| minimal surfaces | sphere packings | Dirichlet domains | normalizers |
| lattice complexes | structure description | International Tables |
Mathematical crystallography is that branch of crystallography that forms the bridge to mathematics by making available mathematical concepts, methods and ideas for the solution of crystallographic problems. This also includes the development of geometrical models for the description and classification of crystal structures. Occasionally, however, mathematical crystallography can conversely also contribute to the solution of mathematical problems by applying crystallographic methods. Significant contributions in this field have emanated and still continue to emanate from German scientists - also at Marburg.
| sphere packings | Dirichlet domains | normalizers | lattice complexes |
| structure description | International Tables | start |
In 1983 a relationship between some crystal structures and certain complicated periodic surfaces was first established by St. Andersson - and shortly after that also by others. These papers stimulated studies of those symmetry conditions required of a 3-periodic surface free of self-intersections in order to partition 3-dimensional space into two congruent parts. The results of these studies have especially been applied to minimal surfaces, i.e. to surfaces formed in each disc-like section like a soap film in a wire frame. With the aid of these symmetry conditions it was possible to derive several new minimal surfaces and to almost treble the number of such surfaces known at that time. In the meantime 3-periodic minimal surfaces intersecting themselves exclusively along straight lines have also been considered. Whereas previously this property was recognised only for one surface since then 77 new types of such surfaces have been derived.W. Fischer & E. Koch (2001): 3-periodic minimal surfaces derivable from Laves nets. - Z. Anorg. Allg. Chem. 627, 2091 - 2094. [abstract]
E. Koch (2001): Self-intersecting three-periodic minimal surfaces forming one-periodic tubes or finite polyhedra. - Z. Kristallogr. 216, 430 - 437. [abstract]
E. Koch (2000b): Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths. - Z. Kristallogr. 215, 386 - 392. [abstract]
E. Koch (2000a): Minimal surfaces with self-intersections along straight lines. II. Surfaces forming three-periodic labyrinths. - Acta Crystallogr. A56, 15 - 23. [abstract]
E. Koch & W. Fischer (1999): Minimal surfaces with self-intersections along straight lines. I. Derivation and properties. - Acta Crystallogr. A55, 58 - 64. [abstract]
W. Fischer & E. Koch (1996b): Spanning minimal surfaces. - Phil. Trans. R. Soc. Lond. A354, 2105 - 2142. [abstract]
W. Fischer & E. Koch (1996a): Two 3-periodic self-intersecting minimal surfaces related to the Cr3Si structure type. - Z. Kristallogr. 211, 1 - 3. [abstract]
E. Koch & W. Fischer (1993b): A crystallographic approach to 3-periodic minimal surfaces. In: Statistical Thermodynamics and Differential Geometry of Microstructured Materials, The IMA Volumes in Mathematics and its Applications, Vol. 51 - New York: Springer, 15 - 48. [abstract]
E. Koch & W. Fischer (1993a): Triply periodic minimal balance surfaces: a correction. - Acta Crystallogr. A49, 209 - 210. [abstract]
W. Fischer & E. Koch (1992): Symmetry aspects of 3-periodic minimal surfaces. In: Bifurcation and Symmetry. Internationale Schriftenreihe zur Numerischen Mathematik, Vol. 104 - Basel, Boston, Berlin: Birkhäuser, 123 - 133. [abstract]
W. Fischer & E. Koch (1990): Crystallographic aspects of minimal surfaces. - Coll. Phys. 51 - C7, 131 - 147. [abstract]
E. Koch & W. Fischer (1990): Flat points of minimal balance surfaces. - Acta Crystallogr. A46, 33 - 40. [abstract]
W. Fischer & E. Koch (1989c): Genera of minimal balance surfaces. - Acta Crystallogr. A45, 726 - 732. [abstract]
E. Koch & W. Fischer (1989b): New surface patches for minimal balance surfaces. IV. Catenoids with spout-like attachments. - Acta Crystallogr. A45, 558 - 563. [abstract]
W. Fischer & E. Koch (1989b): New surface patches for minimal balance surfaces. III. Infinite strips. - Acta Crystallogr. A45, 485 - 490. [abstract]
