AI Minds Think Like Ours Through Hidden Mathematical Shapes

Deep inside artificial intelligence systems, researchers have discovered something remarkable: the same geometric principle that helps humans recognize cats, understand concepts, and learn new skills also governs how machines organize knowledge.

This mathematical property, called convexity, may be the missing link between human and artificial intelligence.

The discovery, published in Nature Communications by researchers at the Technical University of Denmark, reveals that both human brains and AI systems naturally organize concepts into convex regions—curved mental spaces where related ideas cluster together like points inside a stretched rubber band.

A Universal Learning Pattern

When humans learn the concept of “cat,” we don’t simply memorize one image. Instead, our minds create a flexible mental space that encompasses all possible cats—fluffy Persian cats, sleek Siamese, massive Maine Coons, and tiny kittens. This mental region has a special geometric property: if you imagine any two cats in this space, every point on the shortest path between them also represents something we’d recognize as a cat.

This is convexity in action, and cognitive scientist Peter Gärdenfors has long argued it’s fundamental to how humans form natural concepts. Now, DTU researchers have found the same principle operating in the hidden layers of artificial neural networks.

“We found that convexity is surprisingly common in deep networks and might be a fundamental property that emerges naturally as machines learn,” says Lars Kai Hansen, the DTU professor who led the study.

Peering Inside AI’s Hidden Geometry

The research team developed sophisticated mathematical tools to measure convexity within AI systems’ internal representations—the multi-dimensional spaces where algorithms organize their understanding of the world. They tested these tools across five different types of data:

• Images from ImageNet’s vast collection
• Audio recordings from speech command datasets
• Text from newsgroup discussions
• Human activity data from wearable sensors
• Medical images of blood cells

In every domain, they found that AI models naturally develop convex concept regions, especially after a training process called self-supervised learning, where algorithms learn patterns from unlabeled data before being taught specific tasks.

Remarkably, the researchers discovered that convexity actually predicts learning success. AI models with more convex concept regions performed better when later trained on specific classification tasks. It’s as if having well-organized mental geometry makes learning new skills easier—a finding that mirrors theories about human cognition.

Two Types of Mental Geometry

The team measured two distinct types of convexity. Euclidean convexity works like straight-line connections in flat space, similar to how we might blend two similar concepts by averaging their features. Graph convexity follows curved paths through the natural structure of data, like navigating concepts along the winding topology of knowledge itself.

First author Lenka Tetkova, a postdoc at DTU, explains the significance: “We’ve developed new tools to measure convexity within the complex latent spaces of deep neural networks. We found that the same geometric principle that helps humans form and share concepts—convexity—also shapes how machines learn, generalize, and align with us.”

The mathematics proved robust across different AI architectures, from vision transformers that process images to language models that handle text. When models were fine-tuned for specific tasks, their convexity increased—concepts became more sharply defined and geometrically organized.

Implications for Human-Machine Partnership

This geometric commonality between human and artificial minds could have profound implications for building more trustworthy AI systems. If both humans and machines naturally organize knowledge using similar mathematical principles, it suggests a path toward creating AI that thinks in ways we can better understand and predict.

“By showing that AI models exhibit properties like convexity that are fundamental to human conceptual understanding, we move closer to creating machines that ‘think’ in ways that are more comprehensible and aligned with our own,” says Tetkova. “This is vital for building trust and collaboration between humans and machines in critical applications like healthcare, education, and public service.”

The findings also suggest practical applications for improving AI training. Engineers might design algorithms that explicitly encourage convex concept formation, potentially leading to more efficient learning and better generalization from limited examples.

While questions remain about why this geometric principle emerges so consistently, the research opens new avenues for understanding both artificial and human intelligence. As AI systems become increasingly sophisticated, discoveries like these help bridge the gap between silicon and flesh, revealing the shared mathematical foundations of all learning minds.