Showing papers in "Acta Crystallographica Section A in 1989"
345 citations
TL;DR: In this article, a constrained least-squares procedure is used to find a rotation matrix that minimizes or maximizes the sum of the squared distances between corresponding atoms for two structures.
Abstract: Rotation matrices that minimize or maximize the sum of the squared distances between corresponding atoms for two structures are found using a constrained least-squares procedure solved analytically as an eigenvalue problem in quaternion parameters.
268 citations
TL;DR: In this article, a detailed description of the method of crystallographic refinement by simulated annealing is presented, and it has been applied to a 1.5 A resolution X-ray structure of crambin.
Abstract: A detailed description of the method of crystallographic refinement by simulated annealing is presented. To test the method, it has been applied to a 1.5 A resolution X-ray structure of crambin. The dependence of the success of the simulated annealing protocol with respect to the temperature of the heating stage is discussed. Optimal success is achieved at relatively high temperatures. Regardless of the protocol used, the molecular-dynamics refined structure always yields an improved R factor compared with restrained least-squares refinement without manual re-fitting. The differences between the various refined structures and the corresponding electron density maps are discussed.
246 citations
TL;DR: In this paper, it was shown that the infinite periodic minimal surfaces used to describe these structures can be obtained from the hyperbolic plane when some translations are forced to identity, which is indeed formally analogous to the simple process of transformation of a Euclidean plane into a cylinder, when a translation of the plane is forced by rolling the plane onto itself.
Abstract: Infinite periodic minimal surfaces are now being introduced to describe some complex structures with large cells, formed by inorganic and organic materials, which can be considered as crystals of surfaces or films. Among them are the spectacular cubic crystalline structures built by amphiphilic molecules in the presence of water. The crystallographic properties of these surfaces are studied from an intrinsic point of view, using operations of groups of symmetry defined by displacements on their surface. This approach takes advantage of the relation existing between these groups and those characterizing the tilings of the hyperbolic plane. First, the general bases of the particular crystallography of the hyperbolic plane are presented. Then the translation subgroups of the hyperbolic plane are determined in one particular case, that of the tiling involved in the problem of cubic structures of liquid crystals. Finally, it is shown that the infinite periodic minimal surfaces used to describe these structures can be obtained from the hyperbolic plane when some translations are forced to identity. This is indeed formally analogous to the simple process of transformation of a Euclidean plane into a cylinder, when a translation of the plane is forced to identity by rolling the plane onto itself. Thus, this approach transforms the 3D problem of infinite periodic minimal surfaces into a 2D problem and, although the latter is to be treated in a non-Euclidean space, provides a relatively simple formalism for the investigation of infinite periodic surfaces in general and the study of the geometrical transformations relating them.
98 citations
TL;DR: The anisotropic displacements of selected rigid groups in bovine ribonuclease A have been refined from X-ray diffraction data by the application of the rigid-body TLS model and it is shown that the average motion is approximately isotropic with little librational character.
Abstract: The anisotropic displacements of selected rigid groups in bovine ribonuclease A have been refined from X-ray diffraction data by the application of the rigid-body TLS model. The rigid groups chosen were the side chains of tyrosine, histidine and phenylalanine and the planar side chains of aspartic acid, glutamic acid, glutamine, asparagine and arginine. The method has also been applied to the co-crystallizing active-site sulfate anion. This has enabled the description of the motion of the above-mentioned side-chain atoms by anisotropic displacement ellipsoids from a 1.45 A refinement. The hydrophobic side groups in the protein core show mainly translational motion, with mean-square librations of 20 deg2 which are similar to those found in some close-packed crystals of small organic molecules. Librational displacements are much more significant in the hydrophilic side groups where their magnitudes can be correlated with solvent accessibility. Large liberations of some solvent exposed side chains correspond with the breakdown of a simple TLS model and the existence of multiple orientations of the side groups. The TLS model has also been applied to the whole protein molecule and shows that the average motion is approximately isotropic with little librational character.
90 citations
TL;DR: The Subcommittee has attempted to elucidate the nature of problems encountered in the definition and use of statistical descriptors as applied to crystallography and to propose procedural improvements.
Abstract: The Subcommittee has attempted to elucidate the nature of problems encountered in the definition and use of statistical descriptors as applied to crystallography and to propose procedural improvements. The report contains (a) a dictionary of statistical terms established for use by experimentalists; (b) a description of the statistical basis for refinement procedures; (c) sections dealing with defects in the physical model used for refinement, and with the choice and significance of weighting schemes; and (d) recommendations, some of which may be readily implemented, whilst others may require a long-term effort to bring them into general use.
