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Patterns on the 2-Sphere

Published online by Cambridge University Press:  26 February 2010

Branko Grünbaum  and
G. C. Shephard
Branko Grünbaum
Affiliation:
University of Washington, Seattle, Wa 98195, U.S.A.
G. C. Shephard
Affiliation:
University of East Anglia, Norwich NR4 7TJ, England.
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Extract

Problems of classifying and enumerating types of plane patterns, tilings, and other repeating geometrical structures have interested mathematicians, crystallographers and others for many years. Recently we have formulated the general principles that seem to underlie many of the published treatments of these topics, and so have been able to put on a mathematical basis classification criteria often justified mainly on intuitive grounds. In other words, we can now decide whether or not two given patterns are of the same “type”, at one of a number of different possible levels of classification, without relying on some vaguely expressed distinction based on a “feeling” as to whether the objects in question should be regarded as of distinguishable kinds, or not.

Type
Research Article
Information
Mathematika , Volume 28 , Issue 1 , June 1981 , pp. 1 - 35
Copyright
Copyright © University College London 1981

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References

Anonymous, 1960. “Boris Nikolaevič Delone—on the occasion of his seventieth birthday” [in Russian], Kristallografiya, 5 (1960), 339–340. English translation: Soviet Physics—Crystallography, 5 (1960), 321–322.Google Scholar
Bokiĭ, G. B., 1940. “The number of physically distinct simple forms of crystals” [inRussian], Trav. Labor. Crist. Acad. Sci. URSS, 2 (1940), 13–37.Google Scholar
Boldyrev, A. K., 1936. “Are there 47 or 48 simple forms possible on crystals?”, American Mineralogist, 21 (1936), 731–734.Google Scholar
Brückner, M., 1900. Vielecke und Vielflache (Teubner, Leipzig, 1900).Google Scholar
Brun, V., 1972. “Some theorems on the partitioning of the sphere, inspired by virus research” [Norwegian, with summary in English], Nordisk Matem. Tidskrift, 20 (1972), 87–91, 120.Google Scholar
Brun, V., 1976. “On regular packing of equal circles touching each other on the surface of a sphere”, Communic. Pure and Appl. Math., 29 (1976), 583–590.CrossRefGoogle Scholar
Buerger, M. J., 1956. Elementary Crystallography (John Wiley & Sons, New York, 1956).Google Scholar
Burckhardt, J. J., 1967. “Review of Horváth [1965]”, Zentralblatt fur Math., 136 (1967), 155. Reprinted in Math. Reviews, 41 (1971), No. 4389.Google Scholar
Calladine, C. R., 1978. “Buckminster Fuller's ‘Tensegrity’ structures and Clerk Maxwell's rules for the construction of stiff frames”, Int. J. Solids Structures, 14 (1978), 161–172.CrossRefGoogle Scholar
Coxeter, H. S. M. and Moser, W. O. J., 1972. Generators and Relations for Discrete Groups, 3rd edition (Springer-Verlag, Berlin-Heidelberg-New York, 1972).CrossRefGoogle Scholar
Cundy, H. M. and Rollett, A. P., 1961. Mathematical Models, 2nd edition (Oxford Univ. Press, London, 1961).Google Scholar
Delone, B. N. (= Delaunay), 1932. “Neue Darstellung der geometrischen Kristallographie”, Zeitschr. Kristallographie, 84 (1932), 109–149.Google Scholar
Delone, B. N., 1958. “The theory of stereohedra” [in Russian], Uspehi Matem. Nauk 13, No. 4 (1958), 221–223.Google Scholar
