.001

L inferior temporal gyrus −48 50 4 77 13.40 <0.001

L inferior temporal gyrus −48 −61 −11 79 11.80 <0.001

L intraparietal sulcus −30 −67 46 88 10.29 0.002

L lingual/fusiform gyrus −30 −49 49 9.53 0.005

R cerebellum lobule VIII/crus I 6 −79 −29 45 10.10 0.002

R cerebellum dorsal paraﬂocculus 30 −70 −50 13 10.04 0.003

L inferior frontal gyrus −48 14 25 24 9.35 0.006

R intraparietal sulcus 30 −58 43 19 8.80 0.013

DE-ACTIVATIONS—ALL EQUATIONS < BASELINE

L precuneus −12 −46 43 1188 18.22 <0.001

R precuneus 15 −40 43 16.63 <0.001

L precuneus −15 −40 52 14.73 <0.001

R fusifom gyrus 60 −52 16 179 15.39 <0.001

Fusiform gyrus 63 −34 16 10.87 <0.001

Fusiform gyrus 63 −22 16 8.25 0.027

R medial orbito-frontal cortex 332−5 459 14.91 <0.001

L medial orbito-frontal cortex −623−814.25 <0.001

R medial orbito-frontal cortex 12 53 −213.37 <0.0 01

R superior temporal gyrus 57 −16 −5 256 14.53 <0.001

R superior temporal gyrus 63 2 −23 14.39 <0.001

R superior temporal gyrus 45 −13 −811.05 <0.001

L lingual gyrus −12 −70 −22111.81<0.001

L middle temporal gyrus −60 −4 −17 44 11.75 <0.001

L superior medial gyrus

(anterior paracingulate)

0 53 22 33 10.76 0.001

L middle frontal gyrus −24 26 40 16 10.46 0.002

R superior frontal gyrus

(anterior paracingulate)

18 50 19 12 10.38 0.002

L lingual gyrus −24 −46 −2 11 9.41 0.006

R cuneus 12 −88 16 11 8.98 0.010

SPM 2nd level analysis (15 subjects). One sample t-test (df = 14). All loci,

in Montreal Neurological Institute (MNI) space, are signiﬁcant, thresholded at

FWE

< 0.05 with familywise error correction over the whole brain volume and

with an extent threshold of 10 voxels. The peak activation in each cluster is

shown in bold with the cluster size in voxels (k

) with up to two other peaks in

the same cluster listed below.

de-activation, the mOFC de-activation has been posited to be

related to task-unrelated conceptual processing (Shulman et al.,

1997). The overlap between the activation and de-activation sug-

gests that there may be separate compartments or sub-systems

within ﬁeld A1 of mOFC whose activities correlate with general

cognitive tasks on the one hand and the more speciﬁc experience

of beauty on the other.

Parametric “de-activations” with understanding ratings

independent of beauty

As described in the methods, we undertook a second parametric

analysis, with Beauty rating and Understanding rating as ﬁrst and

second parametric modulators, respectively, to isolate activations

due to understanding alone. The result, shown in Figure 5,is

that a large extent of the occipital visual cortex, comprising many

of its subdivisions, was less active for well-understood equations

(or, put another way, more active for less understood equations).

The signiﬁcance of this is discussed below (under Beauty and

Understanding).

DISCUSSION

Art and mathematics are, to most, at polar opposites; the former

has a more “sensible” source and is accessible to many while the

latter has a high cognitive, intellectual, source and is accessible to

few. Yet both can provoke the aesthetic emotion and arouse an

experience of beauty, although neither all great ar t nor all great

mathematical formulations do so. The experience of mathemati-

cal beauty, considered by Plato (1961a,b) to constitute the highest

form of beauty, since it is derived from the intellect alone and

is concerned with eternal and immutable truths, is also one of

the most abstract emotional experiences. In spite of its abstract

nature, there was, for Clive B ell (1914), a strong relation between

mathematical and artistic beauty because the mathematician feels

an emotion for his speculations which “springs... from the heart

of an abstract science. I wonder, sometimes, whether the appre-

ciators of art and of mathematical solutions are not even more

closely allied,” while for Bertrand Russell (1919) “The true spirit

of delight, the exaltation, the sense of being more than Man,

which is the touchstone of the highest excellence, is to be found

in mathematics as surely as poetry.” Given this, we hypothesized

that it would be likely that the experience of beauty derived from

mathematics would correlate with activity in the same par t of the

emotional brain as that derived from other, more sensible and

perceptually based, sources. Although we approached the exper-

iment with difﬁdence, given the profoundly different sources for

these different experiences, we were not surprised to ﬁnd, because

of similarities in the experience of beauty provoked by the differ-

ent sources alluded to above, that the experience of mathematical

beauty correlates with activity in the same brain area(s), princi-

pally ﬁeld A1 of mOFC, that are active during the experience of

visual, musical, and moral beauty. That the activity there is para-

metrically related to the declared intensity of the experience of

beauty, whatever its source, answers afﬁrmatively a critical ques-

tion in the philosophy of aesthetics, namely whether aesthetic

experiences can be quantiﬁed (Gordon, 2005).

