an amphiphilic molecule , such as a lipid and detergent , consists of hydrophilic ( ` water - loving ' ) and hydrophobic ( ` water - fearing ' ) parts . in aqueous solution, these molecules self - assemble into various structures depending on the relative size of the hydrophilic part ; spherical or cylindrical - like micelles , bilayer membranes , and inverted micelles .in particular , the bilayer membrane of phospholipids is the basic structure of the plasma membrane and intracellular compartments of living cells , where the membranes are in a fluid phase and lipid molecules can diffuse in two - dimensional space . a vesicle ( closed membrane )is considered to be a simple model of cells and also has applications as a drug - delivery system .many interesting behaviors of bilayer membranes have been studied such as fluid - to - gel phase transition , lateral phase separation , protein insertion , the lysis of lipid vesicles , and membrane fusion .the thickness of a lipid membrane is nm and cells are m in diameter .thus , the length scale of these phenomena varies from nm to m .to investigate the bilayer structure and molecular interactions , information on a nm scale is necessary .for example , the length mismatch of nm between a protein and lipid can change the stability and interactions between proteins . on the other hand ,the morphologies of lipid vesicles and cells on a m scale can be understood by continuous surface theories .various types of membrane models have been proposed for the simulations .they are classified in two groups depending on whether the bilayer structure is implicitly or explicitly taken into account . in the first group [ figs .[ fig : cg](d ) , ( e ) ] , the bilayer membrane is described as a smoothly - curved mathematical surface .this assumption is valid on length scales greater than the membrane thickness of nm .thus , a unit segment represents not a lipid molecule , but a membrane patch consisting of hundreds to thousands of lipid molecules .the information about the bilayer properties is only reflected in this case in the values of the elastic parameters . a mesh discretization with a triangular or square mesh is typically employed for the simulation .to allow large deformations of a fluid membrane , remeshing is necessary . in particular , a dynamically - triangulated membrane model widely used for the membrane with thermal fluctuations .an alternative model is a meshless membrane , in which particles self - assemble into a membrane by potential interactions .the meshless models are very suitable for studying membrane dynamics accompanied by topological changes .( b ) coarse - grained model with explicit solvent constructed based on the all - atom simulation in ( a )  .( c ) solvent - free molecular model  .( d ) meshless membrane model  .( e ) dynamically - triangulated membrane model  .typical lengths of unit segments are shown in parentheses .figures ( a ) and ( b ) were provided by shinoda . ]  in the second group [ figs .[ fig : cg](a)-(c ) ] , amphiphilic molecules are modeled by atomistic or coarse - grained ( cg ) molecular models , and the solvent is taken into account either explicitly or implicitly .although computer technology is rapidly growing , ns dynamics of hundreds of lipid molecules are typical levels of recent simulations by the all - atom models .thus , cg models have been developed to reduce the computational costs .cg molecular models with the lenard - jones potential and with dissipative particle dynamics ( dpd ) potential have been widely used for the explicit - solvent simulations .recently , the potential parameters in the cg molecular models are tuned by atomistic simulations , see figs .[ fig : cg](a ) and ( b ) . to further reduce the computational costs , larger cg segments ( two or three segment particles per amphiphilic molecule )are employed and the solvent is implicitly represented by an effective attractive potential between the hydrophobic segments , see figs .[ fig : cg](c ) . this type of implicit solvent modelis typically called the solvent - free model .there is a length - scale gap between these molecular models and the curved - surface models . for the molecular models ,the size of the unit segments can be gradually varied from to nm .for the curved - surface models , a unit segment represents nm , and the thermal undulations of smaller lengths are neglected .  recently , the cg molecular simulations and triangulated membrane were reviewed in refs . and in ref . , respectively . the phase separation in multicomponent membraneswas reported by kumar in this jspsj special issue . in this paper , we review the recent developments in the solvent - free molecular models and curved surface models [ figs . [fig : cg](c)-(e ) ] for single - component membranes and also describes some new results . in secs .[ sec : tri ] and [ sec : meshless ] , the triangulated and meshless membrane models are described , respectively .the area - difference elasticity ( ade ) model is newly applied to the dynamically - triangulated membrane .the membrane dynamics in flows are presented for both models . in sec .[ sec : solf ] , the solvent - free molecular models are described .the membrane fusion and the formation of polyhedral vesicles are presented as applications .the curvature energy of a fluid membrane is given by   da \label{eq : helf}\ ] ] where and are the principal curvatures at each point in the membrane .the coefficients and are the bending rigidity and saddle - splay modulus , respectively .the spontaneous curvature vanishes when lipids symmetrically distribute in both monolayers of the bilayer .lipid membranes typically have , where is the thermal energy . for membranes of fixed topology without edges ,the integral over the gaussian curvature is an invariant and their properties are independent of .since the critical micelle concentration ( cmc ) of lipids is very low , the number of lipid molecules in a membrane is essentially constant over the typical experimental time scales .also , the osmotic pressure generated by ions or macromolecules in solution , which can not penetrate the lipid bilayer , keeps the internal volume essentially constant . under the constraints of a constant volume and constant surface area , stomatocyte , discocyte , and prolate shapesgive the energy minimum of with for , , and , respectively , where the reduced volume with .  in a dynamically - triangulated surface model of vesicles and cells ,the membrane is described by vertices which are connected by bonds ( tethers ) to form a triangular network .the vertices have excluded volumes .the curvature energy is discretized using angles of the neighboring faces or using dual lattices of triangulation . to model the fluidity of the membrane, bonds can be flipped to the diagonal of two adjacent triangles using the monte carlo ( mc ) method , see fig .[ fig : bflip ] . the membrane viscosity can be varied by the frequency of the bond flip .this model is widely used to simulate a vesicle with spherical topology .the membrane with open edges was also studied .topological changes can be taken into account by a mesh reconnection .however , this reconnection is a discrete procedure , so that it can not smoothly treat the dynamics .recently , the dynamically - triangulated model were applied to the dynamics of a fluid vesicle in flows by combining with boundary element methods or multi - particle collision ( mpc ) dynamics .mpc is a particle - based simulation method proposed by malevanets and kapral . in a simple shear flow , ,the vesicle shows three types of motions .( i ) tank - treading motion : the vesicle has the constant inclination angle with respect to the flow direction , and instead , the membrane is rotating .( ii ) tumbling : the angle rotates . (iii ) swinging : the angle oscillates around zero . at a low shear rate , the vesicle transits from the tank - treading to tumbling with the increasing viscosity contrast or membrane viscosity , where and are the viscosities of the internal and external fluids . at the higher shear rate , a swinging phase appears between the tank - treading and tumbling phases .these dynamics were also observed in experiments , and are understood by the theory for an ellipsoidal vesicle or the perturbation theory for a quasi - spherical vesicle .we found shear - induced shape transitions ; elongating transitions from stomatocyte and discocyte to prolate , as well as a shrinking transition from a tumbling prolate to a tank - treading discocyte .in capillary flow , the fluid vesicle also transits from discocyte to prolate with the increasing flow velocity .   in experiments , several morphologies of lipid vesicleswere observed in the addition to the above shapes : pear , peal - neckless , and branched starfish - like shapes . to explain them , the area difference elasticity ( ade ) modelis widely used . in a vesicle ,the area of two monolayers of the bilayer membrane are different with , where is the distance between two monolayers .since the flip - flop of lipids ( traverse motion between monolayers ) is very slow , the area difference preferred by lipids is typically different from , where and are the number of lipids in the outer and inner monolayers , respectively , and is the area per lipid . in the ade model ,the energy of this mismatch is given by with the averaged curvature .the curvature normalized by of a spherical vesicle , is denoted by , as .we simulated vesicles using the dynamically - triangulated model for with the harmonic constraint potentials of and .figure [ fig : adesnap ] shows snapshots of the brownian dynamics ( bd ) simulation , where the motions of the membrane vertices are given by the underdamped langevin equations .   and .( a ) three - armed starfish - like shape at and .