71 citations
TL;DR: The genus of a three-periodic intersection-free surface in R3 refers to a primitive unit cell of its symmetry group as mentioned in this paper, and two procedures for the calculation of the genus are described: (1) by means of labyrinth graphs; (2) via the Euler characteristic derived from a tiling on the surface.
Abstract: The genus of a three-periodic intersection-free surface in R3 refers to a primitive unit cell of its symmetry group. Two procedures for the calculation of the genus are described: (1) by means of labyrinth graphs; (2) via the Euler characteristic derived from a tiling on the surface. In both cases new formulae based on crystallographic concepts are given. For all known minimal balance surfaces the genera and the labyrinth graphs are tabulated.
49 citations
TL;DR: An enlarged set of free-atom X-ray atomic scattering (form) factors for high angles [2 ≤ (sin θ)/λ ≤ 6A-1] has been calculated based on those of Doyle & Turner as discussed by the authors.
Abstract: An enlarged set (atomic number Z = 2 to 98) of free-atom X-ray atomic scattering (form) factors for high angles [2 ≤ (sin θ)/λ ≤ 6A-1] has been calculated based on those of Doyle & Turner [Acta Cryst. (1968). A24, 390-397]. Four-parameter 'exponential polynomial' fits for these are presented which give far more accurate estimates of the scattering factors at high angles than the Gaussian fits normally used. The use of the Mott formula in conjunction with these new high-angle X-ray form factors allows the calculation of improved-accuracy high-angle electron scattering factors. The use of these high-accuracy high-angle scattering factors for important applications such as Fourier charge-density analysis and computer simulation of high-resolution electron microscope (HREM) images is discussed.
48 citations
TL;DR: In this article, the relative position of SnS and NbS2 sublattices was determined from the common reflections 0kl: RF = 0.072 for 98 reflections.
Abstract: (SnS)1.17NbS2, Mr = 334.92, is a compound with misfit layer structure consisting of two-atom-thick layers of SnS and three-atom-thick sandwiches of NbS2 which alternate along the c axis. The lattices of SnS and NbS2 both have C-centered orthorhombic unit cells which match along the b and c axes but not along the a axes. The unit cells and space groups are a = 5.673 (1), b = 5.750 (1), c = 11.760 (1) A, space group C2mb (no. 39), Z = 4, for SnS; a = 3.321 (1), b = 5.752 (1), c = 11.763 (1) A, space group Cm2m (no. 38), Z = 2, for NbS2. From the ratio of the lengths of the a axes of the two parts of the complete structure (5.673/3.321 = 1.708 being irrational) one obtains a composition (SnS)1.17NbS2. The structure determination consisted of three parts: the structures of the SnS and NbS2 parts separately and their relative position. Intensities were measured with Mo K{\bar \alpha} radiation (λ = 0.71073 A) at T = 293 K, μ = 102.3 cm-1. For the SnS part RF = 0.088 for 306 unique reflections; for the NbS2 part RF = 0.031 for 329 unique reflections (for both cases 0kl reflections excluded). The relative position of these two lattices was determined from the common reflections 0kl: RF = 0.072 for 98 reflections. The SnS part of the structure consists of deformed slices of SnS with a thickness of half the cell edge of (hypothetical) NaC1-type SnS. Each Sn atom is in this way coordinated by five sulfur atoms; four sulfur atoms are in a plane perpendicular to the c axis with SnS distances 2.74 (1) (1×), 2.913 (1) (2×) and 3.01 (1) A (1×), whereas the fifth Sn-S bond with length 2.695 (9) A is approximately along the c axis. The NbS2 part of the structure is that of NbS2-2H with Nb in trigonal prisms of sulfur; the Nb-S distances are 2.473 (1) A. From refinement of the common 0kl reflections the relative y and z positions of the two sublattices were found; along the common a direction the lattices of SnS and NbS2 are incommensurate.
47 citations
Abstract: The internal strain induced in a crystal structure by lattice deformation was considered. A suitable rotationally invariant representation was introduced and the corresponding contribution to the elastic constants was calculated. The method is based on the use of crystallographic rather than Cartesian atomic coordinates as variables of energy derivatives, with full exploitation of space-group symmetry and no constraint on the lattice geometry. A two-body Born interatomic potential was assumed, for both ionic and molecular crystals; energy derivatives of electrostatic lattice sums were calculated with the Ewald series. Molecular groups are treated within a rigid-body scheme based on Eulerian angles and translations as inner strain variables. Results of computations of Mg2SiO4 (forsterite) are reported, and the importance of optimizing the potential parameters against experimental data is discussed.