Delone, B. N., 1959. “The theory of planigons” [in Russian], Izv. Akad. Nauk SSSR Ser. Mat., 23 (1959), 365–386.Google Scholar
Delone, B. N., 1963. “On regular partitions of spaces” [in Russian], Priroda, 1963, No. 2, 60–63.Google Scholar
Delone, B. N., Dolbilin, N. P. and Stogrin, M. I., 1978. “Combinatorial and metric theory of planigons” [in Russian], Trudy Mat. In-ta AN SSSR, 148 (1978), 109–140.Google Scholar
Delone, B. N., Galiulin, R. V. and Stogrin, M. I., 1979. “The contemporary theory of regular decompositions of Euclidean space” [in Russian]. Pages 235–260 in Fedorov (1979).Google Scholar
Federov, E. S., 1885. “Foundations of the theory of figures” [in Russian], Acad. Sci. Petersburg, 1885. Republished with comments, Akad. Nauk SSSR, 1953.Google Scholar
Fedorov, E. S., 1899. “Regulare Plan– und Raumtheilung”, Abh. K. Bayer. Akad. Wiss. (II. Classe), 20 (1899).Google Scholar
Fedorov, E. S., 1979. “Regular partition of plane and space” [in Russian]. Translation of Fedorov [1899], with comments. Nauka, Leningrad, 1979.Google Scholar
Finsterwalder, S., 1936. “Regelmässige Anordnungen gleicher sich berührender Kreise in der Ebene, auf der Kugel und auf der Pseudosphäre”, Abh. Bayer Akad. Wiss. Math.-Naturwiss. Abt. Neue Folge, 38 (1936), 42.Google Scholar
Fuller, R. B., 1975. Synergetics (MacMillan, New York, 1975).Google Scholar
Grünbaum, B. and Shephard, G. C., 1977. “The eighty-one types of isohedral tilings in the plane”, Math. Proc. Cambridge Phil. Soc, 82 (1977), 177–196.CrossRefGoogle Scholar
Grünbaum, B. and Shephard, G. C., 1978a. “Isohedral tilings of the plane by polygons”, Comment. Math. Helvet., 53 (1978), 542–571.CrossRefGoogle Scholar
Grünbaum, B. and Shephard, G. C., 1978b. “Lectures on Lost Mathematics”, Lecture notes (Univ. of Washington 1975; reissued 1978).Google Scholar
Grünbaum, B. and Shephard, G. C., 1981a. “A hierarchy of classification methods for patterns”, Zeitschr. Kristallographie, 154 (1981), 163–187.Google Scholar
Grünbaum, B. and Shephard, G. C., 1981b. Tilings and Patterns (W. H. Freeman and Co., San Francisco) (to appear).Google Scholar
Grünbaum, B. and Shephard, G. C., 1981c. “Spherical tilings with transitivity properties”. In The geometric vein: Essays presented to H. S. M. Coxeter (Springer-Verlag, Berlin, 1981).Google Scholar
Grünbaum, B. and Shephard, G. C., 1982. “Spherical circle patterns” (to appear).Google Scholar
Heesch, H., 1968. Reguldres Parkettierungsproblem (Westdeutscher Verlag, Koln-Opladen, 1968).CrossRefGoogle Scholar
Henry, N. F. M. and Lonsdale, K., 1965. International Tables for X-ray Crystallography, I (Kynoch Press, Birmingham, 1965).Google Scholar
Hess, E., 1883. Einleitung in die Lehre von der Kugelteilung (Teubner, Leipzig, 1883).Google Scholar
Hinrichs, L. A., 1980. “Prismic tensigrids” (to appear).Google Scholar
Hohenberg, F., 1969. “Einige Geradensysteme der erweiterten Ikosaedergruppe”, Sitz.-Ber. d. öst. Akad. d. Wiss., Mathem.-naturw. Kl., Abt II, 178 (1969), 285–297.Google Scholar
Hohenberg, F., 1970a. “Einige Figuren der erweiterten Oktaedergruppe”, El. d. Math., 25 (1970), 55–60.Google Scholar
Hohenberg, F., 1970b. “Die Hexaeder und Tetraeder im Dodekaeder”, El. d. Math., 25 (1970), 97–101.Google Scholar
Hohenberg, F., 1970c. “Die Geradensysteme der erweiterten Polyedergruppe”, Sitz.-Ber. d. öst. Akad. d. Wiss., Mathem.-naturw. KL, Abt. II, 197 (1970), 63–92.Google Scholar