( b ) two connected buds inside a vesicle at and . the front quarter of the snapshot in ( b )is removed to show the inside . ]   and inclination angle of a vesicle in simple shear flow at , , , , , , and .the vesicle exhibits shape transitions from a budded shape to prolate at and from prolate to the budded shape at .the asphericity is the degree of deviation from a spherical shape , , where are the eigenvalues of the gyration tensor of the vesicle and .the intrinsic time unit is .the other simulation parameters are described in ref .]  vesicles of the ade model also exhibit shape transitions in shear flow .a budded vesicle [ see the bottom snapshots in fig .[ fig : adeshear ] ] shows a tumbling motion at the low shear rate [ see red lines in fig .[ fig : adeshear ] ] , even for and , where prolate and oblate vesicles at any only show tank - treading .a similar tumbling motion was reported for a budded two - component vesicle . at the higher shear rate ,the vesicle is found to transit into a prolate shape , which shows tank - treading [ see blue lines in fig .[ fig : adeshear ] ] . as the shear rate decreases ,the vesicle transits back to a budded shape .the vesicle dynamics of other shapes , such as starfishes , are an interesting problem for further studies .   and , respectively ]   and  .( a ) discocyte shape in slow flow .( b ) parachute shape in fast flow .the upper - front quarter of the snapshot in ( b ) is removed to show the inside .the arrows represent the flow velocities . ]   the membranes ( shells ) of synthetic capsules , virus , and red blood cells ( rbcs ) , have the shear elasticity .we call them elastic membranes. the elastic membrane can be modeled as a fixed mesh ( network ) .the relative importance of the bending and shear elasticities is determined by the dimensionless fppl - von krmn number , where is the two - dimensional young modulus .the elastic membrane with disclinations has a spherical shape at and an icosahedral shape at .the bending or shear elasticity is dominant for the former or latter condition , respectively .the properties of viral capsids are well explained by the elastic membrane models .the models of an rbc membrane have been intensively studied . in the molecular - scale models ,the spectorin network attached on the bilayer membrane is modeled by bonds with a worm - like chain ( wlc ) potential . in the continuum models ,the skalak model is widely used .recently , the comparison with the optical tweezer experiment became a standard method to tune the elastic parameters .figure [ fig : rbcstr ] shows the experimental data and our simulation data . in the simulation , the rbc membrane is modeled by triangular networks with vertices , which are connected by a bond potential with and .the nonlinear term with gives better agreement with the experimental data , but an arbitrariness in the choice of the nonlinear function still remains .the shape transitions of rbc with a stomatocyte - discocyte - echinocyte sequence can be reproduced by the elastic membrane with the ade model .the dynamics of the rbcs and elastic capsules have been simulated by the elastic membrane combined with boundary element methods , immersed boundary methods , mpc , and dpd .figure [ fig : rbccap ] shows the elastic vesicles in capillary flow . at the small flow velocities , the symmetry axis of the discocyte is found not to be oriented perpendicular to the cylinder axis . with the increasing flow velocity , the elastic vesicle transits into a parachute - like shape while the fluid vesicle transits into a prolate shape .the transition velocity linearly depends on the elasticities and .the results are in good agreement with the experimental data . in simple shear flow , elastic capsules and rbcs transit from tumbling to tank - treading with shape oscillation ( swinging ) as the shear rate increases .these dynamics are caused by an energy barrier for the tank - treading rotation .   and dependence of the bending rigidity and the line tension  .the solid ( red ) lines represent data for and .the dashed ( blue ) lines represent data for and .the solid and dashed lines in ( a ) and ( c ) show linear fits . ]  , , , , and  .the membrane stochastically forms one or two vesicles . , width=321 ]  , where is the radius of the initial vesicle of particles .the membrane is in a fluid state with , , , and .( a ) , ( b ) , ( c ) , and ( d ) . ]the meshless models are particle - based methods , which do not need bond connections between neighboring particles .all of the proposed meshless methods are solvent - fee methods , but can be extended to explicit solvent versions .the first meshless model was proposed by drouffe _et al . _ in 1991 .the particles possess an orientational degree of freedom and interact with each other via three potentials : a soft - core repulsion , an anisotropic attraction , and a hydrophobic multibody interaction .the particles self - assemble into a fluid membrane .very recently , the modified models with pairwise potentials were proposed . they also have orientational degrees of freedom and can form fluid membranes .recently , we proposed an alternative meshless model in which particles possess no internal degrees of freedom unlike the other models . in our model ,the particles interact with each other via the potential which consists of a repulsive soft - core potential with a diameter , an attractive potential , and a curvature potential .all three potentials only depend on the positions of the particles .two types of curvature potentials were proposed on the basis of the moving least - squares ( mls ) method  . here , we only describe the simpler potential , which drives the particles onto a plane : .the degree of deviation from a plane , the aplanarity , is defined as where , , and are eigenvalues of the weighted gyration tensor , with and a smoothly - truncated gaussian function  . in this model , and essentially be independently varied , see fig .[ fig : mlspara ] .the bending rigidity linearly increases with and is almost independent of . on the other hand ,the line tension linearly increases with and is almost independent of for .the saddle - splay modulus is roughly estimated as .although the hydrodynamic interactions are not present in the model itself , these interactions can be taken into account by combining with a particle - based hydrodynamic method such as mpc and dpd .thus , the hydrodynamic interactions can be easily switched on or off .static equilibrium properties can be investigated by bd or mc , which requires much less computational time .one of the disadvantages of meshless methods , compared with the mesh models , is that the vesicle volume can not be constrained , since the exact volume is difficult to calculate .however , the volume can be naturally fixed if an explicit solvent is introduced .the effects of the hydrodynamic interactions for the self - assembly and dissolution are investigated by comparing mpc with bd .the hydrodynamic interactions are found to speed up the dynamics in both cases .the particles self - assemble into discoidal clusters , and clusters aggregate , and then large clusters form vesicles .to quantitatively investigate the closing dynamics to a vesicle , the simulations of the initially flat membranes were performed . for a membrane slightly above the critical size , all of them close into vesicles via a bowl - like shape . at ,the line tension energy of a flat disk equals the bending energy of a spherical vesicle . for much larger membranes of , however , the membrane stochastically forms two vesicles via an -shaped conformation , see fig .[ fig : mlsclo ] .the large line tension induces a negative surface tension , and the resulting buckled shape can grow .the closing vesicle has an oblate shape in the bd simulations with large [ see the left - bottom snapshot in fig .[ fig : mlsclo ] ] . with the hydrodynamic interactions , the pressure of the internal fluid makes this closing shape more spherical .when the line tension is reduced , a vesicle dissolves via an opened pore on the membrane with for both mpc and bd .a similar pore - opening was observed in the lysis of a lipid vesicle by detergents .  in simple shear flow ,a spherical vesicle of the meshless membrane elongates into a prolate shape with its decreasing volume .similar elongation is observed for the vesicles consisting of surfactants . at highershears , the vesicle ruptures into two pieces , see fig . [fig : mlsshear ] .the membrane rupture is a fundamental process for the formation of multi - lamellar vesicles in shear flows .dependence of diffusion constant of amphiphilic molecules , consisting of one hydrophilic segment and one hydrophobic segment in the bilayer membrane for various cutoff densities . ]the simulations of lipid molecules with an explicit solvent require the calculation for a large number of water molecules in addition to lipid molecules .a small patch of flat membranes can be simulated with water molecules per lipid by the atomistic models .however , more water molecules are needed for the simulations of vesicles , since the formation of a vesicle ( fig .