42 citations
TL;DR: A vectorizable algorithm for fast computation of structure factors and derivatives during refinement of macromolecular structures is presented, based on fast Fourier transformations on subgrids that cover the unit cell of the crystal.
Abstract: A vectorizable algorithm for fast computation of structure factors and derivatives during refinement of macromolecular structures is presented. It is based on fast Fourier transformations on subgrids that cover the unit cell of the crystal. The use of subgrids allows reduction of the total memory requirements for the computations without producing large overheads. The algorithm is applicable to all space groups. The performance of the algorithm on a conventional scalar computer as well as on supercomputers is discussed.
TL;DR: In this article, anomalous scattering differences have been used with the direct-methods program MULTAN87 to determine the positions of anomalous scatterers in a variety of metalloproteins and a small molecule.
Abstract: Observed anomalous scattering differences have been used with the direct-methods program MULTAN87 to determine the positions of anomalous scatterers in a variety of metalloproteins and a small molecule. The lack of anomalous differences in the centric data did not prevent the determination of the atom positions and the anomalous scatterers were found in all cases. These results show that the method may be useful to determine the positions of anomalous scatterers in the case of multi-site genetically engineered proteins.
TL;DR: The recently developed method of structure factor refinement by molecular dynamics with simulated annealing by Brünger, Kuriyan & Karplus is tested on the 118 residue protein myohemerythrin and results in electron density maps that are easier to interpret and permit identification of the errors in the structure.
Abstract: The recently developed method of structure factor refinement by molecular dynamics with simulated annealing [Brunger, Kuriyan & Karplus (1987). Science, 235, 458-460] is tested on the 118 residue protein myohemerythrin. A highly refined structure for this protein at 1.3/1.7 A resolution has recently been published [Sheriff, Hendrickson & Smith (1987). J. Mol. Biol. 197, 273-296]. This is compared with the results of simulated annealing refinement (with no manual intervention) starting from an earlier model for the protein from a stage in the refinement when conventional least-squares methods could not improve the structure. Simulated annealing reduces the R factor at 2.5 A from 39 to 31%, with uniform temperature factors and no solvent molecules and with similar stereochemistry; the comparable value for the manually refined structure is 27.9%. Errors in backbone and sidechain positions up to about 3 are corrected by the method. The error in backbone positions for roughly 85% of the initial structure is within this range, and in these regions the r.m.s. backbone error is reduced from 1.1 to 0.4 A. For the rest of the structure, including a region which was incorrectly built due to a sequence error, the procedure does not yield any improvement and manual intervention appears to be required. Nevertheless, the overall improvement in the structure results in electron density maps that are easier to interpret and permit identification of the errors in the structure. The general utility of the simulated annealing methodology in X-ray refinement is discussed.
TL;DR: The anomalous high-index faces (hkl) of the mineral calaverite (Au1 - xAgxTe2) measured goniometrically in the year 1931 by Goldschmidt, Palache & Peacock [Neues Jahrb. Mineral (1931), 63, 1-58] are reinterpreted and related to the wave vector q of the displacive incommensurate modulation which was recently found in the crystal structure.
Abstract: The anomalous high-index faces (hkl) of the mineral calaverite (Au1 - xAgxTe2) measured goniometrically in the year 1931 by Goldschmidt, Palache & Peacock [Neues Jahrb. Mineral (1931), 63, 1-58] are re-interpreted and related to the wave vector q of the displacive incommensurate modulation which was recently found in the crystal structure. All crystal faces (including the high-index ones) can be given four low indices (hklm), using q as a fourth basis vector. From this an almost hundred-year-old anomaly in crystal morphology is in principle solved.
TL;DR: The concept of absolute structure was introduced by Jones as mentioned in this paper, and the term "absolute" is extended to cases where the structure has been related to some external macroscopic physical property.
