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.

翻译：本文提出一种新型空间填充多面体（SFPH），名为Josehedron，其来源于Fischer-Koch S三周期极小曲面（TPMS）的极值点。Josehedron是一种具有12个面（4个等腰三角形与8个镜像对称四边形）、12个顶点和22条边的准多面体。它以每立方晶胞12个实例、6种不同取向的方式密铺三维空间。其生成点集在投影至任意坐标平面时，展现出与五边形开罗铺砌的显著关联。本文还描述了若干附加几何性质，包括整数顶点坐标、交织迷宫结构，以及由函数联合最小值和最大值所获得多面体之间的手性对称性。最后，本文提出一种基于任意周期函数、TPMS或其他函数寻找新型SFPH的通用方法。该方法已应用于一系列TPMS，并在附录中展示了7种此前未被记录的额外SFPH。