 Parallelism: Affine GeometryJost-Hinrich Eschenburg
Chapter 2 in Geometry - Intuition and Concepts, 2022, pp 7-20 from Springer Abstract: Abstract 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. Date: 2022 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-658-38640-5_2 Ordering information: This item can be ordered from DOI: 10.1007/978-3-658-38640-5_2 Access Statistics for this chapter More chapters in Springer Books from Springer |
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