Abstract: The concept of 'absolute structure' as introduced by Jones [Acta Cryst. (1984), A40, 660-662] is discussed, and the term 'absolute' is extended to cases where the structure has been related to some external macroscopic physical property. The non-centrosymmetric crystal classes are divided into seven distinct categories for which specific terms are proposed: structural chirality, when the 'absolute structure' is determined for crystals in classes 1, 2, 3, 4, 6, 222, 422, 32, 622, 23 and 432; absolute chirality, when the 'absolute structure' is linked to a chiral property such as optical rotation; absolute polarity, when the 'absolute structure' is linked to a polar property such as pyroelectricity; absolute morphology, when the 'absolute structure' is linked to the crystal habit: in this case there may be two further subdivisions - absolute chiral morphology (or absolute enantiomorphism) to describe a link to the hand of the habit, and absolute polar morphology to describe a link to the polar nature of the habit. It is further recommended that the term absolute configuration should be reserved only for molecular species and not for crystal structures.
TL;DR: In this article, the phase information can be extracted from the three-beam profiles of a Renninger ψ-scan experiment using a special ψ circle diffractometer installed on a rotating Cu-anode generator.
Abstract: Direct measurements of triplet phase relationships for non-centrosymmetric light-atom organic structures with medium-size unit cells are reported. The phase information can be extracted from the three- beam profiles of a Renninger ψ-scan experiment. The measurements were carried out with a special ψ-circle diffractometer installed on a rotating Cu-anode generator. The incident-beam divergence is reduced to 0.02°. The experimental results confirm the theoretical considerations of paper I of this work [Hummer & Billy (1986). Acta Cryst. A42, 127-133]. As triplet phases of +90° can be distinguished, the absolute structure can be determined unambiguously. The measurements show that the triplet-phase-dependent interference effects may be superposed on phase-independent Umweganregung or Aufhellung effects. By a comparison of the ψ-scan profiles of two centrosymmetrically related three-beam cases, the triplet phases of which have opposite signs, it is possible to evaluate the phase-independent effects and to determine the value of the triplet phase with an accuracy of at least 90°.
TL;DR: The 2-hex theorem as mentioned in this paper gives a lower and an upper bound for "min, the minimum value of the multiplicity of specific rotations for a given axial ratio", which is stronger than a similar bound given by Delavignette.
Abstract: Two neighbouring grains of the same phase with a lattice of hexagonal Bravais type are considered which have a three-dimensional lattice of symmetry translations in common, called the coincidence site lattice or CSL. The volume ratio of unit cells for the CSL and the original lattice is called the multiplicity 2. The 2-hex theorem gives 2 in terms of four integral parameters that describe the axis and angle of the rotation connecting the hexagonal lattices of the two neighbouring grains and in terms of their axial ratio c/a. Two types of rotations generating CSL's may be distinguished, viz common rotations, which generate CSL's with the same 2; for every value of c/a, and specific rotations, which generate CSL's with a low value of 2 only for a few values of the axial ratio. The 2-hex theorem makes it possible to determine a lower and an upper bound for "~min, the minimum value of the multiplicity of specific rotations for a given axial ratio. The lower bound can serve to determine systematically all specific rotations with c/a in a given interval and 2 not larger than some given value -~c. The bound is used to complete published tables. The upper bound is stronger than a similar bound given by Delavignette.
TL;DR: In this paper, at wavelengths in the range 0.3-6.2 A, the anharmonicity parameter /3F of the fluorine atoms was refined to 4.19 (30) x 10-19 J ~-3.
Abstract: Measurements have been made, at wavelengths in the range 0.3-6.2 A, of hhl reflections of SrF2 on the single-crystal diffractometer (SXD) at the ISIS Spallation Neutron Source. After application of a variable-wavelength extinction correction to the derived I F,,k,I values, a refinement of the anharmonicity parameter /3F of the fluorine atoms was carried out, yielding a value of-4.19 (30) x 10-19 J ~-3.
TL;DR: Dynamical diffraction within the single molecular layer of these crystals is responsible for the observed Friedel differences and this opens up the possibility that dynamical effects occurring at lower voltages might be used to phase higher-voltage kinematic diffraction intensities.
Abstract: Electron diffraction patterns of 45 A thick two-dimensional crystalline arrays of a cell membrane protein, bacteriorhodopsin, have been recorded at two electron voltages, namely 20 and 120 kV. Significant intensity differences are observed for Friedel mates at 20 kV, but deviations from Friedel symmetry are quite small at 120 kV. It does not seem likely that the measured Friedel differences can be accounted for by complex atomic structure factors, by curvature of the Ewald sphere, or by effects that might occur as a result of inelastic scattering (absorption). It is therefore concluded that dynamical diffraction within the single molecular layer of these crystals is responsible for the observed Friedel differences. The results are useful in estimating the maximum specimen thickness for which the kinematic approximation may be safely used in electron crystallography of biological macromolecules at the usual electron voltage of 100 kV, or even at higher voltages. The results show that the Friedel differences are independent of resolution and this opens up the possibility that dynamical effects occurring at lower voltages might be used to phase higher-voltage kinematic diffraction intensities.
