Geodesic
Symmetric Tessellations on Euclidean Space-Forms
Canadian journal of mathematics, Tome 51 (1999) no. 6, pp. 1230-1239

Voir la notice de l'article provenant de la source Cambridge University Press

It is shown here that, for $n\ge 2$ , the $n$ -torus is the only $n$ -dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if $n\,=\,2$ .
DOI : 10.4153/CJM-1999-055-6
Mots-clés : 51M20, polyhedra and polytopes, regular figures, division of space 
Hartley, Michael I.; McMullen, Peter; Schulte, Egon. Symmetric Tessellations on Euclidean Space-Forms. Canadian journal of mathematics, Tome 51 (1999) no. 6, pp. 1230-1239. doi: 10.4153/CJM-1999-055-6
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[1] [1] Brehm, U., Kühnel, W. and Schulte, E., Manifold structures on abstract regular polytopes. Aequationes Math. 49(1995), 12–35. Google Scholar

[2] [2] Coxeter, H. S. M., Regular Polytopes. 3rd edn, Dover, New York, 1973. Google Scholar

[3] [3] Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups. 4th edn, Springer, 1980. Google Scholar

[4] [4] Davis, M. W., Regular convex cell complexes. In: Geometry and Topology (eds. McCrory, C. and Shifrin, T.), Lecture Notes in Pure and Appl. Math. 105, Marcel Dekker, New York-Basel, 1987, 53–88. Google Scholar

[5] [5] Dress, A. W. M. and Schulte, E., On a theorem of McMullen about combinatorially regular polytopes. Simon Stevin 61(1987), 265–273. Google Scholar

[6] [6] Edmonds, A. L., Ewing, J. H. and Kulkarni, R. S., Regular tessellations of surfaces and (p, q, 2)-triangle groups. Ann. of Math. 116(1982), 113–132. Google Scholar

[7] [7] Garbe, D., Über die regulären Zerlegungen geschlossener orientierbarer Flächen. J. Reine Angew. Math. 237(1969), 39–55. Google Scholar

[8] [8] Gromov, M., Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56(1982), 5–99. Google Scholar

[9] [9] Grünbaum, B., Regularity of graphs, complexes and designs. In: Problèmes combinatoires et théorie des graphes, Colloq. Internat. CNRS 260 (Orsey, 1977), 191–197. Google Scholar

[10] [10] Hartley, M. I., Combinatorially Regular Euler Polytopes. Ph.D. Thesis, University of Western Australia (Nedlands, Perth, 1996). Google Scholar

[11] [11] Kato, M., On combinatorial space-forms. Sci. Papers College Gen. Ed. Univ. Tokyo 30(1980), 107–146. Google Scholar

[12] [12] McMullen, P., Combinatorially regular polytopes. Mathematika 14(1967), 142–150. Google Scholar

[13] [13] McMullen, P., On the Combinatorial Structure of Convex Polytopes. Ph.D. Thesis, University of Birmingham (1968). Google Scholar

[14] [14] McMullen, P. and Schulte, E., Quotients of polytopes and C-groups. Discrete Comput. Geom. 11(1994), 453–464. Google Scholar

[15] [15] McMullen, P. and Schulte, E., Higher toroidal regular polytopes. Adv. Math. 117(1996), 17–51. Google Scholar

[16] [16] McMullen, P. and Schulte, E., Abstract Regular Polytopes. Monograph in preparation. Google Scholar

[17] [17] Schulte, E. and Weiss, A. I., Chiral polytopes. In: Applied Geometry and Discrete Mathematics (The Klee Festschrift) (eds. Gritzmann, P. and Sturmfels, B.), Ser. DiscreteMath. Theoret. Comput. Sci. 4, Amer.Math. Soc., 1991, 493–516. Google Scholar

[18] [18] Sherk, F. A., The regular maps on a surface of genus three. Canad. J. Math. 11(1959), 452–480. Google Scholar

[19] [19] Wilson, S. E., Non-orientable regular maps. Ars Combin. 5(1978), 213–218. Google Scholar

[20] [20] Wolf, J. A., Spaces of Constant Curvature. 5th edn, Publish or Perish, Wilmington, Delaware, 1984. Google Scholar

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