Mathematicians uncover ‘golden rule’ in abstract art

May 14, 2026

5 min read

Add Us On GoogleAdd SciAm

Mathematicians uncover ‘golden rule’ in abstract art

A mathematical ratio could explain why AI-generated art doesn’t evoke awe from viewers

By Stephanie Pappas edited by Jeanna Bryner

Mathematicians found that famous abstract artists, including Wassily Kandinsky, whose "Composition IX" is shown at the Pompidou Center, violate a symmetry called the Alexander duality by a similar.

JORGE GUERRERO/AFP via Getty Images

Love math? Sign up for our weekly newsletter Proof PositiveEnter your email

I agree my information will be processed in accordance with the Scientific American and Springer Nature Limited Privacy Policy. We leverage third party services to both verify and deliver email. By providing your email address, you also consent to having the email address shared with third parties for those purposes.

Sign Up

Mathematicians have calculated a “golden rule” for abstract art that famous artists tend to follow when they compose their works. Artificial intelligence, the team found, does not follow such implicit rules about shape placement, possibly explaining why computer-generated art doesn’t usually evoke awe from viewers.

Scientists and philosophers have long tried to decipher why art moves people: Are there underlying features shared among masterpieces? Do painters unconsciously use similar shapes, contours or compositions to elicit an emotional response? Many of the ways researchers have tried to categorize shapes or complexity in paintings are “arbitrary,” however, says Jacek Rogala, a neuroscientist at the University of Warsaw and co-senior author of the new study.

Rogala, co-senior author Shabnam Kadir, a mathematician at the University of Hertfordshire in England, and their colleagues looked to persistent homology, which is part of topology, a mathematical field that studies shapes as they deform and stretch. In this case, the team analyzed the contours of color in the works of Polish artist Lidia Kot. Then the researchers compared how people responded to Kot’s art and similar AI-generated art both in an art gallery and a lab.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

The team found that viewers were drawn to certain mathematical features of abstract art and that artists created paintings with a remarkably consistent visual balance, whether they were conscious of it or not. At the same time, context matters: the way people responded to human versus AI art depended on whether they saw it on a computer screen, in the lab or in an art gallery.

First, let’s consider the math. Turning a painting into a set of data is no easy feat. Persistent homology accomplishes this by coding each layer of color to a shape. Imagine a painting of a Holstein cow: If you take everything away but the black pixels, you’ll get a few splotches of cow spots, each a unique shape. Now mark down the features of those shapes. Then add in your next-lightest color, dark gray. Your cow is now composed of some spots and a few shadows. Mark that down, too. Keep going, adding lighter and lighter shades, until your whole painting is a white cow with dark spots again.

What you’ve just built is a dataset of all the contours in the painting, or all the ways the artist positioned color in different shapes on the canvas. If you could toggle through the layers, it would morph from spots to cow and back again. And at every stage, you’ve got a snapshot of the features of these shapes, a chunk of data researchers call a “barcode.” (While an untrained human might come up with a barcode that read something like, “Um, squiggly?” mathematicians use more sophisticated numerical descriptors.)

“It is a way to talk more formally about art but without removing the soul of it,” says Barbara Guinti, a mathematician at the University at Albany, State University of New York, who was not involved in the new study.

You can do a lot of things with these barcodes. Vanessa Robins, a mathematician at the Australian National University, who wasn’t involved in the research but was one of the original developers of the method, said she and her colleagues have been using persistent homology to analyze the branching patterns in people’s lungs to see if these shapes influence survival in certain lung diseases. In Rogala and Kadir’s study, the team took the data and first decided to figure out if there were features that defined how shapes were framed in abstract art and, if so, whether they carried over from painter to painter.

“What is abstract art?” Kadir says. “Is it a bunch of nonsense that my child could create?”