golden rule – A math study using persistent homology finds abstract artists share a consistent shape-placement pattern, while AI art often misses it—and perception shifts with the viewing setting.
A surprising “golden rule” may be hiding inside abstract paintings, where mathematics helps explain why some compositions feel more compelling than others.
Mathematicians and neuroscientists report that famous abstract artists tend to follow an implicit pattern governing how shapes relate to the edges of a canvas.. The researchers also found that artificial intelligence systems they tested did not replicate these shape-placement constraints. which they say could help explain why computer-generated abstract art often fails to produce the same sense of awe.. The work was motivated by a long-running question shared by scientists and philosophers: whether masterpieces share measurable features that creators may use—consciously or not—to trigger emotion.
The team. led by Jacek Rogala. a neuroscientist at the University of Warsaw. and co-senior author Shabnam Kadir. a mathematician at the University of Hertfordshire in England. focused on a mathematical tool known as persistent homology.. This approach belongs to topology, a branch of mathematics concerned with how shapes change as they are deformed or stretched.. Rather than treating a painting as a static image, the method studies how contours emerge across multiple layers of color.
To carry out the analysis, the researchers examined the contours of color in works by the Polish artist Lidia Kot.. They then compared people’s responses to Kot’s art with responses to AI-generated pieces intended to resemble the overall color structure.. Importantly. the study also treated viewing conditions as a variable rather than a constant—testing reactions both in an art gallery setting and in a lab.
The starting point is turning a painting into data, which the researchers describe as a nontrivial step.. Persistent homology accomplishes this by coding each layer of color into a corresponding shape.. To build intuition. the paper uses an example of an image simplified to black features and then gradually adds darker and lighter shades. tracking how the shapes evolve as new color layers are introduced.
As layers are added from darkest to lightest. the contours form a hierarchy of features. and at each stage the method captures a snapshot of which structures appear and how they change.. These snapshots are summarized as “barcodes. ” a compact set of numerical descriptors that characterize the persistence of shape features throughout the color layering process.
One reason these barcodes matter is that they provide a way to talk about structural properties in art without discarding what the paintings look like.. A mathematician not involved in the study. Barbara Guinti of the University at Albany. State University of New York. said the approach offers a more formal language for describing artwork while keeping its character intact.. Another researcher who helped develop the method. Vanessa Robins of the Australian National University. has used persistent homology to analyze branching patterns in human lungs. illustrating how the same mathematical machinery can transfer between art analysis and medical research.
In the abstract-art study. persistent homology was used first to identify whether certain features consistently define how shapes are framed in abstract paintings. and then to test whether those features carry across artists.. The question Kadir raises—what truly counts as abstract art—underscores the challenge: without a shared standard. it can be easy to label abstract work as “nonsense” rather than something that might follow recognizable structure.
Mathematically. the researchers argue that the answer is no: the team reports that Kot and several widely known abstract artists—Mark Rothko. Wassily Kandinsky. Kazimir Malevich. Jackson Pollock. and Maria Jarema—shared a common structural relationship in how their compositions sit relative to the boundaries of a canvas.. The pattern is tied to a symmetry called Alexander duality. which links how features appear when a representation is toggled from one set of colors to the opposite.
Alexander duality, however, fails in the presence of shapes that cross the edges of the painting.. The researchers found that the human artists consistently violated this symmetry by a similar ratio, reported as 0.4 in the study.. While the specific numerical value is not framed as a quiz-like takeaway. the result is presented as evidence that artists were effectively following a repeatable rule about how much of the visual structure extends toward—and breaks at—the frame.
Kadir also compares the discovery to other well-known art constants.. The “golden ratio. ” for example. has long been associated with aesthetically pleasing arrangements in nature and design. with a value commonly given as 1.618 to 1.. The researchers argue that their ratio functions in a comparable way within the logic of shape placement—suggesting that deep constraints governing form may be more common in abstract painting than previously assumed.
To test whether AI systems would reproduce the same constraint. the researchers created AI-generated “art” that matched the color intensity found in Kot’s work. but did so without the same kind of human artistic intention.. According to the study’s findings. the AI-generated pieces did not adhere to the shape-placement ratio the team observed in human paintings. indicating that the method can distinguish between art that follows the implicit structural pattern and art that merely reproduces surface-level color behavior.
Perception tests were designed to explore whether these mathematical differences show up in how people experience the works.. The researchers recruited 58 participants, either college-age art students or people with similar age and socioeconomic backgrounds.. One group viewed Kot’s art first in a gallery and then in a lab. while the other group viewed the AI art under the same two conditions without being told it was AI-made.
The study reports that in the lab environment. participants rated the human art higher than the AI imagery and spent more time looking at the human paintings.. In the gallery. however. the ratings between human and AI art became more similar. while participants fixated on the AI imagery for longer.. The researchers describe this as a pattern that may depend on the viewing environment.
Rogala suggests topology may offer part of the explanation.. In an art gallery. color gradients and layered shapes can “pop” as viewers move and change their angles under real lighting. creating a dynamic visual experience.. By contrast. screen viewing in the lab is more static and may flatten subtle differences—potentially giving AI works an unexpected boost if lighting and motion cues are favorable.
While the results point to a hidden rule linking compositional balance and shape placement, Rogala stresses that more questions remain.. One open area is whether artists outside Western traditions follow the same mathematical patterns of form and edge-related symmetry breaking. or whether different visual cultures encode different structural “rules” that influence perception.
abstract art math persistent homology topology shapes AI generated art neuroscience perception art gallery study golden rule
So basically artists are doing math without knowing it? Kinda wild.
I don’t get how AI “misses” some golden rule when it can copy anything… maybe they just picked the worst AI models. Also perception shifts with where you stand? That seems obvious.
Wait so this is saying the “edges of the canvas” like the frame matters more? I swear I’ve felt that with some paintings, like it pulls you in, but I thought that was just vibes not… topology stuff.
Persistent homology sounds like a video game cheat code. But if the claim is AI art fails to trigger awe because of shape placement, then artists should just follow a checklist and call it a day. Museums really about to start hanging with rulers now?