E. Koch & W. Fischer (1989a): New surface patches for minimal balance surfaces. II. Multiple catenoids. - Acta Crystallogr. A45, 169 - 174. [abstract]
W. Fischer & E. Koch (1989a): New surface patches for minimal balance surfaces. I. Branched catenoids. - Acta Crystallogr. A45, 166 - 169. [abstract]
E. Koch & W. Fischer (1988): On 3-periodic minimal surfaces with non-cubic symmetry. - Z. Kristallogr. 183, 129 - 152. [abstract]
W. Fischer & E. Koch (1987): On 3-periodic minimal surfaces. - Z. Kristallogr. 179, 31 - 52. [abstract]
| minimal surfaces | Dirichlet domains | normalizers | lattice complexes |
| structure description | International Tables | start |
The atomic arrangement of many inorganic crystal structures may be understood by imagining the atoms as rigid spheres in mutual contact. It would be desirable to know in advance at least all those patterns of mutually contacting spheres in which all spheres are symmetrically equivalent, the so-called homogeneous sphere packings. Up to now all such sphere packings have been derived that may be generated with cubic (199 cases), tetragonal (394 cases), hexagonal (170 cases), trigonal (225 cases) or triclinic symmetry (30 cases). The derivation of the orthorhombic sphere packings is not yet finished. In addition, all homogeneous sphere packings with three contacts per sphere - irrespective of there symmetry - have been determined, and the related problem of the homogeneous sphere packing with minimal density has been dealt with. H. Sowa, E. Koch & W. Fischer: Orthorhombic sphere packings. II. Bivariant lattice complexes. - Acta Crystallogr. A63 (2007), 354-364. [abstract]
H. Sowa & W. Fischer (2006): Orthorhombic sphere packings. I. Invariant and univariant lattice complexes. - Acta Crystallogr. A62 (2006), 413-418. [abstract]
H. Sowa & E. Koch (2006): Hexagonal and trigonal sphere packings. IV. Trivariant lattice complexes of trigonal space groups. - Acta Crystallogr. A62 (2006), 379-399. [abstract]
T. E. Dorozinski & W. Fischer (2006): A novel series of sphere packings with arbitrarily low density.- Z. Kristallogr. 221 , 563-566. [abstract]
E. Koch, W. Fischer & H. Sowa (2006): Interpenetration of homogeneous sphere packings and of two-periodic layers of spheres. - Acta Crystallogr. A62, 152-167.[abstract]
W. Fischer (2005): Tetragonal sphere packings: minimal densities and subunits. - Acta Crystallogr. A61, 435 - 444. [abstract]
E. Koch, H. Sowa & W. Fischer (2005): On the density of homogeneous sphere packings. - Acta Crystallogr. A61, 426 - 434. [abstract]
W. Fischer (2005): On sphere packings of arbitrarily low density. - Z. Kristallogr. 220, 657 - 662. [abstract]
H. Sowa & E. Koch (2005): Hexagonal and trigonal sphere packings. III. Trivariant lattice complexes of hexagonal space groups. - Acta Crystallogr. A61, 331 - 342. [abstract]
W. Fischer (2004): Minimal densities of cubic sphere-packing types. - Acta Crystallogr. A60, 246 - 249. [abstract]
E. Koch & H. Sowa (2004): Exceptional properties of some sphere packings in the general position of P6222. - Acta Crystallogr. A60, 239 - 245. [abstract]
H. Sow a & E. Koch (2004): Quantification of the deviations from closest sphere packings. - Eur. J. Mineral. 16, 255 - 260. [abstract]
H. Sowa & E. Koch (2004): Hexagonal and trigonal sphere packings. II. Bivariant lattice complexes. - Acta Crystallogr. A60, 158 - 166. [abstract]
H. Sowa, E. Koch & W. Fischer (2003): Hexagonal and trigonal sphere packings. I. Invariant and univariant lattice complexes. - Acta Crystallogr. A59, 317 - 326. [abstract]
W. Fischer & E. Koch (2002): Homogeneous sphere packings with triclinic symmetry. - Acta Crystallogr. A58, 509 - 513. [abstract]
H. Sowa & E. Koch (2002): Group-theoretical and geometrical considerations of the phase transition between the high-temperature polymorphs of quartz and tridymite. - Acta Crystallogr. A58, 327 - 333. [abstract]
H. Sowa & E. Koch (2001): A proposal for a transition mechanism from the diamond to the lonsdaleite type. - Acta Crystallogr. A57, 406 - 413. [abstract]
E. Koch & W. Fischer (1999): Sphere packings and packings of ellipsoids. In: International Tables for Crystallography, Vol. C (Second revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 738 - 743.
