Discovering and Working with 3D isometries

Abstract

In this mini-course, we will use the recently released Cabri 3D software to discover and explore some properties of 3D isometries. Through the observation of the symmetries of the well known five Platonic solids, we will establish a classification of 3D isometries, Euclidean transformations preserving distances. This classification is the following. Direct isometries (preserving orientation): translation, rotation, and screwing. Indirect isometries (changing orientation): plane symmetry (reflection), central symmetry, glide-symmetry, and rotation-symmetry. Using the isometries proposed by Cabri 3D (translation, plane symmetry, central symmetry, half-turn, and rotation), we will show how to obtain and decompose all types of isometries. The half-turn is a special case of rotation, with angle 180 degrees. Our objective is to establish, visualize, and experiment the following properties: - any direct isometry is the product of at most two half-turns. - any rotation is the product of two reflections. - any indirect isometry is the product of one or three reflections.