Abstract
The concept of regular incidence-complexes generalizes the notion of regular polyhedra in a combinatorial sense. A regular incidence-complex is a partially ordered set with regularity defined by certain transitivity properties of its automorphism group. The concept includes all regular d-polytopes and all regular complex d-polytopes as well as many geometries and well-known configurations.
Similar content being viewed by others
Literaturverzeichnis
Buekenhout, F.: ‘Diagrams for Geometries and Groups’. J. Comb. Th., Ser. A 27 (1979), 121–151.
Buekenhout, F.: ‘The Basic Diagram of a Geometry’, in Geometries and Groups (eds. Aigner and B. Jungnickel). Lecture Notes in Math. 893, Springer, Berlin, 1981.
Coxeter, H. S. M.: ‘Configurations and Maps’. Rep. Math. Colloq. (2), 8 (1948), 18–38.
Coxeter, H. S. M.: ‘Self-Dual Configurations and Regular Graphs’. Bull. Amer. Math. Soc. 56 (1950), 413–455.
Coxeter, H. S. M.: Twelve Geometric Essays. Southern Illinois University Press, Carbondale, 1968.
Coxeter, H. S. M.: Twisted Honeycombs. Reg. Conf. Ser. in Math., No. 4, Amer. Math. Soc., Providence, R.I., 1970.
Coxeter, H. S. M.: Regular Polytopes. London, 1948; 3. Auflage, Dover, New York, 1973.
Coxeter, H. S. M.: Regular Complex Polytopes. Cambridge University Press, Cambridge, 1974.
Coxeter, H. S. M. and Shephard, G. C.: ‘Regular 3-Complexes with Toroidal Cells’. J. Comb. Theory (B) 22 (1977), 131–138.
Coxeter, H. S. M. and Moser, W. O. J.: Generators and Relations for Discrete Groups. Berlin, 1957; 4. Auflage, Springer, Berlin, 1980.
Coxeter, H. S. M.: ‘A Symmetrical Arrangement of Eleven Hemi-Icosahedra’ (to appear).
Coxeter, H. S. M.: Ten Toroids and Fifty-Seven Hemi-Dodecahedra’. Geom. Dedicata 13, 87–99 (1982).
Danzer, L.: ‘Regular Incidence-Complexes and Dimensionally Unbounded Sequences of Such I’. Proc. of the Int. Conf. on Convexity and Graph Theory, held in Israel, 1981.
Danzer, L.: ‘Regular Incidence-Complexes and Dimensionally Unbounded Sequences of Such II’ (in preparation).
Dembowski, P.: Finite Geometries. Springer, Berlin, 1968.
Dress, A. W. M.: ‘On the Classification and Generation of Two- and Higher-Dimensional Regular Patterns, in Proc. of the ZiF Conf. of Crystallographic Groups. Match, vol. 9 (1980), 81–100.
Dress, A. W. M.: ‘Regular Polytopes and Equivariant Tessellations from a Combinatorial Point of View’ (to appear).
Fejes Toth, L.: Reguläre Figuren. Verlag der Ungarischen Akademie der Wissenschaften, Budapest, 1965.
Grünbaum, B.: Convex Polytopes. John Wiley and Sons, London, 1967.
Grünbaum, B.: ‘Regular Polyhedra-Old and New’. Aequationes Mathematicae 16 (1977), 1–20.
Grünbaum, B.: ‘Regularity of Graphs, Complexes and Designs’, in Coll. Int. C.N.R.S. No. 260 — Problemès Combinatoire et Théorie des Graphes. Orsay, 1977, pp. 191–197.
Grünbaum, B., in Combinatorics (eds. A. Hajnal and V.T. Sós), Vol. 2, North-Holland Publ. Co., New York, 1978, pp. 1199–1200.
Larman, D. G. and Rogers, C. A.: Durham Symposium on the Relations between Infinite-Dimensional and Finitely-Dimensional Convexity. Bull. Lond. Math. Soc. 8 (1976), 1–33.
McMullen, P.: ‘Combinatorially Regular Polytopes’. Mathematika 14 (1967), 142–150.
McMullen, P.: ‘Affinely and Projectively Regular Polytopes’. J. London Math. Soc. 43 (1968), 755–757.
Schläfli, L.: ‘Theorie der vielfachen Kontinuität’. Denksch. Schweiz. naturforsch. Gesell. 38 (1901), 1–237.
Schulte, E.: Reguläre Inzidenzkomplexe. Dissertation, Dortmund, 1980.
Schulte, E.: ‘Reguläre Inzidenzkomplexe II’ (bei der Geom. Dedicata in Druck).
Schulte, E.: ‘Reguläre Inzidenzkomplexe III’ (bei der Geom. Dedicata in Druck).
Schulte, E.: ‘On Arranging Regular Incidence-Complexes as Faces of Higher-Dimensional Ones’ (preprint).
Shephard, G. C.: ‘Regular Complex Polytopes’. Proc. London Math. Soc. (3), 2 (1952), 82–97.
Sommerville, D. M. Y.: An Introduction to the Geometry of n Dimensions. London, 1929; 2. Auflage, Dover, New York, 1958.
Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Springer, Berlin, 1974.
Tits, J.: ‘Buildings and Buekenhout Geometries’, in Finite Simple Groups II (ed. M. J. Collins). Academic Press, 1980.
Vince, A.: ‘Combinatorial Maps’ (to appear).
Vince, A.: ‘Regular Combinatorial Maps I’ (to appear).
Vince, A.: ‘Regular Combinatorial Maps II’ (to appear).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Danzer, L., Schulte, E. Reguläre Inzidenzkomplexe I. Geom Dedicata 13, 295–308 (1982). https://doi.org/10.1007/BF00148235
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00148235