Geometrical Realizations of Shadow Geometries

Contents Introduction 615 General definitions and notation 618 § 1. Shadow geometries of incidence geometries 619 § 2. Shadow geometries of chamber systems and their geometrical reali-zations 626 § 3. Interpreting the shadow geometries as tessellations. .. . 6 3 1 § 4. Transitive tessellations for reflection groups 636 § 5. The Delaney symbols of the tessellations with a transitive reflection group 642 References 655 Introduction This paper deals with geometries and geometrical objects which are usually symbolized by diagrams of the following kind: O—@—O FIG. 1 Such symbols occur in the literature in two different contexts; on the one hand, they represent shadow geometries of incidence geometries in the sense of Tits, on the other hand, they are used in connection with Wythojfs construction for regular polytopes [7,8,9,10,11]. An example of a shadow geometry is the 'space' which is obtained from a three-dimensional projective space in the following way: the 'points' are the lines of the original space, there is one 'line' for each flag {point/? cz plane z} of the original space, consisting of all 'points' / such that p <= / c z , and there are two classes of 'planes', a 'plane' consisting of all 'points' containing a given point or contained in a given plane of the original space. The associated diagram is of course the second example given above. This example of a shadow geometry generalizes to a projective n-space and any distinguished set / o c {0, ..., n — 1}, of dimensions. The 'points' are the flags (totally ordered sets of subspaces) P of the distinguished dimension type, and each subspace by definition comes from a flag F and consists of all P such that F U P is again a flag in the original projective space. (Not all types of flags F are actually needed to produce all subspaces.) In the case of a general incidence geometry, the definition is the same, just replacing {0,..., n-1} by the set of types of that geometry. An example of a 'semiregular' polytope obtained by WythofPs construction is the truncated cube (Fig. 2). The vertices of this figure are by definition all the transforms of some given point under the symmetry group of the cube. The group being fixed, different choices of the starting vertex lead to different polytopes. Combinatorially, there are seven different choices, corresponding to the subsets of the set of 'types' …