Recent publications

Stephen T. Hyde

(Applied Mathematics Dept, Physics, Australian National University)

... like life, this page mutates in fits and starts ...

(last updated October 2014)

A chronological list of papers, book chapters, books etc, is available here as a pdf file.

A rough classification by topic is shown below

Relevant graphical logos are appended to each paper

tiling logo biomorph logo Minimal surfaces logo hard materials logo liquid crystals logo theoretical crystallography logo
Embedded Graphs
Tangled Patterns
Biominerals and  biomaterials


living vs. dead matter
Cellular structures, surfaces "Hard" framework materials:
Metallo-organic frameworks
Novel carbon nanofoam etc.
Soft matter:
Lyotropic liquid crystals microemulsions
block copolymers

Membranes in vivo
Theoretical crystallography

PDF files are available for private use only as they are copyrighted

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Reactions of the flexible N,N′-di(3-pyridyl)suberoamide (L) with Cu(II) salts in the presence of the isomeric phenylenediacetic acids under hydrothermal conditions afforded three new coordination networks, {[Cu(L)(1,2-pda)]·H2O}n (1,2-H2pda = 1,2-phenylenediacetic acid), 1, {[Cu(L)(1,3-pda)]·2H2O}n (1,3- H2pda = 1,3-phenylenediacetic acid), 2, and {[Cu(L)(1,4-pda)]·2H2O}n (1,4-H2pda = 1,4-phenylenediacetic acid), 3, which have been structurally characterized by X-ray crystallography. Complex 1 forms a single 3,5-coordinated 3D net with the (42.65.83)(42.6)-3,5T1 topology, which can be further simplified as a 6-coordinated (412.63)-pcu topology. Complex 2 is a 5-fold interpenetrated 3D structure with the (65.8)-cds topology, which exhibits the maximum number of interpenetration presently known for cds and complex 3 is the first example of a tangled 1D net that contains neither knots, links nor ravels, thus exemplifying a new mode of entanglement. The ligand-isomerism of the phenylenediacetate ligands is important in determining the structural types of the Cu(II) coordination networks based on the flexible L ligands.

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Self-assembly remains the most efficient route to the formation of ordered nanostructures, including the double gyroid network phase in diblock copolymers based on two intergrown network domains. Here we use self-consistent field theory to show that a tricontinuous structure with monoclinic symmetry, called 3ths(5), based on the intergrowth of three distorted ths nets, is an equilibrium phase of triblock star-copolymer melts when an extended molecular core is introduced. The introduction of the core enhances the role of chain stretching by enforcing larger structural length scales, thus destabilizing the hexagonal columnar phase in favor of morphologies with less packing frustration. This study further demonstrates that the introduction of molecular cores is a general concept for tuning the relative importance of entropic and enthalpic free energy contributions, hence providing a tool to stabilize an extended repertoire of self-assembled nanostructured materials.

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We present coarse-grained simulations of the self-assembly of 3-armed ABC star polyphiles. In systems of star polyphiles with two arms of equal length the simulations corroborate and expand previous findings from related miktoarm star terpolymer systems on the formation of patterns containing columnar domains whose sections are 2D planar tilings. However, the systematic variation of face topologies as the length of the third (unequal) arm is varied differs from earlier findings regarding the compositional dependence. We explore 2D 3-colored foams to establish the optimal patterns based on interfacial energy alone. A generic construction algorithm is described that accounts for all observed 2D tiling patterns and suggests other patterns likely to be found beyond the range of the simulations reported here. Patterns resulting from this algorithm are relaxed using Surface Evolver calculations to form 2D foams with minimal interfacial length as a function of composition. This allows us to estimate the interfacial enthalpic contributions to the free energy of related star molecular assemblies assuming strong segregation. We compare the resulting phase sequence with a number of theoretical results from particle-based simulations and field theory, allowing us to tease out relative enthalpic and entropic contributions as a function of the chain lengths making up the star molecules. Our results indicate that a richer polymorphism is to be expected in systems not dominated by chain entropy. Further, analysis of corresponding planar tiling patterns suggests that related two-periodic columnar structures are unlikely hypothetical phases in 4-arm star polyphile melts in the absence of sufficient arm configurational freedom for minor domains to form lens-shaped di-gons, which require higher molecular weight polymeric arms. Finally, we discuss the possibility of forming a complex tiling pattern that is a quasi- crystalline approximant for 3-arm star polyphiles with unequal arm lengths.

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Numerical simulations reveal a family of hierarchical and chiral multicontinuous network structures self-assembled from a melt blend of Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to have equal-sized A/B and C/D chains, respectively. The C and D majority domains within these patterns form a pair of chiral enantiomeric gyroid labyrinths (srs nets) over a broad range of compositions. The minority A and B components together define a hyperbolic film whose midsurface follows the gyroid minimal surface. A second level of assembly is found within the film, with the minority components also forming labyrinthine do- mains whose geometry and topology changes systematically as a function of composition. These smaller labyrinths are well de- scribed by a family of patterns that tile the hyperbolic plane by regular degree-three trees mapped onto the gyroid. The labyrinths within the gyroid film are densely packed and contain either graphitic hcb nets (chicken wire) or srs nets, forming convoluted intergrowths of multiple nets. Furthermore, each net is ideally a single chiral enantiomer, induced by the gyroid architecture. However, the numerical simulations result in defect-ridden achiral patterns, containing domains of either hand, due to the achiral terpolymeric starting molecules. These mesostructures are among the most topologically complex morphologies identified to date and represent an example of hierarchical ordering within a hyper- bolic pattern, a unique mode of soft-matter self-assembly.

