Chemical GeometryDOI: 10.1002/anie.200801519

A Short History of an Elusive Yet Ubiquitous Structurein Chemistry, Materials, and MathematicsStephen T. Hyde,* Michael O�Keeffe, and Davide M. Proserpio

gyroid · liquid crystals · materials science · nets ·polymers

““Geometry…supplied God with pat-terns for the creation of the world.””Johannes Kepler[1]

1. Introduction

Herein we describe some propertiesand the occurrences of a beautiful geo-metric figure that is ubiquitous inchemistry and materials science, how-ever, it is not as well-known as it shouldbe. We call attention to the need formathematicians to pay more attentionto the richly structured natural world,and for materials scientists to learn alittle more about mathematics. Ouraccount is informal and eschews anypretence of mathematical rigor, but doesstart with some necessary mathematics.

Regular figures such as the fiveregular Platonic polyhedra are an en-during part of human culture and havebeen known and celebrated for thou-sands of years. Herein we consider themas the five regular tilings on the surfaceof a sphere (a two-dimensional surfaceof positive curvature). A flag of a tilingof a two-dimensional surface consists of

a combination of a coincident tile, edge,and vertex. A generally accepted defi-nition of regularity is flag transitivity,which means that all flags are related bysymmetries of the tiling (i.e. there is justone kind of flag). In addition to the fivePlatonic solids, there are three regulartilings of the plane (a surface of zerocurvature), and these are the familiarcoverings of the plane by triangles,squares, or hexagons tiled edge-to-edge.The corresponding regular tilings ofthree-dimensional space are also well-known. Flags are now a polyhedron(tile) with a coincident face, edge, andvertex, and the regular tilings of thethree-sphere are the six nonstellatedregular polytopes of four dimensionalspace. We remark that four dimensionsis the richest space in this regard; higherdimensions have only three regularpolytopes (and of course three dimen-sions has five). However, in flat three-dimensional (Euclidean) space, thespace of our day-to-day experience,there is disappointingly only one regulartiling—the familiar space filling bycubes sharing faces (face-to-face). Theclassic reference to these figures isCoxeter-s Regular Polytopes, in whichhe remarks on the tilings of three-dimensional Euclidean space: “For thedevelopment of a general theory, it is anunhappy accident that only one honey-comb [tiling] is regular…”.[2]

Unhappy indeed, because, perhaps asa consequence, the rich world of periodicgraphs, which are the underlying top-ology of crystal structures, has beenlargely neglected by mathematicians.The graph associated with (carried by)the regular tiling by cubes is the set ofedges and vertices. It is notably thestructure of a form of elemental poloni-um, and chemists often refer to it as the

a-Po net. Recently a system of symbolsfor nets has been developed and this nethas the symbol pcu.[3] Our review isconcerned with another such periodicgraph, and an associated surface.

2. Nets and Tilings

Some years ago in an effort todevelop a taxonomy of three-periodicnets (which are special kinds of aperiodic graph), it was decided to focuson the nets, rather than the tilings thatcarried them.[3,4] Regular nets weredefined as those for which the symmetryrequired the figure (vertex figure), de-fined by the vertices neighboring a givenvertex, to be a regular polygon orpolyhedron. Five such nets were identi-fied; these have vertex figures that arean equilateral triangle, square, tetrahe-dron, octahedron, and cube (see Fig-ure 1). For each of the nets there is aunique natural tiling, but the tiles arenot necessarily polyhedra in the usualsense, as they may have vertices at whichonly two edges meet (Figure 1). We notethat in the case of the regular nets, thenatural tiling is the unique tiling that hasthe full symmetry of the net. Interest-ingly these five nets appear to be theonly ones with natural tilings that haveone kind of vertex, one kind of edge, onekind of face, and one kind of tile(transitivity 1111).[3] These five nets playan essential role in the geometry ofcrystals and periodic materials in gen-eral.

