The Josehedron: A space-filling plesiohedron based on the Fischer-Koch S Triply Periodic Minimal Surface

TL;DR

This paper introduces the Josehedron, a new space-filling polyhedron derived from the Fischer-Koch S TPMS, with unique geometric properties and a general method for discovering similar structures.

Contribution

The paper presents the Josehedron, a novel space-filling polyhedron based on a specific TPMS, and introduces a general method for finding new SFPHs from various functions.

Findings

Josehedron tiles space with 12 orientations

Connection to pentagonal Cairo tiling in projections

Discovered 7 new SFPHs using the proposed method

Abstract

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.