An Amazing Saddle Polyhedron

1. An Amazing Saddle Polyhedron

2. A truncated tetrahedron (Friauf polyhedron) can be partitioned into tetrahedra and octahedra The ‘Petrie polygon’ of a tetrahedron is a skew quadrilateral; that of an octahedron is a skew hexagon. The angles of these polygons are all 60°. Span them by minimal surfaces and we get:

5. The resulting saddle polyhedron is ‘space filling’. This can be seen by noting the small truncated octahedron: The packing together of the saddle polyhedra corresponds to the bcc packing of the truncated octahedra (the Voronoi regions of a bcc lattice. Coxeter and Petrie discovered new regular polyhedra, which (unlike the Platonic regular solids) are infinite. They partition space into two labyrinthine regions. The faces of {6, 4| 4} are the hexagons (yellow) of this space filling:

6. The triply periodic polyhedron {6, 4| 4} consists of half of the truncated octahedra of the bcc packing. The corresponding structure formed from the saddle polyhedra is Schwarz’s triply periodic minimal surface P (yellow). The whole structure, filling half of space, is obtained by placing only the blue faces of the ‘tiles’ together

8. Schwarz’s D surface (blue) is obtained if the tiles are combined by placing only the yellow faces (hexagons) together: The labyrinths formed this way have the 4-connected diamond network structure. One labyrinth is the solid portion occupied by the polyhedra. The other is the void. They are congruent

9. P surface (yellow). Symmetry of surface Im3m Symmetry of labyrinth Pm3m D surface (blue) Symmetry of surface Pn3m Symmetry of labyrinth Fd3m