Mathematical and artistic beauty have been written of in the

same breath by mathematicians and humanists alike, as arous-

ing the “aesthetic emotion.” This implies that there is a com-

mon and abstract nature to the experience of beauty derived

from very different sources. Viewed in that light, the activ-

ity in a common area of the emotional brain that correlates

with the experience of beauty derived from different sources

merely mirrors neurobiologically the s ame powerful and emo-

tional experience of beauty that mathematicians and artists alike

have spoken of.

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Zeki et al. Neural correlates of mathematical beauty

FIGURE 4 | Conjunction of activations with beauty rating and

de-activations with equations. Conjunction-Null of two contrasts,

“Parametric beauty rating” and “De-activations with Equations,” both

thresholded at P

unc

< 0.001. De-activations are shown in red, overlapping the

area revealed by the parametric rating, shown in yellow. Numerals refer to

MNI co-ordinates.

FIGURE 5 | Parametric “De-activations” with Understanding. (A) Second

level parametric analysis derived from 15 subjects, to show parametric

modulation by understanding rating (after orthogonalization to scan-time

beauty rating). One-sample t-test (df = 14) thresholded at P

unc

< 0.001 with

an extent threshold of 10 voxels, revealed two peak “de-activations”

signiﬁcant (P

FWE

< 0.05) with familywise error correction over the whole

brain volume, at (39, −52, −8) (T

= 9.01, P

FWE

= 0.010) and at (30, −91,

19) (T

= 8.05, P

FWE

= 0.032). A third peak at (−27, −85, 7) (T

= 7.65,

FWE

= 0.050) was just above threshold. The location of each peak is

indicated by sections along the three principal axes, pinpointed with blue

crosshairs and superimposed on an anatomical image which was averaged

over all 15 subjects. (Note: A “de-activation” in this case relates to a negative

parametric relationship; i.e., activity decreased as understanding rating

increased. Overall, as can be seen in (B), these locations were signiﬁcantly

active above baseline). (B) A separate categorical analysis based on

understanding ratings alone was used to generate contrast estimates for the

four understanding categories U0, U1, U2, and U3 vs. baseline at each of the

peak locations in (A).

mOFC AND THE EXPERIENCE OF BEAUTY, PREFERENCE, PLEASURE

AND REWARD

The mOFC is active in a variety of conditions, of which experi-

ences relating to pleasure, reward and hedonic states are the most

interesting in our context. The relationship of the experience of

beauty to that of pleasure and reward has been commonly dis-

cussed in the philosophy of aesthetics, without a clear conclusion

(Gordon, 2005). This is perhaps not surprising , because the three

merge into one another, without clear boundaries between them;

neurologically, activity in mOFC correlates with all three expe-

riences thus reﬂecting, perhaps, the difﬁculty of separating these

experiences subjectively. The imperfect distinction between the

three is also reﬂected in the positive post-scan answers given by

the mathematical subjects to the question whether they experi-

enced pleasure, satisfaction or happiness when viewing equations

that they had rated as beautiful. Whether one can ever experi-

ence beauty without at the same time experiencing a sense of

pleasure and/or reward is doubtful. The converse is not true, in

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Zeki et al. Neural correlates of mathematical beauty

