Discovering and Working
with 3D isometries
Eric
Bainville Eric.Bainville@cabri.com Cabrilog France
Abstract
In this minicourse, 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, glidesymmetry, and rotationsymmetry. Using the
isometries proposed by Cabri 3D (translation, plane symmetry,
central symmetry, halfturn, and rotation), we will show how to
obtain and decompose all types of isometries. The halfturn 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 halfturns.  any
rotation is the product of two reflections.  any indirect isometry
is the product of one or three reflections.