[ fig : mlsclo ] ) requires large solvent space to prevent membrane interactions through the periodic boundary conditions of the simulation box .the self - assembly of amphiphilic molecules in dilute solutions requires many more water molecules .the solvent - free models are more efficient tools for the simulations which require a larger solvent space .a similar solvent - free approach is also frequently used in the simulations of polymers and proteins .the first solvent - free molecular model , which self - assembles into a bilayer membrane without an explicit solvent , was proposed by us in  .later , the modified models were proposed for membranes and for micelles .an amphiphilic molecule is modeled as rigid or flexible chains with single or double hydrophobic tails .the molecules interact with each other with pairwise or multibody potentials . in particular , the potentials in refs . are constructed from the radial distribution function of atomistic simulations .a common feature of the models is the requirement of an attractive potential between hydrophobic segments .this attraction mimics the `` hydrophobic '' interaction ( hydrophobic segments dislike to contact with water ) .one of the simple ideas to estimate this interaction is that the hydration energy is assumed to be proportional to the solvent accessible surface area ( asa ) . since the calculation of the asa is a numerically time - consuming task , an effective potential , which is a function of the local density of hydrophobic segments ,were proposed for protein and lipid molecules . at a low density ,the potential acts as a pairwise attractive potential , but at a high density , this attraction vanishes , in which the segments are assumed to be completely surrounded by other hydrophobic segments . this multibody potential can produce a very fast lateral diffusion of molecules and a wide fluid - phase range ( ) .a fluid membrane is also generated by pairwise attractive potentials , but their cmc is very high , and the membrane often coexists with isolated monomers . although the range of the fluid phase becomes larger for wider pairwise potentials , their ranges are still relatively small .figure [ fig : dif ] shows that the lateral diffusion on the membrane becomes faster with the decreasing , where corresponds to a pairwise potential .the attractive pairwise potential of other solvent - free models can be extended to density - dependent potentials in order to obtain faster dynamics .the solvent - free molecular models have been applied to a variety of phenomena : self - assembly to vesicles , membrane fusion , membrane fission , the formation of polyhedral vesicles , pore formation , the adhesion of nanoparticles , the fluid - gel phase transition , phase separation of lipids , protein inclusion in membrane , and dna - membrane complexes .figure [ fig : hedo ] shows a polyhedral vesicle .when the bending rigidity is large compared with the vesicle radius , the lined defects at the edges of polyhedrons are induced by the mismatch of the membrane curvature with the spontaneous curvature of the monolayer .the number of edges and vertices increases with the increasing  . recently , a similar defect was proposed as a kink structure in a symmetric ripple phase .  in early years , an alternative implicit - solvent molecular model , in which the head segment is constrained on a plane ,was employed . in this model ,the membrane can not be deformed or self - assembled . however , the diffusive motion of molecules is still present on a fixed membrane geometry and it may provide a good reference state to investigate the effects of the thermal undulations of a membrane . recently ,a phantom - solvent model was proposed , where an explicit but very simple solvent is employed . in this model ,solvent particles have a repulsive interaction with amphiphilic molecules , but do not interact with each other .thus , the solvent is in the ideal - gas equation of state , , so that its pressure is easily controlled .  , and  .the red spheres and yellow cylinders represent the hydrophilic and hydrophobic segments of amphiphilic molecules , respectively . ]the membrane fusion and fission are key events in various intra- and intercellular processes , such as protein trafficking , fertilization , and viral infection . in an endocytosis pathway , a small vesicle pinches off from the plasma membrane and fuses with a lysosome .several fusion mechanisms and fusion intermediate structures have been proposed for the fusion of biological and lipid membranes  . among them , the stalk model  is widely accepted and is qualitatively supported by experimental studies .the first intermediate in this model , a stalk , is a hourglass - like structure that connects only the outer monolayers of the vesicles [ fig .[ fig : fuspath](b ) ] . in the stalk models ,two types of pore - opening pathways from the stalk intermediate were previously proposed .radial expansion of the stalk results in contact between the inner monolayers inside the stalk , a trans - monolayer contact state [ fig .[ fig : fuspath](c ) ] .in the original model  , expansion of the contact area results in a disk - shaped bilayer consisting of both inner monolayers , called a hemifusion diaphragm [ fig . [fig : fuspath](e ) ] , and a fusion pore is formed in the hemifusion diaphragm [ fig .[ fig : fuspath]:(b)(c)(e)(f ) ] .the original model is based only on the calculation of the bending energy of the monolayers in fusion intermediates .siegel  also took into account the interstice ( void ) energy of junctions of monolayers . in the model modified by siegel , the trans - monolayer contact results in pore formation , rather than expansion of the contact area [ fig[ fig : fuspath]:(b)(c)(f ) ] .although the free energy barrier was estimated to be too high ( ) in ref . , the barrier is reduced by a more quantitative estimation of the monolayer geometry .the molecular tilt in the monolayers was proposed as the mechanism to fill the void space .recently , the free energy landscape of the fusion was also studied using the self - consistent field theory .the first molecular simulation of membrane fusion was performed by a solvent - free model .small vesicles spontaneously fuse via the modified stalk pathway [ fig .[ fig : fuspath]:(b)(c)(f ) ] at low temperature . however , at high temperature , the vesicles fuse via a new pathway , where a pore opens on the side of the stalk and then the elongated stalk bends to form a fusion pore [ fig .[ fig : fuspath]:(b)(d)(f ) ] .the adhesion of a nanoparticle was found to promote this fusion process .recently , the leakage between the interior and exterior of a vesicle was observed in an experiment .this leakage supports the side - pore pathway .the vesicles pinched by particles form the trans - monolayer contact [ fig .[ fig : fussnap](a ) ] and then fusion directly occurs [ fig .[ fig : fuspath]:((c)(f ) ] or via the hemifusion diaphragm [ fig .[ fig : fuspath]:(c)(e)(f ) ] .membrane fission occurs via pathways opposite of the fusion . in a stretched vesicle , a stalk structure [ fig .[ fig : fussnap](b ) ] is formed via the trans - monolayer contact [ fig .[ fig : fuspath]:(f)(c)(b ) ] .on the other hand , the particle adhesion induces pore opening on the necked membrane and the pore expansion leads to the stalk formation [ fig .[ fig : fuspath]:(f)(d)(b ) ] . in large spontaneous curvature of the monolayersinduces the vesicle fission via pore - opening at the lined defects .recently , membrane fusion was simulated by various molecular models such as the bond - fluctuation lattice model , explicit - solvent cg models , dpd models , and the atomistic model .these simulations also show fusion pathways as shown in fig .[ fig : fuspath ] .the hemifusion diaphragm was found to also be formed from the side - pore of the stalk .the membrane fission was also simulated in two - component vesicles .although the dependence on external forces and surface tension was investigated , conditions to determine the pathways are not fully understood .the membrane fusion and fission are induced by proteins in living cells .their molecular mechanisms are not well understood and have not yet been simulated .we have presented three types of membrane models : the triangulated membrane , meshless membrane , and solvent - free molecular models .the first two models are constructed for large scale ( m ) and the last model is for a molecular scale ( nm ) . to study the bending deformation of membranes ,the details on a molecular scale are typically not necessary , therefore , the mesh and meshless models are suitable and much more efficient than the molecular models . on the other hand , to study the phenomena including the non - bilayer structure , such as membrane fusion and protein insertion , the molecular scale can not be neglected .  when compared with the phenomena in thermal equilibrium ,the nonequilibrium phenomena are not well understood . in this paper, we presented several dynamic behaviors on a - scale in flows . on the other hand , the dynamics on a molecular scale were much less explored while a few simulation studies were reported .the electric field is another interesting external field , which can open pores on a membrane and induce shape deformation of a vesicle . in living cells ,the biomembrane is in a complex nonequilibrium environment .the membrane models presented in this paper are powerful tools to study the membrane behaviors under nonequilibrium as well as under equilibrium conditions .the author would like to thank w. shinoda ( aist ) for providing the figures , and g. gompper ( jlich ) for the helpful discussions .* hiroshi noguchi * was born in hyogo prefecture , japan , in 1973 .he received his b. sc .( 1995 ) , m. ph .( 1997 ) , and d. ph .( 2000 ) from nagoya university .he was a postdoc at the institute for molecular science ( okazaki ) ( 2000 - 2003 ) , and a postdoc ( 2003 - 2006 ) and a research staff member ( 2006 - 2008 ) at forschungszentrum jlich .he has been an associate professor at the university of tokyo since may 2008 .he has studied soft matter physics , in particular , membrane physics , using theories and simulations .

Abstract: recent developments in lipid membrane models for simulations are reviewed . to reduce computational costs , various coarse - grained molecular models have been proposed . among them , implicit solvent ( solvent - free ) molecular models are relatively more coarse - grained and efficient for simulating large bilayer membranes . on a m scale , the molecular details are typically negligible and the membrane can be described as a continuous curved surface . the theoretical models for fluid and elastic membranes with mesh or meshless discretizations are presented . as examples of applications , the dynamics of vesicles in flows , vesicle formation , and membrane fusion are presented .