that something can be experienced as being pleasurable, reward-

ing, or preferred without it being also experienced as beautiful

or arousing the “aesthetic emotion.” Neurobiologically, the issue

resolves itself around the question of whether a ctivity in the same

or different parts of the mOFC correlates with these different

experiences. The mOFC is a relatively large expanse of cortex

with several cytocrachitectonic subdivisions, including BA 10,

11, 12, and 32 (see Kringelbach, 2005 for a review) as well as

BA 24 (which may more properly be considered as part of the

rostral anterior cingulate cortex) and BA 25. Brodmann maps

are useful general anatomical guides but do not delimit func-

tional areas and major functional subidvisions can be found

within single cytoarchitectonic ﬁelds such as, for example, BA

18 (see Zeki, 1979). It is possible that, as with lateral orbito-

frontal cortex, future studies may subdivide mOFC into further

subdivisions based on resting-state connectivity or other criteria

(Kahnt et al., 2012). Hence any reference to Brodmann maps is no

more than a rough guide and must be tentative. Our delineation

(Ishizu and Zeki, 2011) of a ﬁeld within mOFC, ﬁeld A1, has the

virtue of delimiting a speciﬁc cortical zone with relatively precise

co-ordinates and dimensions, whose activity correlates with the

“aesthetic emotion” and to which other functional parcellations,

both past and future, can be referred with relative precision. Field

A1 of mOFC has MNI co-ordinates of (−3, 41, −8) and a diam-

eter of between 15 and 17 mm (see Ishizu and Zeki, 2011). This

falls in the middle of the active region of mOFC in the present

study, and hence must be located within what we deﬁned as ﬁeld

A1.Infact,ifoneweretotaketheaverageofthetwohotspots

in the mOFC cluster of the parametric beauty contrast of this

study (namely −3, 45, −8), it will be found to be located 4 mm

fromthecenteroftheﬁeldA1at(−3, 41, −8). A survey of the

published evidence shows that functional parcellations cannot be

restricted to single cytoarchitectonic subdivisions comprising the

medial orbital wall of the frontal cortex. Not all rewarding and

pleasurable experiences activate ﬁeld A1 of mOFC, which occu-

pies mainly BA 32 but probably extends to BA 12 inferiorly and

BA 25 anteriorly. While preference for drinks (McClure et al.,

2004), abstract and predictive rewards and kinetic patterns do

(O’Doherty et al., 2001; Gottfried et al., 2003; Zeki and Stutters,

2011), the hedonic experience of food appears to correlate with a

morelateralpartofmOFCthanﬁeldA1ofmOFC(Kringelbach

et al., 2003) while the experience of erotic pleasure appears to cor-

relate with a region dorsal to it (Sescousse et al., 2010) (see also

Berridge and Kringelbach, 2013) for a review. These are approxi-

mations that do not allow more deﬁnitive conclusions at present,

even with meta-analyses such as the ones provided by Peters and

Büchel (2010) or Kühn and Gallinat (2012). The latter help locate

the general brain regions active with speciﬁc experiences but are

not presently capable of pinpointing whether the identical regions

are active, which would require far more detailed conjunction

analyses in studies employing several different categories of hedo-

nic experience. This is even true of the experience of visual and

musical beauty. Though a conjunction analysis reveals that there

is a common part of A1 of mOFC that correlates with both expe-

riences (Ishizu and Zeki, 2011), the overlap is not perfect and it is

possible that different subdivisions, or perhaps different group-

ings of cells within the same general subdivision, are recruited

during different experiences (see Figure 1 of Ishizu and Zeki,

2011). The same is true of the present results in relation to pre-

vious activations of mOFC that correlate with the experience of

visual and musical beauty. Very close correspondence in the active

regions for all three experiences does not preclude that there may

yet be different distributions or subdivisions within ﬁeld A1 of

mOFC. It is also useful to emphasize here, as we have done in

the past, that the localization of activity in mOFC that correlates

with the experience of mathematical beauty does not imply that

this area alone is responsible for, or underlies, the experience.

As we have shown here and elsewhere, the viewing of mathe-

matical equations also results in activity in cortical areas that are

distinct from the areas engaged w hen viewing paintings which, in

turn, differ from those active when listening to musical excerpts.

ThecommonfactorisactivityinmOFCwhenbeautyinthese

domains is experienced, but mOFC cannot act alone; rather it is

active in concert with other areas which, we suppose, collectively

correlate with the experience of beauty derived from different

sources.

BEAUTY AND UNDERSTANDING

Perhaps one of the most awkward, and at the same time chal-

lenging, aspects of this work was trying to separate out beauty

and understanding. Because the correlation between the two,

though signiﬁcant, was also imperfect, we were able to do so for

mathematicians. T his of course leaves the question of whether

non-mathematicians, with no understanding whatsoever of the

equations, would also ﬁnd the equations beautiful. Ideally, one

would want to have subjects who are mathematically totally illiter-

ate, a search that proved difﬁcult. We relied, instead, on a different

approach. Most of our “non-mathematician” subjects had a very

imperfect understanding of the equations, even though they had

rated some of them as beautiful; we supposed that they did so on

the basis of the formal qualities of the equations—the forms dis-

played, their symmetrical distribution, etc. We surmised that we

could demonstrate this indirectly, by showing that less well under-

stood equations in our mathematical subjects will, when viewed,

lead to more intense activity in the visual areas. This is what we

found (see Figure 5), and it implies that some combinations of

form are more aesthetically pleasing than others, even if they are

not “understood” cognitively (see also quote from Dirac, given

below). What these formal qualities may be requires a s eparate

and detailed study of the beauty of forms, well beyond the scope

of this study. But a parallel may be found in our earlier study,

which shows that there are some conﬁgurations of kinetic stimuli

which activate the motion areas of the visual brain more intensely

(Zeki and Stutters, 2011).