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The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway’s orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler ‘crystallographic hyperbolic groups’ are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e. the P, D, Gyroid, CLP and H surfaces) as well as the genus-four I-WP surface.

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Three isostructural interwoven (3,4)-connected mesoporous metal-organic frameworks of pto-a topology (UTSA-28-Cu, UTSA-28-Zn, and UTSA-28-Mn) were synthesized and structurally characterized. Because of their meta-stable nature, their gas sorption properties are highly dependent on the metal ions and activation profiles. The most stable UTSA-28a-Cu exhibits high gas storage capacities.

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Nets in which different vertices have identical barycentric coordinates (i.e. have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termed unforced.

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The 3-periodic nets of genus 3 (‘minimal nets’) are reviewed and their symmetries re-examined. Although they are all crystallographic, seven of the 15 only have maximum-symmetry embeddings if some links are allowed to have zero length. The connection between the minimal nets and the genus-3 zero- mean-curvature surfaces (‘minimal minimal’ surfaces) is explored by deter- mining the surface associated with a net that has a self-dual tiling. The fact that there are only five such surfaces but 15 minimal nets is rationalized by showing that all the minimal nets can serve as the labyrinth graph of one of the known minimal minimal surfaces.

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Recent advances in the cataloguing of three-dimensional nets mean a systematic search for framework structures with specific properties is now feasible. Theoretical arguments about the elastic deformation of frameworks suggest characteristics of mechanically isotropic networks. We explore these concepts on both isotropic and anisotropic networks by manufacturing porous elastomers with three different periodic net geometries. The blocks of patterned elastomers are subjected to a range of mechanical tests to determine the dependence of elastic moduli on geometric and topological parameters. We report results from axial compression experiments, three-dimensional X-ray computed tomography imaging and image-based finite-element simulations of elastic properties of framework-patterned elastomers.

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Ordered arrays of cylinders, known as rod packings, are now widely used in descriptions of crystalline structures. These are generalized to include crystal- lographic packed arrays of filaments with circular cross sections, including curvilinear cylinders whose central axes are generic helices. A suite of the simplest such general rod packings is constructed by projecting line patterns in the hyperbolic plane (H^2) onto cubic genus-3 triply periodic minimal surfaces in Euclidean space (E^3): the primitive, diamond and gyroid surfaces. The simplest designs correspond to ‘classical’ rod packings containing conventional cylindrical filaments. More complex packings contain three-dimensional arrays of mutually entangled filaments that can be infinitely extended or finite loops forming three-dimensional weavings. The concept of a canonical ‘ideal’ embedding of these weavings is introduced, generalized from that of knot embeddings and found algorithmically by tightening the weaving to minimize the filament length to volume ratio. The tightening algorithm builds on the SONO algorithm for finding ideal conformations of knots. Three distinct classes of weavings are described.

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High-symmetry free tilings of the two-dimensional hyperbolic plane (H^2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

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D’Arcy Thompson’s views on the forms of biomaterials are assessed in the light of current thinking on biomorphology in selected areas of biology. It is clear that his guiding concepts — that biological materials are structured in response to physical forces, and that the biological and abiotic realms share many common features — remain valid. Advances in the physical and biological sciences are discussed, from quantum mechanics and molecular biology to liquid crystalline materials and macroscopic forms. These reveal Thompson’s clear-sighted view of the role of physical and mathematical sciences in biology, as well as his blind-spots.

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The geometrization of physics, which views physical phenomena through the prism of geometry and topology, has left a lasting imprint on many areas of physics, from Einstein’s revolutionary adoption of Riemannian geome- try to build his theories of relativity, to the rapidly multiplying zoo of topological phases in quantum physics. It is therefore not surprising that newer areas of condensed matter research, particularly synthetic and biological soft liquid crystalline matter and related materials, are best explored using the tools of low-dimen- sional geometry and topology. This realization is not new.

Twenty years ago, Elisabeth Dubois-Violette and Brigitte Pansu, both then at the Laboratoire de Physique des Solides at Orsay, organized a seminal meeting in Aussois, ‘Geometry and Interfaces’, which brought together physicists, chemists, biologists and mathemati- cians and resulted in a useful volume that summarized the state of things in 1990 [1]. The Orsay group had an impeccable pedigree in condensed materials research (including, for example, liquid crystal research that led to the award of the Physics Nobel Prize to de Gennes). Their scientific culture recognized the importance of crossing traditional disciplinary borders, and the enrichment of conventional condensed matter physics drawn from studies in other areas, from biology to pure mathematics. That approach now pervades many aspects of contemporary physics research into materials, where it is recognized that biology and materials chemistry offers fertile domains for explora- tion. Another approach to materials research remains however less developed: the exploration of the funda- mental science of biomaterial self-assembly and function using the tools of low-dimensional geometry and topology. Few biologists concern themselves with more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?

In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’ [2]. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?
In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.