except for a finite list of interesting exceptions , richard roth has classified isonemal periodic prefabric designs into 39 infinite species falling into three more general classes as well as the previously defined genera .species 110 have reflection or glide - reflection symmetries with parallel axes and no rotational symmetry , not even half - turns .species 1132 have reflection or glide - reflection symmetries with perpendicular axes , hence half - turns , but no quarter - turns .species 3339 have quarter - turn symmetries but no mirror or glide - reflection symmetries .this taxonomy has been refined slightly and used in , to which reference needs to be made , to determine the feasible symmetry groups and hence isonemal prefabrics . as roth observes beginning his subsequent paper on perfect colourings , ` [ r]ecent mathematical work on the theory of woven fabrics ' begins with , which remains the fundamental reference . in that paper roth determines which fabrics actually prefabrics can be perfectly coloured by striping warp and weft .this paper is intended to reconsider that topic in terms of roth s taxonomy as refined in and also to consider further the related question which of isonemal prefabrics fall apart .a _ prefabric , _ as defined by grnbaum and shephard , consists of two or more congruent layers ( here only two ) of parallel strands in the same plane together with a preferential ranking or ordering of the layers at every point of that does not lie on the boundary of a strand .the points not on the boundary of a strand are naturally arranged into what are called here _ cells _, in each of which one strand is uppermost .the ( parallel ) strands of each layer are perpendicular to those of the other layer , making the cells square .these square cells are taken here to be of unit area .the mathematical literature on weaving has concerned exclusively _ periodic _ arrangements in the plane in the standard two - dimensional sense explained by schattschneider .there exists a non - unique finite region and two linearly independent translations such that the set of all images of the region , when acted upon by the group generated by these translations , reproduces the original configuration , which is assumed to be infinite in all directions for convenience .schattschneider gives the name _ unit _ to a smallest region of the plane having the property that the set of its images under this translation group covers the plane .such units are all of the same area , the _ period _ , but in general can be of a variety of shapes . since our prefabric layers meet at right angles and the symmetry groups with which we shall be concerned here are all rotational ( ) or have axes of reflection or glide - reflection in only parallel or in two perpendicular directions , the _ lattice units _, that is , period parallelograms whose vertices are all images of a single point under the action of the translation subgroup , can be either rectangular or rhombic ( figure 1 ) .   with an outline of no geometrical significance but its dimensions .10 pt b. symmetry group of type with mirrors that become axes of glide - reflection in the side - preserving subgroup .10 pt c. type- symmetry group with mirrors and axes of side - reversing glide - reflection alternating .10 pt d. type- symmetry group with alternative lattice units outlined .centres of quarter - turns without side reversal and with side reversal are illustrated .also centres of side - reversing half - turns .10 pt e. side - preserving subgroup of type with an outline of no geometrical significance through centres of half - turns .10 pt f. type- symmetry group with mirrors and side - reversing glide - reflections .10 pt g. the rhombic lattice unit of a symmetry group of type with four centres of half - turns outside it marking the corners of the dashed outline of a lattice unit of the side - preserving subgroup of type .,title="fig : " ] ( a ) 0.75 in ( b ) 0.75 in ( c ) 0.75 in ( d )  with an outline of no geometrical significance but its dimensions .10 pt b. symmetry group of type with mirrors that become axes of glide - reflection in the side - preserving subgroup .10 pt c. type- symmetry group with mirrors and axes of side - reversing glide - reflection alternating .10 pt d. type- symmetry group with alternative lattice units outlined .centres of quarter - turns without side reversal and with side reversal are illustrated .also centres of side - reversing half - turns .10 pt e. side - preserving subgroup of type with an outline of no geometrical significance through centres of half - turns .10 pt f. type- symmetry group with mirrors and side - reversing glide - reflections .10 pt g. the rhombic lattice unit of a symmetry group of type with four centres of half - turns outside it marking the corners of the dashed outline of a lattice unit of the side - preserving subgroup of type .,title="fig : " ] ( e)1.5 in ( f)1.5 in ( g )  some of the rectangles have one set of parallel boundaries defined by the group but the perpendicular boundaries arbitrary in position , only the distance between them being dictated by the group ( e.g. , figures 1b and 6 ) . the notion of symmetry groupallows the definition of the term isonemal ; a prefabric is said to be _ isonemal _ if its symmetry group is transitive on the strands , whose directions are conventionally chosen to be vertical , called _ warps _ , and horizontal , called _wefts_. the behaviour of each strand in a prefabric is , by isonemality , the same as the behaviour of every other strand ; the periodicity in two dimensions entails periodicity in one dimension along each strand . to distinguish the linear periodicity from the planar, the period along each strand will be called _order_. fabrics were catalogued by grnbaum and shephard , sorted first under order , and then by binary index , which is the sequence of pale and dark cells of order length represented by 0s and 1s and chosen to be a minimum , then given an arbitrary sequence number .the first catalogue listed fabrics other than twills for orders 2 to 8 ( plus 13 ) and an extension likewise up to order 12 ( plus 15 and 17 ) .prefabrics that fall apart were catalogued by hoskins and thomas for orders 4 to 16numerically the same way but with an asterisk to indicate that they are not fabrics .  the standard way to representthe preferential ranking of the strands is to regard the plane as viewed from one side , from which viewpoint one or the other strand is visible in each cell . by the _normal colouring _ of warps dark and wefts pale , the visual appearance of the strands from a particular viewpoint becomes an easily understood code for which strand is uppermost .this visual appearance from one side ( arbitrarily chosen and called the _ obverse _ ) of the strands we refer to as the _ pattern_. such a pattern , which consists of an array of dark and pale congruent cells tessellating the plane is given a topological meaning , the _ design _ of the prefabric .as we shall be considering patterns different from the design of a prefabric , the distinction is important ; a design is the pattern of a prefabric that is normally coloured .so a pattern can be interpreted as a design or not .normal colouring has a consequence that other colourings of the strands need not have . because at every point not on the boundary of a strand the two strands preferentially ranked are of different colours , a design s colour complement ( switching dark and pale ) is the appearance of the prefabric from behind as though viewed from the obverse side in a mirror set up behind itthis we shall call the _ reverse _ of the prefabric .when a prefabric is coloured normally the reflected reverse pattern is the colour complement of the obverse pattern .the distinction between prefabrics and fabrics can now be explained ; a _ fabric _ is a prefabric that _ hangs together _, that is , that does not _ fall apart _ in the sense that some warps or some wefts or some of each can be lifted off the remainder because they are not bound into a coherent network by the interleaving defined by the preferential ranking .the most extreme example of a prefabric that falls apart is the trivial prefabric in which all warps pass over all wefts ( uniformly dark diagram ) or all wefts pass over all warps ( uniformly pale diagram ) . in an obvious extension of the notation adopted in the catalogue of genuinely periodic isonemal prefabrics that fall apart , the trivial prefabric would be denoted 1 - 0 - 1*.  for isonemalprefabrics other than his short list of exceptions , all of order less than 5 , roth showed that the symmetry group is a layer group with two - dimensional projection of crystallographic type , , , , , , , or , that is , it has respectively glide - reflection axes or mirrors or both in parallel directions or glide - reflection axes in perpendicular directions , or glide - reflection axes and axes of reflection in directions perpendicular to each other , or just perpendicular axes of reflection , or both ( alternating ) in both perpendicular directions or just quarter - turns and half - turns . in order for a glide - reflection or quarter - turn to be a symmetry of the prefabric , it may or may not have to be combined with reversal of the sides of the prefabric , , i.e. , reflection in the plane .mirror symmetry must always be planar reflection combined with because any cell through which the mirror passes must be its own image in the symmetry but the reflection alone , reversing warp and weft , would reverse its conventional colour .so is needed to restore it .this means that there is never mirror symmetry in the side - preserving subgroup , which really is two - dimensional ( no ) .it is that determines the two - dimensional period under translation alone . for these species of prefabric, is of type , which is generated by translations only , or , which is a group generated by half - turns only , or , , or already characterized .when is of type , , or it may be the same group as , or it may be a proper subgroup of , or it may be just of the same type as .there are too many possible geometrical configurations of the symmetry groups and side - preserving subgroups just mentioned to illustrate their lattice units here .most of them are illustrated in .there is are tables in of which figures in illustrate which types of group .figure 1 illustrates a selection in order to show the conventions for the symmetry - group operations .glide - reflection axes without ( side - preserving ) are hollow dashed lines ( making the central rectangle of figure 1 g ) and with ( side - reversing ) are filled dashed lines ( figures 1c and f ) .mirrors ( always with ) are filled double lines ( figures 1c , f , and g ) , but if the site of a mirror is also that of an axis of side - preserving glide - reflection the filling of the double lines is dashed ( figure 1b ). axes of glide - reflection can run through the centres and corners of cells ( _ mirror position , _figures 1b , 10 , and 20 ) or through the sides of cells instead ( where a mirror can not go , figure 1c ) .centres of half - turns are represented as and centres of half - turns with as ; centres of quarter - turns as and centres of quarter - turns with as ( figure 7a ) .thin lines are just boundaries of lattice units , outlining or completing the outline of rectangles or rhombs .the alternative lattice units outlined in figure 1d are related by having the corners of each as centres of the other .as the diagram illustrates with a group of roth type 39 ( ) , the corners of the two sorts of lattice unit can be of different kinds .finally , in figure 1 g four centres of half - turns are illustrated outside the lattice unit to complete a lattice unit with dashed outline of a type- subgroup inside the symmetry group of type .the subgroup is inside the group , but the lattice unit of the group is inside the lattice unit of the subgroup .the longer and shorter diagonals of a rhomb will be called its _ length _ and _width_.  the subject matter now having been introduced , the plan of the paper can be given .striping , thin and thick , is explained in 2 , and which isonemal prefabrics can be thinly striped to be perfect colourings is determined in terms of richard roth s taxonomy . in 3 it is determined which of these stripings must always be the design of an _ isonemal _ prefabric that falls apart .the catalogue of designs of isonemal prefabrics that fall apart is extended to order 20 in 4 .then in 5 it is shown that not all such designs can be produced by thinly striping an isonemal fabric , correcting an error in .thick striping will be further considered elsewhere .we turn now to the matter of perfect colourings of the strands of a prefabric with two colours , the subject of roth s later weaving paper .an introduction to the colouring topic with emphasis on two colours is .since the normal colouring of strands , which allows the visual pattern of the coloured strands ( its design ) to represent their topological structure , is a perfect colouring , this idea was more drawn attention to than introduced in .a symmetry operation is called a _ colour symmetry _ ` if it permutes the colors consistently ' .all the pale strands must be mapped either to pale strands or to dark strands , and correspondingly the dark strands .if all of the symmetries of a prefabric with coloured strands are colour symmetries , then the choice of the strand colours is said to be _ perfect _ or _ symmetric _ , where i shall use exclusively the former term .the interaction of design and pattern is important because the relevant symmetry group is that of the prefabric represented by the pattern that is its design , the group of the design for short , but the permutation ( identity or reversal ) of colours occurs only in the pattern that represents the colouring of the strands , identical to the design only with normal colouring of the strands .there is potential for confusion .fortunately , since there are only two non - normal ways to colour a prefabric that can result in perfect colouring , it is easy to make this discussion concrete .warps and wefts can be striped , that is vary pale and dark , either _ thinly _ , that is alternately , or _thickly _ , that is alternating in pairs : pale , pale , dark , dark , pale , pale . adapting a device from , the colouring of a prefabric can be represented by seeming to extend strands outside the pattern to indicate which strands are pale or dark .i adopt this convention as long as it is not completely obvious which strands are which .   10 pt 10 pt 10 pt 10 pt10 pt  \(a ) 32 pt ( b ) 21 pt ( c ) 30 pt ( d ) 32 pt ( e ) 21 pt ( f )  consider the thin striping of the trivial prefabric 1 - 0 - 1 * , whose design is all pale or all dark . in figures 2a and 2d, the design of 1 - 0 - 1 * appears all pale with an indication of how the prefabric is to be coloured , to the left of the resulting patterns 2b and 2e , where it is seen that the colouring of the invisible warps is not relevant , and the patterns 2c and 2f , which are the appearances from behind as in a mirror held beyond the the plane of the prefabric . on the reverse side, it is the colouring of the wefts that is irrelevant , but even this trivial example alerts one to the fact that the reverse is not just the colour reversal of the obverse .neither can be more important than the other , and so their systematic relation is of interest . where the crossing strands are of the same colour , as in the upper - left and lower - right cells of figure 2a , both obverse and reverse have that colour .elsewhere the obverse colour is that of the design in pale rows ( short for predominantly pale rows where the weft is pale ) and its complement in dark rows ( short for predominantly dark rows where the weft is dark ) .the thin striping creates a checkerboard of cells that may be called redundant and irredundant , where the _ redundant _ are those where a colour meets itself and the _ irredundant _ are those where the warp and weft have different colours .the appearance in irredundant cells is dependent on the design and colouring , in redundant cells only on the colouring .in this language , the irredundant cells on the reverse have the complement of the design in pale rows and the colour of the design in dark rows , in both cases the complement of the obverse.   10 pt 10 pt 10 pt 10 pt 10 pt  \(a ) 32 pt ( b ) 21 pt ( c ) 30 pt ( d ) 32 pt ( e ) 21 pt ( f )  consider the thin striping of the warp and weft of plain weave . because of the interlacing , the different colourings have different results in figures 3b , c , e , f , where again the reverse is shown on the right .but results are the same three as those for the trivial prefabric in figure 2 , indicating general non - uniqueness of the results of striping strands .this striping illustrates that the patterns obtained by striping warp and weft of isonemal fabrics are the designs of prefabrics that fall apart ( * ? ? ?* lemma 3 ) . in consequence of this fact together with the known conditions for isomenal fabrics that fall apart ,if the pattern of a fabric obtained by striping warp and weft is the design of an isonemal prefabric , then the prefabric is of genus ii , iv , or v with no overlap with genus i or iii and with one quarter of the cells in half the rows dark .the pattern arising from the striping of warp and weft of an isonemal fabric does not need to _ be _ the design of an isonemal prefabric as we shall see .the effect of striping warp and weft thickly is to produce a checkerboard of redundant cells that is plain weave doubled ( box weave 4 - 3 - 1 ) .if there is to be any hope of perfect colouring , then the colouring , striping , must be chosen so that the colour symmetries of the fabrics map redundant cells to redundant cells and irredundant cells to irredundant cells , or as roth puts it ` preserve ' them .which half of the cells are to be reduntant and which half are to be irredundant is a choice to be made .there are therefore two ways to stripe the same fabric thinly . preserving the two classes of cell rules out as symmetries for any striping ,  glide - reflections with axes not in mirror position ,  translations with and not integers of the same parity , and  half - turns with centres not at the centre or corner of a cell  butnot half - turns with centres in those two positions , not other translations and not mirror symmetries .it rules out , for thin striping , quarter - turns with centres not at the centre of a cell .roth has shown that these modest necessary conditions are also sufficient to allow the two sorts of striping , but not in terms of his symmetry - group types .when we turn to species without quarter - turns , the glide - reflection constraint alone eliminates species , , , , 4 , , 10 , 12 , 14 , 16 , , , 20 , 24 , , and 32 from being perfectly coloured by striped strands .perfect colouring by thin striping is possible for the remainder of the species with only parallel axes of symmetry : , , 3 , , , 6 , , , , and 9 . as roth points out, types with of type can have strands thinly striped to produce perfect colouring provided that twice each glide is an even multiple of the cell diagonal .this is always the case when its axis is in mirror position . as observed in , the position of glide - reflection axes and the size of glidesare related in such a way that it is unsurprising that the glide - reflections eliminated on the basis of position , those in species 12 , 14 , and 16 , also have glides fractional in .accordingly , species 11 , 13 , and 15 are always stripable .roth also points out that designs with of type allow thin striping with the same constraint on glides , since reflection is always a colour symmetry .this permits thin striping for species 17 , ( s for stripable thinly ) , 19 , 21 , 22 , and 23 , but not , , 20 , and 24 , already banned .a similar blanket permission is for of type , that is types 25 and 26 , since they have only reflections .when is of type , the stripability depends on the translations generated by the mirrors and glide - reflections , which are correctly located except in species 32 , already banned .the translations are the diagonals of the central rectangle as in figure 1f , and their components must have the same parity .both the odd - odd and even - even spacings of roth type 27 , which are used also in species 29 and 30 , and the odd - even spacing of species and , the former used again in species 31 , allow the use of striping .so we have seen what can be stated as a theorem , essentially proved in .isonemal periodic prefabrics of order greater than and of every species with symmetry axes that is not ruled out by the placement of glide - reflections can be perfectly coloured by thin striping : , , and   isonemal periodic prefabrics of order greater than with quarter - turn symmetry can be perfectly coloured by thin striping if and only if they are of species .  