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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:

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

This is an open-access article 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 publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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.

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”

The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity. Many—among them the mathematicians Bertrand Russell (

Plato (

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 normal or corrected to normal vision. One subject was eliminated from the study after it transpired that he suffered from attention 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 questionnaires 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 the experience of visual and musical beauty (Kawabata and Zeki,

Stimuli consisting of equations were generated using Cogent 2000 (

Subjects viewed the formulae during four functional scanning sessions, with breaks between sessions which gave them an opportunity to take a rest if required and us to correct any anomalies, 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′ 14″). An echo-planar imaging (EPI) sequence was applied for functional scans, measuring BOLD (Blood Oxygen Level Dependent) signals (echo time

Fifteen equations were displayed during each session (Figure

Each trial (Figure ^{−2}; the 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.,

In a similar way, we undertook another parametric analysis with the scan-time beauty rating as the first parametric modulator 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 second 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

A categorical analysis differs from the corresponding parametric 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 session 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 categorical 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 categorical beauty, we report activation at cluster-level significance (_{Clust-FWE}_{unc.}

For other contrasts vs. baseline which are not of principal interest to this study we report activations that survive a peak voxel threshold of _{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.,

The formula most consistently rated as beautiful (average rating of 0.8667), both before and during the scans, was Leonhard Euler's identity

Other highly rated equations included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler's formula for complex analysis, and the Cauchy-Riemann equations (Data Sheet

In pre-scan beauty ratings, each subject rated each of the 60 equations according to beauty on a scale of −5 (Ugly) through 0 (Neutral) to +5 (Beautiful) while during scan time ratings, subjects rated each equation into the three categories of Ugly, Neutral or Beautiful.

An excel file containing raw behavioral data is provided as Data Sheet

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 ratings 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 equations into three groups, one containing 20 low-rated, another 20 medium-rated, and a third 20 high-rated equations, individually 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

The frequency distribution of post-scan understanding ratings is given in Figure

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: “

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, usually 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 minority 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 “

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,

_{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

The mOFC was the only brain region that showed a BOLD signal that was parametrically related to beauty ratings, significant at cluster level (see Table

_{E} |
_{Clust-FEW} |
_{14} |
_{FWE} |
||||
---|---|---|---|---|---|---|---|

mOFC | |||||||

mOFC | 0 | 35 | −14 | 4.90 | 0.733 | ||

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 |

_{Clust-FWE} < 0.05, background threshold _{unc.} < 0.001, extent threshold 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 _{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.

A categorical analysis of Beauty ratings vs. Baseline is less sophisticated 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

Table

_{E} |
_{Clust-FWE} |
_{14} |
_{FWE} |
||||
---|---|---|---|---|---|---|---|

L angular gyrus | |||||||

mOFC | < |
||||||

mOFC | −9 | 50 | −5 | 5.16 | 0.576 | ||

mOFC | −9 | 38 | −5 | 4.70 | 0.790 | ||

L superior temporal gyrus | |||||||

L superior temporal gyrus | −3 | 17 | 67 | 4.65 | 0.809 | ||

L superior temporal gyrus | −18 | 38 | 46 | 4.52 | 0.858 |

_{Clust-FWE} < 0.05, background threshold _{unc.} < 0.001, extent threshold 10 voxels). Co-ordinates of hotspots within each cluster are given in MNI space. 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

While the experience of mathematical beauty correlated parametrically with activity in mOFC, the contrast All equations > Baseline showed that many sites were generally active when subjects viewed the equations (Table

