The Josehedron: A space-filling plesiohedron based on the Fischer-Koch S Triply Periodic Minimal Surface
This work advances geometric design and materials science by providing a method to generate novel space-filling structures, though it is incremental as it builds on existing TPMS concepts.
The paper tackles the problem of discovering new space-filling polyhedra by introducing the Josehedron, derived from the Fischer-Koch S triply periodic minimal surface, which tiles 3D space with 12 instances per cubic unit cell and exhibits connections to pentagonal Cairo tilings.
This paper presents a novel space-filling polyhedron (SFPH), here named the Josehedron, derived from the extremal points of the Fischer-Koch S triply periodic minimal surface (TPMS). The Josehedron is a plesiohedron with 12 faces (4 isosceles triangles and 8 mirror-symmetric quadrilaterals), 12 vertices, and 22 edges. It tiles three-dimensional space with 12 instances per cubic unit cell in 6 distinct orientations. The generating point set exhibits a remarkable connection to the pentagonal Cairo tiling when projected onto any coordinate plane. Several additional geometric properties are described, including integer vertex coordinates, interwoven labyrinths, and chiral symmetry between the polyhedra obtained from the combined minima and maxima of the function. Finally, the paper presents a general method for finding novel SFPHs based on any periodic function, TPMS, or other functions. The described method is applied to a selection of TPMS, and 7 additional, previously undocumented SFPH are shown in the Appendix.