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Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight generalisations of these bi- continuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bi- layer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing charac- teristics to inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global ge- ometry to form these multi-network phases is quantified by the surfactant shape parameter v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187,193) structure of hexagonal sym- metry P63/mcm with three aqueous domains, the 3dia(24,220) structure of cubic symmetry I43d composed of three distorted Diamond networks, the cubic chiral 4srs*(24,208) with cubic symmetry P4232 and the achiral 4srs(5,133) structure of symmetry P42/nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the QGII Gyroid phase). Structural homogene- ity is analysed by a medial surface approach assuming that the head group in- terfaces are constant mean curvature surfaces. To facilitate future experimen- tal identification, we provide simulated SAXS scattering patterns that, for the 4srs*(24,208) and 3dia(24,220) structures, bear remarkable similarity to those of bicontinuous QGII-Gyroid and QDII-Diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the well-established structures. While polycontinuous lipid phases have to date not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure termed IBN-9 formed in the presence of surfactants [Han et al., Nat. Chem., 2009, 1, 123].

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We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S^2), Euclidean (E^2) and hyperbolic (H^2) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geo- metrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions. 

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We discuss the identification of untangled graph embeddings for finite planar and non-
planar graphs as well as infinite crystallographic nets. Two parallel approaches are discussed:
explicit 3-space embeddings and reticulations of 2-manifolds. 2D and 3D energies are pro-
posed that allow ranking of (un)tangled embedded graphs.

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Nature provides impressive examples of chiral photonic crystals, with the notable example of the
cubic srs network structure realized in wing-scales of several butterfly species. By a novel circular
polarization analysis of the band structure of such networks, we demonstrate strong circular dichro-
ism effects: The butterfly srs micro-structure, of cubic I 41 32 symmetry, shows significant circular
dichroism for blue to ultra-violet light, that warrants a search for biological receptors sensitive to
circular polarization. A derived synthetic structure based on four like-handed silicon srs nets ex-
hibits a large circular polarization stop band of width exceeding 30%. These findings offer design
principles for chiral photonic devices.

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The structure of the porous three-dimensional reticulated pattern in the wing scales of the butterfly C. rubi (the Green Hairstreak)
is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a fragment of
this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry
bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid
is bounded to an excellent approximation by a parallel/cmc surface to the Gyroid, a three-periodic minimal surface with cubic
crystallographic symmetry I 41 32, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with
the wavelength of visible light, with an edge of the conventional cubic unit cell of the cmc-Gyroid of approximately 310 nm. The
genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft
matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is
achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely
form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework
is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during
growth) promotes the formation of a single enantiomer.

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We suggest constructive definitions for the determination of untangled finite graphs and three-periodic
nets, using barycentric embeddings in two and three dimensions. The possibility of deliberately con-
structing tangled graphs and nets is canvassed, and we conclude that tangled patterns offer a novel class
of nano- and meso-structured materials with useful features, including high internal surface area and
volume and chirality.

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Triphilic star-polyphiles are short-chain oligomeric molecules with a radial arrangement of 
hydrophilic, hydrocarbon and fluorocarbon chains linked to a common centre. They form a number
of liquid crystalline structures when mixed with water. In this contribution we focus on a
hexagonal liquid crystalline mesophase found in star-polyphiles as compared to the corresponding
double chain surfactant to determine whether  the hydrocarbon and fluorocarbon chains are in fact
demixed in these star-polyphile systems, or whether both hydrocarbon and fluorocarbon chains are
miscible, leading to a single hydrophobic domain, making the star-polyphile effectively
amphiphilic. We report SANS contrast variation data that is compatible only with the presence of
three distinct immiscible domains within this hexagonal mesophase, confirming that these star-
polyphile liquid crystals are indeed hydrophilic/oleophilic/fluorophilic 3-phase systems.
Quantitative comparison with scattering simulations shows that the experimental data are in very
good agreement with an underlying 2D columnar (12.6.4) tiling. As in a conventional amphiphilic
hexagonal mesophase, the hexagonally packed water channels (dodecagonal prismatic domains)
are embedded in a hydrophobic matrix, but that matrix is split into oleophilic hexagonal prismatic
domains and fluorophilic quadrangular prismatic domains.

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I explore the terrain that lies between architecture and geometry, from the perspective of a structural scientist with no professional architectural expertise. The divide between these disciplines perhaps stems from an ancient dichotomy between the art vs. engineering schools of architecture, fertilised by the current dogma that art and science can never meet. Architects stand to gain much from study of the spectacular advances in geometry in recent decades, such as the growing understanding of cellular patterns in space, tiles, nets and curved surfaces. Some examples of those advances are discussed in detail. I conclude that both architecture and geometry would benefit from a renewed mutual interest.

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We investigate the possibility of forming achiral knottings of polyhedral (3-connected) graphs

whose minimal embeddings lie in the genus-one torus. Various analyses to show that all examples

are chiral. This result suggests a simple route to forming chiral molecules via templating on a

toroidal substrate.

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We construct simple models to compare ordered tricontinuous patterns that are topologically

consistent with the constraints imposed by three-arm star polyphile self-assembly, analogous to steric

packing and elastic bending models used to analyse bicontinuous mesophases in amphiphilic

assemblies. We find a number of competing low-energy ordered structures, composed of threading of
three identical labyrinths, with three-fold infinite branch lines, that are likely to be of comparable

energy for polyphile shapes with moderately splayed arms. These patterns are triply-periodic analogues

of the hexagonal honeycomb, which is most favoured for unsplayed three-arm polyphile architectures.

