Previously, I discussed how to construct a locally hyperbolic tetrahedral building block by puncturing four holes on a tetrahedral C84 along its four tetrahedral axes. This basic unit consists of 12 heptagons, which make the local curvature negative. We can create many different kinds of interesting structures by using this kind of building blocks. One possibility is a dodecahedron. The angle between two axes emanating from the center to two any two vertices (the "tetrahedral angle") of a tetrahedron is 109.47°, which is slightly larger than the inner angle of a dodecahedron (108°). This means that we can use them to construct a dodecahedron without introducing too much local strain. Below is a dodecahedral bead model consisting of 20 such units I just made today. It took me about one week to bead all twenty units and connect them together. More than 2500 8mm faceted beads are used to make this model.
German mathematician Herman Schwarz first proposed P- and D-types triply periodic minimal surfaces (TPMS) in the 19th century. Later, about twenty years ago, Lenosky et al. theoretically suggested graphitic structures with suitable arrangement of seven-membered rings decorated in P- and D-TPMS as possible model structures of sponge carbon. Now, chemists and physicists call this kind of graphitic structures with negative Gaussian curvatures as Schwarzites. The one I have here satisfies this criteria, so we might call it the dodecahedral carbon Schwarzite.