3. The srs Net and K4

Independent of the work on regularnets, Toshikazu Sunada recently asked,

[*] Prof. S. T. HydeDepartment of Applied MathematicsResearch School of Physical SciencesAustralian National UniversityACT 0200 (Australia)Fax: (+61)2-61254553E-mail: stephen.hyde@anu.edu.au

Prof. M. O’KeeffeDepartment of Chemistry and BiochemistryArizona State University, TempeArizona 85287-1604 (USA)

Prof. D. M. ProserpioDipartimento di Chimica Strutturale eStereochimica Inorganica (DCSSI)Universit? di Milano, Via G. Venezian 2120133 Milano (Italy)

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in a stimulating paper, what three-peri-odic nets were strongly isotropic, that is,which permutation of vertices neigh-bouring a vertex can be realized by anoperation that is a symmetry of thepattern, so that all such permutationsextend to isometries of the net.[5] As thenumber of permutations is n! for an n-coordinated vertex, the restriction ofcrystallographic symmetry in a three-periodic structure (in particular no sym-metry elements of order 5) limits thepossible coordination in a strongly iso-tropic net to three or four, and only twoof the regular nets mentioned above arestrongly isotropic. The two nets (Fig-

ure 1) have the symbols srs (three-coor-dinated) and dia (four-coordinated).[3]

The latter is the familiar net of thediamond structure and the former is themain topic of our essay. As Sunadanoticed, the quotient graph (the graphwith the translations factored out) ofthis structure is the complete graph withfour vertices, K4, so he named it the K4crystal.

Sunada-s paper caused a considera-ble stir that was started by a pressrelease from the American Mathemat-ical Society about its “stunning beauty”under the heading “A Crystal that Na-ture May Have Missed”, stating that “itis tempting to wonder whether it mightoccur in nature or could be synthe-sized”.[6] It was described in Science,Nature Materials, and many other pla-ces in similar terms. The fact that astructure, well-known to crystallogra-phers and crystal chemists for almost ahundred years and to materials scientistsand solid state physicists for over fiftyyears, could be described in this waydramatically illustrates the gap thatexists between much of the physicalsciences and mathematics.

In its most symmetrical embedding,the srs net is cubic and the vertices are atfixed sites of 32 (D3) symmetry wherethreefold and twofold axes intersect.This fixes the coordinates, and thestructure is known to crystallographersas an invariant lattice complex. All suchstructures are documented in the Inter-national Tables for Crystallography(which should be required reading formathematicians interested in periodicstructures) where srs has the symbolY*.[7] It appears in a celebrated 1933paper by Heesch and Laves, which isconcerned with rare sphere packings,[8]

and is also known as the Laves net. Italso appears as “Net 1” [later as (10,3)-a] in the first of a series of pioneeringpapers on the geometrical basis ofcrystal chemistry by A. F. Wells whonoted some occurrences in crystalchemistry.[9] The most conspicuous ofthese occurrences is the family of com-pounds with structures related to SrSi2,where the net describes the topology ofthe Si substructure, hence the symbolsrs. A recent spectacular occurrence isthe high-pressure, three-coordinatedform of elemental nitrogen.[10] With theresurgence of interest in coordination

polymers and metal-organic frame-works in the 1990s examples having thesrs topology were soon found,[11,12] and itis now common in this area. As Sunadaremarked, it has also been considered asa possible allotrope of carbon (andpredicted to be metallic).[13] To addi-tionally appreciate the beauty and otherproperties of the net, illustrated inFigure 2, one should consider the sym-metry. In its most symmetrical embed-ding the symmetry is I4132, isomorphicwith the combinatorial symmetry (auto-morphism group) of the graph. Thissymmetry is the most complex of thespace groups with only proper symmetryoperations (rotations and translations).It is relevant that there are noninter-secting fourfold axes, which are either 41

or 43 screws. There are also noninter-secting threefold axes which are eitherthreefold rotations or 31 or 32 screws. Asall the symmetry operations are proper,the structure is chiral—it comes in left-and right-handed forms. We remark thatthe srs net is the only three-coordinated,three-periodic net with threefold sym-metry at the vertices, and accordingly,the only such net with equivalent edges(edge-transitive).

There is an interesting way in whichthe srs net had arisen 50 years earlier inmathematics. The regular polyhedra{n,3} with n= 3, 4, or 5 have 3 n-gonsmeeting at each vertex (tetrahedron,cube, and dodecahedron respectively).{6,3} is the tiling of the plane withregular hexagons and {n,3} with n> 6are tilings of the hyperbolic plane. But if,as suggested by H. S. M. Coxeter,[14] andtaken up later by B. GrJnbaum,[15] weallow faces to have infinitely manyedges and to be skew polygons, then{1,3} corresponds to a generalizedpolyhedron defined now as a family ofpolygons, such that any two polygonshave in common either one vertex orone edge (two adjacent vertices) or haveno vertices in common, and each edge iscommon to exactly two polygons.