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We have simulated the self-assembly of a novel class of three-arm molecules,
ABC star-architecture polyphiles, using coarse-grained bead simulations. A number of

topologically complex liquid crystalline mesostructures arise that can be related to the

better-known bicontinuous mesophases of lyotropic amphiphilic systems. The simulations reveal

3D self-assemblies whose structural variations follow those expected assuming a simple steric

molecular packing model as a function of star polyphile splay and relative volumes of each arm

in the polyphile. The splay of each arm, characterised by the 3D wedge-shape emanating from the

core of each molecule to its exterior induces torsion of the interfaces along the triple lines,

whereas differences in the relative volumes of arms induce curvature of the triple lines. Three

distinct mesostructures are described, characterised by their micro-domain topologies, which are

unknown in simpler amphiphilic systems, but resemble in some respects bicontinuous mesophases.

These three- (or more) arm polyphilic systems offer an interesting extension to the better-known

self-assembly of (two-arm) amphiphiles in solution.

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Precipitation of barium or strontium carbonates in alkaline silica-rich

environments leads to crystalline aggregates that have been named

 silica/carbonate biomorphs because they resemble the morphology of

 primitive organisms. These aggregates are self-assembled materials of

 purely inorganic origin, with an amorphous phase of silica intimately

 intertwined with a carbonate nanocrystalline phase. We propose a

 mechanism that explains all the morphologies described for biomorphs.

 Chemically coupled co-precipitation of carbonate and silica leads to

 fibrillation of the growing front and to laminar structures that experience

 curling on their growing rim. These curls propagate surf-like along the rim

 of the laminae. Observed morphologies with smoothly varying curvatures

 can be explained by the combined growth of counter-propagating curls

 and growing laminae.

Comments on this paper can be found at:

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Crystallisation of barium carbonate from alkaline silica solutions results in the

 formation of extraordinary micron-scale architectures exhibiting

 non-crystallographic curved shapes, such as helical filaments and worm-like braids.

 These so-called “silica biomorphs” consist of a textured assembly of uniform

 elongated witherite nanocrystallites, which is occasionally sheathed by a skin of

 amorphous silica. Although great efforts have been devoted to clarifying the

 physical origin of these fascinating materials, to date little is known about the

 processes underlying the observed self-organisation. Herein, we describe the effect

 of two selected additives, a cationic surfactant and a cationic polymer, on the

 morphology of the forming crystal aggregates, and relate changes to experiments

 conducted in the absence of additives. Minor amounts of both substances are shown

 to exert a significant influence on the growth process, leading to the formation of

 predominantly flower-like spherulitic aggregates. The observed effects are discussed

 in terms of feasible morphogenesis pathways. Based on the assumption of a

 template membrane steering biomorph formation, it is proposed that the two

 additives are capable of performing specific bridging functions promoting the

 aggregation of colloidal silica which constitutes the membrane. Morphological

 changes are tentatively ascribed to varying colloid coordination effecting distinct

 membrane curvatures.

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We present a method for geometric construction of periodic 3D Euclidean nets by

projecting 2D hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMS).

Our techniques extend the combinatorial tiling theory of Dress, Huson, and

Delgado-Friedrichs to enumerate simple reticulations of these TPMS.

We include a taxonomy of all networks arising from kaleidoscopic hyperbolic

tilings with up to two distinct tile types (and dually two vertex types), mapped

to three related TPMS, namely Schwarz's Primitive (P) and Diamond (D) surfaces,

and Schoen's Gyroid (G)


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Herein we describe some properties and the occurrences of a beautiful geometric

figure that is ubiquitous in chemistry and materials science, however, it is not as

well-known as it should be. We call attention to the need for mathematicians to pay

more attention to the richly structured natural world, and for materials scientists to

learn a little more about mathematics. Our account is informal and eschews any

pretence of mathematical rigor, but does start with some necessary mathematics.


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Molecular and extended framework materials, from proteins to catenanes and metal–organic

frameworks, can assume knotted configurations in their bonding networks (the chemical graph).

Indeed, knot theory and structural chemistry have remained closely allied, due to those

connections. Here we introduce a new class of graph entanglement: ‘‘ravels’’. These ravels—often

chiral—tangle a graph without the presence of knots. Just as knots lie within cycles in the graph,

ravels lie in the vicinity of a vertex. We introduce various species of ravels, including fragile

ravels, composite ravels and shelled ravels. The role of ravels is examined in the context of finite

and infinite graphs—analogous to molecular and extended framework nets—related to the

diamond net.

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Cubic membranes are soft three-dimensional crystals found within cell organelles in a variety

of living systems, despite the aphorism of Fedorov: ‘crystallization is death’. They consist of

multi-bilayer lipid–protein stacks, folded onto anticlastic surfaces that resemble triply

periodic minimal surfaces, forming highly swollen crystalline sponges. Although cubic

membranes have been observed in numerous cell types and under different pathophysiolo-

gical conditions, knowledge about the formation and potential function(s) of non-lamellar,

cubic structures in biological systems is scarce. We report that mitochondria with this cubic

membrane organization isolated from starved amoeba Chaos carolinense interact sufficiently

with short segments of phosphorothioate oligonucleotides ( PS-ODNs) to give significant

ODNs uptake. ODNs condensed within the convoluted channels of cubic membrane by an

unknown passive targeting mechanism. Moreover, the interaction between ODNs and cubic

membrane is sufficient to retard electrophoretic mobility of the ODN component in the gel

matrix. These ODN–cubic membrane complexes are readily internalized within the

cytoplasm of cultured mammalian cells. Transmission electron microscopic analysis confirms

ODNs uptake by cubic membranes and internalization of ODN–cubic membrane complexes

into the culture cells. Cubic membranes thus may offer a new, potentially benign medium for

gene transfection.

