mattergeometry

Mattergeometry is an interdisciplinary field that studies how the distribution of matter and the geometry of space influence each other. It encompasses the ways in which geometric structures—such as lattices, manifolds, curvatures, and topological defects—govern material properties, while conversely the arrangement and dynamics of matter induce geometric changes in space and constraints on possible configurations.

Key ideas include differential geometry and topology applied to materials science, geometric frustration in magnetic and glassy systems, packing and tessellation in granular and colloidal materials, and the design of metamaterials whose macroscopic behavior is dictated by microstructure geometry. Methods combine mathematical modeling with computational tools: differential geometry, topology, graph theory, computational geometry, finite element analysis, and data-driven approaches.

Applications span condensed matter physics, photonic and phononic metamaterials, soft matter and colloidal systems, tissue engineering, and nanostructured materials. For example, curvature design in curved-space metamaterials, defect engineering in crystals and foams, and topology-driven phase transitions.

While not always named "mattergeometry," the approach appears across disciplines and represents a unifying perspective for understanding how geometry and matter co-create material properties. It draws on insights from mathematics, physics, and engineering, and is associated with studies of crystal geometry, topological matter, and geometric frustration. See also differential geometry, topology, condensed matter physics, metamaterials, and materials science.