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.
EPINETrelevant publications

S.J. Ramsden, V. Robins and S.T. Hyde,
“Threedimensional Euclidean nets from twodimensional 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 symmetrypreserving 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 twodimensional 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, 3545, (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. 35174, 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 twodimensional crystalline patterns using orbifolds"
Acta Cryst. A: Foundations of Crystallography 70, 4(2014) 319337
[DOI link]

M.C. Pedersen & S.T. Hyde, “Polyhedra and packings from hyperbolic honeycombs.” Proceedings of the National Academy of Sciences, 115(27), 69056910. (2018).
[DOI link]

S.T. Hyde and M.C. Pedersen, “Schwarzite nets: a wealth of 3valent 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]

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 extralarge pores”,
Science 291:1021–994 (2001).
[DOI link]
Hyperbolic geometry, symmetry, and tessellations

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).

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 GoodmanStrauss,
“The Symmetries of Things”
(A.K. Peters, 2008).
A beautifully illustrated book about symmetry and tiling.

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.

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 triplyperiodic minimal surfaces”, Forthcoming in SIAM Journal on Applied Algebra and Geometry (2022) [arXiv link]

B. Kolbe and V. Robins, “Tiletransitive 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

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, 17821789. [DOI link]
Minimal surfaces and differential geometry

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 triplyperiodic minimal surfaces" Interface Focus 2, 658–668 (2012)
[DOI link]

A. Goetz, “Introduction to Differential Geometry” (Addison–Wesley, 1970)

M. Do Carmo, “Differential Geometry of Curves and Surfaces” (Prentice Hall, 1976)

M. Spivak, “A Comprehensive Introduction to Differential Geometry” (Publish or Perish, 197075). 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).