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The precipitation of calcium carbonate in alkaline silica solutions results in the formation of complex curvilinear

forms if aragonite formation is encouraged by growth at an elevated temperature (80 °C). The resulting coralline self-assembled

silica-calcium carbonate particles are “biomorphs”, bearing a striking resemblance to natural coral forms. These materials, comprised

of calcium carbonate nanocrystals and an amorphous silica matrix, have a complex ultrastructure, made of clusters of gathered

sheets of variable curvatures formed by successive curling. The nanocrystals within these “ruled surfaces” are thin, elongated,

densely packed needles of aragonite. These clusters are outgrowths from central saddlelike cores that resemble developable petaloid

surfaces. The size, shape, crystallography, and chemical composition of the resulting biomorphs were examined by optical microscopy,

field emission scanning electron microscopy (FE-SEM), powder X-ray diffractometry (XRD), Fourier transform infrared spectroscopy

(FT-IR), transmission electron microscopy (TEM and HRTEM), and energy dispersive X-ray analysis (EDX).

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This article explores global geometric features of bicontinuous space-partitions and their rele-

vance to self-assembly of block-copolymers. Using a robust definition of ‘local channel radius’, based on the

concept of a medial surface [1], we relate radius variations of the space-partition to polymolecular chain

stretching in bicontinuous diblock- and terblock copolymer assemblies. We associate local surface patches

with corresponding cellular volume elements, to define local volume-to-surface ratios. The distribution of

these v/a ratios and of the channel radii are used to quantify the degree of packing frustration of molecular

chains as a function of the specific bicontinuous geometry, modelled by triply-periodic minimal surfaces

and related parallel interfaces. The Gyroid geometry emerges as the most nearly homogeneous bicontinu-

ous form, with the smallest heterogeneity of channel radii, compared to the cubic Primitive and Diamond

surfaces. We clarify a geometric feature of the Gyroid geometry: the three-coordinated nodes of the graph

are not the widest points of the labyrinths; the widest points are at the midpoints of the edges. We also ex-

plore a more complex cubic triply-periodic surface, the I-WP surface, containing two geometrically distinct

channel subdomains. One of the two channel systems is nearly as homogeneous in local channel diameters

as the Gyroid, the other is more heterogeneous than the Primitive surface. Its hybrid nature suggests the

possibility of an “alternating I-WP” phase in polydisperse linear ABC-terpolymer blends, with monodis-

perse molecular weight distributions (MWD) in the A and B blocks and a more polydisperse C block.

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We explore the use of tetraethoxysilane (TEOS) as a silica source for the formation of

carbonate-silica composite materials known as ‘biomorphs’. The basic hydrolysis of

TEOS furnishes silica in a controllable fashion, allowing a significantly higher reproducibility

of the obtained silica–barium and silica–strontium carbonate co-precipitates compared to

commercial water glass silica used so far. We further discuss the influence of ethanol used

as a co-solvent on the morphologies of biomorphs, which are examined by optical microscopy,

field emission scanning electron microscopy (FESEM) and energy dispersive X-ray analysis (EDX).


pdf version (2.9 MB)


The ‘simplest’ entanglements of the graph of edges of the cube are enumerated,

forming two-cell {6, 3} (hexagonal mesh) complexes on the genus-one two-

dimensional torus. Five chiral pairs of knotted graphs are found. The examples

contain non-trivial knotted and/or linked subgraphs [(2, 2), (2, 4) torus links and

(3, 2), (4, 3) torus knots].

pdf version (730 KB)


Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond

(D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties.

These properties are central to understanding the formation of mesophases in amphiphile and copolymer

molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG,

rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation

pathways between the three cubic surfaces. We introduce ‘packing (or global) homogeneity’, defined as

the standard deviation ∆d of the distribution of the channel diameter throughout the labyrinth, where

the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine

domains. Curvature homogeneity is defined similarly as the standard deviation ∆K of the distribution of

Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume

ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and

second moments of the distribution of channel diameter for all members of these surfaces complementing

curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces

are deep local minima of ∆d along the surface families (with G more homogeneous than D), whereas the

cubic P surface is an inflection point of ∆d with adjacent, more homogeneous surface members. Both

curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces)

in preference to the rhombohedral route (via rG and rPD).

pdf version (1.1 MB)


Crystalline frameworks in 3D Euclidean space can be constructed by projecting tilings

of 2D hyperbolic space onto three-periodic minimal surfaces, giving surface reticulations.

The technique involves Delaney–Dress tiling theory, group theory, differential and

non-Euclidean geometry. Preliminary results of this approach, found at, are discussed and compared with other approaches.

pdf version (276 kB)


Our study of the gyroid minimal surface has revealed that there are two distinct covering

maps from the hyperbolic plane onto the surface that respect its intrinsic symmetries.

We show that if a decoration of H2 is chiral, the pro jection of this pattern via the two

covering maps gives rise to distinct structures in E3 .


pdf version (308 kB)


We explore a novel method to generate and characterize complex networks by means of

their embedding on hyperbolic surfaces. Evolution through local elementary moves allows the

exploration of the ensemble of networks which share common embeddings and consequently

share similar hierarchical properties. This method provides a new perspective to classify

network-complexity both on local and global scale. We demonstrate by means of several

examples that there is a strong relation between the network properties and the embedding



pdf preprint (560 KB)


We propose a general method to study the hierarchical organization of financial data.

