Tessellations       Orbifolds       Polyhedra      
Nanotubes       Zeolites       Networks and space fillings      
Surfaces       Structure types       Space groups      
Clusters       General








Tesselations - 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-level 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.

Online version of 'Symmetry and Ornament' by S. V. Jablan [1995]. An aesthetically pleasing introduction to the wallpaper groups.

Carroll K. Johnson and Michael N. Burnett. Crystallographic Topology: The Topology of Crystallographic Groups and Simple Crystal Structures. An introduction on the topic of crystallographic topology. This site is extremely technical, and is aimed at professional crystallographers.

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).


Silvio Levy. Geometry Formulas and Facts. An excerpt of CRC Press' reference work, including sections on plane symmetry and wallpaper groups. The latter explains the orbifold notation.

The Topology of Crystallographic Groups and Simple Crystal Structures, by C. K. Johnson & M. N. Burnett. An introduction to two- and three-dimensional orbifolds.


Mathworld: polyhedra
Mathworld: Archimedean polyhedra
Mathworld: Platonic polyhedra

Links to geometry websites, including tessellations, polyhedra and polytopes.

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.


The Nanotube Site. Very large number of links to sites dealing with nanotubes, compiled by David Tomanek.


Database of Zeolite Structures. The Structure Commission of the International Zeolite Association (IZA-SC) was established to evaluate zeolite structural data in the literature and to prepare compilations of those data for the benefit of the zeolite community.

International Committee on Natural Zeolites. Listing of selected internet links.

Goeffrey L. Price's zeolite page. A short introduction 'what is a zeolite?' and some good graphics illustrating a few zeolite structures.

M. Hewat. Zeolite catalysts and sieves: microscopic architecture.

University of Colorado Mineral Structure Data Base.

Networks & Space Fillings

Homepage of Olaf Delgado Friedrichs. Combinatorial tiling theory. Some beautiful graphics illustrating space filling by saddle polyhedra.

Daniel Huson's Home Page. Combinatorial tiling theory. Papers & software.

Microporous solids: cages, channels and voids.

Hyde & Ramsden's 'Polycontinuous morphologies and interwoven helical networks' (Europhys. Lett., 50 (2000) 135-141).


Ken Brakke's website. Surface Evolver can be downloaded. An introduction to triply periodic minimal surfaces is given, with numerous illustrations, and source codes for use with Surface Evolver.

Parametrisation of the Costa surface, for Mathematica.

Graphitic triply periodic minimal surfaces.

Gozdz & Holyst's home page. Triply periodic surfaces.

Stephen Hyde's home page. List of publications. Downloadable reprints. Differential geometry, topology and non-euclidean geometry of 2d surfaces, and applications to condensed atomic and molecular crystals and liquid crystals, etc.

Indiana University Minimal Surface Gallery. 29 artistic ray traced images.

Konrad Polthier's home page. Books on mathematical visualization. Mathematical visualization software. Videos on minimal surfaces, soap films, and geodesics.

An introduction to minimal surfaces, by Hermann Karcher and Konrad Polthier.

An introduction to minimal surfaces with some nice graphics.

Stewart Ramsden. Mesoscale research: Crystal frameworks and hyperbolic tilings; embeddings of 3D nets in triply periodic surfaces.

University of Granada. Minimal surfaces, constant mean curvature and related topics.

PDF and HTML versions of Wohlgemuth et al. Triply periodic bicontinuous cubic microdomain morphologies by symmetries. Macromolecules 34 (2001) 6083-6089.

The wonderful fabrications of Sho Takahashi - sculpture inspired by minimal surfaces and other geometrical concepts.

Structure Types

Basic mineral structure types described and illustrated.

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

Space Groups

The Bilbao Crystallographic Server is a web site with crystallographic programs and databases accessible via Internet, created by the Materials Laboratory at University of Basque Country, Spain.


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.


'Connections in Space'. A very extensive website with visualisations of a wide range of geometrical concepts. Divided into 'themes': Spatial Visions, Form & Structure, Randomness & Order, Inner & Outer Space, Mapping, Patterns & Space Filling, Higher Dimensions.

The Geometry Center is now no longer active, but its extensive archives are maintained. Downloadable software includes Surface Evolver.

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.

Werner Fischer
Elke Koch
The contributions of Fischer & Koch to mathematical crystallography. Periodic minimal surfaces, sphere packings, Dirichlet domains, normalizers, lattice complexes, description of crystal structures, International Tables. The information on TPMS in particular is very thorough and extensive.

Morwen Thistlethwaite's home page. Beautiful computer generated images of symmetrical knots.

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.