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 publications in the area as well as a list of recommended 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 should include here, 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.

EPINET-relevant publications

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). [DOI link]
• S.T. Hyde and S.J. Ramsden, “Polycontinuous morphologies and interwoven helical networks”, Europhys. Lett., 50, 135–141, (2000). [DOI 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).
• S.T. Hyde, S.J. Ramsden, V. Robins. "Unification and classification of two-dimensional crystalline patterns using orbifolds" Acta Cryst. A: Foundations of Crystallography 70, 4(2014) 319-337 [DOI link]
• M.C. Pedersen & S.T. Hyde, “Polyhedra and packings from hyperbolic honeycombs.” Proceedings of the National Academy of Sciences, 115(27), 6905-6910. (2018). [DOI link]
• S.T. Hyde and M.C. Pedersen, “Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries”, Proceedings of the Royal Society A, 477 (2021). [DOI link]
• M.C. Pedersen, V. Robins and S.T. Hyde, “The intrinsic group–subgroup structures of the Diamond and Gyroid minimal surfaces in their conventional unit cells”, Acta Cryst. A: Foundations and Advances 78(1) 56–58(2022). [DOI link]
• M. Evans, V. Robins, S.T. Hyde. “Periodic entanglement II: Weavings from hyperbolic line patterns” Acta Crystallographica A: Foundations of Crystallography 69, 262–275 (2013). [DOI link]
• M. Evans, V. Robins, S.T. Hyde. “Periodic entanglement I: Networks from hyperbolic reticulations” Acta Cryst. A 69, 241–261 (2013). [DOI link]

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]

Hyperbolic geometry, symmetry, and tessellations

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)
• NonEuclid provides another overview of 2D hyperbolic geometry with interactive graphics aimed at high school and undergraduate students.
• A gallery of hyperbolic tessellation 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

• Olaf Delgado–Friedrichs's introduction to 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.
• B. Kolbe and M.E. Evans, “Isotopic tiling theory for hyperbolic surfaces” Geom. Dedicata 212, 177–204 (2021). [DOI link]
• B. Kolbe, and M.E. Evans, “Enumerating isotopy classes of tilings guided by the symmetry of triply-periodic minimal surfaces”, Forthcoming in SIAM Journal on Applied Algebra and Geometry (2022) [arXiv link]
• B. Kolbe and V. Robins, “Tile-transitive tilings of the Euclidean and hyperbolic planes by ribbons”, to appear in Research in Computational Topology, vol 2. Springer (2022) [preprint]
• Z. Lucic, E. Molnar and N. Vasiljevic, “An algorithm for classification of fundamental polygons for a plane discontinuous group.”. In: M.D.E. Conder, A. Deza, A.I. Weiss (eds.) Discrete Geometry and Symmetry, pp. 257–278, Springer (2018).

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.
• The Samara Topological Data Center database topcryst. And see also Vladislav Blatov's TOPOS package for the analysis of crystal structures including interpenetrating nets.
• MaThCryst: an international workgroup on mathematical and theoretical crystallography.
• For information on the 3D Euclidean space groups, see the Bilbao Crystallographic Server and the International Tables for Crystallography.
• Omar Yaghi's database for naming and numbering new crystalline extended framework materials.

Books and papers

• M. O'Keeffe and B.G. Hyde. “Crystal structures: patterns and symmetry”, Dover Publications (2020). An excellent text on crystallography and inorganic crystal structure, now updated from the original (1996) version.
• 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]

Minimal surfaces and differential geometry

Minimal surfaces and applications in the natural sciences

• Ken Brakke's extensive catalogue of periodic mimimal surfaces from his surface evolver package.
• A collection of physical models of minimal surfaces created in honour of Hermann Karcher.
• 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]
• Alan Schoen. "Reflections concerning triply-periodic minimal surfaces" Interface Focus 2, 658–668 (2012) [DOI 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).