TL;DR: In this article, closed-form expressions for three-beam dynamical transmission electron diffraction are compared and a comparison of many-beam calculations with experimental non-systematic CBED patterns is used to determine a threephase invariant for CdS with an accuracy of ±5° in the electron structure-factor phase.
Abstract: Closed-form expressions for three-beam dynamical transmission electron diffraction are compared. These are used as a guide to determine the best experimental conditions for the determination of structure-factor phases by convergent-beam electron diffraction in the general non-systematic case. The validity domains of Kambe's [J. Phys. Soc. Jpn (1957), 12, 1-13] 'strong coupling' approximation and Bethe's [Ann. Phys. (Leipzig) (1928), 87, 55-129] second approximation are compared, and these approximations reconciled. A comparison of many-beam calculations with experimental non-systematic CBED patterns is used to determine a three-phase invariant for CdS with an accuracy of ±5° in the electron structure-factor phase. If it is assumed that two of the phases are known exactly, the error in the third (00\bar 2) X-ray structure-factor phase would be ±0.75°. The accuracy of the method for determining phases, atomic position parameters and bonding charge distributions is discussed.
TL;DR: In this article, it was shown that the multiplicities of the CSL's generated by a given rotation in a hexagonal and in a rhombohedral lattice with the same value of c/a differ by at most a factor 3.
Abstract: In experimental investigations and computer simulations of the structure and properties of grain boundaries, the results are usually discussed with reference to the special case of coincidence boundaries, where the two neighbouring grains have a three-dimensional lattice of symmetry translations in common. For historical reasons this lattice is called the coincidence site lattice or CSL. A systematic determination of CSL's for the case of grains with a lattice of rhombohedral Bravais type is presented. It is shown that a number of investigations of the structure of grain boundaries in alumina (α-Al2O3) have to be reinterpreted in the light of the present results. A central result is the Σ-rhomb theorem, which expresses the ratio Σ of unit-cell volumes of the CSL and the rhombohedral crystal lattice in terms of four integral parameters that describe the axis and angle of the rotation connecting the rhombohedral lattices of the two neighbouring grains and in terms of their axial ratio c/a. Two types of coincidence rotations, i.e. of rotations generating CSL's, may be distinguished, viz common rotations, which generate CSL's with the same Σ for every value of c/a, and specific rotations, which generate CSL's with a low value of Σ only for a few values of the axial ratio. The Σ-rhomb theorem makes it possible to determine systematically not only all common rotations with Σ up to a given maximum value Σc but also all specific rotations with Σ ≤ Σc and with c/a in any given interval about the experimental value of c/a for the material in question. It is shown that the multiplicities of the CSL's generated by a given rotation in a hexagonal and in a rhombohedral lattice with the same value of c/a differ by at most a factor 3.
TL;DR: In this paper, Meetsma, Wiegers, Haange & de Boer used high-resolution imaging to study the remarkable incommensurable misfit layer compound (SnS)1.17NbS2, 'SnNbStS3'.
Abstract: Electron diffraction and high-resolution imaging are used to study the remarkable incommensurable misfit layer compound (SnS)1.17NbS2, 'SnNbS3'. Electron diffraction patterns along the zone axis perpendicular to the layer planes are analysed in detail based on the mixed-layer assumption proposed in paper I [Meetsma, Wiegers, Haange & de Boer (1989). Acta Cryst. A45, 285-291]. The relative intensities of the hk0 reflections due to the two types of constituent layers are strongly influenced by the foil thickness. Stacking variants along the c axis are frequently observed and strongly complicate the diffraction patterns. Under particular conditions of thickness and defocus, high-resolution images are shown to reflect the varying degree of coincidence of the atom columns in the misfit layers and thus they directly reveal the incommensurate misfit.
TL;DR: In this article, a multislice theory is proposed to solve Yoshioka's coupling equations for elastic and inelastic scattered high-energy electrons in a solid, which is capable of including the non-periodic crystal structures and the electron multiple scattering among all the excited states in the calculations.