The statistical, geometrical and topological properties of such data are analyzed by

embedding the structure of their correlations in metric graphs in multi-dimensional spaces.

We show an application to two different sets of interest rates data. In this case we

construct triangular embeddings on the sphere. The resulting graph contains the

minimum spanning tree as sub-graph and it preserves its hierarchical structure. This results

in a clear cluster differentiation and allows to compute new local and global topological

quantities. A three dimensional representation of this embedding is constructed together

with its projection on a plane by using the Pelting method and a relaxation procedure to

converge on the correct metric geometry.

[Also available at LANL archive].

pdf version (500 KB)


We report production of nanostructured magnetic carbon foam by a high-repetition-rate,

high-power lase ablation of glassy carbon in Ar atmosphere. A combination of

characterization techniques revealed that the system contains both sp2 and sp3 bonded

carbon atoms. The material is a form of carbon containing graphitelike sheets with

hyperbolic curvature, as proposed for "schwarzite." The foam exhibits ferromagnetic-like

behavior up to 90 K, with a narrow hysteresis curve and a high saturation magnetization.

Such magneticproperties are very unusual for a carbon allotrope. Detailed analysis

excludes impurities as the origin of themagnetic signal. We postulate that localized

unpaired spins occur because of topological and bonding defects associated with the

sheet curvature, and that these spins are stabilized due to the steric protection offered

by the convoluted sheets.


pdf version (1.5 MB)


We investigate the wealth evolution in a system of agents that ex-

change wealth through a disordered network in presence of an additive stochastic

Gaussian noise. We show that the resulting wealth distribution is shaped by the de-

gree distribution of the underlying network and in particular we verify that scale free

networks generate distributions with power-law tails in the high-income region. Nu-

merical simulations of wealth exchanges performed on two different kind of networks

show the inner relation between the wealth distribution and the network properties

and confirm the agreement with a self-consistent solution. We show that empirical

data for the income distribution in Australia are qualitatively well described by our

theoretical predictions.


pdf version (1.2 MB)


Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces.

Such surfaces are covered by the hyperbolic plane, in the same way that the euclidean

plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an

appropriate minimal surface to obtain a 3d periodic net. This requires symmetries of the

hyperbolic net to match the symmetries of the minimal surface. We describe a systematic

algorithm to find all the hyperbolic symmetries that are commensurate with a given

minimal surface, and the generation of simple 3d nets from these symmetry groups.

pdf version (500 KB)


We describe self-assembled silica-carbonate aggregates that show a diverse range of

morphologies, all of which display complex internal structure, orientational ordering

of components, and well-organised, curved global morphologies that bear a strong

resemblance to biogenic forms. The internal order is described as a liquid-crystallike

organisation of colloidal particles. We discuss possible causes for the striking

morphologies of these inorganic materials, including local nanocrystal packing constraints

and global silica membrane templating.

pdf version (350 KB)


This article presents a medial surface analysis of the rhombohedral infinite periodic

minimal surface rPD. This one-parameter family of labyrinth-forming, bicontinuous

surfaces has been suggested as a continuous pathway for transitions between its

two cubic members, the Primitive and the Diamond surface, e.g. in mesophases in

liquid-crystalline self-assembly processes. By providing a definition of a pointwise

channel diameter, the MS allows for the analysis of packing properties, stretching

frustration and homogeneity of such surfaces that cannot be deduced from curvature

characteristics alone. The medial surface (MS) is a representation of a labyrinth

structure as an embedded and centered 2D skeleton, and is a geometrically equivalent

description of the labyrinth as the labyrinth itself. It can be further reduced to a

well-defined 1D line graph. For the rPD surface, we show that variations of the channel

diameter are locally minimal for the member corresponding to the D surface,

and a horizontal inflection point in the case of the P surface. This may have implications

for the phase stability of corresponding liquid-crystalline mesophases. We also

demonstrate that a 1D line graph, if geometrically centered within the labyrinth,

contains curved edges and cannot be deduced from symmetry considerations alone.

pdf version (340 KB)


Small angle x-ray (SAXS) scattering and 14N NMR relaxation were

determined for microemulsion samples formed from didodecyl dimethyl

ammonium bromide (DDAB), water and tetradecane to deduce the associated

microstructures. The swelling features within the tetradecane

microemulsion are unusual compared with DDAB/water/alkane analogs formed

with shorter n-alkanes: tetradecane-containing microemulsions do not show the

characteristic anti-percolation transition seen for the latter microemulsions.

Experimental data along tetradecane dilution lines are consistent with a

continuous transition from a bilayer to monolayer structure of the surfactant interface.

The evolution is topologically complex. It involves the annealing of bilayer punctures

that occur on oil dilution. A quantitative model that allows continuous transformation

from multihandled bilayers (typical of L3 sponge mesophases) to multi-handled

monolayers (typical of microemulsions modelled with shorter chained alkanes)

is proposed that fits well the observed behaviour.

pdf version (336 KB)


A variety of life-like "biomorphs" can be grown by co-precipitation of silica

and alkaline-earth carbonates at high pH. Forms include twisted filaments

and sheets, that are indistinguishable from microscopic inclusions in

ancient rocks, commonly identified as ancient microfossils. Biomorphs are

spectacular examples of self-assembled inorganic colloids, forming

composites of nm-sized rod-shaped carbonate nanocrystals and colloidal

amorphous silica spheres. The structural complexity of these materials, with

orientational and translational order/disorder at distinct length scales, is

reminiscent of many hard biomaterials, such as bone.