There are in fact two such polyhedrain which the faces are helices that havetrivalent vertices: {1,3}a and {1,3}bwith faces that are threefold or fourfoldhelices, respectively (Figure 3) and theirnets (the set of edges and vertices) arethe srs nets in both cases. This character-istic was recognized and clearly illus-trated by GrJnbaum,[15] who in fact also

Figure 1. Fragments of the regular nets withtheir symbols and vertex figures. The naturaltiles for the nets are shown shrunken forclarity; in reality they fill space.

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cites Wells-[9] paper. In his earlier workCoxeter referred to the net as “Laves-graph”.[14]

In the natural tiling of the net thereis just one kind of tile with three 10-sided faces (Figure 1).[3] These 10 rings

are the only rings in the structure (a ringin this context is a cycle that is not thesum of smaller cycles) and 15 of thesedecagons meet at each vertex.

4. Interthreaded srs

For every tiling one can define a dualtiling moti obtained by putting newvertices in the middle of the old tilesand joining them by new edges throughthe old faces to new vertices in adjacenttiles. It should be clear that the net of thedual tiling will have a coordinationnumber equal to the number of facesof the original tiling. In fact in the caseof the srs net, the dual tiling is just srsagain but now of the opposite hand.Accordingly, the two enantiomorphs canelegantly intergrow (Figure 2), as wasalready known to Wells,[16] and indeedsubsequently observed for coordinationpolymers.[17] It is this intergrowth andthe periodic surface dividing them thatleads to the most interesting part of ourstory.

Wells- geometric patterns (nets)used to describe arrangements in crys-tals, soon appeared in a very different

context: soft, atomically disordered liq-uid crystals. The gulf between atomicand molecular (liquid) crystallographyand pure geometry was narrowingthanks to the efforts of Vittorio Luzzatiand colleagues in France, who spentmany years in the 1960s investigatingthe self-assembly of organic amphi-philes, including metallic soaps andlipids. These materials form orderedsupramolecular structures, whose atom-ic ordering is often no different to astructureless melt, yet collectively formstructures that, like atomic crystals, giverise to diffraction. Luzzati-s group un-covered a rich array of phases, depend-ing on temperature and water content,heralding a new class of condensedmaterials which are now collectivelytermed soft matter.

One of the structures, first reportedby Spegt and Skoulios in 1964 for drysoaps, was later described by Luzzatiand Spegt as a pair of interpenetratingnets, whose edges are composed of Srrods, embedded in an nonpolar (hydro-carbon) continuum.[18] This pattern isthe left- and right-handed pair of srs netsshown in Figure 2. By the late 1960-s,Luzzati et al. had found this structure ina variety of soaps and lipid/water sys-tems.[19] Remarkably, the structure wasdeduced on the basis of powder diffrac-tion patterns alone (the scientists delib-erately destroying large single liquidcrystals), without any knowledge ofprior work on those nets.

5. The Gyroid

Similar investigations into liquidcrystals were also being carried out atthe same time by Swedish and Finnishphysical chemists, pioneered by Per Ek-wall and furthered by Krister Fontell,KMre Larsson, and colleagues in Swe-den. Their own work suggested that themolecules forming these liquid crystalstend to aggregate into sheet-like struc-tures, such as the molecular bilayers thatsheath cells.[20] How then can thosemolecules aggregate to form the pair ofchiral srs networks of threaded edges(re)discovered by Luzzati? The connec-tion was not definitively made until1984, when Larsson and co-workersrecognized the link between the pair ofsrs nets and a sponge-like surface,

Figure 2. Aspects of the srs net. a) Viewalmost along a fourfold axis. Note the 43helices. b) View almost along a threefold axis.Note the 31 helices. c) Two srs nets of oppo-site handedness intergrown.

Figure 3. Fragments of the (1,3) polyhedra.The faces shown are 32 and 41 helices withaxes shown as cylinders. The net, srs, is thesame in both cases.

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known as the gyroid (or G surface).[21]

This surface, discovered in the 1960-s byNASA scientist Alan Schoen[22,23] (Fig-ure 4), bisects space into a pair ofinterthreaded channels, whose axes co-incide precisely with edges of the srsnets (Figure 5). So the net description byLuzzati was reconciled with the mem-brane description: the net described thesponge-like channels of the intricatelyfolded membrane.