the type specifications correspond exactly to the conditions for thin striping of a prefabric with symmetry .( this theorem is a second reason for the choice of subscript . )it is a curiosity that the only perfect colourings illustrated in are normally coloured designs .it is surely of interest to see what non - normal perfect colourings look like . to satisfy this curiosity , fabrics of each of the species with only parallel axes of symmetrywill be described or displayed perfectly coloured by thin striping .the fabrics shown in figures 12a , 14c , and 2a of to illustrate species , , and respectively become non - isonemal designs both ways they are striped thinly .the two ways are illustrated for each example in figure 4a to f.   in figure 12a of .10 pt c , d. example of species in figure 14c of .10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig:"]10 pt  in figure 12a of . 10 pt c , d. example of species in figure 14c of . 10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig : " ]  ( a)117 pt ( b )   in figure 12a of . 10 pt c , d. example of species in figure 14c of . 10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig:"]5 pt  in figure 12a of . 10 pt c , d. example of species in figure 14c of .10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig : " ]  ( c)122 pt ( d )   in figure 12a of . 10 pt c , d. example of species in figure 14c of . 10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig:"]10 pt  in figure 12a of . 10 pt c , d. example of species in figure 14c of . 10 pt e , f. example ( 12 - 35 - 1 ) of species in figure 2a of .,title="fig : " ]  ( e)109 pt ( f )  fabric 12 - 79 - 1 ( figure 8a of ) , showing species 3 , becomes 12 - 65 - 4 * when striped one way and a non - isonemal design the other way ( figure 5a ) .fabric 8 - 11 - 1 ( figure 6a of ) , showing species 6 , becomes 8 - 5 - 3 * when striped one way and a non - isonemal design the other way ( figure 5b ) .fabric 8 - 19 - 5 ( figure 17a of ) , showing species , becomes 4 - 1 - 1 * when striped one way and a non - isonemal design the other way ( figure 5c ) .fabric 8 - 11 - 2 ( figure 7a of ) , showing species 9 , becomes 8 - 5 - 1 * when striped one way and a non - isonemal design the other way ( figure 5d ) .  ) .10 pt d. 8 - 11 - 2 ( species 9).,title="fig:"]10 pt ) .10 pt d. 8 - 11 - 2 ( species 9).,title="fig : " ]  ( a)108 pt ( b ) ) .10 pt d. 8 - 11 - 2 ( species 9).,title="fig:"]10 pt ) .10 pt d. 8 - 11 - 2 ( species 9).,title="fig : " ]  ( c)90 pt ( d )  fabric 12 - 183 - 1 in figure 4a of , showing species , becomes 12 - 69 - 2 * or 12 - 21 - 2 * when thinly striped .the fabrics in figures 15a and 16a of , showing species and respectively ( figures 10a , b , here ) both become designs of isonemal prefabrics that fall apart when thinly striped either way ( figures 6a and b , c and d , respectively ) .   in figure 10a.10 pt c , d. fabric of species in figure 10b . ]\(a )  in figure 10a.10 pt c , d. fabric of species in figure 10b.,title="fig : " ]  \(b )  in figure 10a.10 pt c , d. fabric of species in figure 10b.,title="fig:"]5 pt  in figure 10a.10 pt c , d. fabric of species in figure 10b.,title="fig : " ]  ( c)138 pt ( d )  what is perhaps most obvious in these examples , in particular figures 4 and 5 , is that there is no tendency to the production of designs of _ isonemal _ prefabrics that fall apart .more generally , the strand striping rather often produces stripy patterns that are unattractive given isonemal expectations .this is even true for fabrics with symmetry , of which only one species can be perfectly coloured by thin striping .the example that roth mentions of a design of species is 10 - 1 - 1 , the satin .it is illustrated , with its two thin stripings , in figures 7 and 8 ., 10 - 1 - 1 , the satin with symmetriesnoted.10 pt a. specification.10 pt b. obverse.10 pt c. reverse.,title="fig:"]10 pt  ( a)82 pt ( b)60 pt ( c )  10 pt  ( a)82 pt ( b)60 pt ( c )  one notes that figures 7 and 8 emphasize that means rotate _ and reflect ; _ without the the pattern is changed quite seriously by a quarter turn , as those of figures 4 and 5 are by their symmetries , which include , if is forgotten .the predominance of stripiness persists in species 1132 , where a similar analysis of examples illustrated in gives the following results .the example of species in figure 9b of ( 12 - 135 - 1 ) , of species 26 in figure 4b of ( 12 - 619 - 1 ) , of species 27 in figure 12a of ( 6 - 1 - 1 ) , of species in figure 13a of ( 8 - 1 - 1 , the satin ) , of species 29 in figure 15a of ( 16 - 2499 of ) , and of species 31 in figure 15b of ( order 24 ) , when thinly striped , have patterns neither of which is the design of an isonemal prefabric .this behaviour is analogous to that illustrated in figure 4 .these are just examples and not chosen for this purpose ; it is quite possible for other examples to behave differently .the largest class of examples are the next .the example of species 13 in figure 7a of , becomes the design of an unknown order-24 isonemal prefabric when striped thinly one way and the design of a non - isonemal prefabric when striped thinly the other way . the same is true of further examples as follows , where the isonemal result is indicated in parentheses .species 15 in figure 8a of ( 12 - 47 - 5 12 - 21 - 2 * ) .species 17 in figure 9a of ( 8 - 19 - 7 4 - 1 - 1 * ).  species 19 in figure 9c of ( 8 - 19 - 4 4 - 1 - 1 * ). species 21 in figure 10a of ( 8 - 7 - 2 8 - 5 - 1 * ) .species 23 in figure 11b of ( 12 - 31 - 1 12 - 21 - 1 * ) .species 25 in figure 2a of ( 8 - 19 - 6 4 - 1 - 1 * ) .species in figure 13b of ( 8 - 5 - 1 8 - 5 - 3 * ) .examples of designs of only three species produce designs of isonemal prefabrics when striped each way , those of species 11 in figure 5a of , species 22 in figure 11a of , and species 30 in figure 16 of .they are illustrated in figure 9 .10 pt ( a ) 133 pt ( b )  5 pt ( c ) 137 pt ( d )  10 pt  ( e)138 pt ( f )  these patterns , all of species 15 , would be mere curiosities if they did not illustrate a general fact . andthey _ are _ just examples .( other examples would give different indications , some of them unhelpful .species could have been illustrated by 8 - 27 - 5 , which becomes 4 - 1 - 1 * and 8 - 5 - 3 * when striped both ways . ) but _ all _ fabrics of species 11 , 22 , and 30 produce designs of isonemal prefabrics when striped both ways , as do those of species and species 7 from the species with only parallel axes .the examples given above show that _ only _ those species always do so .the pattern of a fabric of order greater than perfectly coloured by thinly striping strands is the design of an isonemal prefabric that falls apart if the side - preserving subgroup of the fabric s symmetry group is generated by side - preserving glide - reflections and is transitive on strands .the side - preserving subgroups of species , 3 , 6 , , , and 9 are of type . and those of species 11 , 15 , 19 , 21 , 22 , 23 , 29 , and 30 are of type . can not be transitive on strands in these species other than , 7 , 11 , 22 , and 30 , for fabrics in the other species are of genus ii , iv , both ii and iv , or v. so the theorem excludes the species that we know need excluding .it is known that the prefabric falls apart ; it is required only to prove that it is isonemal .the side - preserving glide - reflections that generate the side - preserving subgroup of , transitive on strands , have even or odd glides , and their axes pass through or between the imposed redundant cells , giving four situations to consider .they will be considered in pairs .there are two patterns involved here .the first , which is the design of the fabric that will be subjected to the striping , and the second , which is to be shown to be the design of an isonemal prefabric .the first will be called the design , and the second will be called the pattern .  _ first pair . _each even glide - reflection with axis through the redundant cells and odd glide - reflection with axis between the redundant cells preserves the colour of the redundant cells while setting up a correspondence between  predominantly dark columns and  predominantly dark rows in which the irredundant cells have been complemented  and between  predominantly pale columns in which the irredundant cells have been complemented and  predominantly pale rows .  in the design of the fabric on which the pattern ( not yet shown to be the sort of design desired )is based , the side - preserving glide - reflections mapped the irredundant cells of ( now ) predominantly dark columns to irredundant cells of predominantly dark rows , their colour being reversed by the colouring convention for designs .their colours did not match . in the pattern ,the irredundant cells of predominantly dark columns are mapped to irredundant cells of predominantly dark rows , the colours of which rows have been reversed by the complementation . in the pattern there is no colouring convention .their colours now do match .the colours of the corresponding irredundant cells match , and the colours of the redundant cells match .exactly the same is true for the predominantly pale columns mapped to predominantly pale rows except that the complementation occurred in the columns instead of the rows .this consistency means that the pattern has a colour - preserving glide - reflection for each such side - preserving glide - reflection of the fabric design .each such colour - preserving glide - reflection is a side - reversing glide - reflection of the pattern when it is regarded as a design .  _second pair ._ each even glide - reflection with axis between the redundant cells and odd glide - reflection with axis through the redundant cells reverses the colour of the redundant cells while setting up a correspondence between  predominantly dark columns and  predominantly pale rows  and between  predominantly pale columns in which the irredundant cells have been complemented and  predominantly dark rows in which the irredundant cells have been complemented .  