Distance: Euclidean Geometry

In the third century BC, EuclidEuclid compiled the mathematical knowledge of the time in his book “The Elements”. In his geometrical considerations, measurements play a role from the very beginning: distances, angles, areas, volumes. Distance is the fundamental concept. Similarly as before, we do not want to define this concept axiomatically, but derive it from intuitive considerations and only then incorporate it into our framework of linear algebra, namely with the help of the scalar product. The basis for this is the Pythagorean theorem. We will study the group of structure-preserving transformations, the “isometries”, in much more detail this time; their discrete and finite subgroups will also be considered. These, in turn, have to do with crystals and with the Platonic solids; we also introduce the latter in higher dimensions. Next to straight lines, the simplest entities of plane geometry are conic sections, which we will now also examine in terms of lengths and distances. Interestingly enough, this problem becomes more accessible to intuition if we take the term “conic section” literally and also consider the cone in space.