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High-symmetry free tilings of the two-dimensional hyperbolic plane (
) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.
) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.Keywords: periodic entanglement; networks; hyperbolic reticulations; tilings; ideal knots; tight knots; crystal nets; frameworks; surfaces; graphs; catenation; enumeration.
Supporting information
 | Text file https://doi.org/10.1107/S0108767313001670/eo5019sup1.txt |
| Text file https://doi.org/10.1107/S0108767313001670/eo5019sup2.txt |
