Mathematicians are uncovering hidden geometric shapes through positive geometry that may link particle physics and cosmology, offering a new framework to understand particle collisions, the Big Bang, and the fundamental laws shaping the universe.
New Shapes in Physics
Mathematicians uncover hidden shapes through positive geometry that may connect particle physics with cosmology, reshaping our view of the universe.
Mathematicians are exploring unusual new shapes that may offer a common language for both particle physics and cosmology. This approach, known as positive geometry, could provide fresh insights into how subatomic particles interact and how the universe itself took shape.
Linking the Smallest and the Largest
The research, led by Claudia Fevola (Inria Saclay) and Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences), was recently published in the Notices of the American Mathematical Society. Their work asks a big question: Can the same mathematical concepts explain the behavior of both elementary particles and the entire cosmos?
Their answer lies in sophisticated branches of mathematics—algebraic geometry, D-module theory, and combinatorics—that are now being applied to physics in new ways.
Beyond Traditional Feynman Diagrams
Physicists usually rely on Feynman diagrams—drawings that illustrate how particles scatter and interact. But these diagrams only go so far. Positive geometry introduces a broader perspective by expressing these interactions as volumes of high-dimensional geometric objects.
One striking example is the amplituhedron, a geometric structure proposed in 2013 that allows certain scattering probabilities to be calculated more directly than through conventional methods. These ideas don’t replace Feynman diagrams but expand them, offering a new toolkit for physicists.
From Particle Collisions to the Early Universe
The reach of positive geometry extends far beyond particle accelerators. In cosmology, researchers are applying similar methods to understand the cosmic microwave background—the faint glow left over from the Big Bang. By studying structures called cosmological polytopes, scientists hope to reconstruct the physical laws that governed the universe’s earliest stages.
Why This Matters
If developed further, positive geometry could unify concepts across physics. It might help:
Simplify particle collision calculations.
Provide deeper insights into how the early universe formed.
Connect mathematics and physics more tightly, driving new discoveries in both fields.
A Field in Motion
This work is part of a broader international effort supported by the ERC synergy grant UNIVERSE+, led by leading physicists including Nima Arkani-Hamed and Daniel Baumann. Although still in its early stages, positive geometry is already reshaping how researchers think about fundamental questions.
As Fevola and Sattelberger point out, the challenge now is to refine these mathematical frameworks and confirm their usefulness in real-world physics. But the promise is clear: positive geometry may be more than just a new tool—it could be a new language of nature.