Acta Crystallographica Section A

Foundations of Crystallography

Volume 69, Part 3 (May 2013)

research papers

Acta Cryst. (2013). A69, 241-261    [ doi:10.1107/S0108767313001670 ]

Periodic entanglement I: networks from hyperbolic reticulations

M. E. Evans, V. Robins and S. T. Hyde

Abstract: High-symmetry free tilings of the two-dimensional hyperbolic plane ({\bb H}^2) 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.

txtdisplay filedownload file

Text file (13.5 kbytes)
[ doi:10.1107/S0108767313001670/eo5019sup1.txt ]
TOPOS input files with minimal degree-2 vertices

txtdisplay filedownload file

Text file (13.5 kbytes)
[ doi:10.1107/S0108767313001670/eo5019sup2.txt ]
TOPOS input files with additional degree-2 vertices


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