in the design of the fabric on which the pattern is based , the side - preserving glide - reflections mapped the irredundant cells of predominantly dark columns to irredundant cells of predominantly pale rows , their colour being reversed by the colouring convention for designs .their colours did not match . in the pattern ,the irredundant cells of predominantly dark columns are mapped to irredundant cells of predominantly pale rows as before .their colours do not match .the colours of the corresponding irredundant cells are opposite , and the colours of the redundant cells are opposite .exactly the same is true for the predominantly pale columns mapped to predominantly dark rows except that both have been complemented , which is immaterial to oppositeness of colour .this consistency opposite to that of the previous paragraph means that the pattern has a colour - reversing glide - reflection for each such side - preserving glide - reflection of the fabric design .each such colour - reversing glide - reflection is a side - preserving glide - reflection of the pattern when it is regarded as a design .if the pattern is now regarded as a design , it has among its symmetry operations a subgroup generated by side - preserving and/or side - reversing glide - reflections ( depending on spacing of axes and placement of the redundant cells ) that is transitive on the strands because the glide - reflections from which they came were transitive on the strands .the symmetry group of the prefabric defined by the design is accordingly transitive on the strands .and so the prefabric is isonemal as required .  .10 pt b. figure 16a : subspecies .,title="fig : " ] 10 pt .10 pt b. figure 16a : subspecies .,title="fig : " ]  ( a)116 pt ( b )  it may be helpful to consider the examples available in figure 6 :  figure 6a , one colouring of the species- example of figure 10a drops the bottom row of that figure .the dark redundant cells begin in the bottom left corner , as does the central axis of side - preserving glide - reflection with glide .because it runs through redundant cells , it becomes the axis of side - preserving glide - reflections in the prefabric of figure 6a . because the axes distant run between redundant cells , they become axes of side - reversing glide - reflections in figure 6a .figure 6b , the other colouring , drops the first two columns of figure 10a .the dark redundant cells begin in the bottom left corner .the central ( side - preserving ) glide - reflection axis with odd glide begins on the left in the cell four up .because it runs between redundant cells , it becomes an axis of side - reversing glide - reflection in figure 6b . because the axes distant run through redundant cells , they become axes of side - preserving glide reflection in figure 6b .figure 6c , one colouring of the species- example of figure 10b drops the first column of that figure and has pale redundant cells and the central axis of ( side - preserving ) glide - reflection begin one cell above the bottom left corner .since that axis runs through redundant cells with even glide , it becomes the axis of side - reversing glide - reflections marked in figure 6c .the other marked axes , distant and running between redundant cells , become axes of side - preserving glide - reflection in figure 6c .figure 6d , the other colouring , has its bottom left corner match that of figure 10b .dark redundant cells begin one cell up , and the central axis of ( side - preserving ) glide - reflection with glide runs between redundant cells .accordingly , it becomes the axis of side - preserving glide - reflection in figure 6d .the axes distant run through redundant cells , and so they become axes of glide - reflections that are side - reversing .if the side - preserving subgroup of a fabric s symmetry group is generated by side - preserving glide - reflections and is transitive on strands and the fabric is of order greater than and is perfectly coloured by thinly striping strands , then the pattern is the design of a prefabric having a side - preserving glide - reflection where the fabric had a side - preserving glide - reflection either between redundant cells with even glide or through redundant cells with odd glide and having a side - reversing glide - reflection where the fabric had a side - preserving glide - reflection either through redundant cells with even glide or between redundant cells with odd glide .this is exactly what the mechanics of the above proof show .no similar result is true for side - reversing glide - reflections .for this reason one might expect that thinly stripable species for which there are no side - preserving glide - reflections , namely ( figure 5c ) , 13 , 17 , 25 , and , in addition to those listed above as failing to produce two isonemal designs , might also fail . indeed , larger examples from species , 13 , 17 , 25 , and do fail to give designs of isonemal prefabrics when striped either way , so that it is only those that have side - preserving glide - reflections that have dependably even one isonemal design as a result ; they have _ one _ because it is impossible for the axes to avoid the positions covered by the corollary .the pattern of a fabric of order greater than , perfectly coloured by thinly stiping strands , can not be the design of an isonemal prefabric that falls apart if the symmetry group of the fabric has quarter - turn symmetry .the isonemal fabric coloured , being of species by theorem 2.2 , is of level 2 . because the movement from the corner of a level-2 ( in the terms of ) lattice to its centre is an even number of cell widths in one vertical or horizontal direction and then an odd number of cell widths in a perpendicular direction , the lattice - unit corners and centres fall in both the centres of redundant cells and the centres of irredundant cells .the - 0.1 pts in irredundant cells are in predominantly pale rows and dark columns or dark rows and pale columns .these rows and columns can not be related by but by as in figures 7b , c ( central - 0.1 pt ) and 8b , c ( non - central - 0.1 pts ) .but can not occur within a cell as a symmetry .this contradiction shows that the pattern resulting from colouring by thin striping can not be the design of an isonemal prefabric .what goes wrong in the example of figures 7 and 8 always goes wrong .theorem 3.3 , together with the examples before theorem 3.1 , show the sufficient condition of theorem 3.1 to be necessary as well as sufficient for whole species of fabrics . striping individual fabrics beyond those classes may happen to produce isonemal designs .what species can be produced by thin stiping ? to what species do isonemal designs that fall apart belong ?the latter must be of genera pure ii , pure iv , or ii and iv with no i , iii , or v on account of the 3/4 dark , 1/4 pale feature .these species are 3 , 6 , and 9 ( genus ii ) , 15 and 19 ( genus iv ) , and 23 and 31 ( genera ii and iv ) , including none from 3339 .the last fact corresponds to the impossibility of producing an isonemal design by thinly striping a non - exceptional isonemal fabric with quarter - turn symmetry .all of the possible species are represented in the catalogue : 12 - 69 - 2 * ( 3 ) , 12 - 69 - 1 * ( 6 ) , 16 - 277 - 4 * ( 9 ) , 12 - 21 - 4 * ( ) , 16 - 277 - 2 * ( ) , 12 - 69 - 3 * ( ) , 16 - 85 - 3 * ( ) , 12 - 21 - 1 * ( ) , 8 - 5 - 1 * ( ) , 8 - 5 - 3 * ( 31 ) .it is not difficult to determine all of the isonemal designs of prefabrics that fall apart of any species and order by the method indicated in ( * ? ? ?* section 5 ) using the data collected in ( * ? ? ?* section 8) and ( * ? ? ?* section 12 ) .  to take the most obvious case , order with and for species 3 and 6 and with and for 6 alone , and with and for species , , and and with and for .nothing allows species 9 or 31 at order 20 .the lattice units for the 9 possibilities are shown in figure 11 ..10 pt f. species one way.10 pt row 3 .g. species the other way.10 pt h and i. species . ]  one has to fit the axes and centres of the group to the lattice of redundant cells .for example , for species 3 axes of side - preserving glide - reflection with odd glide would have to lie through redundant cells and the side - reversing glide - reflection axes through irredundant cells , but if the glides are even as in figure 11a , then the axes of side - preserving glide - reflection must pass instead through _ irredundant _ cells and the axes of side - reversing glide - reflection through the _ redundant _ cells .that these arrangements are possible is due to the distance in between neighbouring axes , half an odd multiple of .the usual nuisance occurs by which the smaller - order 4 - 1 - 1 * appears at all nine opportunities because the symmetry groups examined are subgroups of that of 4 - 1 - 1*. and then because of the smallness of one dimension of the lattice units used and because of the extra symmetry imposed by the redundant cells , each set of designs produced contains other designs with more symmetry and therefore of a different species than that intended .in particular , species 3 ( figure 11a ) , the second set from species 6 ( figure 11c ) , the first set from species 15 ( figure 11d ) , and the second set from species 19 ( figure 11 g ) all contained the first set of designs of species 23 ( figure 18 ) , and the second set from species 6 ( figure 11c ) , the second set from species 15 ( figure 11e ) and the first set from species 19 ( figure 11f ) contained the second set of designs of species 23 ( figure 19 ) .the first set for species 6 ( figure 11b ) , in fact , contained nothing but designs that were not of species 6 .  