Parallelism: Affine Geometry

Straight line and incidence are the simplest geometric notions. In our opening chapter, however, we will add the notion of parallelism, which will later be recognized as a special case of incidence. This brings us to affine geometry, which is very close to our vision. Even more important: From it, linear algebra can be founded descriptively, because descriptive vector addition has to do with parallelograms, scalar multiplication with homotheties and ray theorems. Thus, in the second step, we can embed affine geometry into linear algebra and express geometric facts algebraically. This concerns especially the symmetry group Symmetry , the group of all transformations that preserve straight lines and parallels: We can characterize them algebraically. The algebraic point of view allows us, without additional effort, to go beyond our spatial intuition in two respects, and thus to apply geometry to non-visual situations: The number of dimensions may be arbitrary, even larger than two or three, and the field of real numbers describing the one-dimensional continuum may be replaced by an arbitrary field.