Aesthetic Measure G.D.Birkhoff

Shortly after deciding on a research topic, I got to know one book titled "Aesthetic Measure" (Harvard Press, 1933) by an American mathematician named George David Birkhoff (1884 – 1944) . At the university library, the book is treated with much care as it is part of the cultural assets.

At the book counter on the second basement, the woman wearing a white bag handed me a huge book wrapped in thin translucent paper saying that 'Viewing is limited to 2 hours and copying is limited to 20 pages. Please specify the part you want to copy and we will copy it for you."

The book was bigger and heavy than assumed, looked like an encyclopedia. It is already out of print and used books are sometimes available on amazon.

Aesthetic Measure (1933)

In this book, Birkhoff quantified the beauty of basic shapes, music, textiles, etc. with the formula M = O/C. thus Measure = Order / Complexity

For example, to measure the beauty of a polygon, the equation evolves as follows

O = V + E + R + HV -F

V : vertical symmetry (+)

E : equilibrium (+)

R : rotational symmetry (+)

HV : relation to a horizontal - vertical network (+)

F : unsatisfactory form (-)

==> too small distances from vertices to vertices or to sides,

or between parallel sides;

angles too near 0 ̊ or 180̊;

other ambiguities;

unsupported reentrant sides;

inclined or horizontal symmetry.

According to his proposed formula, the beauty of a square would be 1.50, a rectangle 1.25, and a star 0.90.

In recent years, there has been a lot of research on using machine learning to measure the beauty of a painting, an approach based on Birkhoff's Aesthetic Measure. It is amazing to see an idea that is about 100 years old implemented in AI.

Conceptualizing Birkhoff's aesthetic measure using Shannon entropy and Kolmogorov complexity | Proceedings of the Third Eurographics conference on Computational Aesthetics in Graphics, Visualization and Imaging (acm.org)

Informational Aesthetics Measures, IEEE computer graphics and applications, 03/2008

Later, I found that Birkhoff is highly regarded not because his proposed formula is correct, but because of his exploits to express ambiguous beauty in such a concise manner.