H. Sowa & E. Koch (1999): Sphere configurations with the symmetry R-3m 18(h) .m. - Z. Kristallogr. 214, 316 - 323. [abstract]
W. Fischer (1997): Comments on Novel regular quinquehedral packing obtained by the local approach by V.V.Manzhar. - Acta Crystallogr. A53, 402. [abstract]
E. Koch & W. Fischer (1995): Sphere packings with three contacts per sphere and the problem of the least dense sphere packing. - Z. Kristallogr. 210, 407 - 414. [abstract]
W. Fischer (1993): Tetragonal sphere packings. III. Lattice complexes with three degrees of freedom. - Z. Kristallogr. 205, 9 - 26. [abstract]
E. Koch & W. Fischer (1992): Sphere packings and packings of ellipsoids. In: International Tables for Crystallography, Vol. C - Dordrecht, Boston, London: Kluwer Academic Publishers, 654 - 659.
W. Fischer (1991b): Tetragonal sphere packings. II. Lattice complexes with two degrees of freedom. - Z. Kristallogr. 194, 87 - 110. [abstract]
W. Fischer (1991a): Tetragonal sphere packings. I. Lattice complexes with zero or one degree of freedom. - Z. Kristallogr. 194, 67 - 85. [abstract]
E. Koch (1985): The geometrical characteristics of the alpha-ThSi2 structure type and of its parameter field. - Z. Kristallogr. 173, 205 - 224. [abstract]
E. Koch (1984): A geometrical classification of cubic point configurations. - Z. Kristallogr. 166, 23 - 52. [abstract]
E. Koch & W. Fischer (1978): Types of sphere packings for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 148, 107 - 152. [abstract]
W. Fischer (1976): Eigenschaften der Heesch-Laves-Packung und ihres Kugelpackungstyps. - Z. Kristallogr. 143, 140 - 155. [abstract]
W. Fischer & E. Koch (1976): Durchdringungen von Kugelpackungen mit kubischer Symmetrie. - Acta Crystallogr. A32 225 - 232. [abstract]
W. Fischer (1974): Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit drei Freiheitsgraden. - Z. Kristallogr. 140, 50 - 74. [abstract]
W. Fischer (1973): Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. - Z. Kristallogr. 138, 129 - 146. [abstract]
E. Koch & W. Fischer (1972): Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharakter. - Z. Kristallogr. 135, 73 - 92. [abstract]
W. Fischer (1971): Existenzbedingungen homogener Kugelpackungen in Raumgruppen tetragonaler Symmetrie. - Z. Kristallogr. 133, 18 - 42. [abstract]
W. Fischer (1968): Kreispackungsbedingungen in der Ebene. - Acta Crystallogr. A24, 67 - 81. [abstract]
| minimal surfaces | sphere packings | normalizers | lattice complexes |
| structure description | International Tables | start |
Within a crystal structure an area of the surrounding space may be related to each atom in such a way that no gaps and no overlaps occur. A special construction for this starts from the midpoints of the atoms and assigns a polyhedron, the Dirichlet domain, to each atomic centre: the faces of the polyhedron run perpendicular to the lines connecting a centre to its neighbours and halve the corresponding distance. The vertices of the Dirichlet domains then constitute the midpoints of the voids within the original atomic arrangement, i.e. the preferred positions for further atoms. So far only part of all conceivable subdivisions of space into Dirichlet domains has systematically been derived and some general properties have been studied. L. Bessais, C. Djéga-Mariadassou & E. Koch (2002): Structural and Mössbauer spectral study of the metastable phase Sm(Fe,Co,Ti)10. - J. Phys.: Condens. Matter 14, 1-10. [abstract]
E. Koch & W. Fischer (1996): DIDO95 and VOID95 - programs for the calculation of Dirichlet domains and coordination polyhedra. - Z. Kristallogr. 211, 251 - 253. [abstract]
W. Fischer (1986): Geometrical aspects of the patterns of conduction paths in fast ion conductors. - Cryst. Res. Techn. 21, 499 - 503. [abstract]
E. Koch (1985): The geometrical characteristics of the alpha-ThSi2 structure type and of its parameter field. - Z. Kristallogr. 173, 205 - 224. [abstract]
E. Koch (1984): A geometrical classification of cubic point configurations. - Z. Kristallogr. 166, 23 - 52. [abstract]
W. Fischer (1980b): On a space-filling polyhedron with 26 faces. - match (communications in mathematical chemistry) 9, 103. [abstract]
W. Fischer (1980a): Normal homogeneous partitions of three-dimensional Euclidean space which are not partitions into fundamental regions of a space group. - match (communications in mathematical chemistry) 9, 101. [abstract]