Holden, A., 1971. Shapes, Space and Symmetry (Columbia Univ. Press, New York, 1971).CrossRefGoogle Scholar
Hortobagyi, I., 1975. “Review of Lampert and Coka [1973]”, Zentralblatt fur Math., 287 (1975), 311.Google Scholar
Horváth, J., 1965. “Bemerkungen zur Theorie der Planigone”, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 8 (1965), 147–153.Google Scholar
Killingbergtrø, H. G., 1976. “Remarks on Brun's spherical tessellations” [in Norwegian, with English summary], Nordisk. Matem. Tidskrift, 24 (1976), 53–55, 75.Google Scholar
Koch, E. and Fischer, W., 1978. “Complexes for crystallographic point groups, rod groups and layer groups”, Zeitschr. Khstatlographie, 147 (1978), 21–38.Google Scholar
Lalvani, H., 1977. Transpolyhedra. Dual Transformations by Explosion-Implosion (Haresh Lalvani, New York, 1977).Google Scholar
Lampert, D. and Čoka, G., 1973. “The density of a regular system of discs” [in Russian], Ann. Univ. Sci. Budapest. Eütvüs Sect. Math., 16 (1973), 69–85.Google Scholar
Miller, G. A., Blichfeldt, H. F. and Dickson, L. E., 1916. Theory and Applications of Finite Groups (G. E. Stechert and Co., New York, 1916; reprinted 1938).Google Scholar
Molnár, J., 1966. “Collocazioni di cerchi con esigenza di spazio”, Ann. Univ. Sci. Budapest Eotvos Sect. Math., 9 (1966), 71–86.Google Scholar
Niggli, A., 1963. “Zur Topologie, Metrik und Symmetrie der einfachen Kristallformen”, Schweiz. Mineral. und Petrograph. Mitt., 43 (1963), 49–58.Google Scholar
Nowacki, W., 1933. “Die nichtkristallographischen Punktgruppen”, Zeitschr. Kristallographie, 86 (1933), 19–31.Google Scholar
Pearce, P. and Pearce, S., 1978. Polyhedra Primer (Van Nostrand-Reinhold, New York, 1978).Google Scholar
Pedersen, J. J., 1981. “Isonemal weaving on polyhedral surfaces.” In The geometric vein: Essays presented to H. S. M. Coxeter (Springer-Verlag, Berlin, 1981).Google Scholar
Pugh, A., 1976. An Introduction to Tensegrity (University of California Press, Berkeley, 1976).CrossRefGoogle Scholar
Robertson, S. A. and Carter, S., 1970. “On the Platonic and Archimedean solids”, J. London Math. Soc. (2), 2 (1970), 125–132.CrossRefGoogle Scholar
Rogers, A. F., 1935. “A tabulation of crystal forms and discussion of form-names”, American Mineralogist, 20 (1935), 838–851.Google Scholar
Šafranovskiá, I. I., 1948. “The forms of crystals” [in Russian], Trudy Inst. Kristallog. Akad. Nauk SSSR, 4 (1948), 13–166.Google Scholar
Schattschneider, D. and Walker, W., 1977. M. C. Escher Kaleidocycles (Ballantine Books, New York, 1977).Google Scholar
Schoenflies, A., 1891. Krystallsysteme und Krystallstructur (Leipzig, 1891).Google Scholar
Schwerdtfeger, H., 1962. “Review of Delone [1959]”, Math. Reviews, 23 (1962), No. A3503.Google Scholar
Senechal, M., 1981. “Colored spherical tessellations” (to appear).Google Scholar
Šubnikov, A. V. (= Schubnikov), 1940. “Thirty-nine types of crystallographic point symmetry” [in Russian, with English summary], Trav. Labor. Crist. Acad. Sci. URSS, 2 (1940), 7–11.Google Scholar
Wenninger, M. J., 1979. Spherical Models (Cambridge Univ. Press, Cambridge, 1979).Google Scholar
Wollny, W., 1974. “Die 36 regularen Pärketts mit dem Quadratnetz”, Geometriae Dedicata, 3 (1974), 41–60.CrossRefGoogle Scholar
Zamorzaeva, E. A., 1979. “The classification of tilings of the plane for groups of similarity symmetries” [in Russian], Doklady Akad. Nauk SSSR, 247 (1979), 276–279.Google Scholar