EPINET Project Bibliography

The material provided in the EPINET Project website touches on a number of important topics. If you'd like to explore these topics further, here's a collection of our publications in the area as well as a list of recommended publications, books, and websites you might find useful.

Disclaimer: This is not intended to be an exhaustive bibiliography, but if you think there is a reference we have overlooked, please contact us and we'll do our best to include it. For publications by the EPINET project researchers we've included (where possible) a link to the DOI or other online version of each paper. If any of these links are unavailable please contact us to arrange reprints.

Our Recent Papers on The Hyperbolic Surface Tiling Approach

• S.J. Ramsden, V. Robins and S.T. Hyde, “Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples”, Acta Cryst. A 65, 81–108, (2009). [DOI link]
  This is the main reference for the techniques we use to generate the tilings and nets in the EPINET database.
• S.T. Hyde, O. Delgado Friedrichs, S.J. Ramsden, V. Robins. “Towards enumeration of crystalline frameworks: the 2D hyperbolic approach”, Solid State Sciences 8, 740–752 (2006). [DOI link]
• V. Robins, S. Ramsden, and S.T. Hyde, “A note on the two symmetry-preserving covering maps of the gyroid minimal surface”, Euro. Phys. J. B 48, 107–111 (2005). [DOI link]
• V. Robins, S.J. Ramsden and S.T. Hyde, “2D hyperbolic groups induce three–periodic euclidean reticulations”, Eur. Phys. J. B, 39, 365–375, (2004). [DOI link] Supplementary online material.
• V. Robins, S.J. Ramsden, S.T. Hyde, “Symmetry groups and reticulations of the hexagonal H surface”, Physica A, 339, pp. 173–180 (2004). [DOI link] Supplementary online material.
• S.T. Hyde, A.–K. Larsson, T. Di Matteo, S.J. Ramsden and V. Robins, “Meditation on an Engraving of Fricke and Klein (The Modular Group and Geometrical Chemistry)”, Aust. J. Chem., 56, 981–1000, (2003). [DOI link]
• S.T.Hyde and S.J. Ramsden, “Some novel three–dimensional euclidean crystalline networks derived from two-dimensional hyperbolic tilings”. Eur. Phys. J. B, 31, 273–284, (2003). [DOI link]
• S.T. Hyde, S.J. Ramsden, T. Di Matteo and J. Longdell, “Ab–initio construction of some crystalline 3D euclidean networks”, Solid State Sciences, 5, 35-45, (2003). [DOI link]
• S.T. Hyde and C. Oguey, “From hyperbolic 2D forests to 3D euclidean entangled thickets”, Eur. Phys. J. B, 16, 613–630 (2000). [EDP sciences link]
• S.T. Hyde and S.J. Ramsden, “Polycontinuous morphologies and interwoven helical networks”, Europhys. Lett., 50, 135–141, (2000). [EDP sciences link]
• S.T. Hyde and S.J. Ramsden, “Chemical frameworks and hyperbolic tilings”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 51, “Discrete Mathematical Chemistry”, Pierre Hansen, Patrick Fowler and Maolin Zheng (eds), 203–224 (2000).
• S.T. Hyde and S.J. Ramsden, “Crystals. Two–dimensional non–euclidean geometry and topology”, pp. 35-174, in “Chemical Topology: Applications and Techniques”, Mathematical Chemistry Series, vol. 6, D. Bonchev and D.H. Rouvray (eds.), Gordon and Breach Science Publishers, N.Y. (1999).

Earlier papers on the role of hyperbolic surfaces in crystalline structure

• S.T. Hyde and S. Andersson, “A systematic net description of saddle polyhedra and periodic minimal surfaces”, Z. Kristallogr., 168, 221–254, (1984).
• S.T. Hyde, “Hyperbolic surfaces in the solid–state and the structure of ZSM–5 zeolites”, Acta Chem. Scand., 45, 860–863, (1991).
• A. Fogden and S.T. Hyde, “Parametrisation of triply periodic minimal surfaces. 1. Mathematical basis of the construction algorithm for the regular class”, Acta Cryst., A48, 442–451, (1992). [DOI link]
• A. Fogden and S.T. Hyde, “Parametrisation of triply periodic minimal surfaces. 2. Regular class solutions”, Acta Cryst., A48, 575–591, (1992). [DOI link]
• S.T. Hyde, “Crystalline frameworks as hyperbolic films”, in "Defects and processes in the solid state: Geoscience applications", J.N. Boland, J. D. FitzGerald, Editors, Elsevier: Amsterdam, (1993).
• S.T. Hyde, “Hyperbolic and elliptic layer warping in some sheet alumino–silicates”, Phys. Chem. Minerals, 20, 190–200, (1993).
• S.T. Hyde and M. O'Keeffe, “Elastic warping of graphitic carbon sheets: relative energies of some fullerenes, schwarzites and buckytubes”, Phil. Trans. R. Soc. Lond. A, 354, 1999–2008, (1996). [JSTOR link]
• “The Language of Shape”, S.T. Hyde, S. Andersson, Z. Blum, S. Lidin, K. Larsson, T. Landh and B.W. Ninham, Elsevier Science B.V. (Amsterdam), 1997. (Reviewed in Nature, 387, p. 249, 1997.) [Nature link]
• Banglin Chen, M. Eddaoudi, S.T. Hyde, M. O'Keeffe and O.M. Yaghi, “Interwoven metal–organic framework on a periodic minimal surface with extra-large pores”, Science 291:1021–994 (2001). [DOI link]