5 pt 5 pt   1109 - 1 * 4373 - 1 * 5141 - 1 * 5 pt 5 pt   17477 - 1 * 17489 - 1 * 17669 - 1 *  the appropriate results from each species are displayed .figures 12 ( 6 designs of species 3 ) and 13 ( 6 designs of species 6 of the second kind shown in figure 11c ) have the peculiarity that , lacking all rotational symmetry , they are an arbitrary choice of which way the design is displayed .the same prefabric could have its axes rotated from the displayed positions through any multiple of including upside down to produce four genuinely different pictures .the other prefabrics look different from the given diagram when rotated either way but the same upside down so that there are only two genuinely different pictures of each pattern in figures 1419 .apart).,title="fig:"]5 pt  apart).,title="fig:"]5 pt  apart).,title="fig : " ]  1109 - 2 * 4373 - 2 * 5141 - 2 *  apart).,title="fig:"]5 pt  apart).,title="fig:"]5 pt  apart).,title="fig : " ]  17477 - 2 * 17489 - 2 * 17669 - 2 *   and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig : " ]  341 - 1 * 341 - 2 * 4433 - 1 *  and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with negative slope apart).,title="fig : " ]  4433 - 2 * 16709 - 1 * 16709 - 2 *   and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig : " ]  1109 - 3 * 4373 - 3 * 5141 - 3 *  and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( side - preserving glide - reflection axes with positive slope apart).,title="fig : " ]  17477 - 3 * 17489 - 3 * 17669 - 3 *   and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig : " ]  341 - 3 * 341 - 4 * 4433 - 3 *  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig : " ]  4433 - 4 * 16709 - 3 * 16709 - 4 *   and order 20 that fall apart ( mirrors with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope apart).,title="fig : " ]  1109 - 4 * 4373 - 4 * 5141 - 4 *  and order 20 that fall apart ( mirrors with negative slope ).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope ).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope apart).,title="fig : " ]  17477 - 4 * 17489 - 4 * 17669 - 4 *   and order 20 that fall apart ( mirrors with negative slope ).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope ).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with negative slope apart).,title="fig : " ] + 341 - 5 * 4433 - 5 * 16709 - 5 *   and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig:"]5 pt  and order 20 that fall apart ( mirrors with positive slope apart).,title="fig : " ]  341 - 6 * 65 pt 4433 - 6 * 65 pt 16709 - 6 *  collectively figures 1219 extend the catalogue of isonemal prefabrics that fall apart in to order 20 for those whose alternate warps and wefts lift off ( as though thinly striped , even genus ) but not those where warps and wefts lift off in adjacent pairs ( as though thickly striped , genus v ) .the 42 patterns / designs are based on only 9 sequences .the 3 palindromes appear 6 times each , and the 6 non - palindromes 4 times .can designs of isonemal prefabrics that fall apart always be produced by thinly striping isonemal fabrics ?this question was answered in the affirmative in , but the answer depended on an algorithm for producing a fabric for colouring by thin striping that does not always work . if one wants to go from the design of a prefabric that falls apart to the design of a fabric that would colour by striping to look like it , one will obviously reverse the colours of the irredundant cells in the predominantly dark rows and not those in the predominantly pale rows precisely so that the striping would give the starting design .the only question really is what to do with the redundant cells , and the proposed algorithm made a wrong choice of making them all pale .since a number of symmetry groups require side - preserving hence colour - reversing glide reflections and half - turns , that can not be done consistently , and the right thing to do is to colour them orbit by orbit in accordance with the requirements of the relevant symmetry group .this works in lots of examples , helped by the fact that side - preserving glide - reflections with odd glides and axes through redundant cells or even glides and axes through irredundant cells and side - reversing glide - reflections with odd glides and axes through irredundant cells or even glides and axes through redundant cells in the given design turn into side - preserving glide - reflections of what were irredundant cells on account of the colour reversals in the predominantly dark rows .but there seems to be no reason to think that every design so produced needs to be isonemal .a way in which it can fail to be isonemal is illustrated in figure 20 by an isonemal prefabric of species 6 that falls apart ( a ) and a specification ( b ) of the irredundant cells of a partial design for a fabric that could produce the design ( a ) by thin striping .lattice units = one lattice unit of an order-40 isonemal prefabric of species 6 that falls apart.10 pt b. partial construction of a fabric that could be coloured to look like the design of ( a ) .dark and pale cells as required before striping , formerly redundant cells neutral.,title="fig:"]10 pt  lattice units = one lattice unit of an order-40 isonemal prefabric of species 6 that falls apart.10 pt b. partial construction of a fabric that could be coloured to look like the design of ( a ) .dark and pale cells as required before striping , formerly redundant cells neutral.,title="fig : " ]   the partial design has been produced by reversing the thin - striping algorithm for irredundant cells , i.e. , the irredundant cells in the predominantly dark rows were complemented .the appearance of the partial construction of a fabric to colour , a pseudofabric because of the missing strands belonging in cells coloured neutral in figure 20b , suggests that it should be a twillin with the same row - to - row offset 9 as the prefabric in figure 20a but without the complementation that is a feature of the row - to - row translation that is a symmetry of the prefabric .it does still have the glide - reflection of the prefabric although the complementing of alternate rows has destroyed the mirror symmetry . while it looks like a simple matter to colour the formerly redundant cells in keeping with the glide - reflection symmetry , doing just that to a single cell reveals that any fabric thereby produced is not isonemal .consider the bottom cell on the central axis within the marked lattice units ( dark in figure 20a , neutral in figure 20b ) .let its colour be .five cells along that axis , the cell , again neutral in figure 20b , must be because the glide - reflection is side - preserving .four cells downward and to the right , the neutral cell must have colour by the periodicity of both diagrams in that direction . but that cell is the image of the initial cell with colour under the row - to - row translation with offset 9 that is the row - to - row symmetry of the formerly redundant cells of figure 20b .such a row - to - row translation can not be a symmetry of any completed construction of the fabric .a glance at the lattice unit of the symmetry group of the irredundant cells plus the two cells coloured and in it confirms by its dimensions , , that it is not a lattice unit of an isonemal fabric , since its parameters violate the isonemal constraint . to be of species ,the width in units would have to be odd and relatively prime to 10 .  in order to be consistent with the side - preserving glide - reflections of the irredundant cells with axes along the former axes ,the redundant cells along those axes must be filled in with each set of five consecutive cells the colour complement of the previous set of five consecutive cells .other formerly redundant cells too need to be filled in in accordance with the group to make a design with group of roth type .one observes that irredundant cells are invariant under translations of multiples of parallel to the axes as well as glide - reflections with glide , but the redundant cells containing the axes are invariant under translations of only multiples of along the axes because moving they are complemented .the period rectangle ( lattice unit ) must be by , and so , no matter how the rest of the redundant cells are filled in , the period rectangle can not be made smaller .the species-6 design is isonemal because the lattice unit of is by , 4 and 5 being relatively prime .the design partly produced can not be isonemal because it is shaping up to be of species with lattice unit being by .the standard isonemality constraint of is violated .( the translations under which it is invariant having those sizes in perpendicular directions , there is no transformation from strand to adjacent strand , as can be seen for the irredundant cells of , for example , the bottom two rows of figure 20a . )we must conclude that theorem 2 of is false and that the algorithm on which it is based does not always work either as stated there or improved here .the improved algorithm works when it can work , but the counterexample of figure 20 shows that no procedure can always work .work on this material has been done at home and at wolfson college , oxford .richard roth helped with the understanding of his papers .will gibson made it possible for me to draw the diagrams with surprising ease from exclusively keyboard input . to them both and wolfson college i make grateful acknowledgement .grnbaum , b. and shephard , g. c. [ 1980 ] : `` satins and twills : an introduction to the geometry of fabrics '' _ math . mag ._ 53 , 139161 .grnbaum , b. and shephard , g. c. [ 1985 ] : `` a catalogue of isonemal fabrics '' in discrete geometry and convexity , jacob e. goodman _ et al .new york acad ._ 440 , 279298 .grnbaum , b. and shephard , g. c. [ 1986 ] : an extension to the catalogue of isonemal fabrics _ discrete math ._ 60 , 155192 .grnbaum , b. and shephard , g. c. [ 1988 ] : `` isonemal fabrics '' _ amer . math .monthly _ 95 , 530 .hoskins , w. d. and thomas , r.s.d .[ 1984 ] : `` conditions for isonemal arrays on a cartesian grid '' _ linear algebra appl . _ 57 , 87103 .hoskins , j.a . and thomas , r.s.d .[ 1991 ] : `` the patterns of the isonemal two - colour two - way two - fold fabrics '' _ bull .austral ._ 44 , 3343 .roth , r.l .[ 1993 ] : `` the symmetry groups of periodic isonemal fabrics '' _ geom .dedicata _ 48 , 191210 .roth , r.l .[ 1995 ] : `` perfect colorings of isonemal fabrics using two colors '' _ geom .dedicata _ 56 , 307326 .schattschneider , d. [ 1978 ] : `` the plane symmetry groups : their recognition and notation '' _ amer .monthly _ 85 , 439450 .schattschneider , d. [ 1986 ] : `` in black and white : how to create perfectly colored symmetric patterns '' _ comput ._ 12b , 673695 .thomas , r.s.d .[ 2010b ] : `` isonemal prefabrics with no axes of symmetry '' _ discrete math ._ , to appear .online at http://dx.doi.org/10.1016/j.disc.2009.12.015 .available at arxiv.org/abs/0911.1467v2 [ math.co ] .

Abstract: perfect colouring of isonemal fabrics by thin striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored . the catalogue of isonemal prefabrics that fall apart is extended to order 20 for those of even genus .