Twisted, Tiled, and Woven: Geometric Modeling of Space-filling Interlocked Shapes Through Partitive Geometry

This talk introduces partitive geometry as a paradigm for geometric modeling and the systematic design of space-filling interlocked shapes. Partitive geometry draws on the principles of growth observed in physical, chemical, and biological systems, offering a novel representation for interlocked, space-filling forms. Central to this approach is Voronoi tessellation generated by arrangements of higher-dimensional Voronoi sites governed by spatial symmetry groups. We begin with the scutoid, a geometric form discovered in biological tissues, which exemplifies how interlocking arises during the growth and packing of animal skin cells. Building on this insight, we discuss computational schemes for capturing such growth to design a family of shapes called Delaunay Lofts, which interpolate between extremal tessellated surfaces through transitions induced by wallpaper symmetries. Using this as an example, we develop the foundational principles of partitive geometry and present design methods for space-filling topologically interlocked tiles by integrating insights from historical masonry and textile fabrics. Extending the discussion to three dimensions, we investigate how 3D lattice isometries and their associated Voronoi (Wigner-Seitz) cells yield a natural framework for generating volumetric, cell-transitive honeycombs composed of handlebody tiles of arbitrary genus. Finally, we introduce Bravais Weaves, a vast and rich design space of volumetric entanglements generated by thread topologies induced by Wigner-Seitz cells. Together, these interconnected methodologies demonstrate how systematic spatial partitioning generates architecturally versatile interlocked structures with applications ranging from metamaterials design to architectural assemblies. We conclude by positioning partitive geometry within the broader context of geometric representations for interlocked forms, inviting further reflection on the fundamental nature of interlocking itself.