IMPLICATIONS FOR FUTURE WORK

The experience of beauty derived from mathematical formula-

tions represents the most extreme case of the experience of beauty

that is dependent on learning and culture. The fact that the expe-

rience of mathematical beauty, like the experience of musical and

visualbeauty,correlateswithactivityinA1ofmOFCsuggests

that there is, neurobiologically, an abstract quality to beauty that

is independent of c ulture and learning. But that there was an

imperfect correlation between understanding and the experience

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Zeki et al. Neural correlates of mathematical beauty

of beauty and that activity in the mOFC cannot be accounted for

by understanding but by the experience of beauty alone, raises

issues of profound interest for the future. It leads to the capital

question of whether beauty, even in so abstract an area as math-

ematics, is a pointer to what is true in nature, both within our

natureandintheworldinwhichwehaveevolved.Paul Dirac

(1939) put it like this: “There is no logical reason why the (method

of mathematical reasoning should make progress in the study of

natural phenomena) but one has found in practice that it does

work and meets with reasonable success. This must be ascribed

to some mathematical quality in Nature, a qualit y which the

casual observer of Nature would not suspect, but which never-

thelessplaysanimportantroleinNature’sscheme...Whatmakes

the theory of relativity so acceptable to physicists in spite of its

going against the principle of simplicity is its great mathematical

beauty. This is a quality which cannot be deﬁned, any more than

beauty in art can be deﬁned, but which people who study math-

ematics usually have no difﬁculty in appreciating. The theory of

relativity introduced mathematical beauty to an unprecedented

extent into the description of Nature...We now see that we have

to change the principle of simplicity into a principle of math-

ematical beauty. The research worker, in his efforts to express

the fundamental laws of Nature in mathematical form, should

strive mainly for mathematical beauty. He should still take sim-

plicity into consideration in a subordinate way to beauty. It often

happens that the requirements of simplicity and of beauty are

the same, but where the y clash the latter must take precedence”

(ellipses added). In similar vein, Hermann Weyl is recorded as

having said, “My work always tried to unite the true with the

beautiful; but when I had to choose one or the other, I usually

chose the beautiful” (Dyson, 1956). Relevant here is the story

of Weyl’s mathematical formulations, which tried to reconcile

electromagnetism with relativity. Rejected at ﬁrst (by Einstein)

because it was thought to conﬂict with experimental evidence,

it came subsequently to be accepted but only after the advent of

quantum mechanics, which led to a new interpretation of Weyl’s

equations. Hence the perceived beauty of his mathematical for-

mulations ultimately predicted truths even before the full facts

were known.

If the experience of mathematical beauty is not strictly related

to understanding (of the equations), what can the source of math-

ematical beauty be? That is perhaps more difﬁcult to account for

in mathematics than in visual art or music. Whereas the source

for the latter can be accounted for, at least theoretically, by pre-

ferred harmonies in nature or preferred distribution of forms or

colors (see Bell, 1914; Zeki and Stutters, 2011; Zeki, 2013), it is

more difﬁcult to make such a correspondence in mathematics.

The Platonic t radition would emphasize that mathematical for-

mulations are experienced as beautiful because they give insights

into the fundamental structure of the universe (see Breitenbach,

2013). For Immanuel Kant, by contrast, the aesthetic experience

is as well grounded in our own nature because, for him, “Aesthetic

judgments may thus be regarded as expressions of our feeling that

something makes sense to us” (Breitenbach, 2013). We believe

that what “makes sense” to us is grounded in the workings of

our brain, which has evolved within our physical environment.

