Tilings, coverings, clusters and quasicrystals

The picture demonstates the three ways in which a pair of triacontahedral clusters can interpenetrate in a quasicrystal (Lord, Ranganathan & Kulkarni. Tilings, coverings, clusters & quasicrystals. Curr. Sci. 78 (2000) 64-72).



Picture gallery

Slide shows

   The rhombic triacontahedron consists of ten oblate and ten prolate units


NB: The PDF files are copyrighted material. They are made available here for private study only.

Mackay, A. A dense non-crystalloraphic packing of equal spheres. Acta Cryst. 15 (1962) 916-918.
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Ramakrishnan, K., Lord, E. A. & Ranganathan, S. An algorithm for generating quasiperiodic patterns & their approximants. Bull. Materials Sci. 23 (2000), 119-123.
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Lord, E. A., Ranganathan, S. & Kulkarni, U. D., Tilings, coverings, clusters & quasicrystals. Curr. Sci. 78 (2000) 64-72.

Lord, E. A., Ranganathan, S. & Kulkarni, U. D., Quasicrystals: tiling versus clustering. Phil. Mag. A 81(2001) 2645-2651.
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Lord, E. A., Ranganathan, S. & Subramaniam, A., Stacking sequences and symmetry properties of trigonal vacancy-ordered phases. Phil. Mag. A 82 (2002) 255-268.
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Lord, E. A., Ranganathan, S., Sphere packing, helices and the polytope {3,3,5} EPJ D 15 (2001) 335-343.
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Lord, E. A. & Ranganathan, S., The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001) 531-539.
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Miracle, D. B., Lord, E. A. & Ranganathan, S. (2006) Candidate atomic cluster configurations in metallic glass structures. Materials Transactions 47, 1737-1742.
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Tessellations       Polyhedra      
Structure types     

















Bonner Design Consultancy. Traditional and innovative ornamental design for contemporary Islamic architecture,inspired by the great design traditions of the past

Tessellations.com - manufacturer of various puzzles based on plane tessellations. Website includes the rather Escher-like mathematical art of Robert Fathauer, animated tilings, etc.

Annotated links to over 150 web sites that illustrate symmetry and tessellations.

Hyperbolic tessellations.

Doris Schattschneider's bibliography of tessellations.

Geometry Center resources. Wallpaper groups, periodic tilings, non-periodic tilings, Penrose tilings. Links.

Patterns illustrating the 17 wallpaper groups, with rotational and reflection symmetries shown.

Xah Lee's website. Mathematical introduction to the wallpaper groups, including orbifold notation.

Xah Lee's bibliography and links to other websites.

Xah Lee The Discontinuous Groups of Rotation and Translation in the Plane. (An undergraduate exposition on wallpaper groups, with illustrations of the 17 wallpaper groups and a wallpaper gallery. Technically, the derivation of all discontinuous groups of rotation and translation in the plane.)

David Joyce. Wallpaper Groups. (Patterns illustrating the 17 wallpaper groups, with rotational and reflection symmetries shown. Includes a short history, and a bibliography.)

Sebastian Truchet's tiling patterns.

An article by D. L. D. Caspar & E. Fontano 'pentilings' (arrangements of regular pentagons in the plane, making edge-to-edge contact with no gaps in the arrangement large enough to contain another pentagon.


Mathworld: polyhedra
Mathworld: Archimedean polyhedra
Mathworld: Platonic polyhedra

V. Bulatov's very extensive collection of models of polyhedra.

George Hart is a sculptor. His sculptures are inspired by geometrical concepts, especially icosahedral symmetry, and are fascinating.

George Hart's 'Encyclopoedia of Polyhedra': the largest online reference source for polyhedra.

Some fascinating constructions made from the Zometool construction kit.

A biblography: books in English dealing with polyhedra.

Magnus Wenninger's homepage. Wenninger has constructed hundreds of polyhedron models, and published books describing his methods. This website displays photographs of selection of over a hundred models.

Buckminster Fuller's Synergetics [1975].

Lectures on the mathematics of symmetry in art, biology and mathematics. The research group on Computation, Vision and Geometry (CVG) was formed by Amir Assadi in 1988 to promote the role of geometric methods and visualization of geometric structures in scientific research and education.

Invisible Architecture: the Nano World of Buckminster Fuller by Bonnie Goldstein DeVarco. An appraisal of the relevance, especially in biology and materials science. of Fuller's ideas. Contains numerous links to other relevant websites.

Izidor Hafner & Tomislav Zitko. 'From dissection of the cube to space filling with prolate rhombohedra and rhombic dodecahedra of the second kind'.

VRML polyhedron models.

Structure Types

Basic mineral structure types described and illustrated.

Crystal lattice structures. Structure types clearly described, classified and extensively illustrated.

The Structure of matter Group: A research group in the Institute of Physics at Universidad Nacional Autónoma de México. Quasicrystals, Fullerenes and Curved Structures. Mathematical Crystallography.

3D modelling of crystalline and molecular structures. In particular: excellent introduction to minerals based on tetrahedral and octahedral clusters.

Structures of simple inorganic solids; oxide structures & networks, by S.J. Heyes. Fourth of Four Lectures in the 1st Year Inorganic Chemistry Course. Nice graphics.

Steffen Weber's home page. Extensive VRML displays. Nanotubes. Fullerenes. Wire frame polyhedra. Structure types.


Johnathan Doye's homepage. Extensive investigation, by simulation techniques, of cluster growth.

The Cambridge Cluster Database. Rare gas clusters. Metal clusters. Molecular clusters (including C60 clusters and water clusters). Ionic clusters. Results of simulation using a variety of potentials. Extensive data, and lists of published papers.

Birmingham Cluster Web.

Some quite remarkable graphics illustrating complex intermetallic phases.