* -- National Center for Science Education,, S -- Project Steve "Steve",, N -- Nobel laureate, % -- Appeared on The Simpsons, $ -- Ordered 16 Project Steve t-shirts in order to outfit all of the staff in his lab, ! -- Not to be confused with the preceding Steve with the identical name.

"The Morphology of Steve", Annals of Improbable Research, July-August (2004)

pdf version (480 KB)


This report is part of Project Steve. Project Steve is, among other things, the

first scientific analysis of the sex, geographiclocation, and body size of scientists

named Steve. We performed this research for the best of all reasons: we

discovered that we had lots of data. No scientist can resist the opportunity to

analyze data, regardless of where that data came from or why it was gathered.

paper available here


We have synthesized inorganic micron-sized filaments, whose microstucture

consists of silica-coated nanometer-sized carbonate crystals, arranged with

strong orientational order. They exhibit noncrystallographic, curved, helical

morphologies, reminiscent of biological forms. The filaments are similar to

supposed cyanobacterial microfossils from the Precambrian Warrawoona chert

formation in Western Australia, reputed to be the oldest terrestrial microfossils.

Simple organic hydrocarbons, whose sources may also be abiotic and indeed

inorganic, readily condense onto these filaments and subsequently polymerize

under gentle heating to yield kerogenous products. Our results demonstrate

that abiotic and morphologically complex microstructures that are identical to

currently accepted biogenic materials can be synthesized inorganically.


(see also comments in Science (News of the Week), 302, 1134, 2003; see also New Scientist, 22 November, 14-15, 2003; Chemical & Engineering News,17 November, 58, 2003.)

pdf version (600 KB)


We introduce a robust algorithm for numerical computation of a medial surface and

an associated medial graph for three-dimensional shapes of arbitrary topological and

geometric complexity, bounded by oriented triangulated surface manifolds in three-

dimensional Euclidean space (domains). We apply the construction to particularly complex

``bicontinuous'' domain shapes found in molecular self-assemblies, the cubic infinite

periodic minimal surfaces of genus three: Gyroid (G), Diamond (D) and Primitive (P) Surfaces.

The medial surface is the locus of centers of maximal spheres wholly contained

within the domains, i.e.~spheres which graze the surface tangentially and are not

contained in any other such sphere. The construction of a medial surface is a natural

generalization of Voronoi diagrams to continuous surfaces. The medial surface algorithm

provides an explicit construction of the bounding surface patch associated with a volume

element, giving a robust measure of surface to volume ratios for complex forms. It also

allows for sensible definition of a line graph (the medial graph), particularly useful for

volumes consisting of connected channels, and not reliant on symmetries of the volumes.

In addition, the medial surface construction produces a length associated with any point

on the surface. Variations of this length give a useful global measure of homogeneity of the

volumes. Comparison of medial surfaces for the P, D and G surfaces reveal the Gyroid to be

the most homogeneous of these cubic bicontinuous forms (of genus three). This result is

compared with the ubiquity of the G surface morphology in soft mesophases, including

lyotropic liquid crystals and block copolymers.


pdf version (720 KB)


We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to

generate three-dimensional Euclidean (E3) networks. The technique involves projection

from tilings of the hyperbolic plane (H2) onto three-periodic minimal surfaces, embedded

in E3. Given the extraordinary wealth of symmetries commensurate with H2, we can

generate networks in E3 that are difficult to construct otherwise. In particular, we form

four-, five- and seven-connected (E3) nets containing three- and five-rings, viz. (3,7),

(5,4) and (5,5) tilings in H2. These examples are of fundamental interest, as they present

"topological symmetries" that are incompatible with the isometries of E3.


pdf version (880 KB)


We describe a technique for construction of 3D Euclidean (E3) networks with

partially-prescribed rings. The algorithm starts with 2Dhyperbolic (H2) tilings,

whose symmetries are commensurate with the intrinsic 2D symmetries of triply

periodic minimal surfaces (orinfinite periodic minimal surfaces, IPMS). The 2D

hyperbolic pattern is then projected from H2 to E3, forming 3D nets. Examples of

cubic and tetragonal 3-connected nets with up to 288 vertices per unit cell, each

linking a pair of 6-rings and a single 8-ring, are derived by projection onto the P, D,

Gyroid and I-WP IPMS. A single example of a projection from close-packed trees in

H2 to E3 (via the D surface) is also shown, that leads to a quartet of interwoven

equivalent chiral nets. The configuration describes the channel system of a novel

quadracontinuous branched minimal surface that is a chiral foam with four identical,

open bubbles.

pdf version (1.7 MB)


Recent developments in theoretical and experimental studies of amphiphilic lyotropic

"intermediate" mesophase formation are summarized. For the purposes of the review, 

we consider intermediate mesophases to be self-assemblies with novel geometries and

topologies, excluding lamellar, sponge, columnar (hexagonal) and micellar mesophases.

Intermediates include novel branched bilayer topologies, enclosing multiple

interwoven channel systems and inclined rod packings , and punctured bilayer morphologies,

including mesh phases and bicontinuous monolayers.

pdf file (29 MB)


This paper deals with the difficulty of decoding the origins of natural structures through

the study of their morphological features. We focus on the case of primitive life detection,

where it is clear that the principles of comparative anatomy cannot be applied. A range of

inorganic processes are described, that result in morphologies emulating biological shapes.