Suggested Further Reading

Minimal surfaces

• Ken Brakke's extensive catalogue of periodic mimimal surfaces from his Evolver package.
• Jim and David Hoffman (MSRI) have an image gallery of many surfaces.
• An archive of Mathematica notebooks is maintained by Mathias Weber.
• The surface geometry group at Granada also maintain a list of links.
• The crystallographers Elke Koch and Werner Fischer have described many balanced periodic minimal surfaces via 3D space groups.
• Eric Lord has a novel view of the area, based more in "experimental mathematics".
• A collection of physical models of minimal surfaces created in honour of Hermann Karcher.

Work applying periodic minimal surfaces in the natural sciences

• A key antecedent to our work is the paper “Infinite Periodic Minimal Surfaces and their Crystallography in the Hyperbolic Plane”, by J.–F. Sadoc and J. Charvolin. Acta Cryst. A45:10–20 (1989). [DOI link]
• Reinhard Nesper (ETH Zürich) and Stefano Leoni have used the concept of Periodic Nodal Surfaces (developed by Nesper and Hans Georg von Schnering) to derive hyperbolic surface structures and reticulations.
• A more general site, Sandforsk, maintained by Sten Andersson and dealing with the description of structures in nature.
• For a collection of papers see the special issue on curved surfaces in chemical structure published as volume 354, Series A of the Philosophical Transactions of the Royal Society of London (1996). [JSTOR link]

Standard texts on differential geometry include discussions of minimal surfaces

• A. Goetz, “Introduction to Differential Geometry” (Addison–Wesley, 1970)
• M. Do Carmo, “Differential Geometry of Curves and Surfaces” (Prentice Hall, 1976)

More detailed differential geometry references

• M. Spivak, “A Comprehensive Introduction to Differential Geometry” (Publish or Perish, 1970-75). For the complete differential geometer, in five volumes!
• Dierkes, Hildebrandt, Küster and Wohlrab, “Minimal Surfaces” (Springer, 1992). Two volumes with excellent historical notes.
• J. Nitsche, “Lectures on minimal surfaces” (Cambridge, 1989).

Hyperbolic geometry, symmetry, and tesselations

The hyperbolic plane online

• An excellent introduction to 2D hyperbolic geometry without formal mathematics is available from The Institute for Figuring.
• The Hyperbolic Geometry section of the Geometry and the Imagination in Minneapolis handouts by John Conway, Peter Doyle, Jane Gilman, Bill Thurston. (from 1991, revised 1994)
• An illustrated and animated tutorial on 2D Euclidean and Hyperbolic geometry.
• NonEuclid provides another overview of 2D hyperbolic geometry with interactive graphics aimed at high school and undergraduate students.

Hyperbolic tesselations

• A gallery of images from Don Hatch.
• David Eppstein has some good links on his Geometry Junkyard page about hyperbolic tilings.

General reference books for the mathematically inclined

• Roberto Bonola, “Non–Euclidean geometry” (Dover, 1955). Contains translations of the original papers by Bolyai and Lobachevski.
• John Stillwell, “Sources of hyperbolic geometry” (American Mathematical Society/ London Mathematical Society, 1996). Introduces hyperbolic geometry through translations of original papers by Beltrami, Klein, and Poincaré.
• John Stillwell, “Geometry of surfaces” (Springer, 1996 corrected reprint)
• D. Brannan, M.F. Esplen, and J. Gray. “Geometry” (Cambridge, 1999).

Tiling patterns

• A general introduction to the mathematical description of tilings and symmetries, mostly of the euclidean plane.
• A tutorial essay by Kevin Mitchell on tilings of the sphere, euclidean, and hyperbolic planes.
• Dror Bar–Natan has a delightfully eclectic gallery of tiling patterns.
• B. Grünbaum and G.C. Shephard. “Tilings and Patterns” (W.H. Freeman, 1987).
  This is the definitive source on tilings of the euclidean plane.
• Doris Schattschneider, “M.C. Escher: Visions of Symmetry” (H.N. Abrams, 2004 2nd ed.).
  A beautiful book that explains the mathematics behind Escher's tiling pictures.
• John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss, “The Symmetries of Things” (A.K. Peters, 2008).
  A beautifully illustrated book about symmetry and tiling.