In 1992 there were two reports froma group at the Mobil Research andDevelopment Corporation showing thatordered liquid crystal structures couldserve as templates for silica-based mes-oporous materials in which the glass-likeinorganic material was confined to theinterface and after calcination orderedempty channel systems remained.[24,25]

In the cubic material, subsequentlynamed MCM-48, the interface is thegyroid surface, and the channels are apair of interpenetrating srs nets.[26]

The impact of those two papers canbe gauged from the fact that they havebeen cited over 10000 times in the15 years since publication. Very manyinorganic materials based on the MCM-48 structure have since been reported,including a recent hierarchical mesopo-rous inorganic material with crystallineordering on the atomic scale; the order-ing is observed simultaneously on theatomic and meso scales, thus closing thegap between atomic and liquid crys-tals.[27]

The gyroid is closely related to otherimportant triply periodic minimal surfa-ces (surfaces with zero mean curvatureeverywhere). The natural tilings of theregular nets, dia and pcu, are also self-dual (the remaining pair nbo and bcuhave mutually dual tilings), and forthese structures pairs of nets can like-wise interpenetrate. The periodic mini-mal surfaces separating the pairs areknown as the D and P surfaces respec-tively. The D surface was parameterizedby Bernhard Riemann around 1860 andpublished posthumously;[28] it was redis-covered shortly thereafter by Schwarz(splendid engravings of it can be foundin his papers), who also introduced the Psurface.[29] Both surfaces share identicalintrinsic two-dimensional geometry andit is only their three-dimensional em-bedding in space that makes themdifferent. Just as a sheet of paper canbe morphed into a cylinder by simplebending, patches of theD and P surfacesare interchangeable by theBonnet trans-formation.[30] Schoen built beautifulplastic models of the P and D surfacesthat were able to bend ' la Bonnet.[31]

His mathematical and physical manipu-lations revealed a third triply-periodicminimal surface hidden among theaperiodic intermediates to the D andP : the gyroid. Yet the gyroid remainedunknown to mathematics until Schoen-sdiscovery—announced only in a NASApatent and accompanying report.[22,23]

This discovery was a bona fide exampleof an important structure that had beenoverlooked by mathematicians for overa century! Indeed, each labyrinth of thegyroid is centered by the net thatintrigued Sunada.

One wonders if this discovery mighthave been recognized earlier if theregular nets and their tilings had beenrecognized (although not all pairs of

nets of dual tilings necessarily havecorresponding minimal surfaces). As itis, perhaps we should rephrase Sunada-scharacterisation of srs (K4) as the struc-ture that “nature might miss creating” to“the structure that mathematiciansmissed—from Riemann on”?

The identification of the srs net withthe gyroid, one of the most uniformfoldings of a saddle-like sheet into three-dimensional euclidean space, paved theway for a host of other links between thenet and materials. The gyroid structureis now known to underlie many othersupramolecular “soft” materials, fromlipids in cells to synthetic polymericmolecular melts.[32] It also affords anelegant description of the director fieldof molecules within the lowest temper-ature (blue phase), a class of tunableelectro-optic materials of some interestin both academic and practical cir-cles.[33,34]

The gyroid is a supreme example ofa regular partition of space, and it is thatregularity that explains why it is found inso many different materials.[35] Onemeasure of its universality can begauged from the range of crystal sizesin gyroid structures. Bicontinuous phas-es in amphiphiles have a lattice repeatspacing of the order of 100 R. In largerpolymeric systems, that spacing can beten times larger, giving (liquid) crystals aperiod roughly one hundred times great-er than the spacings in hard atomiccrystals. It is remarkable that a soft,quasi-molten molecular assembly suchas polymeric melts, can sustain suchlong-range structural ordering, typicallycontaining 103–104 molecules within asingle unit cell of the membrane.