We focus on geochemically plausible processes, in particular, the formation of inorganic

biomorphs in alkaline silica-rich environments, which are described in detail.


pdf file (733 Kbytes)

see also Comment by Férey in Science.


Interpenetration (catenation) has long been considered a major impediment in

the achievement of stable and porous crystalline structures. A strategy for the

design of highly porous and structurally stable networks makes use of metal-organic

building blocks that can be assembled on a triply periodic P-minimal geometric

surface to produce structures that are interpenetrating--more accurately considered

as interwoven. We used 4,4',4"-benzene-1,3,5-triyl-tribenzoic acid (H3BTB),

copper(II) nitrate, and N,N'-dimethylformamide (DMF) to prepare

Cu3(BTB)2(H2O)3·(DMF)9(H2O)2 (MOF-14), whose structure reveals a pair of interwoven

metal-organic frameworks that are mutually reinforced. The structure contains

remarkably large pores, 16.4 angstroms in diameter, in which voluminous

amounts of gases and organic solvents can be reversibly sorbed.

pdf file (1.3 Mbytes) Fig 35 colour plate


The Chapter is a comprehensive account of the current state of awareness of

lyotropic liquid crystalline mesophases, including lamellar, cubic (bicontinuous

and discrete), hexagonal and intermediate mesophases, as well as sponge phases

and microemulsions. The emphasis is on a rigorous identification of the bilayer topology .

Some discussion of possible &endash; as yet unknown &endash; novel mesophases is

present. The chapter also contains a survey of practical techniques to identify distinct

mesophases in the lab, including SAXS, optical microscopy and other techniques.


pdf file (700 Kb)


We describe a construction procedure for polycontinuous structures, giving generalisations

of bicontinuous morphologies to more than two equivalent, continuous and interwoven

sub-volumes. The construction gives helical windings of disjoint graphs on triply periodic

hyperbolic surfaces, whose universal cover in the hyperbolic plane consists of packed, parallel

trees. The simplest tri-, quadra- and octa-continuous morphologies consist of three $(8,3)-c$,

four $(10,3)-a$ and eight $(10,3)-a$ interwoven networks respectively. The quadra- and

octa-continuous cases are chiral. A novel chiral bicontinuous structure is also derived, closely

related to the well-known cubic gyroid mesophase.


pdf file (971 Kbytes)


A method is developed to construct and analyse a wide class of graphs embedded in

Euclidean 3D space, including multiply-connected and entangled examples. The

graphs are derived via embeddings of infinite families of trees (forests) in the

hyperbolic plane, and subsequent folding into triply periodic minimal surfaces,

including the P, D, gyroid and H surfaces. Some of these graphs are natural

generalisations of bicontinuous topologies to bi-, tri-, quadra- and octa-continuous

forms. Interwoven layer graphs and periodic sets of finite clusters also emerge

from the algorithm. Many of the graphs are chiral. The generated graphs are

compared with some organo-metallic molecular crystals with multiple frameworks

and molecular mesophases found in copolymer melts.


pdf version (13.7 Mbytes)


Many complex crystalline 3-dimensional graphs are of interest to solid-state chemists,

in order to relate atomic and molecular crystal structures.Some of those graphs can be

generated from tessellations of the 2d hyperbolic plane, and then mapped onto triply-

periodic hyperbolic surfaces. The properties of such graphs, including their densities,

can be analyses in terms of topological and non-euclidean geometric concepts.

Examples include a variety of zeolite frameworks and novel graphitic carbon structures.


pdf file (495 Kbytes)


Although the primitive (P), diamond (D) and gyroid (G) minimal surfaces form the

structural basis for a multitude of self-assembling phases, such as the bicontinuous

cubics, relatively little is known regarding their geometrical transformations, beyond

the existence of the Bonnet isometry. Here their highest symmetry deformation modes,

the rhombohedral and tetragonal distortions, are fully elucidated to provide a unified

description of these simplest minimal surface families, with all quantities expressed in

terms of complete elliptic integrals. The rhombohedral distortions of the gyroid are

found to merge continuously with those which bridge the P and D surfaces, furnishing

direct transformations between all three cubics, preserving both topology and zero mean

curvature throughout. The tetragonal distortions behave analogously, offering an

alternative route from the gyroid to the D surface. The cell axis ratios, surface areas and

Gaussian curvature moments of all families are given, supplying the necessary geometrical

input to a curvature energy description of cubic and intermediate phase stability.


Sorry, I have not posted  all earlier papers. You can find a few below.
If you really want a copy of something not posted here, contact me.


pdf file (495 Kbytes)


Although the primitive (P), diamond (D) and gyroid (G) minimal surfaces form the

structural basis for a multitude of self-assembling phases, such as the bicontinuous

cubics, relatively little is known regarding their geometrical transformations, beyond

the existence of the Bonnet isometry. Here their highest symmetry deformation modes,

the rhombohedral and tetragonal distortions, are fully elucidated to provide a unified

description of these simplest minimal surface families, with all quantities expressed in

terms of complete elliptic integrals. The rhombohedral distortions of the gyroid are

found to merge continuously with those which bridge the P and D surfaces, furnishing

direct transformations between all three cubics, preserving both topology and zero mean

curvature throughout. The tetragonal distortions behave analogously, offering an

alternative route from the gyroid to the D surface. The cell axis ratios, surface areas and

Gaussian curvature moments of all families are given, supplying the necessary geometrical

input to a curvature energy description of cubic and intermediate phase stability.