Extending Goldberg's method to parametrize and control the geometry of Goldberg polyhedra.

Goldberg polyhedra have been widely studied across multiple fields, as their distinctive pattern can lead to many useful applications. Their topology can be determined using Goldberg's method through generating topologically equivalent structures, named cages. However, the geometry of Goldberg polyhedra remains underexplored. This study extends Goldberg's framework to a new method that can systematically determine the topology and effectively control the geometry of Goldberg polyhedra based on the initial shapes of cages. In detail, we first parametrize the cage's geometry under specified topology and polyhedral symmetry; then, we manipulate the predefined independent variables through optimization to achieve the user-defined geometric properties. The benchmark problem of finding equilateral Goldberg polyhedra is solved to demonstrate the effectiveness of the proposed method. Using this method, we have successfully achieved nearly exact spherical Goldberg polyhedra, with all vertices on a sphere and all faces being planar under extremely low numerical errors. Such results serve as strong numerical evidence for the existence of this new type of Goldberg polyhedra. Furthermore, we iteratively perform k-means clustering and optimization to significantly reduce the number of different edge lengths to benefit the cost reduction for architectural and engineering applications.

Curved-crease origami face shields for infection control.

The COVID-19 pandemic has created enormous global demand for personal protective equipment (PPE). Face shields are an important component of PPE for front-line workers in the context of the COVID-19 pandemic, providing protection of the face from splashes and sprays of virus-containing fluids. Existing face shield designs and manufacturing procedures may not allow for production and distribution of face shields in sufficient volume to meet global demand, particularly in Low and Middle-Income countries. This paper presents a simple, fast, and cost-effective curved-crease origami technique for transforming flat sheets of flexible plastic material into face shields for infection control. It is further shown that the design could be produced using a variety of manufacturing methods, ranging from manual techniques to high-volume die-cutting and creasing. This demonstrates the potential for the design to be applied in a variety of contexts depending on available materials, manufacturing capabilities and labour. An easily implemented and flexible physical-digital parametric design methodology for rapidly exploring and refining variations on the design is presented, potentially allowing others to adapt the design to accommodate a wide range of ergonomic and protection requirements.