Perhaps the most spectacular exam-ple of the srs structure in nature can befound in the summer fields of Europe,rather than in the labs of materialsscientists. The Green Hairstreak (Cal-lophrys rubi) is a splendid butterfly,readily identified by its metallic greenwings. Those painted wings contain amyriad of overlapping scales, easily seenin an optical microscope. Electron mi-croscopy reveals an extraordinary three-dimensional matrix of the hard skeletalmaterial within many scales[36] whosemorphology of which is related to thegyroid.[37] Recent structural studies haverevealed that the matrix structure isaccurately described by the srs net,

Figure 4. Alan Schoen, on the roof of theCourant Institute in Manhattan, with his mod-el of the gyroid. (Picture courtesy of StefanHildebrandt. Reproduced by permission ofSpringer-Verlag.)

Figure 5. The gyroid, a three-periodic minimalsurface discovered by Alan Schoen. The sur-face is illustrated together with its labyrinthgraphs (red and green), which describe thechannel structure. Each graph is the srsstructure as in Figure 2c. (Picture courtesy ofGerd SchrFder-Turk).

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where edges of the net are thickened, sothat the material resembles a smoothsponge formed by filling one labyrinthof the gyroid surface (Figure 6).[38,39]

The length of this giant srs materialis around 3000 R—on the same order asthe wavelength of visible light. Indeed,the lovely coloring of the flapping wingin the sunshine is very likely because ofthe scattering of visible light within thesrs matrix. Optical physicists are keen toexploit these materials as they may benatural three-dimensional photoniccrystals that have been long sought afterto advance optical computing. This ma-terial forms within the butterfly chrys-alis and is a living example of supra-molecular self-assembly of soft matter,composed of membrane and proteinmaterials that form a soft matrix thattemplates the later formation of thehard skeleton in the emergent butterfly.The matrix forms by condensation oflipid bilayers in the smooth endoplasmicreticulum, induced by proteins, giving afoliated multilayered sponge. The ho-mogeneity of the gyroid partition allowsuniform bending and packing of thelipid membranes and the interveningproteins. Here too, the extreme regular-ity of the srs/gyroid structures explainstheir genesis in this complex set ofchemical crucibles.

We end this progress report (for it isnot the end of the story) with a plea formore awareness of the work by mathe-maticians and materials scientists. Thebeautiful and complex periodic struc-tures found in nature would surely be

fertile ground for more in-depth mathe-matical studies. At the same time we begmathematicians to make their workmore accessible to physical scientists(e.g. by including pictures such as thebeautiful one presented by Sunada[5]).

Received: April 1, 2008Published online: September 2, 2008

[1] Geometry, which before the origin ofthings was coeternal with the divinemind and is God himself (for what couldthere be in God which would not be Godhimself?), supplied God with patternsfor the creation of the world, and passedover to Man along with the image ofGod. J. Kepler in Harmonices Mundi,Linz, 1619, translated as “The Harmonyof the World” E. J. Aiton, A. M. Dun-can, J. V. Field, Mem. Am. Philos. Soc.1997, 304.

[2] H. S. M. Coxeter, Regular Polytopes,3rd. ed., Dover, New York, 1973.

[3] O. Delgado-Friedrichs, M. O-Keeffe,O. M. Yaghi, Acta Crystallogr. Sect. A2002, 59, 22 – 27. See also the ReticularChemistry Structural Resource, http:/rcsr.anu.edu.au.

[4] O. Delgado-Friedrichs, M. O-Keeffe,O. M. Yaghi, Phys. Chem. Chem. Phys.2007, 9, 1035 – 1043.

[5] T. Sunada,Not. Am.Math. Soc. 2008, 55,208 – 215.

[6] American Mathematical Society, ACrystal that Nature May Have Missed2008, to be found under www.ams.org/ams/press/sunadanots-feb08.html.

[7] “Lattice complexes”: W. Fischer, E.Koch in International Tables for Crystal-lography A (Ed.: T. Hahn), Kluwer,Dordrecht, 2005, chap. 14.

[8] H. Heesch, F. Laves, Z. Kristallogr. 1933,85, 443 – 453.

[9] A. F. Wells, Acta Crystallogr. 1954, 7,535 – 544.

[10] M. I. Eremets, A. G. Gavriliuk, I. A.Trojan, D. A. Dzivenko, R. Boehler,Nat. Mater. 2004, 3, 558 – 563.

[11] L. Carlucci, G. Ciani, D. M. Proserpio,A. Sironi, J. Am. Chem. Soc. 1995, 117,12861 – 12862.

[12] O. M. Yaghi, C. E. Davis, G. Li, H. Li, J.Am. Chem. Soc. 1997, 119, 2861 – 2868.

