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Related Experiment Video

Updated: Jun 7, 2026

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates

Published on: February 15, 2016

Simple tilings by polyhedra with five- and six-sided faces.

Olaf Delgado-Friedrichs¹, Michael O'Keeffe

¹Supercomputer Facility, The Australian National University, Canberra, ACT 2600, Australia.

Acta Crystallographica. Section A, Foundations of Crystallography

|October 22, 2010

Summary

Researchers discovered 13 unique space-filling arrangements using fullerene-like shapes, including dodecahedra. These novel fullerene tilings and their duals reveal new Frank-Kasper structures with higher coordination numbers.

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Last Updated: Jun 7, 2026

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates

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Published on: February 15, 2016

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Fabrication of Three-Dimensional Graphene-Based Polyhedrons via Origami-Like Self-Folding

Fabrication of Three-Dimensional Graphene-Based Polyhedrons via Origami-Like Self-Folding

Published on: September 23, 2018

Area of Science:

Crystallography
Materials Science
Geometry

Background:

Space-filling with polyhedra is fundamental in crystallography and materials science.
Fullerenes, simple polyhedra with 5- and 6-sided faces, offer unique geometric properties.
Frank-Kasper phases represent complex intermetallic structures with specific coordination polyhedra.

Purpose of the Study:

To systematically explore and report tilings of space using fullerene-type polyhedra.
To investigate the diversity of vertex configurations (up to 11-transitive) in these fullerene tilings.
To analyze the dual structures and their relationship to known and novel Frank-Kasper phases.

Main Methods:

Enumeration and classification of space-filling polyhedral tilings.
Identification of constituent polyhedra, focusing on those with 5- and 6-sided faces (fullerenes).
Analysis of vertex transitivity and dual graph structures.

Main Results:

Thirteen distinct tilings of space by simple polyhedra (fullerenes) were identified.
All identified tilings incorporate dodecahedra and at least one of nine other tile types.
The duals of these tilings correspond to tetrahedral tilings, including four known Frank-Kasper intermetallic phases.
A novel fifth Frank-Kasper structure, composed solely of Frank-Kasper coordination polyhedra, exhibits an unprecedentedly high average coordination number.

Conclusions:

The study presents a comprehensive set of fullerene-based space-filling tilings.
These findings expand the known repertoire of polyhedral tilings and their geometric diversity.
The discovered dual structures provide new insights into Frank-Kasper phases, including a novel structure with a record coordination number, potentially impacting intermetallic compound research.