and Working with 3D isometries
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
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
- 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.