Annals of Combinatorics 1 (1997) 261-278

Flat Regular Polytopes

Peter McMullen and Egon Schulte

Department of Mathematics, University College London Gower Street, London WCIE 6BT, England

Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Received April 11, 1997

AMS Subject Classification: 51M20

With best wishes to H.S.M. Coxeter for his 90th birthday.

Abstract. At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regular n-polytopes P1 and P2, when does there exist a regular (n+1)-polytope P whose facets are isomorphic to P1 and whose vertex-figures are isomorphic to P2? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.

Keywords: polyhedra and polytopes, regular figures, division of space


1.  U. Brehm, W. Kühnel, and E. Schulte, Manifold structures on abstract regular polytopes, Aequationes Math. 49 (1995) 12–35.

2.  F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory, Ser. A 27 (1979) 121–151.

3.  H.S.M. Coxeter, Regular Polytopes, (3rd edition), Dover, New York, 1973.

4.  H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, (4th edition), Springer-Verlag, Berlin, 1980.

5.  L. Danzer and E. Schulte, Reguläre Inzidenzkomplexe I, Geom. Ded. 13 (1982) 295–308.

6.  B. Grünbaum, Regularity of graphs, complexes and designs, In: Problèmes combinatoire et thèorie des graphes, Coll. Int. CNRS No.260, Orsay, 1977, pp.191–197.

7.  P. McMullen and E. Schulte, Locally toroidal regular polytopes of rank 4, Comment. Math. Helvetici 67 (1992) 77–118.

8.  P. McMullen and E. Schulte, Higher toroidal regular polytopes, Adv. Math. 117 (1996) 17–51.

9.  P. McMullen and E. Schulte, Abstract Regular Polytopes, monograph in preparation.

10.  B. Monson and A.I. Weiss, Regular 4-polytopes related to general orthogonal groups, Mathematika 37 (1990) 106–118.

11.  A.Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.

12.  E. Schulte, Amalgamations of regular incidence-polytopes, Proc. London Math. Soc. (3) 56 (1988) 303–328.

13.  R.P. Stanley, Enumerative Combinatorics 1, Wadsworth \& Brooks/Cole, Monterey, Calif., 1986.

14.  J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics 386, Springer-Verlag, Berlin, 1974.

15.  A.I. Weiss, Incidence-polytopes of type {6, 3, 3}, Geom. Ded. 20 (1986) 147–155.

Get the LaTex | DVI | PS file of this abstract.