A K-hedral tiling of a 2D finite domain is a covering of the domain with tiles without gaps or overlaps, where each tile is congruent to one of the K distinct shapes called prototiles. K, the number of prototiles, is preferred to be as small as possible for congruent tiling appearance and reducing fabrication cost, e.g., by molding. Typically, a forward approach is adopted to produce K-hedral tilings by prescribing a set of prototiles and placing prototile instances (i.e., tiles) to cover the input domain. However, the prescribed prototile set may not be sufficient to tile the domain (for small K) or may lead to tiling results with excessive prototiles more than needed (for large K).

Abstract A body‐supporting assembly is an assembly of parts that physically supports a human body during activities like sitting, lying, or leaning. A body‐supporting assembly has a complex global shape to support a specific human body posture, yet each component part has a relatively simple geometry to facilitate fabrication, storage, and maintenance. In this paper, we aim to model and design a personalized body‐supporting assembly that fits a given human body posture, aiming to make the assembly comfortable to use. We choose to model a body‐supporting assembly from scratch to offer high flexibility for fitting a given body posture, which however makes it challenging to determine the assembly's topology and geometry. To address this problem, we classify parts in the assembly into two categories according the functionality: supporting parts for fitting different portions of the body and connecting parts for connecting all the supporting parts to form a stable structure. We also propose a geometric representation of supporting parts such that they can have a variety of shapes controlled by a few parameters. Given a body posture as input, we present a computational approach for designing a body‐supporting assembly that fits the posture, in which the supporting parts are initialized and optimized to minimize a discomfort measure and then the connecting parts are generated using a procedural approach. We demonstrate the effectiveness of our approach by designing body‐supporting assemblies that accommodate to a variety of body postures and 3D printing two of them for physical validation.

We study a problem of modeling wireframe meshes where the vertices and edges fall into a set of discrete equivalence classes, respectively. This problem is motivated by the need of fabricating large wireframe structures at lower cost and faster speed since both nodes (thickened vertices) and rods (thickened edges) can be mass-produced. Given a 3D shape represented as a wireframe mesh, our goal is to compute a set of template vertices and a set of template edges, whose instances can be used to produce a fabricable wireframe mesh that approximates the input shape. To achieve this goal, we propose a computational approach that generates the template vertices and template edges by iteratively clustering and optimizing the mesh vertices and edges. At the clustering stage, we cluster mesh vertices and edges according to their shape and length, respectively. At the optimization stage, we first locally optimize the mesh to reduce the number of clusters of vertices and/or edges, and then globally optimize the mesh to reduce the intra-cluster variance for vertices and edges, while facilitating fabricability of the wireframe mesh. We demonstrate that our approach is able to model wireframe meshes with various shapes and topologies, compare it with three state-of-the-art approaches to show its superiority, and validate fabricability of our results by making three physical prototypes.

This paper proposes a method to model masonry shell structures where the shell elements fall into a set of discrete equivalence classes. Such shell structure can reduce the fabrication cost and simplify the physical construction due to reuse of a few template shell elements. Given a freeform surface, our goal is to generate a small set of template shell elements that can be reused to produce a seamless and buildable structure that closely resembles the surface. The major technical challenge in this process is balancing the desire for high reusability of template elements with the need for a seamless and buildable final structure. To address the challenge, we define three error metrics to measure the seamlessness and buildability of shell structures made from discrete equivalence classes and develop a hierarchical cluster-and-optimize approach to generate a small set of template elements that produce a structure closely approximating the surface with low error metrics. We demonstrate the feasibility of our approach on various freeform surfaces and geometric patterns, and validate buildability of our results with four physical prototypes. Code and data of this paper are at https://github.com/Linsanity81/TileableShell.

Interlocking puzzles are intriguing geometric games where the puzzle pieces are held together based on their geometric arrangement, preventing the puzzle from falling apart. High-level-of-difficulty , or simply high-level , interlocking puzzles are a subclass of interlocking puzzles that require multiple moves to take out the first subassembly from the puzzle. Solving a high-level interlocking puzzle is a challenging task since one has to explore many different configurations of the puzzle pieces until reaching a configuration where the first subassembly can be taken out. Designing a high-level interlocking puzzle with a user-specified level of difficulty is even harder since the puzzle pieces have to be interlocking in all the configurations before the first subassembly is taken out. In this paper, we present a computational approach to design high-level interlocking puzzles. The core idea is to represent all possible configurations of an interlocking puzzle as well as transitions among these configurations using a rooted, undirected graph called a disassembly graph and leverage this graph to find a disassembly plan that requires a minimal number of moves to take out the first subassembly from the puzzle. At the design stage, our algorithm iteratively constructs the geometry of each puzzle piece to expand the disassembly graph incrementally, aiming to achieve a user-specified level of difficulty. We show that our approach allows efficient generation of high-level interlocking puzzles of various shape complexities, including new solutions not attainable by state-of-the-art approaches.