Geometries
The fundamental structure in this book is a geometry. We look at a geometry as an incidence system: abstract objects that are related by means of incidence. The concept of a point lying on a line is carried over to the more abstract notion of an element o
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Geometries
The fundamental structure in this book is a geometry. We look at a geometry as an incidence system: abstract objects that are related by means of incidence. The concept of a point lying on a line is carried over to the more abstract notion of an element of type point being incident to an element of type line. The basic ideas and definitions are given in this chapter. The usual related concepts like homomorphisms and subgeometries, and less general concepts such as connectedness and residues, are introduced. The important notion of residual connectedness is described in different but equivalent ways (Corollary 1.6.6). Many of the examples we give display a lot of symmetry. In the later sections of this chapter, we show how the automorphism group of such a highly symmetric geometry can be used for a complete description of the geometry in terms of this group and some of its subgroups. Towards the end of the chapter, we describe how properties like residual connectedness can be expressed in term of these subgroups (Corollary 1.8.13).
1.1 The Concept of a Geometry In this book, the word geometry is used in a technical sense, just as words like topology and algebra. It provides a generalization of the concept of incidence. In a broader context, it would be appropriate to speak of an incidence geometry, but in this book, there is no danger of confusion. A geometry consists of elements of different types such as points, lines, planes (or vertices, edges, faces, cells, or subspaces of dimension i where i is an integer). In this context, the reader should momentarily abandon the usual physical viewpoint according to which a line is a set of points, an edge is a set of two vertices, and so on. The same status will be given to each of the different types of elements. Afterwards, we can assign the role of basic elements, traditionally played by points and lines, to any of these types. F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_1, © Springer-Verlag Berlin Heidelberg 2013
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1 Geometries
Fig. 1.1 The cube geometry
Example 1.1.1 See Fig. 1.1 for a picture of the cube. Let ε1 , ε2 , ε3 be the standard basis of the Euclidean vector space R3 and consider the cube Γ whose 8 vertices are the vectors ±ε1 ± ε2 ± ε3 . The edges (faces) of Γ can be viewed as pairs (respectively, quadruples) of vertices. By replacing each edge and face by its barycentric vector (up to a suitable scaling by a scalar multiple) in the physical cube, we can visualize Γ by 26 vectors: the 8 vectors corresponding to vertices, the 12 vectors ±2εi ± 2εj (1 ≤ i < j ≤ 3) corresponding to edges, and the 6 vectors ±2εi (1 ≤ i ≤ 3) corresponding to faces. Incidence between elements of Γ can now be visualized as a line segment connecting two vectors. There are 72 line segments representing incident pairs of elements and 48 triangles representing incident triples. In general, incidence is a symmetric, reflexive rel
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