2D Orbifolds and surface topology

• J.H. Conway, “The orbifold notation for surface groups.” In Groups, combinatorics, and geometry, vol. 165 of LMS Lecture Notes, pp. 438–447. (Cambridge, 1992).
• J.H. Conway and D.H. Huson. “The orbifold notation for two–dimensional groups.” Structural Chemistry, 13:247–256, 2002. [DOI link]
• John Conway, Peter Doyle, Jane Gilman, Bill Thurston. “Geometry and the Imagination in Minneapolis”, June 17–28 1991. The Symmetry and Orbifolds section of the course notes.
• G.K. Francis and J.R. Weeks. “Conway's ZIP proof.” American Mathematical Monthly, 106:393–399, 1999. The paper and figures are available online.

Combinatorial Tiling Theory

• Daniel Huson's essay on Combinatorial Tiling Theory.
• Olaf Delgado–Friedrichs's introduction to Data structures and algorithms for tilings. Published as “Data structures and algorithms for tilings I”. Theoretical Computer Science 303:431–445 (2003). [DOI link]
• A.W.M. Dress. “Presentations of discrete groups, acting on simply connected manifolds”. Advances in Mathematics 63:196–212 (1987).
  This paper contains the formal definitions and proofs regarding Delaney–Dress symbols.
• A.W.M. Dress and D.H. Huson. “On tilings of the plane.” Geometriae Dedicata 24:269–296 (1987).
• D.H. Huson. “The generation and classification of tile–k–transitive tilings on the Euclidean plane, sphere, and hyperbolic plane.” Geometriae Dedicata 47:295–310 (1993).
  Describes the split and glue operations for deriving more complex tilings from fundamental ones.
• Tilings of the sphere by Elizaveta Zamorzaeva.

Periodic networks and crystal structures

Databases and online resources

• Olaf Delgado's collection of software for the Generation, Analysis and Visualization of Reticular Ornaments using GAVROG.
• Michael O'Keeffe's Reticular Chemistry Structure Resource, RCSR.
• IZA Databases of Zeolite Structures, including Framework types and hypothetical zeolites.
• The Inorganic crystal structure database.
• Vladislav Blatov's TOPOS package for the analysis of crystal structures including interpenetrating nets.
• A table of examples of interpenetrating structures compiled by Stuart Batten and Richard Robson.
• MaThCryst: an international workgroup on mathematical and theoretical crystallography.
• Elke Koch summarises aspects of her research into crystallography, in particular minimal surfaces and sphere packings.
• For information on the 3D Euclidean space groups, see the Bilbao Crystallographic Server and the International Tables for Crystallography.

Books and papers

• M. O'Keeffe and B.G. Hyde. “Crystal Structures: I Patterns and Symmetry” (Mineralogical Society of America, 1996). Now out of print, but an excellent text on crystallography and inorganic crystal structure.
• A.F. Wells “Three–dimensional nets and polyhedra” (Wiley, 1977). The first book to systematically describe nets and polyhedra of interest in solid state chemistry.
• O. Delgado–Friedrichs and M. O'Keeffe, “Identification of and symmetry computation for crystal nets”, Acta Cryst. A 59: 351–360 (2003). This paper describes a canonical form for euclidean periodic nets without collisions. [DOI link]
• O. Delgado–Friedrichs, A.W.M. Dress, D.H. Huson, J. Klinowski, and A.L. Mackay. “Systematic enumeration of crystalline networks” Nature 400: 644–647 (1999). An application of 3D euclidean tiling theory to the enumeration of 4–coordinated 3–periodic nets. [DOI link]
• Georg Thimm, “Crystal Structures and Their Enumeration via Quotient Graphs” Zeitschrift für Kristallographie 219: 528–536 (2004). [DOI link]
• W.E. Klee, “Crystallographic nets and their quotient graphs”. Cryst. Res. Technol., 39:959–968 (2004). [DOI link]
• J.-G. Eon, “Infinite geodesic paths and fibers, new topological invariants in periodic graphs” Acta Cryst. A 63:53–65 (2007). [DOI link]
• M. O'Keeffe, M. A. Peskov, S. J. Ramsden, O. M. Yaghi, “The Reticular Chemistry Structure Resource (RCSR) Database of, and Symbols for, Crystal Nets” Accts. Chem. Res. 2008, 41, 1782-1789. [DOI link]