E. Koch & W. Fischer (1980): Calculation of volume increments for organic compounds by means of Dirichlet domains. - Z. Kristallogr. 153, 255 - 263. [abstract]
W. Fischer & E. Koch (1979): Geometrical packing analysis of molecular compounds. - Z. Kristallogr. 150, 245 - 260. [abstract]
E. Koch & W. Fischer (1974): Zur Bestimmung asymmetrischer Einheiten kubischer Raumgruppen mit Hilfe von Wirkungsbereichen. - Acta Crystallogr. A30, 490 - 496. [abstract]
E. Koch & W. Fischer (1973): Über den Einfluß der Kugelradien auf heterogene Wirkungsbereichsteilungen. - N. Jb. Min. Mh. 1973, 361 - 380. [abstract]
W. Fischer & E. Koch (1973): Über heterogene Wirkungsbereichsteilungen in Abhängigkeit von zwei Parametern. - N. Jb. Min. Mh. 1973, 252 - 273. [abstract]
E. Koch (1973): Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. - Z. Kristallogr. 138, 196 - 215. [abstract]
E. Koch & W. Fischer (1972): Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharakter. - Z. Kristallogr. 135, 73 - 92. [abstract]
W. Fischer, E. Koch & E. Hellner (1971): Zur Berechnung von Wirkungsbereichen in Strukturen anorganischer Verbindungen. - N. Jb. Min. Mh. 1971, 227 - 237. [abstract]
| minimal surfaces | sphere packings | Dirichlet domains | lattice complexes |
| structure description | International Tables | start |
The solutions to some problems treated on an ad hoc and isolated basis by crystallographers in the past may be explained by the group-theoretical concept of normalizers. Problems of this kind are: ambiguities in the description of crystal structures (which complicate comparisons); the number of subgroups of the same kind; the choice of a starting set of signs or phases in the course of a crystal-structure determination by means of direct methods; the reduction of the asymmetric unit for geometrical studies. The Euclidean and the affine normalizers of the space groups and of the plane groups have been tabulated in the "International Tables for Crystallography".
E. Koch & W. Fischer (2005): Normalizers of space groups: A useful tool in crystal-structure description, comparison and determination. - Z. Kristallogr. 220, in the press. [abstract]
E. Koch, W. Fischer & U. Müller (2002): Normalizers of space groups and their use in crystallography. In: International Tables for Crystallography, Vol. A (Fifth revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 877 - 905.
E. Koch & U. Müller (1990): Euklidische Normalisatoren für trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. - Acta Crystallogr. A46, 826 - 831. [abstract]
E. Koch & W. Fischer (1987): Euclidean and affine normalizers of space groups and their use in crystallography. In: International Tables for Crystallography, Vol. A (Second revised edition) - Dordrecht, Boston, Lancaster, Tokyo: D. Reidel Publishing Company, 855 - 869.
E. Koch (1986): Implications of the Euclidean normalizers of space groups in reciprocal space. - Cryst. Res. Techn. 21, 1213 - 1219. [abstract]
E. Koch (1984): The implications of normalizers on group-subgroup relations between space groups. - Acta Crystallogr. A40, 593 - 600. [abstract]
W. Fischer & E. Koch (1983): On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. - Acta Crystallogr. A39, 907 - 915. [abstract]
E. Koch & W. Fischer (1975): Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. - Acta Crystallogr. A31, 88 - 95. [abstract]
| minimal surfaces | sphere packings | Dirichlet domains | normalizers |
| structure description | International Tables | start |
In analogy to the face forms of crystal polyhedra Paul Niggli introduced the concept of "lattice complexes" in order to characterize relationships between patterns of points with space-group symmetry. Carl Hermann included this concept in a modified form into "Internationale Tabellen zur Bestimmung von Kristallstrukturen" (1935). A mathematically satisfactory definition was missing, however, until 1974. The lattice-complex concept is important for the recognition of structural relationships, in connection with subgroup relations and with respect to possible phase transitions. For geometrical studies it is sufficient to consider only one representative Wyckoff position per lattice complex.
