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Updated: Oct 8, 2025
06:35
Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
Published on: February 15, 2016
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Symmetric tangled Platonic polyhedra.
Stephen T Hyde1, Myfanwy E Evans2
1School of Chemistry, The University of Sydney, Sydney, New South Wales 2006, Australia; stephen.hyde@sydney.edu.au.
Summary
This study untangles Platonic polyhedra embeddings, revealing that maximally symmetric tangled polyhedra are topologically constrained as knots or links, with simpler forms appearing in nature.
Area of Science:
- Geometric Topology
- Graph Theory
- Materials Science
- Structural Biology
Background:
- Conventional embeddings of Platonic polyhedra edge-graphs are untangled, allowing crossing-free spherical representations analogous to unknotted loops.
- The symmetries of classical Platonic polyhedra are denoted as *2fz in Conway's 2D orbifold notation.
Purpose of the Study:
- To investigate the construction and symmetries of tangled Platonic polyhedra.
- To analyze the topological constraints and potential applications of these complex structures.
Main Methods:
- Construction of tangled polyhedra by winding helices on multigenus surfaces derived from Platonic polyhedra.
- Analysis of symmetries using Conway's 2D orbifold notation and comparison with existing polyhedral structures.
- Topological classification of tangled polyhedra as self-entangled graphs (knots) or catenated compounds (links).
Main Results:
- Tangled Platonic polyhedra, with symmetries 2fz, are maximally symmetric embeddings; more symmetric ones are untangled.
- These structures exhibit constrained topologies, either as knots or links, with curvilinear edges due to helicity.
- Simpler entangled polyhedra exhibit patterns found in synthetic organometallic materials and clathrin coats.
Conclusions:
- Maximally symmetric polyhedral embeddings can be tangled, exhibiting topological constraints analogous to knots and links.
- The study provides a framework for understanding complex polyhedral structures and their potential natural occurrences.
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