[13] M. O-Keeffe, G. B. Adams, O. F. Sankey,Phys. Rev. Lett. 1992, 68, 2325 – 2328.

[14] H. S. M. Coxeter, Can. J. Math. 1955, 7,18 – 23.

[15] B. GrJnbaum, Aequationes Math. 1977,16, 1 – 20.

[16] A. F. Wells, Acta Crystallogr. 1954, 7,849 – 853.

[17] a) L. Carlucci, G. Ciani, D. M. Proser-pio, A. Sironi, Chem. Commun. 1996,1393 – 1394; b) S. R. Batten, R. Robson,

Angew. Chem. 1998, 110, 1558 – 1595;Angew. Chem. Int. Ed. 1998, 37, 1460 –1494.

[18] V. Luzzati, P. A. Spegt,Nature 1967, 215,701 – 702.

[19] V. Luzzati, A. Tardieu, T. Gulik-Krzy-wicki, E. Rivas, F. Reiss-Huson, Nature1968, 220, 485 – 486.

[20] G. Lindblom, K. Larsson, L. Johansson,S. ForsTn, K. Fontell, J. Am. Chem. Soc.1979, 101, 5465 – 5470.

[21] S. T. Hyde, S. Andersson, B. Ericsson, K.Larsson, Z. Kristallogr. 1984, 168, 213 –219.

[22] A. H. Schoen, NASA Technical Note,TN D-5541, 1970.

[23] A. H. Schoen, U.S. patent 3663347, filedJul. 16, 1970, issued May 16., 1972.

[24] C. T. Kresge, M. E. Leonowicz, W. J.Roth, J. C. Vartuli, J. S. Beck, Nature1992, 359, 710 – 712.

[25] J. S. Beck, et al., J. Am. Chem. Soc.1992, 114, 10834 – 10843.

[26] V. Alfredsson, M. Anderson, Chem.Mater. 1996, 8, 1141 – 1146.

[27] X. Zou, T. Conradsson, M. Klingstedt,M. S. Dadachov, M. O-Keeffe, Nature2005, 437, 716 – 719.

[28] B. Riemann, K. Hattendorf, Abh. Ges.Wiss. Goettingen Math.-Phys. Kl. 1868,33, 3. Reprinted in B. Riemann, K.Hattendorf, Gesammelte MathematischeWerke, Wissenschaftlicher Nachlass undNachtr=ge (Eds.: H. Weber, R. Dede-kind, R. Narasimhan), Springer, Berlin,1996.

[29] H. A. Schwarz, Bestimmung einer spe-ciellen MinimalflWchen (inkl. Nachtragu. Anhang) in Gesammelte Mathemati-sche Abhandlungen, Vol. 1, Verlag vonJulius Springer, Berlin, 1865, pp. 6 – 125.

[30] O. Bonnet, C. R. Hebd. Seances Acad.Sci.1853, 37, 529 – 532.

[31] A. H. Schoen, Geometric Analysis andComputer Graphics, Mathematical Sci-ence Research Institute Publication #17,Springer, New York, 1991.

[32] S. T. Hyde, et al., The Language ofShape Elsevier, Amsterdam, 1997.

[33] B. Pansu, E. Dubois-Violette, Europhys.Lett. 1989, 10, 43 – 48.

[34] H. J. Coles, M. N. Pivnenko, Nature2005, 436, 997 – 1000.

[35] G. E. SchrYder, S. J. Ramsden, A. G.Christy, S. T. Hyde, Eur. Phys. J. B2003, 35, 551 – 564.

[36] H. Ghiradella, J. Morphol. 1989, 202,69 – 88.

[37] K. Michielsen, D. G. Stavenga, J. R. Soc.Interface 2008, 5, 85 – 94.

[38] S. Wickham, M.Sc. thesis, SydneyUniversity (Australia), 2006.

[39] S. Wickham, H. Averdunk, S. T. Hyde,M. Large, L. Poladian, G. E. SchrYder-Turk, unpublished results.

Figure 6. Chitin network in the butterfly wingscales of Callophrys rubi. The brown surfacedefines the outer form of the chitin matrixdetermined by electron tomography, so thatthe channels containing blue edges are filled.The blue srs net describes the chitin networkto a remarkable degree of accuracy. (Imagecourtesy of Stuart Ramsden, Gerd SchrFder-Turk).

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