Handlebody Plesiohedra Unchained: Topologically Interlocked Cell-Transitive 3-Honeycombs

Abstract

We present an approach for systematic design of generalized Plesiohedra, a new type of 3D space-filling shapes that can even include unchained handlebodies. We call these handlebody plesiohedra unchained, since they are topologically interlocked, i.e., they can be assembled and disassembled without breaking any of the solids apart and they can keep in place with a set of boundary constraints. These space-filling shapes (i.e. congruent prototiles) are obtained from the Voronoi decomposition of symmetric Delone (Delaunay) point sets. To create this new class of shapes, we generalize the design space of classical Plesiohedra by introducing two novel geometric steps: (a) extension of point sites to piecewise linear approximations of higher-dimensional geometries and (b) extension of symmetries to 3D crystallographic symmetries. We show how these specific collections of higher-dimensional geometries can admit the symmetric Delone property. A Voronoi partitioning of 3D space using these specific collections of higher-dimensional shapes as Voronoi sites naturally results in congruent prototiles. This generalizes the idea of classical Plesiohedra by allowing for piecewise linear approximation of curved edges and faces, non-convex boundaries, and even handlebodies with positive genus boundaries to provide truly volumetric material systems in contrast to traditional planar or shell-like systems. To demonstrate existence of these solid shapes, we produced a large set of unchained congruent space-filling handlebodies as proofs of concept. For this, we focused our investigation using isometries of some space-filling polyhedra, such as a cube and a truncated octahedron with circles, and curve complexes as Voronoi sites. These results point to a rich and vast parametric design space of unchained handlebody plesiohedra making them an excellent representations for engineering applications such as topologically interlocked architectured materials.