Dirac (1939) wrote: “the mathematician plays a game in which

he himself invents the rules while the physicist plays a game in

which the rules are provided by Nature, but as time goes on

it becomes increasingly evident that the rules which the math-

ematician ﬁnds interesting are the same as those which Nature

has chosen” and therefore that in the choice of new branches of

mathematics, “One should be inﬂuenced very much... by con-

siderations of mathematical beauty” (ellipsis added). Hence the

work we report here, as well as our previous work, highlights

further the extent to which even future mathematical formula-

tions may, by being based on beauty, reveal something about

our brain on the one hand, and about the extent to which our

brain organization reveals something about our universe on the

other.

ACKNOWLEDGMENTS

This work was supported by the Wellcome Trust London,

Strategic Award “Neuroesthetics” 083149/Z/07/Z. We are ver y

grateful to Karl Friston for his many good suggestions during the

course of this work.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found

online at: http://www.frontiersin.org/journal/10.3389/fnhum.

2014.00068/abstract

We include four datasheets containing experimental details

which are not essential for general understanding of the article:

Data Sheet 1: EquationsForm.pdf—The equation beauty-rating

questionnaire.

Data Sheet 2: UnderstandingForm.pdf—The equation under-

standing questionnaire.

Data Sheet 3: BehavioralData.xlsx—Tables of behavioral data.

Data Sheet 4: BeauNeutUglyBase.pdf—Tables of categorical

beauty activations vs. baseline.

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Conﬂict of Interest Statement: The authors declare that the research was con-

ducted in the absence of any commercial or ﬁnancial relationships that could be

construed as a potential conﬂict of interest.

Received: 11 November 2013; accepted: 28 January 2014; published online: 13

February 2014.

Citation: Zeki S, Romaya JP, Benincasa DMT and Atiyah MF (2014) The experience

of mathematical beauty and its neural correlates. Front. Hum. Neurosci. 8:68. doi:

10.3389/fnhum.2014.00068

This article was submitted to the journal Frontiers in Human Neuroscience.

cle distributed under the terms of the Creative Commons Attribution License (CC BY).

The use, distribution or reproduction in other forums is permitted, provided the

original author(s) or licensor are credited and that the original publicat ion in this

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

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Discussion

$$1+e^{i\pi}=0$$ Euler's Identity is a special case of Euler's Formula that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Expert mathematicians had an emotional response described as an "aesthetic emotion" when viewing equations they rated as beautiful. This emotional experience was also characterized as satisfying or pleasurable. The subjects demonstrated a deep understanding and appreciation of mathematics by identifying specific equations they found particularly beautiful, and expressing regret when some of their favorite equations were not included in the list. > ***"The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity. [...] 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 musical beauty, which are more “sensible” and perceptually based."*** Art and mathematics are sometimes perceived as polar opposites. They can both provoke aesthetic emotions and arouse an experience of beauty. Despite its abstract nature, there is a strong relation between mathematical and artistic beauty, as both mathematicians and art appreciators feel a similar sense of delight and exaltation. Plato considered mathematical beauty to be the highest form of beauty. You can find the Data Sheet with all of the 60 mathematical equations here: [The Beauty of Mathematics - Equations](https://fermatslibrary.com/p/413aae33) The results show a strong positive correlation between the level of understanding and scan-time beauty ratings, indicating that well-understood equations were generally perceived as more beautiful. The relationship was not perfect, allowing the researchers to separate the effects of beauty from those of understanding. Authors note that for ideal analysis they would prefer equal ratios of the four understanding categories in each scanning session. > ***"Perhaps one of the most awkward, and at the same time challenging, aspects of this work was trying to separate out beauty and understanding. Because the correlation between the two, though significant, was also imperfect, we were able to do so for mathematicians. This of course leaves the question of whether non-mathematicians, with no understanding whatsoever of the equations, would also find the equations beautiful."*** ### TL;DR This paper explores the fascinating connection between the experience of beauty in mathematics and art, and how the brain processes these abstract emotional experiences. The authors used functional magnetic resonance imaging (fMRI) to identify the brain regions activated when mathematicians experience mathematical beauty. The researchers discovered that: - The experience of mathematical beauty correlates with activity in the same brain area as the experience of beauty derived from other sources, such as art or music. - The experience of mathematical beauty is related to activity in brain areas associated with reward, emotion, and cognition, suggesting a complex interplay between these processes. - Mathematical beauty is a combination of the understanding of mathematical concepts and the emotional response they evoke. - There seems to exist a strong link between mathematical and artistic beauty. Understanding the neural basis of mathematical beauty may help educators design teaching methods that foster a deeper appreciation for the subject. > ***"Art and mathematics are, to most, at polar opposites; the former has a more “sensible” source and is accessible to many while the latter has a high cognitive, intellectual, source and is accessible to few."***

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