_{E} |
_{14} |
_{FWE} |
||||
---|---|---|---|---|---|---|

L fusiform gyrus | < |
|||||

L fusiform gyrus | −30 | −91 | −5 | 14.96 | <0.001 | |

R fusiform gyrus | < |
|||||

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 | < |
|||||

L inferior temporal gyrus | < |
|||||

L intraparietal sulcus | ||||||

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

R cerebellum lobule VIII/crus I | 0.002 | |||||

R cerebellum dorsal paraflocculus | ||||||

L inferior frontal gyrus | ||||||

R intraparietal sulcus | ||||||

L precuneus | < |
|||||

R precuneus | 15 | −40 | 43 | 16.63 | <0.001 | |

L precuneus | −15 | −40 | 52 | 14.73 | <0.001 | |

R fusifom gyrus | < |
|||||

Fusiform gyrus | 63 | −34 | 16 | 10.87 | <0.001 | |

Fusiform gyrus | 63 | −22 | 16 | 8.25 | 0.027 | |

R medial orbito-frontal cortex | < |
|||||

L medial orbito-frontal cortex | −6 | 23 | −8 | 14.25 | <0.001 | |

R medial orbito-frontal cortex | 12 | 53 | −2 | 13.37 | <0.001 | |

R superior temporal gyrus | < |
|||||

R superior temporal gyrus | 63 | 2 | −23 | 14.39 | <0.001 | |

R superior temporal gyrus | 45 | −13 | −8 | 11.05 | <0.001 | |

L lingual gyrus | < |
|||||

L middle temporal gyrus | < |
|||||

L superior medial gyrus (anterior paracingulate) | ||||||

L middle frontal gyrus | ||||||

R superior frontal gyrus (anterior paracingulate) | ||||||

L lingual gyrus | ||||||

R cuneus |

_{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 (_{E}) with up to two other peaks in the same cluster listed below

As well, the contrast All equations < Baseline revealed widespread cortical de-activations (Table _{unc} < 0.001, showed that this de-activation overlaps the mOFC activation which correlates parametrically with the experience of mathematical beauty (Figure

_{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.

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

_{unc} < 0.001 with an extent threshold of 10 voxels, revealed two peak “de-activations” significant (_{FWE} < _{14} = 9.01, _{FWE} = 0.010) and at (30, −91, 19) (_{14} = 8.05, _{FWE} = 0.032). A third peak at (−27, −85, 7) (_{14} = 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

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 art nor all great mathematical formulations do so. The experience of mathematical beauty, considered by Plato (

Mathematical and artistic beauty have been written of in the same breath by mathematicians and humanists alike, as arousing the “aesthetic emotion.” This implies that there is a common and abstract nature to the experience of beauty derived from very different sources. Viewed in that light, the activity in a common area of the emotional brain that correlates with the experience of beauty derived from different sources merely mirrors neurobiologically the same powerful and emotional experience of beauty that mathematicians and artists alike have spoken of.

The mOFC is active in a variety of conditions, of which experiences 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 discussed in the philosophy of aesthetics, without a clear conclusion (Gordon,

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. Ideally, one would want to have subjects who are mathematically totally illiterate, 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 displayed, their symmetrical distribution, etc. We surmised that we could demonstrate this indirectly, by showing that less well understood equations in our mathematical subjects will, when viewed, lead to more intense activity in the visual areas. This is what we found (see Figure

The experience of beauty derived from mathematical formulations represents the most extreme case of the experience of beauty that is dependent on learning and culture. The fact that the experience of mathematical beauty, like the experience of musical and visual beauty, correlates with activity in A1 of mOFC suggests that there is, neurobiologically, an abstract quality to beauty that is independent of culture and learning. But that there was an imperfect correlation between understanding and the experience 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 mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved. Paul Dirac (

If the experience of mathematical beauty is not strictly related to understanding (of the equations), what can the source of mathematical 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 preferred harmonies in nature or preferred distribution of forms or colors (see Bell,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Wellcome Trust London, Strategic Award “Neuroesthetics” 083149/Z/07/Z. We are very grateful to Karl Friston for his many good suggestions during the course of this work.

The Supplementary Material for this article can be found online at:

We include four datasheets containing experimental details which are not essential for general understanding of the article:

EquationsForm.pdf—The equation beauty-rating questionnaire.

UnderstandingForm.pdf—The equation understanding questionnaire.

BehavioralData.xlsx—Tables of behavioral data.

BeauNeutUglyBase.pdf—Tables of categorical beauty activations vs. baseline.