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The human quest for discovering mathematical beauty in the arts

The human quest for discovering mathematical beauty in the arts

“The human function is to ‘discover or observe’ mathematics,” said twentieth-century British mathematician G. H. Hardy. Humanity has been searching for beauty and order in the arts and in nature for generations, dating back to the ancient Greeks. This search for mathematical beauty has led to the discovery of recurring mathematical structures such as the golden ratio, Fibonacci numbers, and Lucas numbers, which have captivated the interest of artists and scientists alike.

This quest’s enchantment comes with significant stakes. In truth, art is the ultimate expression of human ingenuity, and comprehending it mathematically would provide us with the keys to decoding human civilization and evolution. However, it wasn’t until later that the scope and size of humanity’s pursuit of mathematical beauty was dramatically broadened by the convergence of three distinct inventions.

The development of robust statistical approaches to capture hidden patterns in massive amounts of data, as well as the mass digitization of large art archives, have made it feasible to disclose the—otherwise imperceptible to the human eye—mathematics hidden in large artistic corpora.

Music, storytelling, language phonology, comedy in jokes, and even equations have all been included in the current growth. Lee et al. extend this quest by looking for statistical signatures of compositional proportions in a quasi-canonical dataset of 14,912 landscape paintings spanning the period from Western renaissance to contemporary art (from 1500 CE to 2000 CE). 

They mathematically investigate how artists organize colors on the canvas across styles and time using an information-theoretical framework based on Rigau’s et al work (Fig. 1). They use a computational approach to divide each painting in their collection into the most color uniform vertical and horizontal areas. Their approach works in steps, maximizing mutual information between colors and areas across all conceivable partitions in both horizontal and vertical dimensions at each phase.

Methodology and main results in Lee et al. The vast majority of landscape paintings feature a first horizontal partition, while the direction of the second partition evolved from vertical (H–V) to horizontal (H–H); this shift is consistent across individual artists’ nationalities (Result 1). The ratio of the compositional proportion rc in horizontally partitioned paintings denotes the height of the horizon line; Lee et al. map its progression into three macro historical periods (Result 2; graph reconstructed from figure 3A). Network analysis reveals the existence of three coherent communities of artists clustered in time and in terms of their horizon choice (Result 3). Painting in the illustration is Seaport with the Embarkation of the Queen of Sheba (1648) by Claude Lorrain (1604–1682), adapted from figure 1. Painting images credit: The National Gallery, London.

Gaining information in this context means becoming more convinced that the palettes of the partitioned parts are chromatically distant, as information “encodes counterfactual knowledge and describes the level of ambiguity or noise in a system”. By comparing the compositional information received by early partitions in abstract and landscape paintings, Lee et al. demonstrate that the information gained by early partitions in landscape paintings is significantly higher than in abstract paintings, which exhibit no directional preference.

The dissection analysis exposes underlying metapatterns of community consensus that are completely removed from aesthetic considerations. Nonetheless, the upshot is a unified macrohistory of Western landscape painting. Furthermore, their research provides a quantitative knowledge of the relationships between artistic styles, trends, and artists.

Is there a set of mathematically defined organizational principles that apply to all genres and artists? Do these values vary by country and culture? What happens to them over time?

C.E. Shannon, The Bandwagon

Mutual information, statistical surprise, and permutation entropy have all been utilized to turn art’s abstract intricacy into a numerical form. “A few exciting words like information, entropy, redundancy, do not solve all our problems,” Claude E. Shannon, the creator of information theory, cautioned us in 1956.

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For example, the Lee et al. approach performs poorly with paintings that require diagonal partitions (such as Paul Cezanne’s Landscape on the Mediterranean) or when large objects are placed in the center of the canvas (like in The Babel Tower by Pieter Bruegel the Elder).

Although Lee’s work does not answer all of our challenges, it is a great place to start and has foundational importance. It invites scholars from other disciplines to push the hunt for mathematical beauty toward broader categories and deeper knowledge by spelling out the enigma of decoding mathematical beauty via the lenses of geometric proportions.

The human quest for discovering mathematical beauty in the arts, Stefano Balietti

Published: October 2020
DOI: 10.1073/pnas.2018652117

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