Amalgamation of Regular Incidence-Polytopes

We discuss the following problem from the theory of regular incidence-polytopes. Given two regular d-incidence-polytopes 9X and !?2 such that the facets of 0>2 °d the vertex-figures of 0\ are isomorphic to some regular (d l)-incidence-polytope 3C, is there a regular (d +1)incidence-polytope X with facets of type ^ and vertex-figures of type &•?. Such amalgamations X of 0\ and 0*2 along 3iT exist (and then in fact very small ones) at least under the assumption that 0>, and 9*2 have the so-called degenerate amalgamation property with respect to 3if. We prove some results on preassigning the (d l)-dimensional medial section-complex 3if for self-dual (d + l)-dimensional regular incidence-polytopes X.

[1]  H. Coxeter.  REGULAR HONEYCOMBS IN HYPERBOLIC SPACE , 1956 .

[2]  Egon Schulte.  On Arranging Regular Incidence-Complexes as Faces of Higher-Dimensional Ones , 1983, Eur. J. Comb..

[3]  A. I. Weiss.  Incidence-polytopes of type {6, 3, 3} , 1986 .

[4]  Charles J. Colbourn,et al.  A census of regular 3-polystroma arising from honeycombs , 1984, Discret. Math..

[5]  Mark A. Ronan,et al.  Coverings and Automorphisms of Chamber Systems , 1980, Eur. J. Comb..

[6]  H. Coxeter,et al.  The Geometric vein : the Coxeter Festschrift , 1981 .

[7]  Asia Ivic Weiss.  Incidence-polytopes with toroidal cells , 1989, Discret. Comput. Geom..

[8]  Andreas W. M. Dress,et al.  Regular polytopes and equivariant tessellations from a combinatorial point of view , 1985 .

[9]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[10]  H. Coxeter.  A Symmetrical Arrangement of Eleven Hemi-Icosahedra , 1984 .

[11]  L. Danzer,et al.  Regular Incidence-Complexes and Dimensionally Unbounded Sequences of Such, I , 1984 .

[12]  Geoffrey C. Shephard,et al.  Regular 3-complexes with toroidal cells , 1977, J. Comb. Theory, Ser. B.

[13]  Francis Buekenhout,et al.  Diagrams for Geometries and Groups , 1979, J. Comb. Theory, Ser. A.

[14]  Peter McMullen,et al.  Infinite series of combinatorially regular polyhedra in three-space , 1988 .

[15]  E. Schulte,et al.  Kepler-Poinsot-Type Realizations of Regular Maps of Klein, Fricke, Gordan and Sherk , 1987, Canadian Mathematical Bulletin.

[16]  Egon Schulte,et al.  Regular Incidence-Polytopes with Euclidean or Toroidal Faces and Vertex-Figures , 1985, J. Comb. Theory, Ser. A.