Abstract: A multislice theory is proposed to solve Yoshioka's coupling equations for elastic and inelastic scattered high-energy electrons in a solid. This method is capable, in principle, of including the non-periodic crystal structures and the electron multiple scattering among all the excited states in the calculations. It is proved that the proposed theory for calculating the energy-filtered inelastic images [Wang (1989). Acta Cryst. A45, 193-199], based on the physical optics approach, is equivalent to the quantum-mechanical theory under some approximations. The basic theory of simulating the energy-filtered inelastic image of core-shell losses and thermal diffuse scattering is outlined.
TL;DR: In this paper, the authors show that the circular arcs and high-order Laue zone (HOLZ) fine lines observed in both Kikuchi-line and convergent-beam electron diffraction (CBED) patterns have been shown to be consistent with the concept of Bloch-wave channeling of fast electrons along rows of atoms.
Abstract: The circular arcs and high-order Laue zone (HOLZ) fine lines observed in both Kikuchi-line and convergent-beam electron diffraction (CBED) patterns have been shown to be consistent with the concept of Bloch-wave channeling of fast electrons along rows of atoms. This interpretation has the advantages that it provides a direct correlation between HOLZ lines and the crystal periodicity along the incident-beam direction, and forms a basis which could be incorporated into a more rigorous electron diffraction theory in the future for understanding the extreme Laue and Bragg cases of electron diffraction, as well as the intermediate reflection and transmission high-energy electron diffraction (RTHEED). Examples are given of all three types of CBED, RHEED and RTHEED patterns from Pt and GaAs single crystals, which show the common dynamical diffraction effects, circular arcs and HOLZ fine lines, resulting from Bloch-wave channeling.
TL;DR: In this paper, a comparative study of Gaussian and exponential charge distribution functions in calculating Coulomb potentials, energies and fields at arbitrary points due to lattice slices, using the Ewald method, is performed.
Abstract: A comparative study has been undertaken of the employment of Gaussian and exponential charge distribution functions in calculating Coulomb potentials, energies and fields at arbitrary points due to lattice slices, using the Ewald method. The APL program SURFPOT has been developed for this purpose, for a general crystal structure with user-defined slice orientations and slice boundary configurations. Results are presented for the (111) face of NaC1, (001) and (112) faces of aragonite (CaCO3) and (001) face of phlogopite (KM3T4O10(OH)2, M = divalent cation, T4 = Si3Al]. The convergence behaviour of the potential sums is consistently and considerably better when the Gaussian form is used.
TL;DR: An alternative convention for solving the ambiguity problem of (3 + 1) superspace group symbols described by Yamamoto, Janssen, Janner & de Wolff [Acta Cryst. (1985), A41, 528-530] is proposed in this article.
Abstract: An alternative convention is proposed for solving the ambiguity problem of (3 + 1) superspace group symbols described by Yamamoto, Janssen, Janner & de Wolff [Acta Cryst. (1985), A41, 528-530] based on the requirement that the condition for equivalence of modulation vectors to be independent on a selection of basis vectors is satisfied.
TL;DR: In this paper, a method of bounding a protein molecule in a very noisy synthesis is considered, which consists of two steps: in the first step, basic points are chosen that are most likely to belong to the region of the molecule and in the second step, a compact region with the maximal concentration of these points is searched.
Abstract: A method of bounding a protein molecule in a very noisy synthesis is considered. It consists of two steps. In the first step ('nonlinear filtration') basic points are chosen that are most likely to belong to the region of the molecule. In the second step ('linear filtration') a compact region with the maximal concentration of these points is searched. Various modifications of the method are analysed. It is shown that the molecular region in a finite-resolution synthesis contains not only the highest positive maxima of the density distribution but also the deepest negative minima.
TL;DR: Eight new families of minimal balance surfaces are described, complementary to other minimal Balance surfaces which are built up from catenoid-like surface patches and have been known before.
Abstract: Eight new families of minimal balance surfaces are described. Their surface patches belong to a new kind, called multiple catenoids. The generating circuits of such a minimal surface are two congruent concave polygons with one point of self-contact each. The new minimal balance surfaces are complementary to other minimal balance surfaces which are built up from catenoid-like surface patches and have been known before.
TL;DR: In this paper, a concave polygon with one point of self-contact and a convex polygon are the two generating circuits of such a minimal balance surface, and three new families of minimal balance surfaces have been derived.
Abstract: Three new families of minimal balance surfaces have been derived. For this a new kind of surface patch, i.e. branched catenoid, has been used. A concave polygon with one point of self-contact and a convex polygon are the two generating circuits of such a minimal balance surface.