E. Koch & H. Sowa: The cubic limiting complexes of lattice complexes with trigonal characteristic symmetry - Z. Kristallogr. 220, accepted. [abstract]
E. Koch & W. Fischer (2003): The cubic limiting complexes of the tetragonal lattice complexes. - Z. Kristallogr. 218, 597 - 603. [abstract]
W. Fischer & E. Koch (2002): Lattice complexes. In: International Tables for Crystallography, Vol. A (Fifth revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 845 - 876.
W. Fischer & E. Koch (1995): Lattice complexes with at most one comprehensive complex. - Acta Crystallogr. A51, 586 - 587. [abstract]
E. Koch & W. Fischer (1985): Lattice complexes and limiting complexes versus orbit types and non-characteristic orbits: a comparative discussion. - Acta Crystallogr. A41, 421 - 426. [abstract]
W. Fischer & E. Koch (1978): Limiting forms and comprehensive complexes for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 147, 255 - 273. [abstract]
E. Koch & W. Fischer (1978): Complexes for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 147, 21 - 38. [abstract]
W. Fischer & E. Koch (1974b): Kubische Strukturtypen mit festen Koordinaten. - Z. Kristallogr. 140, 324 - 330. [abstract]
E. Koch (1974): Die Grenzformen der kubischen Gitterkomplexe. - Z. Kristallogr. 140, 75 - 86. [abstract]
W. Fischer & E. Koch (1974a): Eine Definition des Begriffs "Gitterkomplex". - Z. Kristallogr. 139, 268 - 278. [abstract]
H. Burzlaff, W. Fischer, E. Hellner & A. Niggli (1974): Zur Entwicklung des Begriffs "Gitterkomplex". - Z. Kristallogr. 139, 246 - 251. [abstract]
W. Fischer, H. Burzlaff, E. Hellner & J.D.H. Donnay (1973): Space Groups and Lattice Complexes. - National Bureau of Standards Monograph 134, Washington. [abstract]
E. Koch & E. Hellner (1971): Die Pattersonkomplexe der Gitterkomplexe. - Z. Krist. 133, 242 - 259. [abstract]
H. Burzlaff, W. Fischer & E. Hellner (1969): Bemerkungen zu "Die Gitterkomplexe der Ebenengruppen" (Acta Cryst. A24, 57) (1968) und "Kreispackungsbedingungen in der Ebene" (Acta Cryst. A24, 67) (1968). - Acta Crystallogr. A25, 710. [abstract]
H. Burzlaff, W. Fischer & E. Hellner (1968): Die Gitterkomplexe der Ebenengruppen. - Acta Crystallogr. A24, 57 - 67. [abstract]
| minimal surfaces | sphere packings | Dirichlet domains | normalizers |
| lattice complexes | International Tables | start |
Erwin Hellner developed the concept of "frameworks" for a classifying description of crystal structures: one kind of particle (the anions in most cases) constitute a supporting skeleton of the crystal structure (frequently in form of a sphere packing), the other particles (cations) are situated in its voids. Structural relationships then result from the different occupation of the voids calculated as vertices of Dirichlet domains. This approach was applied to several inorganic crystal structures. In the course of these studies it turned out that in crystal structures the tendency to form a sphere packing as well as to built up an arrangement with ideal voids may play an important role. Both tendencies may counteract each other in which case the crystal structure realizes a compromise between the two ideal cases.
E. Hellner & E. Koch (1982): The garnet-like cyanide framework of ammonium ferrocyanide hydrate with a channel system for ionic conductivity. - Acta Crystallogr. B38, 376 - 379. [abstract]
E. Hellner, E. Koch & A. Reinhardt (1981): The homogeneous frameworks of the cubic crystal structures. - Physik Daten - Physics Data 16-2, 1 - 67. [abstract]
E. Hellner & E. Koch (1981): Cluster or framework considerations for the structures of Tl7Sb2, alpha-Mn, Cu5Zn8 and their variants Li22Si5, Cu41Sn11, Sm11Cd45, Mg6Pd and Na6Tl with octuple unit cells. -Acta Crystallogr. A37, 1 - 6. [abstract]
E. Koch & E. Hellner (1981): The frameworks of sodalite-like structures and of tetrahedrite-like structures. - Z. Kristallogr. 154, 95 - 114. [abstract]
E. Hellner & E. Koch (1980): A comparison of the crystal structures of Sb2Tl7, Cu5Zn8 ( gamma-brass) and Ir3Ge7. - Can. J. Chem. 58, 708 - 713. [abstract]
E. Hellner & E. Koch (1979): The oxygen framework of leucite and analcime. - Miner. Petrogr. Acta 23, 303 - 311. [abstract]
E. Hellner, R. Gerlich, E. Koch & W. Fischer (1979): The oxygen framework in garnet and its occurence in the structures of Na3Al2Li3F12, Ca3Al2(OH)12, RhBi4 and Hg3TeO6. - Physik Daten - Physics Data 16-1, 1 - 31. [abstract]
| minimal surfaces | sphere packings | Dirichlet domains | normalizers |
| lattice complexes | structure description | start |
Already at the beginning of the 60s it became obvious that a completely new version of the crystallographic standard tables was required. An international commission had to work for many years before Volume A "Space-Group Symmetry" of the "International Tables for Crystallography" could be published in 1983. German crystallographers from Aachen, Erlangen, Karlsruhe and Marburg contributed substantially to it. To date the first edition has repeatedly been revised and supplemented.
E. Koch, W. Fischer & U. Müller (2002): Normalizers of space groups and their use in crystallography. In: International Tables for Crystallography, Vol. A (Fifth revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 877 - 905.
W. Fischer & E. Koch (2002): Lattice complexes. In: International Tables for Crystallography, Vol. A (Fifth revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 845 - 876.
W. Fischer & E. Koch (2002): Symmetry operations. In: International Tables for Crystallography, Vol. A (Fifth revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 809 - 816.
E. Koch & W. Fischer (1999): Sphere packings and packings of ellipsoids. In: International Tables for Crystallography, Vol. C (Second revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 738 - 743.
E. Koch (1999): Crystal geometry and symmetry. In: International Tables for Crystallography, Vol. C (Second revised edition) - Dordrecht, Boston, London: Kluwer Academic Publishers, 1 - 14.
P.M. de Wolff, Y. Billiet, J.D.H. Donnay, W. Fischer, R.B. Galiulin, A.M. Glazer, Th. Hahn, M. Senechal, D.P. Shoemaker, H. Wondratschek, A.J.C. Wilson & S.C. Abrahams (1992): Symbols for symmetry elements and symmetry operations. Final report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A48, 727 - 732. [abstract]
E. Koch & W. Fischer (1992): Sphere packings and packings of ellipsoids. In: International Tables for Crystallography, Vol. C - Dordrecht, Boston, London: Kluwer Academic Publishers, 654 - 659.
E. Koch (1992): Crystal geometry and symmetry. In: International Tables for Crystallography, Vol. C - Dordrecht, Boston, London: Kluwer Academic Publishers, 1 - 14.
P.M. de Wolff, Y. Billiet, J.D.H. Donnay, W. Fischer, R.B. Galiulin, A.M. Glazer, M. Senechal, D.P. Shoemaker, H. Wondratschek, Th. Hahn, A.J.C. Wilson & S.C. Abrahams (1989): Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A45, 494 - 499. [abstract]
E. Koch & W. Fischer (1987): Euclidean and affine normalizers of space groups and their use in crystallography. In: International Tables for Crystallography, Vol. A (Second revised edition) - Dordrecht, Boston, Lancaster, Tokyo: D. Reidel Publishing Company, 855 - 869.
P.M. de Wolff, N.V. Belov, E.F. Bertaut, M.J. Buerger, J.D.H. Donnay, W. Fischer, Th. Hahn, V.A. Koptsik, A.L. Mackay, H. Wondratschek, A.J.C. Wilson & S.C. Abrahams (1985): Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A41, 278 - 280. [abstract]
W. Fischer & E. Koch (1983b): Lattice complexes. In: International Tables for Crystallography, Vol. A - Dordrecht, Boston, London: D. Reidel Publishing Company, 819 - 848.
W. Fischer & E. Koch (1983a): Symmetry operations. In: International Tables for Crystallography, Vol. A - Dordrecht, Boston, London: D. Reidel Publishing Company, 787 - 792.
| minimal surfaces | sphere packings | Dirichlet domains | normalizers |
| lattice complexes | structure description | International Tables | start |
Last update: July 2007